## Biography

Professor Eckhard Platen joined UTS in 1997 from ANU. He was a joint appointment between the School of Finance and Economics and the School of Mathematical Sciences to the newly created Chair in Quantitative Finance.

Prior to this appointment he was the Founding Head of the Centre for Financial Mathematics at the Institute of Advanced Studies at the Australian National University in Canberra. He completed a PhD in Mathematics at the Technical University in Dresden in 1975 and obtained in 1985 his Dr.sc. from the Academy of Sciences in Berlin, where he headed the Sector of Stochastics at the Weierstrass Institute.

He is the co-author of three successful books on Numerical Methods for Stochastic Differential Equations, a book on his innovative Benchmark Approach, published by Springer Verlag, another monograph on Functionals of Multidimensional Diffusions with Applications to Finance and he has authored more than 190 research papers in quantitative finance and applicable mathematics. He has also held more than 90 visiting appointments at leading institutions world wide.

He serves on the Editorial Boards of the Bocconi-Springer book series and seven journals including Mathematical Finance as Associate Editor, Asia Pacific Financial Markets as Advisor and Quantitative Finance and previously Finance and Stochastics. He is initiator and co-organizer of the annual Quantitative Methods in Finance conference series, which started in 1993.

Those interested in his more recent research should visit the Benchmark Approach site.

## Professional

- President of the Bachelier Finance Society.
- Honorary Professor at the University of Cape Town.
- Honorary Professor at the Australian National University

**Professional Memberships:**

- Australian Mathematical Society
- Bachelier Finance Society
- Econometric Society

#### Can supervise: YES

#### Research Interests

Numerical Methods in Finance; Financial Market Modelling; Asset Pricing Theory; Estimation of Discretely Observed Financial Markets; Stochastic Differential Equations; Portfolio Optimization; Pensions.

#### Teaching Areas

Stochastic Analysis; Mathematics of Finance; Numerical Methods in Finance..

#### Publications

Baldeaux, JF & Platen, E 2013, *Functionals of Multidimensional Diffusions with Applications to Finance*, 1, Springer, Germany.View/Download from: Publisher's site

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This research monograph provides an introduction to tractable multidimensional diffusion models, where transition densities, Laplace transforms, Fourier transforms, fundamental solutions or functionals can be obtained in explicit form. The book also provides an introduction to the use of Lie symmetry group methods for diffusions, which allows to compute a wide range of functionals. Besides the well-known methodology on affine diffusions it presents a novel approach to affine processes with applications in finance. Numerical methods, including Monte Carlo and quadrature methods, are discussed together with supporting material on stochastic processes. Applications in finance, for instance, on credit risk and credit valuation adjustment are included in the book. The functionals of multidimensional diffusions analyzed in this book are significant for many areas of application beyond finance. The book is aimed at a wide readership, and develops an intuitive and rigorous understanding of the mathematics underlying the derivation of explicit formulas for functionals of multidimensional diffusions.

Platen, E & Bruti Liberati, N 2010, *Numerical Solution of Stochastic Differential Equations with Jumps in Finance*, 1st, Springer, Germany.View/Download from: Publisher's site

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This research monograph concerns the design and analysis of discrete-time approximations for stochastic differential equations (SDEs) driven by ·Wiener processes and Poisson processes or Poisson jump measures, In financial and actuarial modeling and other areas of application I such jump difrusions are often used to dScribe the dynamics of ',.-arious state variables. In finance these may represent, for instance, asset prices, credit ratings, stock indices, luterest rates, exchange rates or commodity prices. The jump component can capture event-driven unC<'xtainties, such as corporato defaults, operational failures or insured events.

Platen, E & Heath, D 2010, *A Benchmark Approach to Quantitative Finance*, 2nd, Springer, Germany.

Platen, E & Heath, DP 2006, *A Benchmark Approach to Quantitative Finance*, 1st, Springer, Berlin, Germany.View/Download from: Publisher's site

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The benchmark approach provides a general framework for financial market modeling, which extends beyond the standard risk-neutral pricing theory. It permits a unified treatment of portfolio optimisation, derivative pricing, integrated risk managemetn and insurance risk modeling. Th existence of an equivalent risk-neutral pricing measure is not required.Instead, it leads to pricing formulae with respect to the real-world probability measure. This yields important modeling freedom which turns out to be necessary for the derivation of realistic, parsimonious market models. The first part of the book describes the necessary tools fromprobability theory, statistics, stochastic calculus and the theory of stochastic differential equations with jumps. The second part of devoted to financial modeling by the benchmark approach. Various quajtitative methods for the real-world pricing and hedging of derivatives are explained. The general framework is used to provide an understanding of the nature of stochastic volatility. The book is intended for a wide audience that includes quantitative analysts, postgraduate students and practitioners in finance, economics and insurance. It aims to be a self- contained, accessible but methematically rigorous introduction to quantitative finance for readers that have a reasonable mathematical or quantitative background. Finally, the book hsould stimulate inetrest in the benchmark approach by describing some of its power and wide applicability.

Kloeden, P, Platen, E & Schurz, H 2003, *Numerical Solution of SDE Through Computer Experiments*, 3rd, Springer, Germany.

Kloeden, P & Platen, E 1999, *Numerical Solution of Stochastic Differential Equations*, 3rd, Springer, Germany.

Barone-Adesi, G, Platen, E & Sala, C 2020, 'On the Use of Equities in Target Date Funds', *Swiss Finance Institute Research Paper*, no. 20.

Rudd, R, McWalter, T, Kienitz, J & Platen, E 2020, 'Robust Product Markovian Quantization'.

Fontana, C, Pelger, M & Platen, E 2019, 'On the existence of sure profits via flash strategies', *Journal of Applied Probability*, vol. 56, no. 2, pp. 384-397.View/Download from: Publisher's site

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In an arbitrage-free financial market, asset prices (including dividends)

should not exhibit jumps of a predictable magnitude at predictable times. We

provide a rigorous formulation of this result in a fully general setting,

without imposing any semimartingale restriction and only allowing for

buy-and-hold positions. We prove that asset prices do not exhibit predictable

jumps if and only if there is no possibility of obtaining sure profits via

high-frequency limits of buy-and-hold trading strategies. We furthermore show

that right-continuity is an indispensable requirement for any price process

that does not admit sure profits. Our results are robust with respect to

transaction costs or model mis-specifications and imply that, under minimal

assumptions, price changes occurring at scheduled dates should only be due to

unanticipated information releases.

Baldeaux, J, Ignatieva, K & Platen, E 2018, 'Detecting money market bubbles', *Journal of Banking and Finance*, vol. 87, pp. 369-379.View/Download from: Publisher's site

McWalter, TA, Rudd, R, Kienitz, J & Platen, E 2018, 'Recursive marginal quantization of higher-order schemes', *Quantitative Finance*, vol. 18, no. 4, pp. 693-706.View/Download from: Publisher's site

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© 2018 Informa UK Limited, trading as Taylor & Francis Group. Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive marginal quantization (RMQ) of the Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This method involves recursively quantizing the conditional marginals of the discrete-time Euler approximation of the underlying process. By generalizing this approach, we show that it is possible to perform RMQ for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. We further extend the applicability of RMQ by showing how absorption and reflection at the zero boundary may be incorporated, when necessary. To illustrate the improved accuracy of the higher-order schemes, various computations are performed using geometric Brownian motion and the constant elasticity of variance model. For both models, we provide evidence of improved weak order convergence and computational efficiency. By pricing European, Bermudan and barrier options, further evidence of improved accuracy of the higher-order schemes is demonstrated.

Kardaras, C, Oblłój, J & Platen, E 2017, 'The numéraire property and long-term growth optimality for drawdown-constrained investments', *Mathematical Finance*, vol. 27, no. 1, pp. 68-95.View/Download from: Publisher's site

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We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We combine the decision criterion of pathwise growth optimality with a flexible specification of attitude toward risk, encoded by a linear drawdown constraint imposed on admissible wealth processes. We define the constrained numéraire property through the notion of expected relative return and prove that drawdown-constrained numéraire portfolio exists and is unique, but may depend on the investment horizon. However, when sampled at the times of its maximum and asymptotically as the time-horizon becomes distant, the drawdown-constrained numéraire portfolio is given explicitly through a model-independent transformation of the unconstrained numéraire portfolio. The asymptotically growth-optimal strategy is obtained as limit of numéraire strategies on finite horizons.

Barkhagen, M, Blomvall, J & Platen, E 2016, 'Recovering the real-world density and liquidity premia from option data', *Quantitative Finance*, vol. 16, no. 7, pp. 1147-1164.View/Download from: Publisher's site

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© 2016 Taylor & Francis In this paper, we develop a methodology for simultaneous recovery of the real-world probability density and liquidity premia from observed S&P 500 index option prices. Assuming the existence of a numéraire portfolio for the US equity market, fair prices of derivatives under the benchmark approach can be obtained directly under the real-world measure. Under this modelling framework, there exists a direct link between observed call option prices on the index and the real-world density for the underlying index. We use a novel method for the estimation of option-implied volatility surfaces of high quality, which enables the subsequent analysis. We show that the real-world density that we recover is consistent with the observed realized dynamics of the underlying index. This admits the identification of liquidity premia embedded in option price data. We identify and estimate two separate liquidity premia embedded in S&P 500 index options that are consistent with previous findings in the literature.

Chan, L & Platen, E 2016, 'Pricing of long dated equity-linked life insurance contracts', *STOCHASTIC ANALYSIS AND APPLICATIONS*, vol. 34, no. 2, pp. 339-355.View/Download from: Publisher's site

Du, K & Platen, E 2016, 'Benchmarked Risk Minimization', *Mathematical Finance*, vol. 26, no. 3, pp. 617-637.View/Download from: Publisher's site

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This paper discusses the problem of hedging not perfectly replicable contingent

claims using the numeraire portfolio. The proposed concept of benchmarked risk min- ´

imization leads beyond the classical no-arbitrage paradigm. It provides in incomplete

markets a generalization of the pricing under classical risk minimization, pioneered

by Follmer, Sondermann, and Schweizer. The latter relies on a quadratic criterion, ¨

requests square integrability of claims and gains processes, and relies on the existence

of an equivalent risk-neutral probability measure. Benchmarked risk minimization

avoids these restrictive assumptions and provides symmetry with respect to all primary

securities. It employs the real-world probability measure and the numeraire portfolio ´

to identify the minimal possible price for a contingent claim. Furthermore, the resulting

benchmarked (i.e., numeraire portfolio denominated) profit and loss is only driven ´

by uncertainty that is orthogonal to benchmarked-traded uncertainty, and forms a

local martingale that starts at zero. Consequently, sufficiently different benchmarked

profits and losses, when pooled, become asymptotically negligible through diversification.

This property makes benchmarked risk minimization the least expensive method

for pricing and hedging diversified pools of not fully replicable benchmarked contingent

claims. In addition, when hedging it incorporates evolving information about

nonhedgeable uncertainty, which is ignored under classical risk minimization.

FERGUSSON, K & PLATEN, E 2015, 'APPLICATION OF MAXIMUM LIKELIHOOD ESTIMATION TO STOCHASTIC SHORT RATE MODELS', *Annals of Financial Economics*, vol. 10, no. 02.

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The application of maximum likelihood estimation is not well studied for stochastic short rate models because of the cumbersome detail of this approach. We investigate the applicability of maximum likelihood estimation to stochastic short rate models. We restrict our consideration to three important short rate models, namely the Vasicek, Cox–Ingersoll–Ross (CIR) and 3/2 short rate models, each having a closed-form formula for the transition density function. The parameters of the three interest rate models are fitted to US cash rates and are found to be consistent with market assessments.

FERGUSSON, K & PLATEN, E 2015, 'APPLICATION OF MAXIMUM LIKELIHOOD ESTIMATION TO STOCHASTIC SHORT RATE MODELS', *Annals of Financial Economics*, vol. 10, no. 02.

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The application of maximum likelihood estimation is not well studied for stochastic short rate models because of the cumbersome detail of this approach. We investigate the applicability of maximum likelihood estimation to stochastic short rate models. We restrict our consideration to three important short rate models, namely the Vasicek, Cox–Ingersoll–Ross (CIR) and 3/2 short rate models, each having a closed-form formula for the transition density function. The parameters of the three interest rate models are fitted to US cash rates and are found to be consistent with market assessments.

FERGUSSON, K & PLATEN, E 2015, 'APPLICATION OF MAXIMUM LIKELIHOOD ESTIMATION TO STOCHASTIC SHORT RATE MODELS', *Annals of Financial Economics*, vol. 10, no. 02.

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The application of maximum likelihood estimation is not well studied for stochastic short rate models because of the cumbersome detail of this approach. We investigate the applicability of maximum likelihood estimation to stochastic short rate models. We restrict our consideration to three important short rate models, namely the Vasicek, Cox–Ingersoll–Ross (CIR) and 3/2 short rate models, each having a closed-form formula for the transition density function. The parameters of the three interest rate models are fitted to US cash rates and are found to be consistent with market assessments.

Baldeaux, J & Platen, E 2015, 'Credit Derivative Evaluation and CVA Under the Benchmark Approach', *Asia-Pacific Financial Markets*, vol. 22, no. 3, pp. 305-331.View/Download from: Publisher's site

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In this paper, we discuss how to model credit risk under the benchmark approach. Firstly we introduce an affine credit risk model. We then show how to price credit default swaps (CDSs) and introduce credit valuation adjustment (CVA) as an extension of CDSs. In particular, our model can capture right-way—and wrong-way exposure. This means, we capture the dependence structure of the default event and the value of the transaction under consideration. For simple contracts, we provide closed-form solutions. However, due to the fact that we allow for a dependence between the default event and the value of the transaction, closed-form solutions are difficult to obtain in general. Hence we conclude this paper with a reduced form model, which is more tractable.

Baldeaux, J, Fung, MC, Ignatieva, K & Platen, E 2015, 'A Hybrid Model for Pricing and Hedging of Long-dated Bonds', *Applied Mathematical Finance*, vol. 22, no. 4, pp. 366-398.View/Download from: Publisher's site

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Long-dated fixed income securities play an important role in asset-liability management, in life insurance and in annuity businesses. This paper applies the benchmark approach, where the growth optimal portfolio (GOP) is employed as numéraire together with the real-world probability measure for pricing and hedging of long-dated bonds. It employs a time-dependent constant elasticity of variance model for the discounted GOP and takes stochastic interest rate risk into account. This results in a hybrid framework that models the stochastic dynamics of the GOP and the short rate simultaneously. We estimate and compare a variety of continuous-time models for short-term interest rates using non-parametric kernel-based estimation. The hybrid models remain highly tractable and fit reasonably well the observed dynamics of proxies of the GOP and interest rates. Our results involve closed-form expressions for bond prices and hedge ratios. Across all models under consideration we find that the hybrid model with the 3/2 dynamics for the interest rate provides the best fit to the data with respect to lowest prices and least expensive hedges.

Baldeaux, JF, Grasselli, M & Platen, E 2015, 'Pricing currency derivatives under the benchmark approach', *Journal of Banking and Finance*, vol. 53, pp. 34-48.View/Download from: Publisher's site

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This paper considers the realistic modelling of derivative contracts on exchange rates. We propose a stochastic

volatility model that recovers not only the typically observed implied volatility smiles and skews

for short dated vanilla foreign exchange options but allows one also to price payoffs in foreign currencies,

lower than possible under classical risk neutral pricing, in particular, for long dated derivatives. The main

reason for this important feature is the strict supermartingale property of benchmarked savings accounts

under the real world probability measure, which the calibrated parameters identify under the proposed

model. Using a real dataset on vanilla option quotes, we calibrate our model on a triangle of currencies

and find that the risk neutral approach fails for the calibrated model, while the benchmark approach still

works.

Chan, L & Platen, E 2015, 'Pricing and hedging of long dated variance swaps under a 3/2 volatility model', *JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS*, vol. 278, pp. 181-196.View/Download from: Publisher's site

Chan, L & Platen, E 2015, 'Pricing volatility derivatives under the modified constant elasticity of variance model', *OPERATIONS RESEARCH LETTERS*, vol. 43, no. 4, pp. 419-422.View/Download from: Publisher's site

FERGUSSON, K & PLATEN, E 2015, 'APPLICATION OF MAXIMUM LIKELIHOOD ESTIMATION TO STOCHASTIC SHORT RATE MODELS', *Annals of Financial Economics*, vol. 10, no. 02, pp. 1-26.View/Download from: Publisher's site

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Platen, E & Tappe, S 2015, 'Real-World Forward Rate Dynamics With Affine Realizations', *STOCHASTIC ANALYSIS AND APPLICATIONS*, vol. 33, no. 4, pp. 573-608.View/Download from: Publisher's site

Du, K & Platen, E 2014, 'Benchmarked risk management', *Mathematical Finance*.

Baldeaux, JF, Ignatieva, K & Platen, E 2014, 'A tractable model for indices approximating the growth optimal portfolio', *Studies in NonLinear Dynamics and Econometrics*, vol. 18, no. 1, pp. 1-21.View/Download from: Publisher's site

Biagini, F, Cretarola, A & Platen, E 2014, 'Local risk-minimization under the benchmark approach', *Mathematics and Financial Economics*, vol. 8, no. 2, pp. 109-134.View/Download from: Publisher's site

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We study the pricing and hedging of derivatives in incomplete financial markets by considering the local risk-minimization method in the context of the benchmark approach, which will be called benchmarked local risk-minimization. We show that the proposed benchmarked local risk-minimization allows to handle under extremely weak assumptions a much richer modeling world than the classical methodology.

Fergusson, K & Platen, E 2014, 'Hedging long-dated interest rate derivatives for Australian pension funds and life insurers', *Australian Journal of Actuarial Practice*, vol. 1, pp. 29-44.

West, J & Platen, E 2014, 'Natural Disasters, Insurance Stocks and the Numeraire Portfolio', *Advances in Quantitative Analysis of Finance and Accounting*, vol. 12, pp. 23-58.View/Download from: Publisher's site

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This study employs a generalized numeraire portfolio to benchmark insurance stocks to detect abnormal returns in the presence of natural disasters and insurable losses. We show that under the benchmark approach the efficient markets hypothesis holds in the presence of extreme insurable loss whereas other common methods such as the market model and Fama- French three factor model often fail due to the accumulation of estimation errors. We construct a portfolio of U.S. insurance firms and observe the market reaction to a set of major insured natural disasters. Numeraire denominated or benchmarked returns are shown to be are natural measures of abnormal returns. Using the benchmark approach we observe no significant trend in the cumulative abnormal returns of insurance securities following a natural disaster. Using both the traditional market model and the Fama-French three factor model however, we observe significantly positive cumulative abnormal returns following an insured event. The errors inherent in the market model and three-factor model for event studies are shown to be eliminated using the benchmark approach.

Kardaras, C & Platen, E 2013, 'Multiplicative approximation of wealth processes involving no-short-sale strategies via simple trading', *Mathematical Finance*, vol. 23, no. 3, pp. 579-590.View/Download from: Publisher's site

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A financial market model with general semimartingale asset-price processes and where agents can only trade using no-short-sales strategies is considered. We show that wealth processes using continuous trading can be approximated very closely by wealth processes using simple combinations of buy-and-hold trading. This approximation is based on controlling the proportions of wealth invested in the assets. As an application, the utility maximization problem is considered and it is shown that optimal expected utilities and wealth processes resulting from continuous trading can be approximated arbitrarily well by the use of simple combinations of buy-and-hold strategies.

Nikeghbali, A & Platen, E 2013, 'A reading guide for last passage times with financial applications in view', *Finance & Stochastics*, vol. 17, no. 3, pp. 615-640.View/Download from: Publisher's site

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In this survey on last passage times, we propose a new viewpoint which provides a unified approach to many different results which appear in the mathematical finance literature and in the theory of stochastic processes. In particular, we are able to improve the assumptions under which some well-known results are usually stated. Moreover we give some new and detailed calculations for the computation of the distribution of some large classes of last passage times. We have kept in this survey only the aspects of the theory which we expect potentially to be relevant for financial applications.

Platen, E & Shi, L 2013, 'On the numerical stability of simulation methods for SDEs under multiplicative noise in finance', *Quantitative Finance*, vol. 13, no. 2, pp. 183-194.View/Download from: Publisher's site

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When simulating discrete-time approximations of solutions of stochastic differential equations (SDEs), in particular martingales, numerical stability is clearly more important than some higher order of convergence. Discrete-time approximations of solutions of SDEs with multiplicative noise, similar to the BlackScholes model, are widely used in simulation in finance. The stability criterion presented in this paper is designed to handle both scenario simulation and Monte Carlo simulation, i.e. both strong and weak approximations. Methods are identified that have the potential to overcome some of the numerical instabilities experienced when using the explicit Euler scheme. This is of particular importance in finance, where martingale dynamics arise frequently and the diffusion coefficients are often multiplicative. Stability regions for a range of schemes are visualized and analysed to provide a methodology for a better understanding of the numerical stability issues that arise from time to time in practice. The result being that schemes that have implicitness in the approximations of both the drift and the diffusion terms exhibit the largest stability regions. Most importantly, it is shown that by refining the time step size one can leave a stability region and may face numerical instabilities, which is not what one is used to experiencing in deterministic numerical analysis.

Nikitopoulos Sklibosios, C & Platen, E 2013, 'Alternative term structure models for reviewing the expectations puzzle', *International journal of economic research*, vol. 10, no. 2, pp. 361-384.

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NA

Guo, ZJ & Platen, E 2012, 'The Small And Large Time Implied Volatilities In The Minimal Market Model', *International Journal of Theoretical and Applied Finance*, vol. 15, no. 08, pp. 1-23.View/Download from: Publisher's site

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This paper derives explicit formulas for both the small and the large time limits of the implied volatility in the minimal market model. It is shown that interest rates do impact on the implied volatility in the long run, even though they are negligible in the short time limit.

Cheridito, P, Nikeghbali, A & Platen, E 2012, 'Processes of class Sigma, last passage times, and drawdowns', *SIAM Journal on Financial Mathematics*, vol. 3, pp. 280-303.View/Download from: Publisher's site

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We propose a general framework for studying last passage times, suprema, and drawdowns of a large class of continuous-time stochastic processes. Our approach is based on processes of class Sigma and the more general concept of two processes, one of which moves only when the other is at the origin. After investigating certain transformations of such processes and their convergence properties, we provide three general representation results. The first allows the recovery of a process of class Sigma from its final value and the last time it visited the origin. In many situations this gives access to the distribution of the last time a stochastic process attains a certain level or is equal to its running maximum. It also leads to recently discovered formulas expressing option prices in terms of last passage times. Our second representation result is a stochastic integral representation that will allow us to price and hedge options on the running maximum of an underlying that are triggered when the underlying drops to a given level or, alternatively, when the drawdown or relative drawdown of the underlying attains a given height. The third representation gives conditional expectations of certain functionals of processes of class Sigma. It can be used to deduce the distributions of a variety of interesting random variables such as running maxima, drawdowns, and maximum drawdowns of suitably stopped processes.

Guo, Z & Platen, E 2012, 'The Small And Large Time Implied Volatilities In The Minimal Market Model', *International Journal of Theoretical and Applied Finance*, vol. 15, no. 08, pp. 1-23.View/Download from: Publisher's site

#### View description

This paper derives explicit formulas for both the small and the large time limits of the implied volatility in the minimal market model. It is shown that interest rates do impact on the implied volatility in the long run, even though they are negligible in the short time limit.

Hulley, H & Platen, E 2012, 'Hedging for the long run', *Mathematics and Financial Economics*, vol. 6, no. 2, pp. 105-124.View/Download from: Publisher's site

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In the years following the publication of Black and Scholes (J Political Econ, 81(3), 637-654, 1973), numerous alternative models have been proposed for pricing and hedging equity derivatives. Prominent examples include stochastic volatility models, jump-diffusion models, and models based on Lévy processes. These all have their own shortcomings, and evidence suggests that none is up to the task of satisfactorily pricing and hedging extremely long-dated claims. Since they all fall within the ambit of risk-neutral valuation, it is natural to speculate that the deficiencies of these models are (at least in part) attributable to the constraints imposed by the risk-neutral approach itself. To investigate this idea, we present a simple two-parameter model for a diversified equity accumulation index. Although our model does not admit an equivalent risk-neutral probability measure, it nevertheless fulfils a minimal no-arbitrage condition for an economically viable financial market. Furthermore, we demonstrate that contingent claims can be priced and hedged, without the need for an equivalent change of probability measure. Convenient formulae for the prices and hedge ratios of a number of standard European claims are derived, and a series of hedge experiments for extremely long-dated claims on the S&P 500 total return index are conducted. Our model serves also as a convenient medium for illustrating and clarifying several points on asset price bubbles and the economics of arbitrage.

Ignatieva, K & Platen, E 2012, 'Estimating the diffusion coefficient function for a diversified world stock index', *Computational Statistics and Data Analysis*, vol. 56, no. 6, pp. 1333-1349.View/Download from: Publisher's site

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This paper deals with the estimation of continuous-time diffusion processes which model the dynamics of a well diversified world stock index (WSI). We use the nonparametric kernel-based estimation to empirically identify a square root type diffusion coefficient function in the dynamics of the discounted WSI. A square root process turns out to be an excellent building block for a parsimonious model for the WSI. Its dynamics allow capturing various empirical stylized facts and long term properties of the index, as well as, the explicit computation of various financial quantities

Kardaras, C & Platen, E 2012, 'On the Dybvig-Ingersoll-Ross theorem', *Mathematical Finance*, vol. 22, no. 4, pp. 729-740.View/Download from: Publisher's site

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The Dybvig-Ingersoll-Ross (DIR) theorem states that, in arbitrage-free term structure models, long-term yields and forward rates can never fall. We present a refined version of the DIR theorem, where we identify the reciprocal of the maturity date as the maximal order that long-term rates at earlier dates can dominate long-term rates at later dates. The viability assumption imposed on the market model is weaker than those appearing previously in the literature.

Platen, E & Rendek, RJ 2012, 'Approximating the numeraire portfolio by naive diversification', *Journal of Asset Management*, vol. 13, no. 1, pp. 34-50.View/Download from: Publisher's site

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Estimation theory has shown, owing to the limited estimation window available for real asset data, that the sample-based Markowitz mean-variance approach produces unreliable weights that fluctuate substantially over time. This article proposes an alternate approach to portfolio optimization, being the use of naive diversification to approximate the numéraire portfolio (NP). The NP is the strictly positive portfolio that, when used as benchmark, makes all benchmarked non-negative portfolios either mean decreasing or trendless. Furthermore, it maximizes expected logarithmic utility and outperforms any other strictly positive portfolio in the long run. The article proves for a well-securitized market that the naive equal value-weighted portfolio converges to the NP when the number of constituents tends to infinity. This result is model independent and, therefore, very robust. The systematic construction of diversified stock indices by naive diversification from real data is demonstrated. Even when taking transaction costs into account, these indices significantly outperform the corresponding market capitalization- weighted indices in the long run, indicating empirically their asymptotic proximity to the NP. Finally, in the time of financial crisis, a large equi-weighted fund carrying the investments of major pension funds and insurance companies would provide important liquidity. It would not only dampen the drawdown of a crisis, but would also moderate the excesses of an asset price bubble.

Ignatieva, K, Platen, E & Rendek, RJ 2011, 'Using Dynamic Copulae for Modeling Dependency in Currency Denominations of a Diversified World Stock Index', *Journal of Statistical Theory and Practice*, vol. 5, no. 3, pp. 425-452.View/Download from: Publisher's site

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The aim of this paper is to model the dependency among log-returns when security account prices are expressed in units of a well diversified world stock index. The dependency in log-returns of currency denominations of the index is modeled using time-varying copulae, aiming to identify the best fitting copula family. The Student-t copula turns generally out to be superior to e.g. the Gaussian copula, where the dependence structure relates to the multivariate normal distribution. It is shown that merely changing the distributional assumption for the log-returns of the marginals from normal to Student-t leads to a significantly better fit. The Student-t copula with Student-t marginals is able to better capture dependent extreme values than the other models considered. Furthermore, the paper applies copulae to the estimation of the Value-at-Risk and the expected shortfall of a portfolio constructed of savings accounts of different currencies. The proposed copula-based approach allows to split market risk into general and specific market risk, as defined in regulatory documents. The paper demonstrates that the approach performs clearly better than the RiskMetrics approach, a widely used methodology for Value-at-Risk estimation.

Kardaras, C & Platen, E 2011, 'On the semimartingale property of discounted asset-price processes', *Stochastic Processes And Their Applications*, vol. 121, no. 11, pp. 2678-2691.View/Download from: Publisher's site

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A financial market model where agents trade using realistic combinations of simple (i.e., finite combinations of buy-and-hold) no-short-sales strategies is considered. Minimal assumptions are made on the discounted asset-price process . in particular, the semimartingale property is not assumed. Via a natural market viability assumption, namely, absence of arbitrage of the first kind, we establish that discounted asset-prices have to be semimartingales. Our main result can also be regarded as reminiscent of the Fundamental Theorem of Asset Pricing.

Platen, E & Hulley, H 2011, 'A visual criterion for identifying Itô diffusions as martingales or strict local martingales', *Seminar on Stochastic Analysis, Random Fields and Applications*, vol. 6, pp. 147-157.

Platen, E & West, J 2011, 'Intraday Empirical Analysis of Electricity Price Behaviour', *Communications on Stochastic Analysis*, vol. 5, no. 4, pp. 721-744.

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This paper proposes an approach to the intraday analysis of the dynamics of electricity prices. The growth optimal portfolio (GOP) is used as a reference unit in a continuous financial electricity price model. A diversified global portfolio in the form a market capitalisation weighted index approx- imates the GOP. The GOP, measured in units of electricity, is normalised and then modelled as a time transformed square root process of dimension four. The dynamics of the resulting process is empirically verified. Intra- day spot electricity prices from the US and Australian markets are used for this analysis. The empirical findings identify a simple but realistic model for examining the volatile behaviour of electricity prices. The proposed model reflects the historical price evolution reasonably well by using only a few ro- bust and readily observable parameters. The evolution of the transformed time is modelled via a rapidly evolving market activity. A periodic, ergodic process with deterministic volatility is used to model market activity.

Ignatieva, K & Platen, E 2010, 'Modelling co-movements and tail dependency in the international stock market via copulae', *Asia-Pacific Financial Markets*, vol. 17, no. 3, pp. 261-302.View/Download from: Publisher's site

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This paper examines international equity market co-movements using time-varying copulae. We examine distributions from the class of Symmetric Generalized Hyperbolic (SGH) distributions for modelling univariate marginals of equity index returns. We show based on the goodness-of-fit testing that the SGH class outperforms the normal distribution, and that the Student-t assumption on marginals leads to the best performance, and thus, can be used to fit multivariate copula for the joint distribution of equity index returns. We show in our study that the Student-t copula is not only superior to the Gaussian copula, where the dependence structure relates to the multivariate normal distribution, but also outperforms some alternative mixture copula models which allow to reflect asymmetric dependencies in the tails of the distribution. The Student-t copula with Student-t marginals allows to model realistically simultaneous co-movements and to capture tail dependency in the equity index returns.

Kardaras, C & Platen, E 2010, 'Minimizing the expected market time to reach a certain wealth level', *SIAM Journal on Financial Mathematics*, vol. 1, no. 1, pp. 16-29.View/Download from: Publisher's site

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In a financial market model, we consider variations of the problem of minimizing the expected time to upcross a certain wealth level. For exponential L´evy markets, we show the asymptotic optimality of the growth-optimal portfolio for the above problem and obtain tight bounds for the value function for any wealth level. In an Ito market, we employ the concept of market time, which is a clock that runs according to the underlying market growth. We show the optimality of the growth-optimal portfolio for minimizing the expected market time to reach any wealth level. This reveals a general definition of market time which can be useful from an investors point of view. We utilize this last definition to extend the previous results in a general semimartingale setting.

Miller, S & Platen, E 2010, 'Real-world pricing for a modified constant elasticity of variance model', *Applied Mathematical Finance*, vol. 1466-4313, pp. 1-29.View/Download from: Publisher's site

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This paper considers a modified constant elasticity of variance (MCEV) model. This model uses the familiar constant elasticity of variance form for the volatility of the growth optimal portfolio (GOP) in a continuous market. It leads to a GOP that follows the power of a time-transformed squared Bessel process. This paper derives analytic real-world prices for zero-coupon bonds, instantaneous forward rates and options on the GOP that are both theoretically revealing and computationally efficient. In addition, the paper examines options on exchange prices and options on zero-coupon bonds under the MCEV model. The semi-analytic prices derived for options on zero-coupon bonds can subsequently be used to price interest rate caps and floors.

Platen, E & Rendek, RJ 2010, 'Quasi-exact approximation of hidden Markov chain filters', *Communications on Stochastic Analysis*, vol. 4, no. 1, pp. 129-142.

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This paper studies the application of exact simulation methods for multi-dimensional multiplicative noise stochastic differential equations to filtering. Stochastic differential equations with multiplicative noise naturally occur as Zakai equation in hidden Markov chain filtering. The paper proposes a quasi-exact approximation method for hidden Markov chain filters, which can be applied when discrete time approximations, such as the Euler scheme, may fail in practice.

Bruti Liberati, N, Nikitopoulos Sklibosios, C & Platen, E 2010, 'Real-world jump-diffusion term structure models', *Quantative Finance*, vol. 10, no. 1, pp. 23-37.View/Download from: Publisher's site

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This paper considers interest rate term structure models in a market attracting both continuous and discrete types of uncertainty. The event-driven noise is modelled by a Poisson random measure. Using as numeraire the growth optimal portfolio, interest rate derivatives are priced under the real-world probability measure. In particular, the real-world dynamics of the forward rates are derived and, for specific volatility structures, finite-dimensional Markovian representations are obtained. Furthermore, allowing for a stochastic short rate in a non-Markovian setting, a class of tractable affine term structures is derived where an equivalent risk-neutral probability measure may not exist

Breymann, W, Luthi, D & Platen, E 2009, 'Empirical behavior of a world stock index from intra-day to monthly time scales', *The European Physical Journal B*, vol. 71, no. 4, pp. 511-522.View/Download from: Publisher's site

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Most of the papers that study the distributional and fractal properties of financial instruments focus on stock prices or foreign exchange rates. This typically leads to mixed results concerning the distributions of log-returns and some multi-fractal properties of exchange rates, stock prices, and regional indices. This paper uses a well diversified world stock index as the central object of analysis. Such index approximates the growth optimal portfolio, which is demonstrated under the benchmark approach, it is the ideal reference unit for studying basic securities. When denominating this world index in units of a given currency, one measures the movements of the currency against the entire market. This provides a least disturbed observation of the currency dynamics. In this manner, one can expect to disentangle, e.g., the superposition of the two currencies involved in an exchange rate. This benchmark approach to the empirical analysis of financial data allows us to establish remarkable stylized facts. Most important is the observation that the repeatedly documented multi-fractal appearance of financial time series is very weak and much less pronounced than the deviation of the mono-scaling properties from Brownian-motion type scaling. The generalized Hurst exponent H(2) assumes typical values between 0.55 and 0.6. Accordingly, autocorrelations of log-returns decay according to a power law, and the quadratic variation vanishes when going to vanishing observation time step size.

Filipovic, D & Platen, E 2009, 'Consistent Market Extensions Under the Benchmark Approach', *Mathematical Finance*, vol. 19, no. 1, pp. 41-52.View/Download from: Publisher's site

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The existence of the growth optimal portfolio (GOP), also known as the Kelly portfolio, is vital for a financial market to be meaningful. The GOP, if it exists, is uniquely determined by the market parameters of the primary security accounts. However, markets may develop and new security accounts become tradable. What happens to the GOP if the original market is extended? In this paper we provide a complete characterization of market extensions which are consistent with the existence of a GOP. We show that a three fund separation theorem applies for the extended GOP. This includes, in particular, the introduction of a locally risk free security, the savings account. We give necessary and sufficient conditions for a consistent exogenous specification of the prevailing short rates.

Ghilarducci, T, Nell, E, Mittnik, S, Platen, E, Semmler, W & Chappe, R 2009, 'Memorandum on A new financial architecture and new regulations', *Investigacion Economica*, vol. 68, no. 267, pp. 147-161.

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NA

Mittnik, S, Nell, E, Platen, E, Semmler, W & Chappe, R 2009, 'Financial market meltdown and a need for new financial regulations', *METU Studies in Development*, vol. 36, no. 1, pp. 253-269.

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The financial crisis, triggered by the subprime and real estate crisis in the US, has become global It is deeply rooted in a decade-long misuse of the financial market for rent-seeking. The financial industry has largely abandoned Its role as a service industry, supposedly charging reasonable fees for the services of spreading risk and allocating capital and credit. Instead it provides a market for speculation, corporate control - mergers and acquisitions -: and a casino for bettmg on or hedging practically any kind of risk - the derivatives market.

Platen, E & Rendek, RJ 2009, 'Exact scenario simulation for selected multi-dimensional stochastic processes', *Communications on Stochastic Analysis*, vol. 3, no. 3, pp. 443-465.

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Accurate scenario simulation methods for solutions of multi - dimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for instance, in finance. Various discrete time simulation methods have been developed over the years. However, the simulation of solutions of some stochastic differential equations can be problematic due to systematic errors and numerical instabilities. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems encountered by those simulation methods that use discrete time approximations. This paper provides a survey of methods for the exact simulation of paths of some multidimensional solutions of stochastic differential equations including Ornstein- Uhlenbeck, square root, squared Bessel, Wishart and L´evy type processes.

Bruti Liberati, N, Nikitopoulos Sklibosios, C, Platen, E & Schlogl, E 2009, 'Alternative defaultable term structure models', *Asia - Pacific Financial Markets*, vol. 16, no. 1, pp. 1-31.View/Download from: Publisher's site

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The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives.

Bruti Liberati, N & Platen, E 2008, 'Strong Predictor-Corrector Euler Methods for Stochastic Differential Equations', *Stochastics and Dynamics*, vol. 8, no. 3, pp. 561-581.View/Download from: Publisher's site

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This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor-corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the new symmetric predictor-corrector Euler methods.

Hardle, WK, Kleinow, T, Korostelev, A, Logeay, C & Platen, E 2008, 'Semiparametric Diffusion Estimation and application to a Stock Market Index', *Quantitative Finance*, vol. 8, no. 1, pp. 81-92.View/Download from: Publisher's site

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The analysis of diffusion processes in financial models is crucially dependent on the form of the drift and diffusion coefficient functions. A new model for a stock market index process is proposed in which the index is decomposed into an average growth process and an ergodic diffusion. The ergodic diffusion part of the model is not directly observable. A methodology is developed for estimating and testing the coefficient functions of this unobserved diffusion process. The estimation is based on the observations of the index process and uses semiparametric and non-parametric techniques. The testing is performed via the wild bootstrap resampling technique. The method is illustrated on S&P 500 index data.

Miller, S & Platen, E 2008, 'Analytic pricing of contingent claims under the real-world measure', *International Journal of Theoretical and Applied Finance*, vol. 11, no. 8, pp. 841-867.View/Download from: Publisher's site

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This article derives a series of analytic formulae for various contingent claims under the real-world probability measure using the stylised minimal market model (SMMM). This model provides realistic dynamics for the growth optimal portfolio (GOP) as a well-diversified equity index. It captures both leptokurtic returns with correct tail properties and the leverage effect. Under the SMMM, the discounted GOP takes the form of a time-transformed squared Bessel process of dimension four. From this property, one finds that the SMMM possesses a special and interesting relationship to non-central chi-square random variables with zero degrees of freedom. The analytic formulae derived under the SMMM include options on the GOP, options on exchange prices and options on zero-coupon bonds. For options on zero-coupon bonds, analytic prices facilitate efficient calculation of interest rate caps and floors.

MILLER, SM & PLATEN, E 2008, 'ANALYTIC PRICING OF CONTINGENT CLAIMS UNDER THE REAL-WORLD MEASURE', vol. 11, no. 08, pp. 841-867.

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This article derives a series of analytic formulae for various contingent claims under the real-world probability measure using the stylised minimal market model (SMMM). This model provides realistic dynamics for the growth optimal portfolio (GOP) as a well-diversified equity index. It captures both leptokurtic returns with correct tail properties and the leverage effect. Under the SMMM, the discounted GOP takes the form of a time-transformed squared Bessel process of dimension four. From this property, one finds that the SMMM possesses a special and interesting relationship to non-central chi-square random variables with zero degrees of freedom. The analytic formulae derived under the SMMM include options on the GOP, options on exchange prices and options on zero-coupon bonds. For options on zero-coupon bonds, analytic prices facilitate efficient calculation of interest rate caps and floors.

Platen, E & Rendek, RJ 2008, 'Empirical evidence on Student-t log-returns of diversified world stock indices', *Journal of Statistical Theory and Practice*, vol. 2, no. 2, pp. 233-251.View/Download from: Publisher's site

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The aim of this paper is to document some empirical facts related to log-returns of diversified workld stock indices when these are denominated in different currencies. Motivated by eaarlier results we jave obtained the estimated distribution of log-returns for a range of world sotckindices over long observation periods. We expand previous studies bya pplying the maximum likelihood ration test to the large class of generalised hyperbolic distributions and investigate the log-returns ofa variety of diversified world stock indices in different currency denominations.

Bruti Liberati, N, Martini, F, Piccardi, M & Platen, E 2008, 'A Hardware Generator of Multi-Point Distributed Random Numbers for Monte Carlo Simulation', *Mathematics and Computers in Simulation*, vol. 77, no. 1, pp. 45-56.View/Download from: Publisher's site

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Monte Carlo simulation of weak approximation of stochastic differential equations constitutes an intensive computational task. In applications such as finance, for instance, to achieve "real time" execution, as often required, one needs highly efficient implementations of the multi-point distributed random number generator underlying the simulations. In this paper, a fast and flexible dedicated hardware solution on a field programmable gate array is presented. A comparative performance analysis between a software-only and the poposed hardware solution demonstrated that the hardware solution is bottleneck-free, retains the flexibility of the software solution and significantly increases the computational efficiency. Moreover, simulations in Applications wuch as economics insurance, physics, population dynamics, epidemiology, structural mechanics, checmistry and biotechnology can benefit from the obtained speedups.

CHRISTENSEN, MM & PLATEN, E 2007, 'SHARPE RATIO MAXIMIZATION AND EXPECTED UTILITY WHEN ASSET PRICES HAVE JUMPS', vol. 10, no. 08, pp. 1339-1364.

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We analyze portfolio strategies which are locally optimal, meaning that they maximize the Sharpe ratio in a general continuous time jump-diffusion framework. These portfolios are characterized explicitly and compared to utility based strategies. We show that in the presence of jumps, maximizing the Sharpe ratio is generally inconsistent with maximizing expected utility, in the sense that a utility maximizing individual will not choose a strategy which has a maximal Sharpe ratio. This result will hold unless markets are incomplete or jump risk has no risk premium. In case of an incomplete market we show that the optimal portfolio of a utility maximizing individual may "accidentally" have maximal Sharpe ratio. Furthermore, if there is no risk premium for jump risk, a utility maximizing investor may select a portfolio having a maximal Sharpe ratio, if jump risk can be hedged away. We note that uncritical use of the Sharpe ratio as a performance measure in a world where asset prices exhibit jumps may lead to unreasonable investments with positive probability of ruin.

Bruti Liberati, N & Platen, E 2007, 'Approximation of Jump Diffusions in Finance and Economics', *Computational Economics*, vol. 29, no. 3-4, pp. 283-312.

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In finance and economics the key dynamics are often specified via stochastic differential equations (SDEs) of jump-diffusion type. The class of jump-diffusion SDEs that admits explicit solutions is rather limited. Consequently, discrete time approximations are required. In this paper we give a survey of strong and weak numerical schemes for SDEs with jumps. Strong schemes provide pathwise approximations and therefore can be employed in scenario analysis, filtering or hedge simulation. Weak schemes are appropriate for problems such as derivative pricing or the evaluation of risk measures and expected utilities. Here only an approximation of the probability distribution of the jump-diffusion process is needed. As a framework for applications of these methods in finance and economics we use the benchmark approach. Strong approximation methods are illustrated by scenario simulations. Numerical results on the pricing of options on an index are presented using weak approximation methods.

Bruti Liberati, N & Platen, E 2007, 'Strong Approximations of Stochastic Differential Equations with Jumps', *Journal of Computational and Applied Mathematics*, vol. 205, no. 2, pp. 982-1001.View/Download from: Publisher's site

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This paper is a survey of strong discrete time approximations of jump-diffusion processes described by stochastic differential equations (SDEs). It also presents new results on strong discrete time approximations for the specific case of pure jump SDEs. Strong approximations based on jump-adapted time discretizations, which produce no discretization error in the case of pure jump processes, are analyzed. The computational complexity of these approximations is proportional to the jump intensity. By exploiting a stochastic expansion for pure jump processes, higher order discrete time approximations, whose computational complexity is not dependent on the jump intensity, are proposed. For the specific case of pure jump SDEs, the strong order of convergence of strong Taylor schemes is established under weaker conditions than those currently known in the literature.

Christensen, MM & Platen, E 2007, 'Sharpe Ratio Maximization and Expected Utility When Asset Prices Have Jumps', *International Journal of Theoretical and Applied Finance*, vol. 10, no. 8, pp. 1339-1364.View/Download from: Publisher's site

Platen, E & Runggaldier, WJ 2007, 'A Benchmark Approach to Portfolio Optimization under Partial Information', *Asia Pacific Financial Markets*, vol. 14, no. 1-2, pp. 25-43.View/Download from: Publisher's site

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This paper proposes a filtering methodology for portfolio optimization when some factors of the underlying model are only partially observed. The level of information is given by the observed quantities that are here supposed to be the primary securities and empirical log-price covariations. For a given level of information we determine the growth optimal portfolio, identify locally optimal portfolios that are located on a corresponding Markowitz efficient frontier and present an approach for expected utility maximization. We also present an expected utility indifference pricing approach under partial information for the pricing of nonreplicable contracts. This results in a real world pricing formula under partial information that turns out to be independent of the subjective utility of the investor and for which an equivalent risk neutral probability measure need not exist.

Breymann, W, Kelly, L & Platen, E 2006, 'Intraday empirical analysis and modeling of diversified world stock indices', *Asia - Pacific Financial Markets*, vol. 12, no. 1, pp. 1-28.

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This paper proposes an approach to the intraday analysis of diversified world stock accumulation indices. The growth optimal portfolio (GOP) is used as reference unit or benchmark in a continuous financial market model. Diversified portfolios, covering the world stock market, are constructed and shown to approximate the GOP, providing the basis for a range of financial applications. The normalized GOP is modeled as a time transformed square root process of dimension four. Its dynamics are empirically verified for several world stock indices. Furthermore, the evolution of the transformed time is modeled as the integral over a rapidly evolving mean-reverting market activity process with deterministic volatility. The empirical findings suggest a rather simple and robust model for a world stock index that reflects the historical evolution, by using only a few readily observable parameters.

Burrage, K, Burrage, P, Higham, DJ, Kloeden, PE & Platen, E 2006, 'Comment on "numerical methods for stochastic differential equations"', *Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 74, no. 6.View/Download from: Publisher's site

Fergusson, KJ & Platen, E 2006, 'On the distributional characterization of daily log-returns of a world stock index', *Applied Mathematical Finance*, vol. 13, no. 1, pp. 19-38.View/Download from: Publisher's site

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In this paper distributions are identified which suitably fit log-returns of the world stock index when these are expressed in units of different currencies. By searching for a best fit in the class of symmetric generalized hyperbolic distributions the maximum likelihood estimates appear to cluster in the neighbourhood of those of the Student t distribution. This is confirmed at a high significance level under the likelihood ratio test. Finally, the paper derives the minimal market model, which explains the empirical findings as a consequence of the optimal market dynamics

Heath, DP & Platen, E 2006, 'Local volatility function models under a benchmark approach', *Quantitative Finance*, vol. 6, no. 3, pp. 197-206.View/Download from: Publisher's site

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Without requiring the existence of an equivalent risk-neutral probability measure this paper studies a class of one-factor local volatility function models for stock indices under a benchmark approach. It is assumed that the dynamics for a large diversif

Le, T & Platen, E 2006, 'Approximating the growth optimal portfolio with a diversified world stock index', *The Journal of Risk Finance*, vol. 7, no. 5, pp. 559-574.View/Download from: Publisher's site

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Purpose This paper aims to construct and compare various total-return world stock indices based on daily data. Design/methodology/approach Because of diversification, these indices are noticeably similar. A diversification theorem identifies any diversified portfolio as a proxy for the growth optimal portfolio. Findings The paper constructs a diversified world stock index that outperforms a number of other indices and argues that it is a good proxy for the growth optimal portfolio. Originality/value The diversified world stock index has applications to derivative pricing and investment management.

Platen, E 2006, 'A benchmark approach to asset management', *Journal of Asset Management*, vol. 6, no. 6, pp. 390-405.View/Download from: Publisher's site

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DP0343913 This paper aims to discuss the optimal selection of investments for the short and long runin a continuous time financial market setting. First, it documents the almost sure pathwise long-run outperformance of all positive portfolios by the growth optimal portfolio. Secondly, it assumes that every investor prefers more rather than less wealth and keeps the freedom to adjust his or her risk aversion at any time. In a general continuous market, a two fund separation result is derived which yields optimal portfolios located on the Markowitz efficient frontier. A optimal portfolio is shown to have a fraction of its wealth invested inthe growth optimal portfolio and the remaining fraction inthe savings account. The risk aversion of the investor at a given time determines the volatility of her/his optimal portfolio. It is pointed out that it is usually not rational to reduce risk aversion further than is necessary to achieve the maximum growth rate. Assuming an optimal dynamics for a global market, the market portfolio turns out to be growth optimal. The discounted market portfolio is shown to follow a particular time transformed diffusion process with explicitly known transition density. Assuming that the drift og yhr discounted market portfolio grows exponentially, a parsimonioous and realistic model for its dynamics results. It allows for efficient portfolio optimisation and derivative pricing.

Platen, E 2006, 'A benchmark approach to finance', *Mathematical Finance*, vol. 16, no. 1, pp. 131-151.View/Download from: Publisher's site

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This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling, and risk measurement. It is based on the natural assumption that investors prefer more rather than less, in the sense that given two portfolios wit

Platen, E 2006, 'Portfolio selection and asset pricing under a benchmark approach', *Physica A: Statistical Mechanics And Its Applications*, vol. 370, no. 1, pp. 23-29.View/Download from: Publisher's site

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The paper presents classical and new results on portfolio optimization, as well as the fair pricing concept for derivative pricing under the benchmark approach. The growth optimal portfolio is shown to be a central object in a market model. It links asse

Platen, E & Heath, D 2006, 'Currency derivatives under a minimal market model', *ICFAI Journal of Derivatives Markets*, vol. 3, pp. 68-86.

Bruti Liberati, N, Nikitopoulos Sklibosios, C & Platen, E 2006, 'First order strong approximations of jump diffusions', *Monte Carlo Methods and Applications*, vol. 12, no. 3-4, pp. 191-209.

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DP0559879

Chiarella, C & Platen, E 2005, 'Special Issue: Introduction To Selected Proceedings From The Quantitative Methods In Finance 2004 Conference (QMF 2004)', *Quantitative Finance*, vol. 5, no. 3, pp. 235-235.

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NA

Christensen, MM & Platen, E 2005, 'A general benchmark model for stochastic jump sizes', *Stochastic Analysis And Applications*, vol. 23, no. 5, pp. 1017-1044.View/Download from: Publisher's site

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Under few technical assumptions and allowing for the absence of an equivalent martingale measure, we show how to price and hedge in a sequence of incomplete markets driven by Wiener noise and a marked point process. We investigate the structure of market

Heath, DP & Platen, E 2005, 'Currency derivatives under a minimal market model with random scaling', *International Journal of Theoretical and Applied Finance*, vol. 8, no. 8, pp. 1157-1177.

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This paper uses an alternative, parsimonious volatility model to describe the dunamics of a currency market for the pricing and hedging of derivatives. Time transformed squared Bessel processes are the basic driving factors of the minimal marketr model. The time transformation is chracterised by a random scaling, wich provides for realistic exchange rate dynamics. The pricing of standard European options is studied. In particular, it is shown that the model produces implied volatility surfaces that are tpically observed in real markets.

Heath, DP & Platen, E 2005, 'Understanding the implied volatility surface for options on a diversified index', *Asia-Pacific Financial markets*, vol. 11, no. 1, pp. 55-77.

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This paper describes a two-factor model for a diveersified index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analhsis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed square Bessel process of dimension four. It turns our that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The prposed model, which includes a random initial condition for market activity, generates implied colatility surfaces for European call and put options that are typically observed in real markets. The paper also examines the price differences of binary options for th epropsed model and their Black-Scholes counterparts.

Hulley, H, Miller, S & Platen, E 2005, 'Benchmarking and fair pricing applied to two market models', *The Kyoto Economic Review*, vol. 74, no. 1, pp. 85-118.

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This paper considers a market containing both continuous and discrete noise. Modest assumptions ensure the existence of a growth optimal portfolio. Non-negative self-financing trading strategies, when benchmarked by this portfolio, are local martingales unde the real-world measure. This justifies the fair pricing approach, which expresses derivative prices in terms of real-world conditional expectations of benchmarked pay-offs. Two models for benchmarked primary security accounts are presentated, and fair pricing formulas for some common contingent claims are derived.

Miller, S & Platen, E 2005, 'A two-factor model for low interest rate regimes', *Asia-Pacific Financial Markets*, vol. 11, no. 1, pp. 107-133.

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This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yielf curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP and is modelled as a aquared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, thsi study focusses in the difficult but important case where the short rate stays close to zero for a prolonged period of time. FOr the proposed model, an equivalent risk neutral martingale measure is neither possile nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, themodel replicates the ket features of the inetrest rate cap market for economies with low inetrest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew.

Platen, E 2005, 'An alternative interest rate term structure model', *International Journal of Theoretical & Applied Finance*, vol. 8, no. 6, pp. 717-735.

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This paper proposes and alternative approach to the modeling of the interest rate twem structure. It suggests that the total market price for risk is an important factor that has to be modeled carefully. The growth optimal portfolio which is characterised by thie factor is used as reference unit or benchmark for obtaining a cosistent price system. Benchmarked derivative prices are tajen as conditional expectations of future bench-marked prices under the real world probability measure. The inverse of the squared total market price for risk is modeled as a square root process and shown to influence the medium and long term forward rates. With constant parameters and constant short rate the model already generates a hump shaped mean for the forward rate curve and other empirical features typically observed.

Platen, E 2005, 'Diversified portfolios with jumps in a benchmark framework', *Asia-Pacific Financial Markets*, vol. 11, no. 1, pp. 1-22.

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This paper considers diversified portfolios in a sequence of jump diffusion market models. Conditions for the approximation of the growth optimal portfolio (GOP) by diversified portfolios are provided. Under realistic assumptions, it is shown that diversified portfolios approximate GOP without requiring any major model specifications. This provides a basis for systematic use of diversified stock indices as proxies for the GOP in derivative pricing, risk management and portfolio optimisation

Platen, E 2005, 'On the role of the growth optimal portfolio in finance', *Australian Economic Papers*, vol. 44, no. 4, pp. 365-388.

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The paper discusses various roles that the growth optimal portfolio (GOP) plays in finance. For the case of a continuous market we show how the GOP can be interpreted as a fundamental building block in financial market modelling, portfolio optimisation, contingent claim pricing and risk measurement. On the basis of aportfolio selection theorem, optimal portfolios are derived. These allocate funds into the GOP and the savings account. A risk aversion coefficient is introduced, controlling the amount invested in the savings account, which allows to characterise portfolio strategies that maximise expected utilities. Natural conditions are formulated under which the GOP appears as the market portfolio. derivation of the intertemporal capital asset pricving model is given without relying on Markovianity, equilibrium arguments or utility functions. Fair contingent claim pricing, with the GOP as numeraire portfolio, is shown to generalise risk neutral and actuarial pricing. Finally the GOPis described in various ways as the best performing portfolio.

Platen, E & Runggaldier, WJ 2005, 'A benchmark approach to filtering in finance', *Asia-Pacific Financial Markets*, vol. 11, no. 1, pp. 79-105.

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The paper propsoed the use of the growth optimal portfolio for pricing and hedging in imcomplete markets when there are unobserved factors that have to be filtered. The proposed filtering framework is applicable also in cases when there does not exist an equivalent risk neutral martingale measure. The reduction of the variance of derivative prices for increasing degrees of available iformation is measured.

Platen, E & West, JM 2005, 'A fair pricing approach to weather derivatives', *Asia-Pacific Financial Markets*, vol. 11, no. 1, pp. 23-53.

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This paper proposes a consistent approach to the pricing of weather derivatives. Since weather derivatives are traded in an incomplete market setting, standard hedging based pricing methods cannot be applied. The growth optimal portfolio, which is interpreted as a world stock index,is used as a benchmark or numeraire such that all benchmarked derivative price processes are martingales. No measure transformation is needed for the proposed fair paricing. For weather derivative payoffs that are independent of the value of the growth optimal portfolio, it is shown that the classical actuarial pricing methodology is a particular case of the fair pricing concepts. A discrete time model is constructed to approximate historical weather characteristics. The fair prices of some partuclar weather derivatives are derived using historical and Gaussian residuals. The question of weather risk as diversifiable risk is also discussed.

Craddock, MJ & Platen, E 2004, 'Symmetry group methods for fundamental solutions', *Journal of Differential Equations*, vol. 207, no. 2, pp. 285-302.View/Download from: Publisher's site

Kelly, L, Platen, E & Sorensen, M 2004, 'Estimation for discretely observed diffusions using transform functions', *Journal Of Applied Probability*, vol. 41, no. A, pp. 99-118.

Platen, E 2004, 'A class of complete benchmark models with intensity-based jumps', *Journal Of Applied Probability*, vol. 41, no. 1, pp. 19-34.View/Download from: Publisher's site

Platen, E 2004, 'Modeling the volatility and expected value of a diversified world index', *International Journal of Theoretical and Applied Finance*, vol. 7, no. 4, pp. 511-529.View/Download from: Publisher's site

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This paper considers a diversified world tock index in a continuous financial market with the growth optimal portfolio (GOP) as reference unit or benchmark. Diversified boradly based indices and portfolios, which include major world stock market indices, are shown to approximate the GOP. It is demonstated that a key financial quantity is the trend of a world index. It turns out tat it can be directly observed since the expected increments of the index equal four times those of the quadratic variation of its square root. Using a world atock index as approximation of the discounted GTOP it is shown that, in reality, the trend of the discounted GOP does not vary greatly in the long term. This leads for a diversified world index to a natural model, where the index is transformed square root process of dimension four. The squared index volatility appears then as the inverse of the square root process. This feature explains most of te properties of an index and its volatility

Buhlmann, H & Platen, E 2003, 'A discrete time benchmark approach for insurance and finance', *Astin Bulletin*, vol. 33, no. 2, pp. 153-172.View/Download from: Publisher's site

Chiarella, C & Platen, E 2003, 'Introduction To Selected Proceedings From Quantitative Methods In Finance 2002', *Quantitative Finance*, vol. 3, no. 1, pp. 0-0.

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Heath, DP & Platen, E 2003, 'Pricing of index options under a minimal market model with log-normal scaling', *Quantitative Finance*, vol. 3, no. 6, pp. 442-450.View/Download from: Publisher's site

Platen, E & Stahl, G 2003, 'A structure for general and specific market risk', *Computational Statistics*, vol. 18, no. 3, pp. 355-373.

HEATH, D & PLATEN, E 2002, 'PERFECT HEDGING OF INDEX DERIVATIVES UNDER A MINIMAL MARKET MODEL', *International Journal of Theoretical and Applied Finance*, vol. 05, no. 07, pp. 757-774.

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The paper presents a financial market model that generates stochastic volatility using a minimal set of factors. These factors, formed by transformations of square root processes, model the dynamics of different denominations of a benchmark portfolio. Benchmarked prices are assumed to be local martingales. Numerical results for the pricing and hedging of basic derivatives on indices are described for the minimal market model. This includes cases where the standard risk neutral pricing methodology fails because of the presence of a strict local martingale measure. However, payoffs can be perfectly hedged using self-financing strategies and a form of arbitrage exists. This is illustrated by hedge simulations. The different term structure of implied volatilities is documented for calls and puts on an index.

Heath, DP & Platen, E 2002, 'Consistent pricing and hedging for a modified constant elasticity of variance model', *Quantitative Finance*, vol. 2, no. 6, pp. 459-467.View/Download from: Publisher's site

Heath, DP & Platen, E 2002, 'Perfect hedging on index derivatives under a minimal model', *International Journal of Theoretical and Applied Finance*, vol. 5, no. 7, pp. 757-774.View/Download from: Publisher's site

Kubilius, K & Platen, E 2002, 'Rate of weak convergence of the Euler approximation for diffusion processes with jumps', *Monte Carlo Methods and Applications*, vol. 8, no. 1, pp. 83-96.

Kuechler, U & Platen, E 2002, 'Weak discrete time approximation of stochastic differential equations with time delay', *Mathematics and Computers in Simulation*, vol. 59, no. 6, pp. 497-507.View/Download from: Publisher's site

Platen, E 2002, 'Arbitrage in continuous complete markets', *Advances in Applied Probability*, vol. 34, no. 3, pp. 540-558.View/Download from: Publisher's site

Platen, E & Heath, DP 2002, 'A variance reduction technique based on integral representations', *Quantative Finance*, vol. 2, no. 5, pp. 362-369.

Heath, D, Hurst, SR & Platen, E 2001, 'Modelling the stochastic dynamics of volatility for equity indices', *Asia Pacific Financial Markets*, vol. 8, no. 3, pp. 179-195.

Heath, DP, Hurst, SR & Platen, E 2001, 'Modelling the stochastic dynamics of volatility for equity indices', *Asia Pacific Financial Markets*, vol. 8, pp. 179-195.

Heath, DP, Platen, E & Schweizer, M 2001, 'A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets', *Mathematical Finance*, vol. 11, no. 4, pp. 385-413.View/Download from: Publisher's site

Craddock, MJ, Heath, DP & Platen, E 2000, 'Numerical inversion of Laplace transforms: a survey of techniques with applications to derivative pricing', *Journal of Computational Finance*, vol. 4, no. 1, pp. 57-81.

Hofmann, N & Platen, E 2000, 'Approximating large diversified portfolios', *Mathematical Finance*, vol. 10, no. 1, pp. 77-88.View/Download from: Publisher's site

Kuechler, U & Platen, E 2000, 'Strong discrete time approximation of stochastic differential equations with time delay', *Mathematics and Computers in Simulation*, vol. 54, no. 0, pp. 189-205.

Platen, E 1999, 'Axiomatic Principles For A Market Model', *Journal Of Applied Probability*, vol. 36, no. 1, pp. 295-300.

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Fischer, P & Platen, E 1999, 'Applications of the balanced method to stochastic differential equations in filtering', *Monte Carlo Methods and Applications*, vol. 5, no. 1, pp. 19-38.View/Download from: Publisher's site

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The paper studies the application of the balanced method in hidden Markov chain filtering, an important practical area that requires the strong numerical solution of stochastic differential equations with multiplicative noise. Numerical experiments are conducted to enable comparisons between the balanced method and standard alternative methods in the context of filtering. Both the mean global error and the sample path properties of the approximate solutions are compared in a numerical study. © 1999 VSP.

Hurst, SK, Platen, E & Rachev, S 1999, 'Option Pricing For A Logstable Asset Price Model', *Mathematical And Computer Modelling*, vol. 29, no. 10-12, pp. 105-119.View/Download from: Publisher's site

#### View description

The paper generalises the celebrated Black and Scholes [1] European option pricing formula for a class of logstable asset price models. The theoretical option prices have the potential to explain the implied volatility smiles evident in the market. (C) 1

Platen, E 1999, 'A short term interest rate model', *Finance and Stochastics*, vol. 3, no. 2, pp. 215-225.

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This paper suggests a short term interest rate model. It incorporates inflation rate, market variance, market net growth rate and market volatility trend. Empirical evidence from different markets supports the model.

Platen, E 1999, 'An introduction to numerical methods for stochastic differential equations', *Acta Numerica*, vol. 8, pp. 197-246.

Platen, E 1999, 'Axiomatic principles for a market model', *Journal of Applied Probability*, vol. 36, no. 1, pp. 295-300.View/Download from: Publisher's site

Milstein, G, Platen, E & Schurz, H 1998, 'Balanced Implicit Methods For Stiff Stochastic Systems', *Siam Journal On Numerical Analysis*, vol. 35, no. 3, pp. 1010-1019.View/Download from: Publisher's site

#### View description

This paper introduces some implicitness in stochastic terms of numerical methods for solving stiff stochastic differential equations and especially a class of fully implicit methods, the balanced methods. Their order of strong convergence is proved. Nume

Platen, E & Schweizer, M 1998, 'On Feedback Effects From Hedging Derivatives', *Mathematical Finance*, vol. 8, no. 1, pp. 67-84.View/Download from: Publisher's site

#### View description

This paper proposes a new explanation for the smile and skewness effects in implied volatilities. Starting from a microeconomic equilibrium approach, we develop a diffusion model for stock prices explicitly incorporating the technical demand induced by h

Platen, E, Hurst, S & Rachev, S 1997, 'Subordinated market index models: A comparison', *Financial Engineering and the Japanese Markets*, vol. 4, no. 2, pp. 97-124.

Hall, P, Matthews, D & Platen, E 1996, 'Algorithms for Analyzing Nonstationary Time Series with Fractal Noise', *Journal of Computational and Graphical Statistics*, vol. 5, no. 4, pp. 351-364.View/Download from: Publisher's site

#### View description

Arguably the best-known applications of fractal methods are in relatively homogeneous, stationary settings, where the environment is controllable by scientists or engineers. For example, in applications to surface science, an unblemished portion of a surface is selected for analysis; and in environmental science, an artificial soil bed of controlled homogeneity is subjected to uniformly distributed water droplets, to model the effect of actual rain on a real soil surface. In some applications, however, the environment is uncontrollable, with the result that measurements are subject to irregular fluctuations that are not so plausibly modeled by fractal processes. The fluctuations may include discontinuities and nonlinear drift in the mean. Some approaches to analysis do not distinguish between this nonstationary contamination and the "background," with the result that a jump process may provide a significantly better explanation of the data than a fractal process. In this article we suggest decomposing an irregular time series into at least three components—a jump process, a nonlinear drift, and a fractal background. We identify the jump process using threshold methods, and subtract it out. Then we estimate the nonlinear drift using local regression. After the jumps and drift have been removed, the fractal background is relatively homogeneous. It may be analyzed using techniques based on the variogram, and its dimension used to quantify the "erraticism" of the environment that produced the data. © 1996 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America.

Heath, D & Platen, E 1996, 'Valuation of FX barrier options under stochastic volatility', *Asia-Pacific Financial Markets*, vol. 3, no. 3, pp. 195-215.View/Download from: Publisher's site

#### View description

This paper describes European-style valuation and hedging procedures for a class of knockout barrier options under stochastic volatility. A pricing framework is established by applying mean self-financing arguments and the minimal equivalent martingale measure. Using appropriate combinations of stochastic numerical and variance reduction procedures we demonstrate that fast and accurate valuations can be obtained for down-and-out call options for the Heston model. © 1996 Kluwer Academic Publishers.

Kloeden, P, Platen, E, Schurz, H & Sorensen, M 1996, 'On Effects Of Discretization On Estimators Of Drift Parameters For Diffusion Processes', *Journal Of Applied Probability*, vol. 33, no. 4, pp. 1061-1076.View/Download from: Publisher's site

#### View description

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modem numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators

Platen, E & Hofmann, N 1996, 'Stability of superimplicit numerical methods for stochastic differential equations', *Fields Institute Communications*, vol. 9, pp. 93-104.

Platen, E & Rebolledo, R 1996, 'Principles For Modelling Financial Markets', *Journal Of Applied Probability*, vol. 33, no. 3, pp. 601-613.View/Download from: Publisher's site

#### View description

The paper introduces an approach focused towards the modelling of dynamics of financial markets. It is based on the three principles of market clearing, exclusion of instantaneous arbitrage and minimization of increase of arbitrage information. The last

Kloeden, P, Platen, E & Hofmann, N 1995, 'Extrapolation Methods For The Weak Approximation Of Ito Diffusions', *Siam Journal On Numerical Analysis*, vol. 32, no. 5, pp. 1519-1534.View/Download from: Publisher's site

#### View description

Higher-order weak extrapolation methods for the approximation of functionals of Ito diffusions are considered. Under appropriate regularity conditions it is shown that extrapolations allow a considerable increase in the weak order of convergence of a dis

Platen, E 1995, 'On Weak Implicit And Predictor-corrector Methods', *Mathematics And Computers In Simulation*, vol. 38, no. 1-3, pp. 69-76.View/Download from: Publisher's site

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The paper gives a short survey on weak schemes for stochastic differential equations and discusses several implicit and predictor-corrector type methods.

Hofmann, N & Platen, E 1994, 'Stability Of Weak Numerical Schemes For Stochastic Differential-equations', *Computers & Mathematics With Applications*, vol. 28, no. 10-12, pp. 45-57.View/Download from: Publisher's site

#### View description

This paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations wi

Platen, E & Rebolledo, R 1994, 'Pricing Via Anticipative Stochastic Calculus', *Advances In Applied Probability*, vol. 26, no. 4, pp. 1006-1021.View/Download from: Publisher's site

#### View description

The paper proposes a general model for pricing of derivative securities. The underlying dynamics follows stochastic equations involving anticipative stochastic integrals. These equations are solved explicitly and structural properties of solutions are st

Kloeden, PE, Platen, E & Hofmann, N 1992, 'Stochastic differential equations: applications and numerical methods'.

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The well known numerical methods for deterministic differential equations are inadequate for stochastic differential equations (SDEs) on account of the subtle differences between the deterministic and stochastic calculi. New higher order numerical methods for SDEs now allow hitherto intractable problems, particularly in the engineering sciences, to be tackled effectively. As a simple illustration the validity of closure schemes is investigated with a comparison of the first and second moments of a nonlinear scalar SDE determined by direct estimation with a second order numerical method and by the exact solution of a truncated system of ordinary differential equations for the moments. (Authors)

Kloeden, PE, Platen, E & Hofmann, N 1992, 'Stochastic differential equations: applications and numerical methods'.

#### View description

The well known numerical methods for deterministic differential equations are inadequate for stochastic differential equations (SDEs) on account of the subtle differences between the deterministic and stochastic calculi. New higher order numerical methods for SDEs now allow hitherto intractable problems, particularly in the engineering sciences, to be tackled effectively. As a simple illustration the validity of closure schemes is investigated with a comparison of the first and second moments of a nonlinear scalar SDE determined by direct estimation with a second order numerical method and by the exact solution of a truncated system of ordinary differential equations for the moments. (Authors)

Hofmann, N, Platen, E & Schweizer, M 1992, 'Option Pricing Under Incompleteness and Stochastic Volatility', *Mathematical Finance*, vol. 2, no. 3, pp. 153-187.View/Download from: Publisher's site

#### View description

We consider a very general diffusion model for asset prices which allows the description of stochastic and past‐dependent volatilities. Since this model typically yields an incomplete market, we show that for the purpose of pricing options, a small investor should use the minimal equivalent martingale measure associated to the underlying stock price process. Then we present stochastic numerical methods permitting the explicit computation of option prices and hedging strategies, and we illustrate our approach by specific examples. Copyright © 1992, Wiley Blackwell. All rights reserved

Kloeden, P & Platen, E 1992, 'Higher-order Implicit Strong Numerical Schemes For Stochastic Differential-equations', *Journal Of Statistical Physics*, vol. 66, no. 1-2, pp. 283-314.View/Download from: Publisher's site

#### View description

Higher-order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed. These are based on a stochastic Taylor expansion and converge strongly to the corresponding solution of the stochastic differential equat

Kloeden, P, Platen, E & Wright, IA 1992, 'The Approximation Of Multiple Stochastic Integrals', *Stochastic Analysis And Applications*, vol. 10, no. 4, pp. 431-441.View/Download from: Publisher's site

#### View description

A method for approximating the multiple stochastic integrals appearing in stochastic Taylor expansions is proposed. It is based on a series expansion of the Brownian bridge process. Some higher order time discrete approximations for the simulation of Ito

Kloeden, PE, Platen, E & Hofmann, N 1992, 'Stochastic differential equations: applications and numerical methods'.

#### View description

The well known numerical methods for deterministic differential equations are inadequate for stochastic differential equations (SDEs) on account of the subtle differences between the deterministic and stochastic calculi. New higher order numerical methods for SDEs now allow hitherto intractable problems, particularly in the engineering sciences, to be tackled effectively. As a simple illustration the validity of closure schemes is investigated with a comparison of the first and second moments of a nonlinear scalar SDE determined by direct estimation with a second order numerical method and by the exact solution of a truncated system of ordinary differential equations for the moments. (Authors)

Kloeden, PE, Platen, E & Hofmann, N 1992, 'Stochastic differential equations: applications and numerical methods', *Stochastic hydraulics '92. Proc. IAHR symposium, Taipei, 1992*, pp. 75-81.

#### View description

The well known numerical methods for deterministic differential equations are inadequate for stochastic differential equations (SDEs) on account of the subtle differences between the deterministic and stochastic calculi. New higher order numerical methods for SDEs now allow hitherto intractable problems, particularly in the engineering sciences, to be tackled effectively. As a simple illustration the validity of closure schemes is investigated with a comparison of the first and second movements of a nonlinear scalar SDE determined by direct estimation with a second order numerical method and by the exact solution of a truncated system of ordinary differential equations for the moments. -Authors

Platen, E 1992, 'Higher-order weak approximation of ito diffusions by markov chains', *Probability in the Engineering and Informational Sciences*, vol. 6, no. 3, pp. 391-408.View/Download from: Publisher's site

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This paper proposes a method that allows the construction of discrete-state Markov chains approximating an Ito-diffusion process. The transition probabilities of the Markov chains are chosen in such a way that functionals converge with a desired weak order with respect to vanishing step size under sufficient smoothness assumptions. © 1992, Cambridge University Press. All rights reserved.

Kloeden, P & Platen, E 1991, 'Relations Between Multiple Ito And Stratonovich Integrals', *Stochastic Analysis And Applications*, vol. 9, no. 3, pp. 311-321.View/Download from: Publisher's site

#### View description

In this paper we investigate relations within and between sets of multiple Ito and Stratonovich integrals which are useful in the application of stochastic Taylor expansions of Ito processes. We obtain formulae expressing multiple Ito integrals in terms

Kloeden, P & Platen, E 1991, 'Stratonovich And Ito Stochastic Taylor Expansions', *Mathematische Nachrichten*, vol. 151, pp. 33-50.View/Download from: Publisher's site

#### View description

The Stratonovich stochastic Taylor formula for diffusion processes is stated and proved. It has a simpler structure and is a more natural generalization of the deterministic Taylor formula than the Ito stochastic Taylor formula.

KLOEDEN, PE, PLATEN, E & SCHURZ, H 1991, 'THE NUMERICAL SOLUTION OF NONLINEAR STOCHASTIC DYNAMICAL SYSTEMS: A BRIEF INTRODUCTION', *International Journal of Bifurcation and Chaos*, vol. 01, no. 02, pp. 277-286.View/Download from: Publisher's site

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The numerical analysis of stochastic differential equations, currently undergoing rapid development, differs significantly from its deterministic counterpart due to the peculiarities of stochastic calculus. This article presents a brief, pedagogical introduction to the subject from the perspective of stochastic dynamical systems. The key tool is the stochastic Taylor expansion. Strong, pathwise approximations are distinguished from weak, functional approximations, and their role in stability with Lyapunov exponents and stiffness is discussed.

Mikulevicius, R & Platen, E 1991, 'Rate Of Convergence Of The Euler Approximation For Diffusion-processes', *Mathematische Nachrichten*, vol. 151, pp. 233-239.View/Download from: Publisher's site

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Platen, E 1990, 'A Stochastic Approach To Hopping Transport In Semiconductors', *Journal Of Statistical Physics*, vol. 59, no. 5-6, pp. 1329-1353.View/Download from: Publisher's site

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Kloeden, P & Platen, E 1989, 'A Survey Of Numerical-methods For Stochastic Differential-equations', *Stochastic Hydrology And Hydraulics*, vol. 3, no. 3, pp. 155-178.View/Download from: Publisher's site

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Platen, E 1989, 'A Law Of Large Numbers For Wide-range Exclusion Processes In Random-media', *Stochastic Processes And Their Applications*, vol. 31, no. 1, pp. 33-49.View/Download from: Publisher's site

#### View description

The paper presents a law of large numbers for the asymptotic macroscopic nonequilibrium dynamics of wide range exclusion processes with births and deaths on a random set of sites.

Mikulevicius, R & Platen, E 1988, 'Time Discrete Taylor Approximations For Ito Processes With Jump Component', *Mathematische Nachrichten*, vol. 138, pp. 93-104.View/Download from: Publisher's site

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Brehmer, L, Platen, E, Fanter, D & Liemant, A 1987, 'A Stochastic Description Of The Nonequilibrium Charge-carrier Transport Process In Polymer Insulators', *IEEE Transactions On Electrical Insulation*, vol. 22, no. 3, pp. 245-248.View/Download from: Publisher's site

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Brehmer, L, Platen, E, Richter, K, Fanter, D & Liemant, A 1987, 'Electronic-structure And Stochastic Hopping Transport In Polymer Insulators', *Acta Polymerica*, vol. 38, no. 6, pp. 374-377.View/Download from: Publisher's site

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Liske, H & Platen, E 1987, 'Simulation Studies On Time Discrete Diffusion Approximations', *Mathematics And Computers In Simulation*, vol. 29, no. 3-4, pp. 253-260.View/Download from: Publisher's site

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Platen, E 1987, 'Derivative Free Numerical-methods For Stochastic Differential-equations', *Lecture Notes In Control And Information Sciences*, vol. 96, pp. 187-193.

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Platen, E & Rebolledo, R 1985, 'Weak-convergence Of Semimartingales And Discretization Methods', *Stochastic Processes And Their Applications*, vol. 20, no. 1, pp. 41-58.View/Download from: Publisher's site

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Platen, E 1983, 'Approximation Of 1st Exit Times Of Diffusions And Approximate Solution Of Parabolic Equations', *Mathematische Nachrichten*, vol. 111, pp. 127-146.View/Download from: Publisher's site

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Platen, E 1982, 'A generalized Taylor formula for solutions of stochastic differential equations', *Sankhya A - Mathematical Statistics and Probability*, vol. 44, no. 2, pp. 163-172.

Platen, E 1982, 'An approximation method for a class of Ito processes with jump component', *Lithuanian Mathematical Journal*, vol. 22, no. 2, pp. 124-136.

Platen, E & Wagner, W 1982, 'On a Taylor formula for a class of Ito processes', *Probability and Mathematical Statistics*, vol. 3, no. 1, pp. 37-51.

Platen, E 1981, 'An approximation method for a class of Ito processes', *Lithuanian Mathematical Journal*, vol. 21, pp. 121-133.

Platen, E 1980, 'Weak Convergence of Approximations of I tǒ Integral Equations', *ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik*, vol. 60, no. 11, pp. 609-614.View/Download from: Publisher's site

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Weak convergence is proved for continuously interpolated approximations of Itǒ integral equations, which were presented in Wagner, Platen [8], under the assumption of existence of the 4th moment of the initial value. Copyright © 1980 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Platen, E 1977, 'Sequentielle Rangauswahlprobleme ‐ eine Erweiterung des "Secretary Problems"', *ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik*, vol. 57, no. 10, pp. 571-577.View/Download from: Publisher's site

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Für sequentielle Rangauswahlprobleme, bei denen es sich um die Auswahl von m aus n (1 ≦ m ≦ n) sequentiell beobachtbaren Objekten ohne Rückgriffsmöglichkeit mit dem Ziel der Maximierung einer vorgegebenen Ertragsfunktion handelt, wird unter Voraussetzung einer bestehenden Rangfolge für die einzelnen Objekte die allgemeine Lösung angegeben. Davon ausgehend erfolgt die Lösung des speziellen Problems der Maximierung der Wahrscheinlichkeit für die Auswahl genau der m besten Objekte. Abschließend wird das asymptotische Verhalten der dabei gewonnenen optimalen Strategien bezüglich wachsender Objektanzahl n untersucht. Copyright © 1977 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Mueller, PH & Platen, E 1974, 'Rank Strategies in Sequential Selection Problems.', *Wiss Z Tech Univ Dres*, vol. 23, no. 5, pp. 1069-1076.

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A Bellman equation for the determination of an optimal strategy is derived for the so-called "optimum selection problem" , under the assumption of an existing rank sequence for individual objects. This "optimum selection problem" involves the selection of m from n sequentially observable objects without the possibility of supplementary examination. the method is based on an additive result functional whose The results, in case of the selection of an object, depend on their rank number, on the number of the already observed and on the number of the already selected objects. In the majority of practical applications, the elementary result is a monotonous function of the rank number. It is shown that optimum strategy is pure strategy and, in the case of a monotonous results functional, it is, moreover, a so-called simple strategy. These optimal simple strategies can be relatively easily represented. Some special sequential selection problems are solved.

Platen, E & Tappe, S, 'The fundamental theorem of asset pricing for self-financing portfolios'.

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Consider a financial market with nonnegative semimartingales which does not

need to have a num\'{e}raire. We are interested in the absence of arbitrage in

the sense that no self-financing portfolio gives rise to arbitrage

opportunities, where we are allowed to add a savings account to the market. We

will prove that in this sense the market is free of arbitrage if and only if

there exists an equivalent local martingale deflator which is a multiplicative

special semimartingale. In this case, the additional savings account relates to

the finite variation part of the multiplicative decomposition of the deflator.

By focusing on self-financing portfolios, this result clarifies links between

previous results in the literature and makes the respective concepts more

realistic.

Platen, E, 'About secretary problems', *Banach Center Publications*, vol. 6, no. 1, pp. 257-266.View/Download from: Publisher's site

Platen, E & Rendek, R, 'Approximating the Growth Optimal Portfolio and Stock Price Bubbles', *International Journal of Theoretical and Applied Finance*.View/Download from: Publisher's site

Platen, E & Rendek, RJ 2012, 'Simulation of diversified portfolios in a continuous financial markets' in Zhang, T & Zhou, X (eds), *Stochastic Analysis and Applications to Finance: Essays in Honour of Jia-an Yan*, World Scientific Publishing, USA, pp. 385-410.

Platen, E & Semmler, W 2012, 'A dynamic portfolio approach to asset markets and monetary policy' in Cohen, SN, Madan, D, Siu, TK & Yang, H (eds), *Advances in Statistics, Probability and Actuarial Science: Stochastic Processes, Finance and Control*, World Scientific Publishing, USA, pp. 347-373.

Platen, E 2011, 'A benchmark approach to investing and pricing' in MacLean, LC, Thorp, EO & Ziemba, WT (eds), *The Kelly Capital Growth Investment Criterion: Theory and Practice*, World Scientific Publishing, USA, pp. 409-426.

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This paper introduces a general market modeling framework, the benchmark appma.chl which assumes the existence of the nume!'raire portfolio. This is the strictly positive portfolio that when used as benchmMk makes all benchmarked non-negati.ve portfolios sllperma.rtinga!es, that is intuitively speaking downward trending or trendless. It can be shQ'Wn to equal the Kelly portfolio, which tna.-"jmiz. es expected logarithmk utility. In several Wa.ys, the KeUy or numeraire portfolio is the "bestll performing portfolio and cannot be outperformed systematically by any other non-negative portfolio. Its use in pricing as nttmeroire leads directly to the real world pricing formula) which employs the real world probability when calculating conditional expectations. In a large regular financial market, the Kelly portfolio is shawn to be approxima.ted by well·divcnrified portfolios.

Bruti Liberati, N & Platen, E 2010, 'Monte Carlo simulation for stochastic differential equations' in al, RCE (ed), *Encyclopedia of Quantitative Finance*, Wiley, US, pp. 1271-1278.

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Bruti Liberati, N & Platen, E 2010, 'Stochastic differential equations with jumps: Simulation' in al, RCE (ed), *Encyclopedia of Quantitative Finance*, Wiley, US, pp. 1693-1697.

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Jackel, P & Platen, E 2010, 'Monte Carlo simulations' in al, RCE (ed), *Encyclopedia of Quantitative Finance*, Wiley, US, pp. 1266-1271.

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Platen, E 2010, 'Stochastic Taylor expansions' in al, RCE (ed), *Encyclopedia of Quantitative Finance*, Wiley, US, pp. 1731-1734.

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Platen, E & Bruti Liberati, N 2010, 'Stochastic differential equations: Scenario simulation' in al, RCE (ed), *Encyclopedia of Quantitative Finance*, Wiley, US, pp. 1697-1706.

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Platen, E 2008, 'Simulation methods for stochastic differential equations' in Seese, D, Weinhardt, C & Schlottmann, F (eds), *Handbook on Information Technology in Finance*, Springer, Germany, pp. 501-514.

Platen, E 2004, 'Simulation methods for stochastic differential equations' in Teugels, J & Sundt, B (eds), *Encyclopedia of Actuarial Science*, John Wiley & Sons, Chichester, UK, pp. 1559-1564.

Heath, DP & Platen, E 2002, 'Pricing and hedging of index derivatives under an alternative asset price model with endogenous stochastic volatility' in Yong, J (ed), *Recent Developments in Mathematical Finance*, World Scientific, Singapore, pp. 117-126.

Elliot, RJ & Platen, E 2001, 'Hidden Markov Chain Filtering for Generalised Bessel Processes' in Hida, T, Karandikar, RL, Kunita, H, Rajput, BS, Watanabe, S & Xiong, J (eds), *Stochastic in Finite & Infinite Dimensions*, Birkhauser Boston, Basel, Germany, pp. 123-143.

Heath, DP, Platen, E & Schweizer, M 2001, 'Numerical Comparison of Local Risk-Minimisation & Mean-Variance Hedging' in Jouini, E, Cvitanic, J & Musiela, M (eds), *Handbooks in Mathematical Finance: Option Pricing, Interest Rates & Risk Management*, Cambridge University Press, Cambridge, UK, pp. 509-537.

Platen, E 1970, 'The Natural Evolution of a Stock Index', Quantitative Methods in Finance, Sydney.

Platen, E 1970, 'Dynamics of a Well-Diversified Equity Index and Martingale Inference', Seminar Presentations, Zurich, Switzerland.

Platen, E 1970, 'Introduction to the Benchmark Approach', Workshop at Leibniz University of Hannover, Hannover, Germany.

Platen, E 1970, 'Investing for the Long Run', International Conference on Mathematical Finance, Seoul.

Platen, E 1970, 'Less Expensive Investing for the Long Run', Symposium on Optimal Stopping, Houston, USA.

Platen, E & Leisen, D 1970, 'Investing for the Long Run', Bachelier Finance Society World Congress, Dublin, Ireland.

Platen, E & Schmutz, M 1970, 'Preparing Financial Regulation for the Next Crises', 31st International Conference of Actuaries, Berlin, Germany.

Platen, E 1970, 'Long-Term Risk of a Well-Diversified Equity Index', 5th Asian Quantitative Finance Conference, Seoul, South Korea.

Platen, E 1970, 'Preparing Financial Regulation for Forthcoming Crises', AUT Mathematical Sciences Symposium, Aukland, New Zealand.

Platen, E & Schmutz, M 1970, 'Preparing Financial Regulation', 5th NUS Workshop on Risk and Regulation, Singapore.

Platen, E 1970, 'An Introduction to the Benchmark Approach', Heriot Whatt University Seminar, Edinburgh.

Platen, E 1970, 'Benchmarked Risk Minimization', 10th Risk and Stochastics Conference, London.

Platen, E 1970, 'Numeraire Portfolio Investing', 9th World Congress of the Bachelier Finance Society, New York.

Platen, E 1970, 'On Interest Rate Term Structure Modelling under the Benchmark Approach', Seminar Institute Bachelier, Paris.

Platen, E 1970, 'A Benchmark Approach to Investing Pricing and Hedging', Workshop at the Actuarial Association Austria, Vienna, Austria.

Platen, E 1970, 'Benchmarked Risk Minimization', Berlin Stochastic Analysis and Mathematical Finance Seminar, Berlin, Germany.

Platen, E 1970, 'Benchmarked Risk Minimization', ETH Mathematical Finance Seminar, Zurich, Switzerland.

Platen, E 1970, 'Numerical Methods in Finance', Workshop USI, Lugano, Switzerland.

Platen, E 2014, 'A benchmark approach to finance', 13th Winter School on Mathematical Finance, Lunteren, The Netherlands.

Platen, E 2014, 'Beyond the classical paradigm', 8th World Congress of the Bachelier Finance Society, Brussels, Belgium.

Platen, E 2014, 'Numerical solution of stochastic differential equations in finance', Current Numerical Methods and Stochastic Volatility Modelling in Quantitative Finance Workshop, Johannesburg, South Africa.

Platen, E 2014, 'Numerical solution of stochastic differential equations with jumps in finance', 7th Summer School in Mathematical Finance, Cape Town, South Africa.

Platen, E 2014, 'The affine nature of aggregate wealth dynamics', 7th Bachelier Colloquium on Mathematical Finance and Stochastic Calculus, Metabief, France.

Baldeaux, JF & Platen, E 2012, 'Computing functionals of square root and Wishart processes under the benchmark approach via exact simulation', *Monte Carlo and Quasi Monte Carlo Methods 2012, Springer Proceedings in Mathematics and Statistics*, International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Springer, Sydney, Australia, pp. 3-22.View/Download from: Publisher's site

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The aim of the paper is to show how Wishart processes can be used flexibly in financial modeling. We explain how functionals, resulting from the benchmark approach to finance, can be accurately computed via exact simulation methods. We employ Lie symmetry methods to identify explicit transition densities and explicitly computable functionals. We illustrate the proposed methods via finance problems formulated under the benchmark approach. This approach allows us to exploit conveniently the analytical tractability of the considered diffusion processes.

Bruti Liberati, N & Platen, E 2011, 'On weak predictor-corrector schemes for jump-diffusion processes in finance', *Topics in Numerical Methods for Finance: Proceedings in Mathematics and Statistics*, Numerical Methods for Finance Conference, Springer, Limerick, Ireland, pp. 1-12.

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Event-driven uncertainties such as corporate defaults, operational failures, or central bank announcements are important elements in the modeling of financial quantities. Therefore, stochastic differential equations (SDEs) of jumpdiffusion type are often used in finance. We consider in this paper weak discrete time approximations of jump-diffusion SDEs which are appropriate for problems such as derivative pricing and the evaluation of risk measures. We present regular and jump-adapted predictorcorrector schemes with first and second order of weak convergence. The regular schemes are constructed on regular time discretizations that do not include jump times, while the jump-adapted schemes are based on time discretizations that include all jump times. A numerical analysis of the accuracy of these schemes when applied to the jump-diffusion Merton model is provided.

Platen, E 2012, 'Benchmarked risk minimization', The Art of Finance 2012 FIRN Annual Conference, Hobart, Tasmania.

Platen, E 2012, 'Benchmarked risk minimization', AJOU Workshop in Financial Economics and Mathematics, Seoul, Korea.

Platen, E 1970, 'Benchmarked risk minimization in incomplete markets', The Sixth Bachelier Colloquium on Mathematical Finance and Stochastic Calculus, Metabief, France.

Platen, E 2012, 'Numerical solution of SDEs', Seminar Presentation, Ecole Centrale, Paris, France.

Platen, E 2012, 'Numerical solution of stochastic differential equations with jumps in finance', Tenth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Sydney Australia.

Baldeaux, J, Chan, L & Platen, E 2010, 'Quasi-Monte Carlo methods for derivatives on realised variance of an index under the benchmark approach', *ANZIAM Journal*, Computational Techniques and Applications Conference, Australian Mathematical Society, Sydney, Australia, pp. C727-C741.View/Download from: Publisher's site

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We apply quasi-Monte Carlo methods to the pricing of derivatives on realised variance of an index under the benchmark approach. The resulting integration problem is shown to depend on the joint density of the realised variance of the index and the terminal value of the index. Employing a transformation mapping for this joint density to the unit square reduces the diffculty of the resulting integration problem. The quasi-Monte Carlo methods compare favourably to Monte Carlo methods when applied to the given problem. © Austral. Mathematical Soc. 2011.

Hulley, H & Platen, E 2008, 'A visual criterion for identifying Ito diffusions as martingales or strict local martingales', *Seminar on Stochastic Analysis, Random Fields and Applications VI*, Seminar on Stochastic Processes, Random Fields and Applications, Springer, Ascona, Switzerland, pp. 147-157.

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It is often important, in applications of stochastic calculus to financial modelling, to know whether a given local martingale is a martingale or a strict local martingale. We address this problem in the context of a time-homogenous diffusion process with a finite lower boundary, presented as the solution of a driftless stochastic differential equation. Our main theorem demonstrates that the question of whether or not this process is a martingale may be decided simply by examining the slope of a certain increasing function. Further results establish the connection between our theorem and other results in the literature, while a number of examples are provided to illustrate the use of our criterion.

Platen, E 2011, 'A benchmark approach to quantitative finance', Seminar Presentation, Humboldt University, Berlin, Germany.

Platen, E 2011, 'A dynamic portfolio approach to monetary policy', Southern Africa Mathematical Sciences Association (SAMSA) 2011 Conference, Livingston, Zambia.

Platen, E 2011, 'On targeted pensions', Quantitative Methods in Finance 2011 Conference, Sydney Australia.

Nikitopoulos Sklibosios, C & Platen, E 1970, 'Alternative term structure models for reviewing expectation puzzles', World Finance Conference, Rhodes, Greece.

Ignatieva, K & Platen, E 2010, 'Modelling co-movements and tail dependency in the international stock market via copulae', 6th World Congress of the Bachelier Finance Society, Toronto, Canada.

Platen, E 2010, 'A benchmark approach to computative finnce', 1st R/Rmetrics Summer School and 4th User/Developer Meeting on Computational Finance and Financial Engineering, Meielisalp, Switzerland.

Platen, E 2010, 'Empirical properties of a diversified global stock index', Workshop on Stochastics, Control and Finance, London, UK.

Platen, E 2010, 'Real world pricing of long term contracts', 14th International Congress on Insurance: Mathematics and Economics, Toronto, Canada.

Platen, E 2010, 'Real world pricing of long term contracts', Seminar Presentation, The Fields Institute, Toronto, Canada.

Platen, E 2010, 'Real world pricing of long term contracts', Actuarial and Financial Mathematics Conference 2010, Brussels, Belgium.

Platen, E 2010, 'Stylized empirical facts on diversified indices', Workshop on Financial Econometrics, Toronto, Canada.

Platen, E 2010, 'The benchmark approach', Workshop on Foundations of Mathematical Finance, Toronto, Canada.

Platen, E 2010, 'Valuing guaranteed minimum death benefits', Seminar Presentation, UNSW Actuarial Studies, Sydney, Australia.

Platen, E & Rendek, RJ 2010, 'Simulation of diversified portfolios in a continuous financial market', 6th World Congress of the Bachelier Finance Society, Toronto, Canada.

Bruti Liberati, N & Platen, E 2009, 'Strong Predictor-Corrector Euler Methods for Stochastic Differential Equations', X Workshop on Quantitative Finance to the Memory of Nicola Bruti-Liberati, Milan, Italy.

Platen, E 2009, 'A benchmark approach beyond semi-martingales', Workshop on Non-Semi-Martingale Modelling in Finance, Helsinki, Finland.

Platen, E 1970, 'A benchmark approach to quantitative finance', Workshop at Enterprise Risk Management Symposium, Chicago, USA.

Platen, E 2009, 'A Variance Reduction Technique Based on Integral Representations', Seminar Presentation, Fraunhofer Institute and the University of Kaiserslauten, Kaiserslauten, Germany.

Platen, E 2009, 'Asset markets and monetary policy', Seminar Presentation, Technical University Munich, Munich, Germany.

Platen, E 2009, 'Asset markets and monetary policy', Seminar Presentation, University of Lugano, Lugano, Switzerland.

Platen, E 2009, 'Asset markets and monetary policy', Seminar Presentation, Imperial College, London, London, UK.

Platen, E 2009, 'Numerical solution of stochastic differential equations with jumps in finance', Seminar Presentation, University of Oxford, Oxford, UK.

Platen, E 2009, 'On interest rate term structure modeling under the benchmark approach', 4th General Conference on Advanced Mathematical Methods in Finance, Alesund, Norway.

Platen, E 2009, 'On interest rate term structure modeling under the benchmark approach', Seminar Presentation, Humboldt University, Berlin, Germany.

Platen, E 1970, 'Quantitative methods - Computing and numerical methods', Workshop, Ajou University, Korea.

Platen, E 1970, 'Valuing guaranteed minimum death benefit options in variable annuities under a benchmark approach', Enterprise Risk Management Symposium, Chicago, USA.

Bruti Liberati, N, Nikitopoulos Sklibosios, C, Platen, E & Schlogl, E 2009, 'Alternative defaultable term structure models', X Workshop on Quantitative Finance to the Memory of Nicola Bruti-Liberati, Milan, Italy.

Thulasiram, RK, Downing, CT, Chiarella, C, Coleman, T, Dempster, M, Dongarra, J, Duan, JC, Gao, G, Appadoo, SS, Atiya, A, Bagchi, A, Birge, J, Brabazon, A, Broadie, M, Campolieti, J, Cincotti, S, Downing, C, Gilli, M, Isaenko, S, Jacoby, G, Kumar, K, Klebaner, F, Li, X, Li, Y, Livdan, D, Lyuu, YD, Nath, GC, Okten, G, Oosterlee, CW, Ouskel, AM, Platen, E, Seco, L, Srinivasan, A, Srinivasan, R, Thenmozhi, M, Thulasiraman, P, Tsang, EPK, Wagner, A, Wang, L, Wilson, C, Wittum, G, Ing, CW & Tanaka-Yamawaki, M 2008, 'Message from PDCoF-08 Workshop Chairs', *IPDPS Miami 2008 - Proceedings of the 22nd IEEE International Parallel and Distributed Processing Symposium, Program and CD-ROM*.View/Download from: Publisher's site

Hulley, H & Platen, E 2007, 'Laplace transform identities for diffusions, with applications to rebates and barrier options', *Banach Centre Publications: Advances in Mathematics of Finance*, General AMaMeF Conference and Banach Centre Conference, Polska Akademia Nauk, Bedlewo, Poland, pp. 139-157.

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Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms of some functions of first-passage times for the diffusion. These results are applied to the special case of squared Bessel processes with killing or reflecting boundaries. In particular, we demonstrate how the above-mentioned integral identity enables us to derive the transition density of a squared Bessel process killed at the origin, without the need to invert a Laplace transform. Finally, as an application, we consider the problem of pricing barrier options on an index described by the minimal market model.

Platen, E 2008, 'A Benchmark Approach to Finance', Mathematics in Finance Conference, Cape Town, South Africa.

Platen, E 2008, 'A benchmark approach to quantitative finance', Summer School: Risk Theory and Related Fields, European Mathematical Society, Bedlewo, Poland.

Platen, E 2008, 'A Unifying Approach to Asset Pricing', International Conference on Price, Liquidity and Credit Risks, Konstanz, Germany.

Platen, E 2008, 'Conditions for Martingales with Applications in Finance', Conference on Stochastic Analysis, Random Fields and Applications, Ascona, Switzerland.

Platen, E 2008, 'Honest Times in Financial Modelling', Monash-Ritsumeikan Symposium on Probability and Related Fields, Melbourne, Australia.

Platen, E 2008, 'Law of the Minimal Price', Workshop on Mathematics in Finance, Cape Town, South Africa.

Platen, E 2008, 'Numerical solution of stochastic differential equations', Workshop at Third International Conference on Mathematics in Finance, Kruger National Park, South Africa.

Platen, E 2008, 'On Interest Rate Term Structure Modelling Under the Benchmark Approach', Third International Conference on Mathematics in Finance, Kruger National Park, South Africa.

Platen, E 2008, 'Portfolio Optimization Under Partial Information', Seminar Paper, Boston University, Boston, USA.

Platen, E 2008, 'Properties of a Diversified World Stock Index', Seminar Presentation, Princeton University, New York, USA.

Platen, E 2008, 'The Law of the Minimal Price', Seminar Presentation, Columbia University, New York, USA.

Platen, E 2008, 'The Law of the Minimal Price', Seminar Paper, University of Santa Barbara, Santa Barbara, USA.

Platen, E 2008, 'The Law of the Minimal Price', Frankfurt MathFinance Conference 2008, Frankfurt, Germany.

Platen, E 2008, 'The Law of the Minimal Price', The 3rd General AMaMeF Conference - Advances in Mathematical Finance, Pitesti, Romania.

Platen, E 2008, 'Valuing Guaranteed Minimum Death Benefit Options', Quantitative Methods in Finance 2008 Conference, Sydney, Australia.

Bruti Liberati, N, Nikitopoulos Sklibosios, C & Platen, E 2008, 'Monte-Carlo simulations of alternative defaultable term structure models', Bachelier Finance Society 5th World Congress, London, UK.

Nikitopoulos Sklibosios, C, Bruti Liberati, N, Platen, E & Schlogl, E 2008, 'Real-world pricing for defaultable term structure models', Bachelier Finance Society 5th World Congress, London, UK.

Platen, E 2007, 'Benchmark Approach to Continuous Time Finance', 3rd Australian Postgraduate Workshop on Stochastic Processes and Applications, Sydney, Australia.

Platen, E 2007, 'Extreme Maturity Options', Advances in Mathematics of Finance, Bedelow, Poland.

Platen, E 2007, 'Numerical Solution of SDEs with Jumps in Finance', International Colloquium on Stochastic and Potential Analysis, Hammamut, Tunisia.

Platen, E 2007, 'Numerical Solutions of Stochastic Differential Equations with Jumps in Finance', Stochastic Processes: Theory and Applications Conference, Bressanone, Italy.

Platen, E 2007, 'Portfolio Optimization Under Partial Information', Advanced Mathematical Methods in Finance Conference, Toulouse, France.

Platen, E 2007, 'Simulation of High-Dimensional Models in Finance', Workshop on High-Dimensional Approximation, Canberra, Australia.

Bruti Liberati, N, Nikitopoulos Sklibosios, C, Platen, E & Schlogl, E 2007, 'Defaultable term structure models under the benchmark approach', Quantitative Methods in Finance 2007 Conference, Sydney, Australia.

Bruti Liberati, N, Nikitopoulos Sklibosios, C, Platen, E & Schlogl, E 2007, 'Real-World Pricing for Defaultable Term Structure Models', CREDIT 2007, Venice, Italy.

Bruti Liberati, N & Platen, E 2007, 'On the weak approximation of jump-diffusion processes with applications in finance', *Proceedings of the VII Workshop on Quantitative Finance*, VII Workshop on Quantitative Finance, University of Perugia, Italy, Perugia, Italy, pp. 1-96.

Bruti Liberati, N & Platen, E 2006, 'Predictor-corrector schemes for jump-diffusion processes.', International Conference on Numerical Methods for Finance, Dublin, Ireland.

Bruti Liberati, N & Platen, E 2006, 'Weak predictor-corrector methods for jump diffusions in finance', 5th National Symposium on Financial Mathematics, Melbourne, Australia.

Le, T & Platen, E 2006, 'Approximating the growth optimal portfolio with a diversified world stock index.', *3rd National Symposium on Financial Mathematics*, 3rd National Symposium on Financial Mathematics, Melbourne, Australia.

Platen, E 2004, 'A benchmark approach to finance', *MATHEMATICAL FINANCE*, Workshop on Mathematical Finance and Insurance, WILEY, Yellow Mt, PEOPLES R CHINA, pp. 131-151.View/Download from: Publisher's site

Platen, E 2006, 'A benchmark approach to portfolio optimization and derivative pricing.', *Statistical Modeling in Finance Conference*, Statistical Modeling in Finance Conference, Philadelphia, USA.

Platen, E 2006, 'A benchmark approach to portfolio optimization and derivative pricing.', *First Conference of Advanced Mathematical Methods for Finance*, First Conference of Advanced Mathematical Methods for Finance, Antalya, Turkey.

Platen, E 2006, 'A parsimonious financial market model in a jump diffusion setting', Workshop on Mathematical Finance and Insurance, Lijiang, China.

Platen, E 2006, 'A unified approach to portfolio optimization and derivative pricing.', *Conference on Risk Management*, Conference on Risk Management, Ascona, Switzerland.

Platen, E 2004, 'Capital asset pricing for markets with intensity based jumps', *Stochastic Finance*, International Conference on Stochastic Finance 2004, Springer, Lisboa, Portugal, pp. 157-182.

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DP0343913 This paper proposes a unified framework for portfolio optimization, derivative pricing, modeling and risk measurement in financial markets with security price processes that exhibit intensity based jumps. It is based on the natural assumption that invest

Platen, E 2006, 'On the pricing and hedging of long dated zero coupon bonds.', *5th National Symposium on Financial Mathematics*, 5th National Symposium on Financial Mathematics, Melbourne, Australia.

Platen, E 2006, 'Pricing and hedging extreme maturity contracts under the benchmark approach.', *Quantitative Methods in Finance 2006 Conference*, Quantitative Methods in Finance 2006 Conference, Sydney, Australia.

Platen, E 2006, 'Pricing and hedging of long dated zero coupon bonds.', *2006 DAIWA International Workshop on Financial Engineering*, DAIWA International Workshop on Financial Engineering, Tokyo, Japan.

Platen, E 2006, 'Sharpe ratio maximization and expected utility when asset prices have jumps.', *2006 Symposium on Credit Risk, Extreme Values and Actuarial Studies*, Symposium on Credit Risk, Extreme Values and Actuarial Studies, Canberra, Australia.

Bruti Liberati, N, Nikitopoulos Sklibosios, C & Platen, E 2006, 'Heath Jarrow Morton equation for jump-diffusions under the benchmark approach', *2nd International Symposium on Economic Theory, Policy & Applications*, 2nd International Symposium on Economic Theory, Policy & Applications, -, Athens, Greece.

Bruti Liberati, N, Nikitopoulos Sklibosios, C & Platen, E 2006, 'On the strong approximation of jump-diffusion processes', Stochastic Calculus and its Applications to Quantitative Finance and Electrical Engineering, Calgary, Canada.

Hulley, H, Heath, DP & Platen, E 1970, 'A comparative study of performance robustness for equity index models', *4th National Symposium on Financial Mathematics*, 4th National Symposium on Financial Mathematics, -, Day Dream Island, Australia.

Hulley, H, Heath, DP & Platen, E 1970, 'A comparative study of performance robustness for equity index models', *Mathematics in Finance International Conference*, Mathematics in Finance International Conference, -, Kruger National Park, South Africa.

Miller, S & Platen, E 1970, 'Analytic pricing of European contingent claims under the real world measure', 4th National Symposium on Financial Mathematics, -, Day Dream Island, Australia.

Platen, E 1970, 'A benchmark approach to finance', *Workshop on Stochastic Analysis and Applications in Finance*, Workshop on Stochastic Analysis and Applications in Finance, -, Leipzig, Germany.

Platen, E 1970, 'A general benchmark model for stochastic jump sizes', CMA National Research Symposium on Levy Process Theory and its Applications in Finance, -, Canberra, Australia.

Platen, E 1970, 'A unified framework for portfolio optimisation and asset pricing', *49th Annual Meeting of the Australian Mathematical Society*, 49th Annual Meeting of the Australian Mathematical Society, -, Perth, Australia.

Platen, E 1970, 'A unified framework for portfolio optimization and asset pricing', Developments in Quantitative Finance, -, Cambridge, UK.

Platen, E 1970, 'Investment for the short and long run', *Past, Present and Future in Investment Management*, Past, Present and Future in Investment Management, -, Taipei, Taiwan.

Platen, E 1970, 'Investment for the short and long run', *Econophysics Colloquium*, Econophysics Colloquium Canberra, -, Canberra, Australia.

Platen, E 2005, 'On the role of the growth optimal portfolio in finance', *7th JAFEE International Conference*, 7th JAFEE International Conference, -, Tokyo, Japan.

Platen, E 2005, 'On the role of the growth optimal portfolio in finance', *Mathematical Finance Workshop*, Mathematical Finance Workshop, -, Frankfurt, Gremany.

Platen, E 1970, 'On the role of the growth optimal portfolio in finance', *4th National Symposium on Financial Mathematics*, 4th National Symposium on Financial Mathematics, -, Day Dream Island, Australia.

Platen, E 1970, 'Portfolio optimization and derivative pricing in a jump diffusion market', *Quantitative Methods in Finance 2005 Conference*, Quantitative Methods in Finance 2005 Conference, -, Sydney, Australia.

Platen, E, Hulley, H & Miller, S 1970, 'Benchmarking and fair pricing applied to two market models', *Conference on Stochastic Calculus and its Applications to Quantitative finance and Electrical Engineering*, Conference on Stochastic Calculus and its Applications to Quantitative finance and Electrical Engineering, -, Calgary, Canada.

Bruti Liberati, N, Platen, E, Martini, F & Piccardi, M 2005, 'A multi-point distributed random variable accelerator for Monte Carlo simulation in finance', *Proceedings of 5th International Conference On Intelligent Systems Design And Applications*, International Conference on Intelligent Systems Designs and Applications, IEEE, Wroclaw, Poland, pp. 532-537.View/Download from: Publisher's site

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The pricing and hedging of complex derivative securities via Monte Carlo simulations of stachastic deferential equations constitutes an intensive computational task. To achive real time execution, as often required by financial institutions, one needs highly efficient implementations of the multi-point distributed random variables underlying the simulations. In this paper a fast and flexible dedicated hardware solution is proposed. A comparative performance analysis demonstrates that the hardware solution is bottleneck-free and flexible, and significantly increases the computational efficiency of the software solution.

Liberati, NB, Platen, E, Martini, F & Piccardi, M 2005, 'An FPGA generator for multipoint distributed random variables', *ACM/SIGDA International Symposium on Field Programmable Gate Arrays - FPGA*, p. 280.

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Multi-point distributed random variables whose moments match those of a Gaussian random variable up to a certain order play an important role in Monte Carlo simulations of weak approximations of stochastic differential equations. In applications such as finance, where "real time" execution is required, there is a strong need for highly efficient implementations. In this paper a fast and flexible dedicated hardware solution on a Field Programmable Gate Array (FPGA) is presented. A comparative performance analysis between a software-only and the proposed hardware solution demonstrates that the FPGA solution is bottleneck-free, retains the flexibility of the software solution and significantly increases the computational efficiency.

Martini, F, Piccardi, M, Bruti Liberati, N & Platen, E 2005, 'A hardware generator for multi-point distributed random variables', *2005 IEEE International Symposium On Circuits And Systems (Iscas), Vols 1-6, Conference Proceedings*, International Symposium on Circuits and Systems, IEEE Computer Society Press, Kobe, Japan, pp. 1702-1705.View/Download from: Publisher's site

#### View description

Monte Carlo simulation of weak approximations of stochastic differential equations constitutes an intensive computational task. In applications such as finance, for instance, to achieve "real time" execution, as often required, one needs highly efficient implementations of the multi-point distributed random number generator underlying the simulations. In this paper a fast and flexible dedicated hardware solution on a field programmable gate array is presented. A comparative performance analysis between a software-only and the proposed hardware solution demonstrates that the hardware solution is bottleneck-free, retains the flexibility of the software solution and significantly increases the computational efficiency. Moreover, simulations in applications such as economics, insurance, physics, population dynamics, epidemiology, structural mechanics, chemistry and biotechnology can benefit from the obtained speedup.

Bruti Liberati, N & Platen, E 2004, 'On the efficiency of simplified weak Taylor schemes for Monte Carlo simulation in finance.', *Computational Science - ICCS 2004*, International Conference on Computational Science, Springer-Verlag, Wroclaw, Poland, pp. 771-778.

Heath, DP & Platen, E 1970, 'Local volatility function models under a benchmark approach.', *Daiwa International Workshop on Financial Engineering*, Daiwa International Workshop on Financial Engineering, DAIWA, Tokyo/ Kyoto, Japan.

Kelly, L, Platen, E & Sorensen, M 1970, 'Estimation for discretely observed diffusions using transform functions.', *National symposium on probability and its applications*, National Symposium on Probability and its Applications, -, Canberra, Australia.

Platen, E 1970, 'A benchmark approach to risk management.', *Stochastic Finance 2004*, Stochastic Finance 2004, -, Lisbon, Portugal.

Platen, E 2003, 'A benchmark framework for risk management', *Proceedings of the Ritsumeikan International Symposium: "Stochastic Processes and Applications to Mathematical Finance"*, International Symposium on "Stochastic Processes and Applications to Mathematical Finance", World Scientific, Kusatsu, Shiga, Japan, pp. 305-335.

Platen, E 1970, 'Modelling the expected value of a diversified world index.', *3rd World Congress, Bachelier Finance Society*, 3rd World Congress, Bachelier Finance Society, -, Chicago, USA.

Platen, E 1970, 'Modelling the volatility and expected value of a diversified world index.', *International workshop on mathematical finance and insurance*, International Workshop on Mathematical Finance and Insurance, -, Huang Shan, China.

Platen, E 1970, 'Numerical solution of stochastic differential equations', *Workshop on mathematical methods in finance*, Workshop on Mathematical Methods in Finance, -, Melbourne, Australia.

Platen, E 2003, 'Pricing and hedging for incomplete jump diffusion benchmark models', *Mathematics of Finance: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance*, AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance, American Mathematical Society, Snowbird, Utah, USA, pp. 287-301.

Heath, DP & Platen, E 2003, 'Pricing of index options under a minimal market model with lognormal scaling', *Workshop on Mathematical Finance*, Workshop on Mathematical Finance, --, St Johns, Canada.

Platen, E 1970, 'A benchmark framework for financial markets', *Conference of Stochastic Processes and Mathematical Finance*, Conference of Stochastic Processes and Mathematical Finance, --, Kyoto, Japan.

Platen, E 2003, 'A class of complete benchmark models', *2nd National Symposium on Financial Mathematics*, 2nd National Symposium on Financial Mathematics, --, Sydney.

Platen, E 2003, 'An incomplete benchmark model', *AMS-SIAM Mathematical Finance Conference*, AMS-SIAM Mathematical Finance Conference, --, Salt Lake City, USA.

Platen, E 1970, 'Modeling the volatility and expected value of a diversified world index', *Quantitative Methods in Finance 2003 Conference*, Quantitative Methods in Finance, --, Sydney.

Platen, E 2003, 'The two-factor interest rate term structure minimal market model', *Japanese Association Financial Econometrics and Engineering Meeting*, Japanese Association Financial Econometrics and Engineering Meeting, --, Tokyo, Japan.

Platen, E 2002, 'Financial markets model', *RISK 2003 Conference*, RISK 2003 Conference.

Heath, D & Platen, E 2001, 'Pricing and hedging of index derivatives under an alternative asset price model with endogenous stochastic volatility', *Recent developments in mathematical finance: Proceedings of the international conference on mathematical finance*, International Conference on Mathematical Finance, World Scientific, Shanghai, China, pp. 117-126.

Platen, E 1970, 'A modified constant elasticity of variance model', *Quantitative Finance Conference*, Quantitative Finance Conference, Singapore.

Platen, E 1970, 'Arbitrage in continuous complete markets', *2nd World Congress of the Bachelier Finance Society*, 2nd World Congress of the Bachelier Finance Society, Crete, Greece.

Platen, E 1970, 'Diversified portfolios in a benchmark framework', *Quantitative Methods in Finance 2002 Conference*, Quantitative Methods in Finance 2002 Conference, Sydney.

Platen, E 1970, 'Diversified portfolios in a benchmark framework', *Conference on Computational Finance*, Conference on Computational Finance, Singapore.

Platen, E 2001, 'A Minimal Financial Market Model', *Proceedings of Mathematical Finance: Workshop of the Mathematical Finance Research Project*, Mathematical Finance: Workshop of the Mathematical Finance Research Project, Birkhausen Verlag, Konstanz, pp. 293-301.

Elliott, R, Fischer, P & Platen, E 1999, 'Hidden Markov filtering for a mean reverting interest rate model', *Proceedings of the IEEE Conference on Decision and Control*, pp. 2782-2787.

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A Hidden Markov Model with mean reverting characteristics is considered as a model for financial time series, particularly interest rates. The optimal filter for the state of the hidden Markov chain is obtained. A number of auxiliary filters are obtained that enable the parameters of the model to be estimated using the EM algorithm. A simulation study demonstrates the feasibility of this approach.

Muller, I, Brehmer, L, Liemant, A & Platen, E 1992, 'Influence of trap distributions on the surface potential kinetics in disordered materials', *Proceedings - International Symposium on Electrets*, pp. 84-89.

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The surface potential kinetics for disordered materials described by Gaussian trap distributions was calculated on the basis of a stochastic hopping model. It is shown that an exact estimation of the transit time is possible only for narrow trap distributions. The transit time increases with the square of the standard deviation of the Gaussian trap distribution.

Mueller, I, Brehmer, L, Geiss, D, Platen, E, Liemant, A & Wagner, W 1988, 'Stochastic hopping transport in disordered polymers', pp. 394-398.

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A stochastic approach to electronic charge transport in disordered materials based on a hopping mechanism is presented. The field-strength dependence of the space charge evolution is calculated for a defined concentration and distribution of localized levels. The numerical solution procedure and first results, taking into consideration the temperature and field strength of the process, are presented. The influence of the field strength on charge-carrier evolution is seen quantitatively.

Brehmer, L, Platen, E, Fanter, D, Liemant, A & Geiss, D 1985, 'STOCHASTIC DESCRIPTION OF THE NON-EQUILIBRIUM CHARGE-CARRIER TRANSPORT PROCESS IN POLYMER INSULATORS.', pp. 310-315.

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A balance equation of the charge carrier transport process in the insulator regime (long-time process, time dependent injection, long-time stable space charges) is established. The equation describes the stochastic hopping process between localized levels having random distribution in space and of energy. The specific aspect of the model is the stochastic treatment of nonequilibrium dynamics of the space charge.

Platen, E 2020, 'Stochastic Modelling of the COVID-19 Epidemic'.

Platen, E & Rendek, R 2019, 'Dynamics of a Well-Diversified Equity Index'.

Sun, J & Platen, E 2019, 'Benchmarked Risk Minimizing Hedging Strategies for Life Insurance Policies'.

Sun, J, Fergusson, K, Platen, E & Shevchenko, PV 2019, 'Fair Pricing of Variable Annuities with Guarantees under the Benchmark Approach'.

Baldeaux, J, Ignatieva, K & Platen, E 2018, 'Detecting money market bubbles'.

Fontana, C, Pelger, M & Platen, E 2018, 'On the Existence of Sure Profits via Flash Strategies'.

Platen, E & Rendek, R 2018, 'Market Efficiency and Growth Optimal Portfolio'.

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The paper predicts an Efficient Market Property for the equity market, where

stocks, when denominated in units of the growth optimal portfolio (GP), have

zero instantaneous expected returns. Well-diversified equity portfolios are

shown to approximate the GP, which explains the well-observed good performance

of equally weighted portfolios. The proposed hierarchically weighted index

(HWI) is shown to be an even better proxy of the GP. It sets weights equal

within industrial and geographical groupings of stocks. When using the HWI as

proxy of the GP the Efficient Market Property cannot be easily rejected and

appears to be very robust.

Rudd, R, McWalter, TA, Kienitz, J & Platen, E 2018, 'Quantization Under the Real-world Measure: Fast and Accurate Valuation of Long-dated Contracts'.

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This paper provides a methodology for fast and accurate pricing of the

long-dated contracts that arise as the building blocks of insurance and pension

fund agreements. It applies the recursive marginal quantization (RMQ) and joint

recursive marginal quantization (JRMQ) algorithms outside the framework of

traditional risk-neutral methods by pricing options under the real-world

probability measure, using the benchmark approach. The benchmark approach is

reviewed, and the real-world pricing theorem is presented and applied to

various long-dated claims to obtain less expensive prices than suggested by

traditional risk-neutral valuation. The growth-optimal portfolio (GOP), the

central object of the benchmark approach, is modelled using the time-dependent

constant elasticity of variance model (TCEV). Analytic European option prices

are derived and the RMQ algorithm is used to efficiently and accurately price

Bermudan options on the GOP. The TCEV model is then combined with a $3/2$

stochastic short-rate model and RMQ is used to price zero-coupon bonds and

zero-coupon bond options, highlighting the departure from risk-neutral pricing.

Fergusson, K & Platen, E 2017, 'Less-Expensive Valuation of Long Term Annuities Linked to Mortality, Cash and Equity'.

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This paper proposes a paradigm shift in the valuation of long term annuities,

away from classical no-arbitrage valuation towards valuation under the real

world probability measure. Furthermore, we apply this valuation method to two

examples of annuity products, one having annual payments linked to a mortality

index and the savings account and the other having annual payments linked to a

mortality index and an equity index with a guarantee that is linked to the same

mortality index and the savings account. Out-of-sample hedge simulations

demonstrate the effectiveness of real world valuation.

In contrast to risk neutral valuation, which is a form of relative valuation,

the long term average excess return of the equity market comes into play.

Instead of the savings account, the num\'eraire portfolio is employed as the

fundamental unit of value in the analysis. The num\'eraire portfolio is the

strictly positive, tradable portfolio that when used as benchmark makes all

benchmarked nonnegative portfolios supermartingales. The benchmarked real world

value of a benchmarked contingent claim equals its real world conditional

expectation. This yields the minimal possible value for its hedgeable part and

minimizes the fluctuations for its benchmarked hedge error. Under classical

assumptions, actuarial and risk neutral valuation emerge as special cases of

the proposed real world valuation. In long term liability and asset valuation,

the proposed real world valuation can lead to significantly lower values than

suggested by classical approaches when an equivalent risk neutral probability

measure does not exist.

Leisen, DPJ & Platen, E 2017, 'Investing for the Long Run'.

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This paper studies long term investing by an investor that maximizes either expected utility from terminal wealth or from consumption. We introduce the concepts of a generalized stochastic discount

factor (SDF) and of the minimum price to attain target payouts. The paper finds that the dynamics of the SDF needs to be captured and not the entire market dynamics, which simplifies significantly practical

implementations of optimal portfolio strategies. We pay particular attention to the case where the SDF is equal to the inverse of the growth-optimal portfolio in the given market. Then, optimal wealth

evolution is closely linked to the growth optimal portfolio. In particular, our concepts allow us to reconcile utility optimization with the practitioner approach of growth investing. We illustrate

empirically that our new framework leads to improved lifetime consumption-portfolio choice and asset allocation strategies.

McWalter, T, Rudd, R, Kienitz, J & Platen, E 2017, 'Appendix: Recursive Marginal Quantization of Higher-Order Schemes'.

Platen, E & Taylor, D 2017, 'Loading Pricing of Long-Dated, Insurance-Type Contracts'.

Rudd, R, McWalter, TA, Kienitz, J & Platen, E 2017, 'Fast Quantization of Stochastic Volatility Models'.

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Recursive Marginal Quantization (RMQ) allows fast approximation of solutions to stochastic differential equations in one-dimension. When applied to two factor models, RMQ is inefficient due to the

fact that the optimization problem is usually performed using stochastic methods, e.g., Lloyd's algorithm or Competitive Learning Vector Quantization. In this paper, a new algorithm is proposed that allows

RMQ to be applied to two-factor stochastic volatility models, which retains the efficiency of gradient-descent techniques. By margining over potential realizations of the volatility process, a significant

decrease in computational effort is achieved when compared to current quantization methods. Additionally, techniques for modelling the correct zero-boundary behaviour are used to allow the new algorithm to

be applied to cases where the previous methods would fail. The proposed technique is illustrated for European options on the Heston and Stein-Stein models, while a more thorough application is considered in

the case of the popular SABR model, where various exotic options are also priced.

Gnoatto, A, Grasselli, M & Platen, E 2016, 'A Penny Saved is a Penny Earned: Less Expensive Zero Coupon Bonds'.

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This paper introduces a more general modeling world than available under the classical noarbitrage paradigm in finance. New research questions and interesting related econometric studies emerge naturally. To explain in this

paper the new approach and illustrate first important consequences, we show how to hedge a zero coupon bond with a smaller amount of initial capital than required by the classical risk neutral paradigm, whose (trivial) hedging

strategy does not suggest to invest in the risky assets. Long dated zero coupon bonds we derive, invest first primarily in risky securities and when approaching more and more the maturity date they increase also more and more the

fraction invested in fixed income. The conventional wisdom of financial planners suggesting investor to invest in risky securities when they are young and mostly in fixed income when they approach retirement, is here made rigorous.

The main reason for the existence of less expensive zero coupon bonds is the strict supermartingale property of benchmarked savings accounts under the real world probability measure, which the calibrated parameters identify under the

proposed model. We provide intuition and insight on the strict supermartingale property. The less expensive zero coupon bonds provide only one first example that is indicative for the changes that the new approach offers in the much

wider modeling world. The paper provides a strong warning for life insurers, pension fund managers and long term investors to take the possibility of less expensive products seriously to avoid the adverse consequences of the low

interest rate regimes that many developed economies face.

Platen, E & Taylor, D 2016, 'Loading Pricing of Catastrophe Bonds and Other Long-Dated, Insurance-Type Contracts'.

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Catastrophe risk is a major threat faced by individuals, companies, and entire economies. Catastrophe (CAT) bonds have emerged as a method to offset this risk and a corresponding literature has developed that attempts to

provide a market-consistent pricing methodology for these and other long-dated, insurance-type contracts. This paper aims to unify and generalize several of the widely-used pricing approaches for long-dated contracts with a focus on

stylized CAT bonds and market-consistent valuation. It proposes a loading pricing concept that combines the theoretically possible minimal price of a contract with its formally obtained risk neutral price, without creating economically

meaningful arbitrage. A loading degree controls how much influence the formally obtained risk neutral price has on the market price. A key finding is that this loading degree has to be constant for a minimally fluctuating contract,

and is an important, measurable characteristic for prices of long-dated contracts. Loading pricing allows long-dated, insurance-type contracts to be priced less expensively and with higher return on investment than under classical

pricing approaches. Loading pricing enables insurance companies to accumulate systematically reserves needed to manage its risk of ruin in a market consistent manner.

Fergusson, K & Platen, E 2015, 'Less Expensive Pricing and Hedging of Long-Dated Equity Index Options When Interest Rates are Stochastic'.

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Many providers of variable annuities such as pension funds and life insurers seek to hedge their exposure to embedded guarantees using longdated derivatives. This paper extends the benchmark approach to price and hedge

long-dated equity index options using a combination of cash, bonds and equities under a variety of market models. The results show that when the discounted index is modelled as a squared Bessel process, as in Platen's minimal market

model, less expensive hedging is achieved irrespective of the short rate model.

Fergusson, K & Platen, E 2014, 'Stylised Properties of the Interest Rate Term Structure Under The Benchmark Approach'.

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Market models which re

ect stylised properties of the interest rate term structure are widely used for modelling and pricing interest rate derivatives. We consider a market model involving the short rate and a diversified global

stock index. We illustrate the stylised properties of the interest rate term structure implied by a system of stochastic differential equations specifying the short rate and the discounted stock index under the benchmark approach.

Comparison with empirical evidence demonstrates the explanatory power of a discounted stock index modelled by a squared Bessel process.

Heath, D & Platen, E 2014, 'A Monte Carlo Method using PDE Expansions for a Diversifed Equity Index Model'.

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This paper considers a new class of Monte Carlo methods that are combined with PDE expansions for the pricing and hedging of derivative securities for multidimensional diffusion models. The proposed method combines the

advantages of both PDE and Monte Carlo methods and can be directly applied to models with more than two state variables. The pricing procedure is illustrated using a three-component index model that captures some of the key features

of a diversified stock index over long time periods. The method is widely applicable and is demonstrated here in the general setting of the benchmark approach, where spatial boundary limiting conditions for the PDE need to be

appropriately chosen and approximated. The PDE expansion is based on a Taylor series approximation for the underlying three-component PDE. A Monte Carlo method with variance reduction is then formulated to approximate the true

solution. Almost exact simulation schemes are described for the given state variables in the model. Numerical results are presented that demonstrate the effectiveness and tractability of the proposed pricing and hedging methodology.

Baldeaux, J & Platen, E 2013, 'Liability Driven Investments under a Benchmark Based Approach'.

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In this paper, we present an alternative approach as a suitable framework under which liability driven investments can be valued and hedged. This benchmark approach values both assets and liabilities consistently under the

real world probability measure using the best performing portfolio, the growth optimal portfolio, as benchmark and numeraire. The benchmark approach identifies the investment strategy which is replicating a given claim at minimal cost.

Should the liability under consideration be subject to nonhedgeable risk, e.g. mortality risk, benchmarked risk minimization identifies with its real world pricing formula the investment strategy which minimizes in a practical sense

the price of a given claim and minimizes the benchmarked profit and loss from hedging. The application of the approach will be demonstrated for pensions. A least expensive pension scheme will be described that allows one in a fair and

transparent manner to hedge in the least expensive way with minimal risk the post retirement payments for its members.

Baldeaux, JF, Fung, MC, Ignatieva, K & Platen, E 2013, 'A Hybrid Model for Equity Indices and Stochastic Interest Rates'.

Fergusson, K & Platen, E 2013, 'Real World Pricing of Long Term Cash-Linked Annuities and Equity-Linked Annuities with Cash-Linked Guarantees'.

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This paper proposes a paradigm shift in the valuation of long term cash-linked annuities and equity-linked annuities with cash-linked guarantees, away from classical no-arbitrage pricing towards pricing under the real world

probability measure. In contrast to risk neutral pricing, which is a form of relative pricing, the long term average excess return of the equity market comes into play. Instead of the savings account, the numeraire portfolio is

employed as the fundamental unit of value in the analysis. The num eraire portfolio is the strictly positive, tradable portfolio that when used as benchmark makes all benchmarked nonnegative portfolios supermartingales. Intuitively,

benchmarked portfolios are in the mean downward trending or trendless. The benchmarked real world price of a benchmarked contingent claim equals its real world conditional expectation. This yields the minimal possible price for its

hedgeable part and minimizes the variance for its hedge error. Classical actuarial and risk neutral pricing emerge as special cases of the proposed real world pricing. In long term liability and asset valuation, the proposed real

world pricing can lead to significantly lower prices than suggested by classical approaches. The existence of an equivalent risk neutral probability measure is not required.

Du, K, Platen, E & Rendek, R 2012, 'Modeling of Oil Prices'.

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The paper derives a parsimonious two-component affine diffusion model with one driving Brownian motion to capture the dynamics of oil prices. It can be observed that the oil price behaves in some sense similarly to the US

dollar. However, there are also clear differences. To identify these the paper studies the empirical features of an extremely well diversified world stock index, which is a proxy of the numeraire portfolio, in the denomination of the

oil price. Using a diversified index in oil price denomination allows us to disentangle the factors driving the oil price. The paper reveals that the volatility of the numeraire portfolio denominated in crude oil, increases at major

oil price upward moves. Furthermore, the log-returns of the index in oil price denomination appear to follow a Student-t distribution. These and other stylized empirical properties lead to the proposed tractable diffusion model, which

has the normalized numeraire portfolio and market activity as components. An almost exact simulation technique is described, which illustrates the characteristics of the proposed model and confirms that it matches well the observed

stylized empirical facts.

Du, K & Platen, E 2011, 'Three-Benchmarked Risk Minimization for Jump Diffusion Markets'.

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The paper discusses the problem of hedging not perfectly replicable contingent claims by using a benchmark, the numerraire portfolio, as reference unit. The proposed concept of benchmarked

risk minimization generalizes classical risk minimization, pioneered by Follmer, Sondermann and Schweizer. The latter relies on a quadratic criterion, requesting the square integrability of contingent

claims and the existence of an equivalent risk neutral probability measure. The proposed concept of benchmarked risk minimization avoids these restrictive assumptions. It employs the real world probability

measure as pricing measure and identifies the minimal possible price for the hedgable part of a contingent claim. Furthermore, the resulting benchmarked profit and loss is only driven by nontraded

uncertainty and forms a martingale that starts at zero. Benchmarked profit and losses, when pooled and sufficiently independent, become in total negligible. This property is highly desirable from a risk

management point of view. It is making a symptotically benchmarked risk minimization the least expensive method for pricing and hedging for an increasing number of not fully replicable benchmarked

contingent claims.

Platen, E & Du, K 2011, 'Benchmarked Risk Minimization for Jump Diffusion Markets'.

Platen, E & Tappe, S 2011, 'Affine Realizations for Levy Driven Interest Rate Models with Real-World Forward Rate Dynamics'.

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We investigate the existence of affine realizations for interest rate term structure models driven by Levy processes. Using as numeraire the growth optimal portfolio, we model the interest rate

term structure under the real-world probability measure, and hence, we do not need the existence of an equivalent risk-neutral probability measure. Furthermore, we include finite dimensional external

factors, thus admitting a stochastic volatility structure.

Breymann, W, LÃ¼thi, D & Platen, E 2009, 'Empirical behavior of a world stock index from intra-day to monthly time scales', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 251 Abstract: Inspired by the theoretically oriented dynamic analysis of moving average rules in Chiarella, He and Hommes (CHH) (2006a) model, this paper conducts a dynamic analysis of a microstructure model of continuous double auctions in which the probability of heterogeneous agents to trade is determined by the rules of either fundamentalists mean-reverting to the fundamental or chartists choosing moving average rules based their relative performance. With such a realistic market microstructure, the model is able not only to obtain the results of the CHH model but also to characterise most of the stylized facts including the power-law behaviour of volatility. The results seem to suggest that a comprehensive explanation of several statistical properties of returns is possible in a framework where both behavioral traits and realistic microstructure have a role.

Craddock, MJ & Platen, E 2009, 'On explicit probability laws for classes of scalar diffusions', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 246 Abstract: This paper uses Lie symmetry group methods to obtain transition probability densities for scalar diffusions, where the diffusion coefficient is given by a power law. We will show that if the drift of the diffusion satisfies a certain family of Riccati equations, then it is possible to compute a generalized Laplace transform of the transition density for the process. Various explicit examples are provided. We also obtain fundamental solutions of the Kolmogorov forward equation for diffusions, which do not correspond to transition probability densities.

Hulley, H & Platen, E 2009, 'A visual criterion for identifying Ito diffusions as martingalesor strict local martingales', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 263 Abstract: It is often important, in applications of stochastic calculus to financial modelling, to know whether a given local martingale is a martingale or a strict local martingale. We address this problem in the context of a time-homogenous diffusion process with a finite lower boundary, presented as the solution of a driftless stochastic differential equation. Our main theorem demonstrates that the question of whether or not this process is a martingale may be decided simply by examining the slope of a certain increasing function. Further results establish the connection between our theorem and other results in the literature, while a number of examples are provided to illustrate the use of our criterion.

Ignatieva, K & Platen, E 2009, 'Modelling co-movements and tail dependency in the international stock market via copulae', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 265 Abstract: This paper examines international equity market co-movements using time-varying copulae. We examine distributions from the class of Symmetric Generalized Hyperbolic (SGH) distributions for modelling univariate marginals of equity index returns. We show based on the goodness-of-ï¬t testing that the SGH class outperforms the normal distribution, and that the Student-t assumption o nmarginals leads to the best performance, and thus, can be used to ï¬t multivariate copula for the joint distribution of equity index returns. We show in our study that the Student-t copula is not only superior to the Gaussian copula, where the dependence structure relates to the multivariate normal distribution, but also out performs some alternative mixture copula models which allow to reï¬ect asymmetric dependencies in the tails of the distribution. The Student-t copula with Student-t marginals allows to model realistically simultaneous co-movements and to capture taild ependency in the equity index returns. From the point of view of risk management, it is a good candidate for modelling the returns arising in an international equity index portfolio where the extreme losses are known to have a tendency to occur simultaneously. We apply copulae to the estimation of the Value-at-Risk and the Expected Shortfall, and show that the Student-t copula with Student-t marginals iss uperior to the alternative copula models investigated, as well the Riskmetics approach.

Platen, E 2009, 'A benchmark approach to investing and pricing', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 253 Abstract: This paper introduces a general market modeling framework, the benchmark approach, which assumes the existence of the numeraire portfolio. This is the strictly positive portfolio that when used as benchmark makes all benchmarked nonnegative portfolios supermartingales, that is intuitively speaking downward trending or trendless. It can be shown to equal the Kelly portfolio which maximizes expected logarithmic utility. In several ways the Kelly or numeraire portfolio is the "best" performing portfolio and can not be out performed systematically by any other nonnegative portfolio. Its use in pricing as numeraire leads directly to the real world pricing formula, which employs the real world probability when calculating conditional expectations. In a large regular financial market, the Kelly portfolio is shown to be approximated by well diversified portfolios.

Platen, E 2009, 'Real world pricing of long term contracts', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 262 Abstract: Long dated contingent claims are relevant in insurance, pension fund management and derivative pricing. This paper proposes a paradigm shift in the valuation of long term contracts, away from classical no-arbitrage pricing towards pricing under the real world probability measure. In contrast to risk neutral pricing, the long term excess return of the equity market, known as the equity premium, is taken into account. Further, instead of the savings account, the numeraire portfolio isused, as the fundamental unit of value in the analysis. The numeraire portfolio is the strictly positive, tradable portfolio that when used as benchmark makes all benchmarked non negative portfolios supermartingales, which means intuitively that these are downward trending or at least trendless. Furthermore, the benchmarked real world price of a benchmarked claimis defined to be its real world conditional expectation. This yields the minimal possible price for its hedgable part and minimizes the variance of the benchmarked hedge error. The pooled total benchmarked replication error of a large insurance company or bank essentially vanishes due to diversification. Interestingly, in long terml iability and asset valuation, real world pricing can lead to significantly lower prices than suggested by classical no-arbitragea rguments. Moreover, since the existence of some equivalent risk neutral probability measure is no longer required, a wider and more realistic modeling framework is available for exploration. Classical actuarial and risk neutral pricing emerge as special cases of real world pricing.

Platen, E & Rendek, RJ 2009, 'Exact scenario simulation for selected multi-dimensional stochastic processes', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 259 Abstract: Accurate scenario simulation methods for solutions of multi-dimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for instance, in finance. They have been playing a crucial role as standard models in various areas and dominate often the communication and thinking in a particular field of application, even that they may be too simple for more advanced tasks. Various discrete time simulation methods have been developed over the years. However, the simulation of solutions of some stochastic differential equations can be problematic due to systematic errors and numerical instabilities. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems encountered by those simulation methods that use discrete time approximations. This paper provides a survey of methods for the exact simulation of paths of some multi-dimensional solutions of stochastic differential equations including Ornstein-Uhlenbeck, square root, squared Bessel, Wishart and Levy type processes.

Platen, E & Rendek, RJ 2009, 'Quasi-exact approximation of hidden Markov chain filters', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 258 Abstract: This paper studies the application of exact simulation methods for multi-dimensional multiplicative noise stochastic differential equations to filtering. Stochastic differential equations with multiplicative noise naturally occur as Zakai equation in hidden Markov chain filtering. The paper proposes a quasi-exact approximation method for hidden Markov chain filters, which can be applied when discrete time approximations, such as the Euler scheme, may fail in practice.

Platen, E & Rendek, RJ 2009, 'Simulation of diversified portfolios in a continuous financial market', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 264 Abstract: In this paper we analyze the simulated behavior of diversified portfolios in a continuous financial market. In particular, we focus on equally weighted portfolios. Wei llustrate that these well diversified portfolios constitute good proxies of the growth optimal portfolio. The multi-asset market models considered include the Black-Scholes model, the Heston model, the ARCH diffusion model, the geometric Ornstein-Uhlenbeck volatility model and the multi-currency minimal market model. The choice of these models was motivated by the fact that they can be simulated almost exactly and, therefore, very accurately also over longer periods of time. Finally, we provide examples, which demonstrate the robustness of the diversification phenomenon when approximating the growth optimal portfolio of a market by an equal value weighted portfolio. Significant outperformance of the market capitalization weighted portfolio by the equal value weightedp ortfolio can be observed for models.

Platen, E & Semmler, W 2009, 'Asset markets and monetary policy', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 247 Abstract: Monetary policy has pursued the concept of inflation targeting. This has been implemented in many countries. Here interest rates are supposed to respond to an inflation gap and output gap. Despite long term continuing growth of the world financial assets, recently, monetary policy, in particular in the U.S. after the subprime credit crisis, was challenged by severe disruptions and a meltdown of the financial market. Subsequently, academics have been in search of a type of monetary policy that does allow to influence in an appropriate manner the investor's behavior and, thus, the dynamics of the economy and its financial market. The paper suggests a dynamic portfolio approach. It allows one to study the interaction between investors` strategic behavior and monetary policy. The article derives rules that explain how monetary authorities should set the short term interest rate in interaction with inflation rate, economic growth, asset prices, risk aversion, asset price volatility, and consumption rates. Interesting is that the inflation rate needs to have a certain minimal level to allow the interest rate to be a viable control instrument. A particular target interest rate has been identified for the desirable optimal regime. If the proposed monetary policy rule is applied properly, then the consumption rate will remain stable and the inflation rate can be kept close to a minimal possible level. Empirical evidence is provided to support this view. Additionally, in the case of an economic crisis the proposed relationships indicate in which direction to act to bring the economy back on track.

Bruti Liberati, N, Nikitopoulos Sklibosios, C, Platen, E & Schlogl, E 2009, 'Alternative Defaultable Term Structure Models', *Quantitative Finance Research Paper Series*.

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The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives.

Bruti Liberati, N & Platen, E 2008, 'Strong predictor-corrector Euler methods for stochastic differential equations', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 222 Abstract: This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor-corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the new symmetric predictor-corrector Euler methods.

Chavez, SA & Platen, E 2008, 'Distributional deviations in random number generation in finance', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 228 Abstract: This paper points out that pseudo-random number generators in widely used standard software can generate severe distributional deviations from targeted distributions when used in parallel implementations. In Monte Carlo simulation of random walks for financial applications this can lead to remarkable errors. These are not reduced when increasing the sample size. The paper suggests to use instead of standard routines, combined feedback shift register methods for generating random bits in parallel that are based on particular polynomials of degree twelve. As seed numbers the use of natural random numbers is suggested. The resulting hybrid random bit generators are then suitable for parallel implementation with random walk type applications. They show better distributional properties than those typically available and can produce massive streams of random numbers in parallel, suitable for Monte Carlo simulation in finance.

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This paper considers the problem of when a local martingale is a martingale or a universally integrable martingale, for the case of time-homogeneous scalar diffusions. Necessary and sufficient conditions of a geometric nature are obtained for answering this question. These results are widely applicable to problems in stochastic finance. For example, in order to apply risk-neutral pricing, one must first check that the chosen density process for an equivalent change of probability measure is in fact a martingale. If not, risk-neutral pricing is infeasible. Furthermore, even if the density process is a martingale, the possibility remains that the discounted price of some security could be a strict local martingale under the equivalent risk-neutral probability measure. In this case, well-known identities for option prices, such as put-call parity, may fail. Using our results, we examine a number of basic asset price models, and identify those that suffer from the above-mentioned difficulties.

Kardaras, C & Platen, E 2008, 'Minimizing the expected market time to reach a certain wealth level', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 230 Abstract: In a financial market model, we consider variations of the problem of minimizing the expected time to upcross a certain wealth level. For exponential Levy markets, we show the asymptotic optimality of the growth-optimal portfolio for the above problem and obtain tight bounds for the value function for any wealth level. In an Ito market, we employ the concept of market time, which is a clock that runs according to the underlying market growth. We show the optimality of the growth-optimal portfolio for minimizing the expected market time to reach any wealth level. This reveals a general definition of market time which can be useful from an investorâs point of view. We utilize this last definition to extend the previous results in a general semimartingale setting.

Kardaras, C & Platen, E 2008, 'Multiplicative approximation of wealth processes involving no-short-sale strategies', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 240

Kardaras, C & Platen, E 2008, 'On Financial Markets where only Buy-And-Hold Trading is Possible'.

Marquardt, T, Platen, E & Jaschke, S 2008, 'Valuing guaranteed minimum death benefit options in variable annuities under a benchmark approach', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 221 Abstract: Variable annuities (VAs) represent a marked change from earlier life products in the guarantees that they offer and it is no longer possible to manage the risks of these liabilities using traditional actuarial methods. Thinking about guarantees as options suggests applying risk neutral pricing in order to value the embedded guarantees, such as guaranteed minimum death benefits (GMDBs). However, due to the long maturities of contracts, stochastic volatility and many other reasons, VA markets are incomplete. In this paper we propose a methodology for pricing GMDBs under a benchmark approach which does not require the existence of a risk neutral probability measure. We assume that the insurance company invests in the growth optimal portfolio of its investment universe and apply real world pricing rather than risk neutral pricing. In particular, we consider the minimal market model and conclude that in this setup the fair price of a roll-up GMDB is lower than the price obtained by applying standard risk neutral pricing. Moreover, we take into account rational as well as irrational lapsation of the policyholder.

Miller, S & Platen, E 2008, 'Analytic Pricing of Contingent Claims Under the Real-World Measure'.

Miller, S & Platen, E 2008, 'Real world pricing for a modified constant elasticity of variance model', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 237 Abstract: This paper considers a modified constant elasticity of variance (MCEV) model. This model uses the familiar constant elasticity of variance form for the volatility of the growth optimal portfolio (GOP) in a continuous market. It leads to a GOP that follows the power of a time-transformed squared Bessel process. This paper derives analytic real-world prices for zero-coupon bonds, instantaneous forward rates and options on the GOP that are both theoretically revealing and computationally efficient. In addition, the paper examines options on exchange prices and options on zero-coupon bonds under the MCEV model. The semi-analytic prices derived for options on zero-coupon bonds can subsequently be used to price interest rate caps and floors.

Nikeghbali, A & Platen, E 2008, 'On honest times in financial modeling', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 229 Abstract: This paper demonstrates the usefulness and importance of the concept of honest times to financial modeling. It studies a financial market with asset prices that follow jump-diffusions with negative jumps. The central building block of the market model is its growth optimal portfolio (GOP), which maximizes the growth rate of strictly positive portfolios. Primary security account prices, when expressed in units of the GOP, turn out to be nonnegative local martingales. In the proposed framework an equivalent risk neutral probability measure need not exist. Derivative prices are obtained as conditional expectations of corresponding future payoffs, with the GOP as numeraire and the real world probability as pricing measure. The time when the global maximum of a portfolio with no positive jumps, when expressed in units of the GOP, is reached, is shown to be a generic representation of an honest time. We provide a general formula for the law of such honest times and compute the conditional distributions of the global maximum of a portfolio in this framework. Moreover, we provide a stochastic integral representation for uniformly integrable martingales whose terminal values are functions of the global maximum of a portfolio. These formulae are model independent and universal. We also specialize our results to some examples where we hedge a payoff that arrives at an honest time.

Platen, E 2008, 'A unifying approach to asset pricing', *REsearch Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 227 Abstract: This paper introduces a general market modeling framework under which the Law of One Price no longer holds. A contingent claim can have in this setting several self-financing, replicating portfolios. The new Law of the Minimal Price identifies the lowest replicating price process for a given contingent claim. The proposed unifying asset pricing methodology is model independent and only requires the existence of a tradable numeraire portfolio, which turns out to be the growth optimal portfolio that maximizes expected logarithmic utility. By the Law of the Minimal Price the inverse of the numeraire portfolio becomes the stochastic discount factor. This allows pricing in extremely general settings and avoids the restrictive assumptions of risk neutral pricing. In several ways the numeraire portfolio is the âbestâ performing portfolio and cannot be outperformed by any other nonnegative portfolio. Several classical pricing rules are recovered under this unifying approach. The paper explains that pricing by classical no-arbitrage arguments is, in general, not unique and may lead to overpricing. In an example, a surprisingly low price of a zero coupon bond with extreme maturity illustrates one of the new effects that can be captured under the proposed benchmark approach, where the numeraire portfolio represents the benchmark.

Platen, E 2008, 'The Law of Minimum Price'.

Platen, E & Shi, L 2008, 'On the numerical stability of simulation methods for SDES', *Research Paper Series, Quantitative Finance Research Centre, UNiversity of Technology, Sydney*.

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Research Paper Number: 234 Abstract: When simulating discrete time approximations of solutions of stochastic differential equations (SDEs), numerical stability is clearly more important than numerical efficiency or some higher order of convergence. Discrete time approximations of solutions of SDEs are widely used in simulations in finance and other areas of application. The stability criterion presented is designed to handle both scenario simulation and Monte Carlo simulation, that is, strong and weak simulation methods. The symmetric predictor-corrector Euler method is shown to have the potential to overcome some of the numerical instabilities that may be experienced when using the explicit Euler method. This is of particular importance in finance, where martingale dynamics arise for solutions of SDEs and diffusion coefficients are often of multiplicative type. Stability regions for a range of schemes are visualized and discussed. For Monte Carlo simulation it turns out that schemes, which have implicitness in both the drift and the diffusion terms, exhibit the largest stability regions. It will be shown that refining the time step size in a Monte Carlo simulation can lead to numerical instabilities.

Chiarella, C & Platen, E 2007, 'The history of the Quantitative Methods in Finance Conference Series. 1992-2007', *Research Paper Series, Qunatitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 207 Abstract: This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

Filipovic, D & Platen, E 2007, 'Consistent market extensions under the benchmark approach', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 189 Abstract: The existence of the growth optimal portfolio (GOP), also known as Kelly portfolio, is vital for a financial market to be meaningful. The GOP, if it exists, is uniquely determined by the market parameters of the primary security accounts. However, markets may develop and new security accounts become tradable. What happens to the GOP if the original market is extended? In this paper we provide a complete characterization of market extensions which are consistent with the existence of a GOP. We show that a three fund separation theorem applies for the extended GOP. This includes, in particular, the introduction of a locally risk free security, the savings account. We give necessary and sufficient conditions for a consistent exogenous specification of the prevailing short rates.

Hulley, H & Platen, E 2007, 'Laplace transform identities for diffusions, with applications to rebates and barrier options', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 203 Abstract: Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms of some functions of first-passage times for the diffusion. These results are applied to the special case of squared Bessel processes with killing or reflecting boundaries. In particular, we demonstrate how the above-mentioned integral identity enables us to derive the transition density of a squared Bessel process killed at the origin, without the need to invert a Laplace transform. Finally, as an application, we consider the problem of pricing barrier options on an index described by the minimal market model.

KÃ¼chler, U & Platen, E 2007, 'Time delay and noise explaining cyclical fluctuations in prices of commodities', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 195 Abstract: This paper suggests to model jointly time delay and random effects in economics and finance. It proposes to explain the random and often cyclical fluctuations in commodity prices as a consequence of the interplay between external noise and time delays caused by the time between initiation of production and delivery. The proposed model is formulated as a stochastic delay differential equation. The typical behavior of a commodity price index under this model will be discussed. Methods for parameter estimation and the evaluation of functionals will be proposed.

Platen, E & Rendek, RJ 2007, 'Empirical evidence on Student-t log-returns of diversified world stock indices', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 194 Abstract: The aim of this paper is to document some empirical facts related to log-returns of diversified world stock indices when these are denominated in different currencies. Motivated by earlier results, we have obtained the estimated distribution of log-returns for a range of world stock indices over long observation periods. We expand previous studies by applying the maximum likelihood ratio test to the large class of generalized hyperbolic distributions, and investigate the log-returns of a variety of diversified world stock indices in different currency denominations. This identifies the Student-t distribution with about four degrees of freedom as the typical estimated log-return distribution of such indices. Owing to the observed high levels of significance, this result can be interpreted as a stylized empirical fact.

Platen, E & Runggaldier, WJ 2007, 'A benchmark approach to portfolio optimization under partial information', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 191 Abstract: This paper proposes a filtering methodology for portfolio optimization when some factors of the underlying model are only partially observed. The level of information is given by the observed quantities that are here supposed to be the primary securities and empirical log-price covariations. For a given level of information we determine the growth optimal portfolio, identify locally optimal portfolios that are located on a corresponding Markowitz efficient frontier and present an approach for expected utility maximization. We also present an expected utility indifference pricing approach under partial information for the pricing of nonreplicable contracts. This results in a real world pricing formula under partial information that turns out to be independent of the subjective utility of the investor and for which an equivalent risk neutral probability measure need not exist.

Bruti Liberati, N, Nikitopoulos Sklibosios, C & Platen, E 2007, 'Pricing Under the Real-World Probability Measure for Jump-Diffusion Term Structure Models', *Quantitative Finance Research Paper Series*.

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This paper considers interest rate term structure models in a market attracting both continuous and discrete types of uncertainty. The event driven noise is modelled by a Poisson random measure. Using as numeraire the growth optimal portfolio, interest rate derivatives are priced under the real-world probability measure. In particular, the real-world dynamics of the forward rates are derived and, for specific volatility structures, finite dimensional Markovian representations are obtained. Furthermore, allowing for a stochastic short rate, a class of tractable affine term structures is derived where an equivalent risk-neutral probability measure does not exist.

Bruti Liberati, N & Platen, E 2006, 'On weak predictor-corrector schemes for jump-diffusion processes in finance (QFRC paper #179)', *Quantitative Finance Research Centre Working Paper Series*.

Le, T & Platen, E 2006, 'Approximating the growth optimal portfolio with a diversified world stock index (QFRC paper #184)', *Quantitative Finance Research Centre Working Paper Series*.

Platen, E 2006, 'On the pricing and hedging of long dated zero coupon bonds (QFRC paper #185)', *Quantitative Finance Research Centre Working Paper Series*.

Platen, E & Bruti Liberati, N 2006, 'Approximation of jump-diffusion in finance and economics (QFRC paper #176)', *Quantitative Finance Research Centre Working Paper Series*.

Bruti Liberati, N & Platen, E 2005, 'On the strong approximation of jump-diffusion processes (QFRC paper #157)'.

Bruti Liberati, N & Platen, E 2005, 'On the strong approximation of pure jump processes (QFRC paper #164)'.

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Bruti-Liberati, N & Platen, E 2005, 'On the Strong Approximation of Pure Jump Processes'.

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This paper constructs strong discrete time approximations for pure jump processes that can be described by stochastic differential equations. Strong approximations based on jump-adapted time

discretizations, which produce no discretization bias, are analyzed. The computational complexity of these approximations is proportional to the jump intensity. Furthermore, by exploiting a stochastic

expansion for pure jump processes, higher order discrete time approximations, whose computational complexity is not dependent on the jump intensity, are proposed. The strong order of convergence of the

resulting schemes is analyzed.

Christensen, MM & Platen, E 2005, 'Sharpe ratio maximization and expected utility when asset prices have jumps (QFRC paper #170)'.

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Fergusson, KJ & Platen, E 2005, 'On the distributional characterisation of log-returns of a world stock index (QFRC paper #153)'.

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Heath, DP & Platen, E 2005, 'Currency derivatives under a minimal market model with random scaling (QFRC paper #154)'.

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Hulley, H, Miller, S & Platen, E 2005, 'Benchmarking and fair pricing applied to two market models (QFRC paper #155)'.

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Platen, E 2005, 'Investments in the short and long run (QFRC paper #163)'.

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Platen, E 2005, 'On the role of the growth optimal portfolio in finance (QFRC paper #144)'.

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Bruti Liberati, N, Martini, F, Piccardi, M & Platen, E 2005, 'A hardware generator of multi-point distributed random numbers for Monte Carlo simulation (QFRC paper #156)'.

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Breymann, W, Kelly, L & Platen, E 2004, 'Intraday empirical analysis and modeling of diversified world stock indices (QFRC paper #125)'.

Bruti Liberati, N & Platen, E 2004, 'On the efficiency of simplified weak Taylor schemes for Monte Carlo simulation in finance (QFRC paper #114)'.

Christensen, MM & Platen, E 2004, 'A general benchmark model for stochastic jump sizes', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 139 Abstract: This paper extends the benchmark framework of Platen (2002) by introducing a sequence of incomplete markets, having uncertainty driven by a Wiener process and a marked point process. By introducing an idealized market, in which all relevant economical variables are observed, but may not all be traded, a generalized growth optimal portfolio (GOP) is obtained and calculated explicitly. The problem of determining the GOP is solved in a general setting which extends existing treatments and provides a clear link to the market prices of risk. The connection between traded securities, arbitrage and market incompleteness is analyzed. This provides a framework for analyzing the degree of incompleteness associated with jump processes, a problem well-known from insurance and credit risk modeling. By staying under the empirical measure, the resulting benchmark model has potential advantages for various applications in finance and insurance.

Heath, DP & Platen, E 2004, 'Local volatility function models under a benchmark approach (QFRC paper #124)'.

Heath, DP & Platen, E 2004, 'Understanding the implied volatility surface for options on a diversified index (QFRC paper #128)'.

Miller, S & Platen, E 2004, 'Two-factor model for low interest rate regimes (QFRC paper #130)'.

Platen, E 2004, 'A benchmark approach to finance (QFRC paper #138)'.

Platen, E 2004, 'Capital asset pricing for markets with intensity based jumps', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 143 Abstract: This paper proposes a unified framework for portfolio optimization, derivative pricing, modeling and risk measurement in financial markets with security price processes that exhibit intensity based jumps. It is based on the natural assumption that investors prefer more for less, in the sense that for two given portfolios with the same variance of its increments, the one with the higher expected increment is preferred. If one additionally assumes that the market together with its monetary authority acts to maximize the long term growth of the market portfolio, then this portfolio exhibits a very particular dynamics. In a market without jumps the resulting dynamics equals that of the growth optimal portfolio (GOP). Conditions are formulated under which the well-known capital asset pricing model is generalized for markets with intensity based jumps. Furthermore, the Markowitz efficient frontier and the Sharpe ratio are recovered in this continuous time setting. In this paper the numeraire for derivative pricing is chosen to be the GOP. Primary security account prices, when expressed in units of the GOP, turn out to be supermartingales. In the proposed framework an equivalent risk neutral martingale measure need not exist. Fair derivative prices are obtained as conditional expectations of future payoff structures under the real world probability measure. The concept of fair pricing is shown to generalize the classical risk neutral and the actuarial net present value pricing methodologies.

Platen, E 2004, 'Diversified portfolios with jumps in a benchmark framework (QFRC paper #129)'.

Platen, E, West, JM & Breymann, W 2004, 'An Intraday Empirical Analysis of Electricity Price Behaviour', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 140 Abstract: This paper proposes an approach to the intraday analysis of the dynamics of electricity prices. The Growth Optimal Portfolio (GOP) is used as a reference unit in a continuous financial electricity price model. A diversified global portfolio in the form of a market capitalisation weighted index approximates the GOP. The GOP, measured in units of electricity, is normalised and then modeled as a time transformed square root process of dimension four. The dynamics of the resulting process is empirically verified. Intraday spot electricity prices from the US and Australian markets are used for this analysis. The empirical findings identify a simple but realistic model for examining the volatile behaviours of electricity prices. The proposed model reflects the historical price evolution reasonably well by using a only a few robust but readily observable parameters. The evolution of the tranformed times is modeled via a rapidly evolving market activity. A periodic, ergodic process with deterministic volatility is used to model market activity.

Craddock, MJ & Platen, E 2003, 'Symmetric group methods for fundamental solution and characteristics functions (QFRC paper #90)'.

Craddock, MJ & Platen, E 2003, 'Symmetry group methods for fundamental solutions and characteristic functions', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 90 Abstract: This paper uses Lie symmetry group methods to analyse a class of partial differential equations of he form It is shown that when the drift function f is a solution of a family of Ricatti equations, then symmetry techniques can be used to find the characteristic functions and transition densities of the corresponding diffusion processes.

Heath, DP & Platen, E 2003, 'Pricing of index options under a minimal market model with lognormal scaling (QFRC paper #101)', *Journal of macroeconomics*.

Kelly, L, Platen, E & Sorensen, M 2003, 'Estimation for discretely observed diffusions using transform functions (QFRC paper #96)'.

Platen, E 2003, 'A benchmark framework for risk management (QFRC paper #113)'.

Platen, E 2003, 'An alternative interest rate term structure model (QFRC paper #97)'.

Platen, E 2003, 'Diversified portfolios in a benchmark framework', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 87 Abstract: This paper considers diversified portfolios in a benchmark framework. A new limit theorem for the approximation of the benchmark, which is the growth optimal portfolio, is obtained. In a diverse market it is shown that there exist approximations for the benchmark that are independent of model specifications. This leads to a robust modeling, calibration and risk management framework. For diversified portfolios with a large number of securities the limit theorem provides significant reductions in the complexity of quantitative applications as statistical inference and Value at Risk calculations.

Platen, E 2003, 'Modeling the volatility and expected value of a diversified world index (QFRC paper #103)'.

Platen, E 2003, 'Pricing and hedging for incomplete jump diffusion benchmark models (QFRC paper #110)'.

Platen, E & Stahl, G 2003, 'A structure for general and specific market risk (QFRC paper #91)'.

Platen, E & West, J 2003, 'Fair Pricing of Weather Derivatives'.

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This paper proposes a consistent benchmark approach to price weather derivatives. The growth optimal portfolio to price weather derivatives. The growth optimal portfolio is used as numeraire

such that all benchmarked fair price processes are martingales. No measure transformation is needed for fair pricing. Since weather derivatives are traded in an incomplete market setting, standard hedging

based pricing methods cannot be applied. For weather derivative payoffs that are independent from the value of the growth optimal portfolio it is shown that the classical actuarial pricing methodology is

a particular case of the fair pricing concept. A discrete time model is constructed to approximate historical weather characteristics assuming Gaussian residuals. For particular weather derivatives their

fair prices are derived.

Platen, E & West, JM 2003, 'Fair pricing of weather derivatives (QFRC paper #106)'.

Buhlmann, H & Platen, E 2002, 'A discrete time benchmark approach for finance & insurance'.

Heath, DP & Platen, E 2002, 'A variance reduction technique based on integral representations', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 75 Abstract: Standard Monte Carlo methods can often be significantly improved with the addition of appropriate variance reduction techniques. In this paper a new and powerful variance reduction technique is presented. The method is based directly on the Ito calculus and is used to find unbiased variance reduced estimators for the expectation of functionals of Ito diffusion processes. The approach considered has wide applicability, for instance, it can be used as a means of approximating solutions of parabolic partial differential equations or applied to valuation problems that arise in mathematical finance. We illustrate how the method can be applied by considering the pricing of European style derivative securities for a class of stochastic volatility models, including the Heston model.

Heath, DP & Platen, E 2002, 'Consistent pricing & hedging for a modified constant elasticity of variance model'.

Platen, E 2002, 'A benchmark framework for integrated risk management'.

Platen, E 2002, 'Benchmark model with intensity based jumps', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 81 Abstract: This paper proposes a class of financial market models with security price processes that exhibit intensity based jumps. Primary security account prices, when expressed in units of the benchmark, turn out to be local martingales. The benchmark model exludes, so called, benchmark arbitrage but permits arbitrage amounts, which arise for benchmarked price processes that are strict local martingales. In the proposed framework, generally, an equivalent risk neutral measure does not exist. Benchmarked fair derivative prices are obtained as conditional expectations of future benchmarked prices under the real world probability measure.

Platen, E & Runggaldier, WJ 2002, 'A benchmark approach to filtering in finance'.

Craddock, MJ & Platen, E 2001, 'Benchmark pricing of credit derivatives under a standard market model', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 60 Abstract: This paper makes use of an integrated benchmark modelling framework that allows us to model credit risk. We demonstrate how to price contingent claims by taking expectations under the real world probability measure in a benchmarked world. Furthermore, put and call options on an index are studied that measure the credit worthiness of a firm.

Hardle, WK, Kleinow, T, Korostelev, A & Platen, E 2001, 'Semiparametric diffusion estimation and application to a stock market model', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 51 Abstract: The analysis of diffusion process in financial models is crucially dependent on the form of the drift and diffusion coefficient functions. A methodology is proposed for estimating and testing coefficient functions for ergodic diffusions that are not directly observable. It is based on semiparametric and nonparametric estimates. The testing is performed via the wild bootstrap resampling technique. The method is illustrated on S&P 500 index.

Heath, DP & Platen, E 2001, 'Perfect hedging of index derivatives under a locally arbitrage free minimal market model', *Research Paper Series, Quantitative Finance Researh Centre, University of Technology,Sydney*.

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Research Paper Number: 61 Abstract: The paper presents a financial market model that generates stochastic volatility using a minimal set of factors. These factors, formed from transformations of square root processes, model the dynamics of different denominations of a benchmark portfolio. Benchmarked prices are assumed to be local martingales. Numerical results for the pricing and hedging of basic derivatives on indices are described. This includes cases where the standard risk neutral pricing methodology fails. However, payoffs can be perfectly hedged using self-financing strategies and a form of arbitrage still exists. This is illustrated by hedge simulations. The term structure of implied volatilities is documented.

Kubilius, K & Platen, E 2001, 'Rate of weak convergence of the Euler approximation for diffusion processes with jumps', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 54

Kuchler, U & Platen, E 2001, 'Weak discrete time approximation of stochastic differential equations with time delay', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 50 Abstract: The paper considers the derivation of weak discrete time approximations for solutions of stochastic differential equations with time delay. These are suitable for Monte Carlo simulation and allow the computation of expectations for functionals of stochastic delay equations. The suggested approximations converge in a weak sense.

Platen, E 2001, 'A benchmark model for financial markets', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 59 Abstract: This paper introduces a benchmark model for financial markets, which is based on the unique characterization of a benchmark portfolio that is chosen to be the growth optimal portfolio. The general structure of risk premia for asset prices as an average of appreciation rates. The benchmark model is shown to be locally arbitrage free, however, it still permits some form of arbitrage. Finally, a subclass of arbitrage free contingent claim prices is derived.

Platen, E 2001, 'A minimal financial market model', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 48 Abstract: The paper proposes a financial market model that generates stochastic volatilities and stochastic interest rates using a minimal number of factors that characterise the dynamics of different denominations of a benchmark portfolio. It models asset prices essentially as functionals of square root and Ornstein-Uhlenbeck processes. The resulting price processes exhibit stochastic volatility with leptokurtic log-return distributions that closely match those observed in reality. The benchmark portfolio is negatively correlated with its volatility which models the well-known leverage effect. The average growth rates of the different denominations of the benchmark portfolio are Ornstein-Uhlenbeck processes which generates the typically observed long term Gaussianity of log-returns of asset prices.

Platen, E 2001, 'Arbitrage in continuous complete markets', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 72 Abstract: This paper introduces a benchmark approach for the modelling of continuous, complete financial markets when an equivalent risk neutral measure does not exist. This approach is based on the unique characterization of a benchmark portfolio, the growth optimal portfolio, which is obtained via a generalization of the mutual fund theorem. The discounted growth optimal portfolio with minimum variance drift is shown to follow a Bessel process of dimension four. Some form of arbitrage can be explicitly measured by arbitrage amounts. Fair contingent claim prices are derived as conditional expectations under the real world probability measure. The Heath-Jarrow-Morton forward rate equation remains valid despite the absence of an equivalent risk neutral measure.

Kuchler, U & Platen, E 2000, 'Strong discrete time approximation of stochastic differential equations with time delay', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 44 Abstract: The paper introduces an approach for the derivation of discrete time approximations for solutions of stochastic differential equations with time delay. The suggested approximations converge in a strong sense. Furthermore, explicit solutions for linear stochastic delay equations are given.

Platen, E 2000, 'Risk premia and financial modelling without measure transformation', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 45 Abstract: This paper describes a financial market modelling framework that exploits the notion of a deflator. The demonstrations of the deflator measured in units of primary assets form a minimal set of basic financial quantities that completely specify overall market dynamics. Risk premia of asset prices are obtained as a natural consequence of the approach. Contingent claim prices are computed under the real world measure both in the case of complete and incomplete markets.

Craddock, MJ, Heath, DP & Platen, E 1999, 'Numerical inversion of Laplace transforms: A survey of techniques with applications to derivative pricing', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 27 Abstract: We consider different approaches to the problem of numerically inverting Laplace transforms in finance. In particular, we discuss numerical inversion techniques in the context of Asian option pricing.

Elliott, R, Fischer, P & Platen, E 1999, 'Filtering and Parameter Estimation for a Mean Reverting Interest Rate Model'.

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A Hidden Markov Model with mean reverting characteristics is considered as a model for financial time series, particularly interest rates. The optimal filter for the state of the hidden Markov

chain is obtained. A number of auxiliary filters are obtained that enable the parameters of the model to be estimated using the EM algorithm. A simulation study demonstrates the feasibility of this

approach.

Hoek, JVD & Platen, E 1999, 'Pricing and Hedging in the Presence of Transaction Costs Under Local Risk Minimisation'.

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The paper considers the continuous time pricing and hedging of European options in the presence of small transaction costs and frequent trading under local risk minimisation. The approach yields

mean-self-financing strategies. The resulting dynamical hedges adapt the trading frequency in dependence on actual asset price and time to maturity. Explicit asymptotic expressions for prices and hedging

strategies are derived.

Hurst, S & Platen, E 1999, 'On the Marginal Distribution of Trade Weighted Currency Indices'.

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In this paper we identify a distribution which suitably fits the marginal distribution for the daily log increments of trade weighted currency indices. By considering the class of symmetric

generalised hyperbolic distributions for these increments the Student t distribution appears to be an excellent candidate. Further well-known asset price models are also studied.

Platen, E 1999, 'A Financial Market Model with Trading Volume and Stochastic Volatility'.

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The paper describes a continuous time financial market model, where the basic factord are trading volumes per unit time. These are modelled by squared Bessel processes. The asset prices are

formed by rations of these trading volumes. They have leptokurtic return distributions and stochastic volatilities with properties that are similar to those observed in practice. For the market index the

model generates naturally the well-known leverage effect due to negative correlation between the index and its volatility.

Platen, E 1999, 'A Minimal Share Market Model with Stochastic Volatility'.

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The paper describes a continuous time share market model with a minimal number of factors. These factors are powers of Bessel processes. The asset prices are formed by ratios of the factors and

have consequently leptokurtic return distributions. In this framework stochastic volatility with properties that are similar to those actually observed arises naturally. The model generates for the market

index the well-known leverage effect due to negative correlation between the index and its volatility. It also incorporates possible default of an asset and thus models credit risk.

Platen, E 1999, 'An introduction to numerical methods for stochastic differential equations'.

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This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems. © 1999, Cambridge University Press. All rights reserved.

Platen, E 1999, 'On the Log-Return Distribution of Index Benchmarked Share Prices'.

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This paper identifies a distribution, which fits the daily log-returns of index benchmarked share prices. For this data the Student t distribution appears to provide the best fit under the

maximum likelihood ratio test within the class of symmetric generalised hyperbolic distributions. A share market model that generates share prices with the observed log-return distribution is also

described.

Heath, D, Platen, E & Schweizer, M 1998, 'Comparison of Some Key Approaches to Hedging in Incomplete Markets'.

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The paper provides a numerical comparison of local risk minimisation and mean-variance hedging for some key variations of stochastic volatility models. A hedging and pricing framework is

established for both approaches. Important quantitative differences become apparent that have implications for the implementation of hedging strategies under stochastic volatility.

Kardaras, C & Platen, E, 'Multiplicative approximation of wealth processes involving no-short-sale strategies via simple trading'.

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A financial market model with general semimartingale asset-price processes

and where agents can only trade using no-short-sales strategies is considered.

We show that wealth processes using continuous trading can be approximated very

closely by wealth processes using simple combinations of buy-and-hold trading.

This approximation is based on controlling the proportions of wealth invested

in the assets. As an application, the utility maximization problem is

considered and it is shown that optimal expected utilities and wealth processes

resulting from continuous trading can be approximated arbitrarily well by the

use of simple combinations of buy-and-hold strategies.

Platen, E & Tappe, S, 'No-arbitrage concepts in topological vector lattices'.

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We provide a general framework for no-arbitrage concepts in topological

vector lattices, which covers many of the well-known no-arbitrage concepts as

particular cases. The main structural condition which we impose is that the

outcomes of trading strategies with initial wealth zero and those with positive

initial wealth have the structure of a convex cone. As one consequence of our

approach, the concepts NUPBR, NAA$_1$ and NA$_1$ may fail to be equivalent in

our general setting. Furthermore, we derive abstract versions of the

fundamental theorem of asset pricing. We also consider a financial market with

semimartingales which does not need to have a num\'{e}raire, and derive results

which show the links between the no-arbitrage concepts by only using the theory

of topological vector lattices and well-known results from stochastic analysis

in a sequence of short proofs.

#### Projects

**Selected projects**