Journal articles
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.
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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.
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.
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.
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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.
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© 2016 Taylor & Francis Group, LLC. Abstract: This article adopts an approach to pricing of equity-linked life insurance contracts, which only requires the existence of the numéraire portfolio. An equity-linked life insurance contract is equivalent to a sum of the guaranteed amount and the value of an option on the equity index with some mortality risk attached. The numéraire portfolio equals the growth optimal portfolio and is used as numéraire or benchmark, where the real-world probability measure is taken as pricing measure. To obtain tractable solutions the short rate is modelled as a quadratic form of some Gaussian factor processes. Furthermore, the dynamics of the mortality rate is modelled as a threshold life table. The dynamics of the discounted equity market index or benchmark is modelled by a time transformed squared Bessel process. The equity-linked life insurance contracts are evaluated analytically.
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.
Baldeaux, J.F., Grasselli, M. & Platen, E. 2015, 'Pricing currency derivatives under the benchmark approach', Journal of Banking and Finance, vol. 53, pp. 34-48.
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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.
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© 2014 Elsevier B.V. This paper investigates the pricing and hedging of variance swaps under a 3/2 volatility model using explicit formulae. Pricing and hedging is performed under the benchmark approach, which only requires the existence of the numéraire portfolio. The growth optimal portfolio is used as numéraire together with the real world probability measure as pricing measure. This real world pricing concept provides minimal prices for variance swaps even when an equivalent risk neutral probability measure does not exist.
Baldeaux, J., Fung, M.C., 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.
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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, J. & Platen, E. 2015, 'Credit Derivative Evaluation and CVA Under the Benchmark Approach', Asia-Pacific Financial Markets, vol. 22, no. 3, pp. 305-331.
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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.
Platen, E. & Tappe, S. 2015, 'Real-World Forward Rate Dynamics With Affine Realizations', STOCHASTIC ANALYSIS AND APPLICATIONS, vol. 33, no. 4, pp. 573-608.
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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.
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© 2015 Elsevier B.V. All rights reserved. This paper studies volatility derivatives such as variance and volatility swaps, options on variance in the modified constant elasticity of variance model using the benchmark approach. The analytical expressions of pricing formulas for variance swaps are presented. In addition, the numerical solutions for variance swaps, volatility swaps and options on variance are demonstrated.
Fergusson, K. & Platen, E. 2015, 'Application of Maximum Likelhood Estimation to Stochastic Short Rate Models', Annals of Financial Economics, vol. 10, no. 2, pp. 1550009-1550009.
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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.
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, pp. 1-26.
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.F., 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.
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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.
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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.
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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.
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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.
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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.
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.
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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. & 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.
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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.
Kardaras, C. & Platen, E. 2012, 'On the Dybvig-Ingersoll-Ross theorem', Mathematical Finance, vol. 22, no. 4, pp. 729-740.
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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, R.J. 2012, 'Approximating the numeraire portfolio by naive diversification', Journal of Asset Management, vol. 13, no. 1, pp. 34-50.
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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.
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.
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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.
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.
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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
Guo, Z.J. & 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. 8, pp. 1-23.
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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.
GUO, Z.H.I.J.U.N. & PLATEN, E.C.K.H.A.R.D. 2012, 'THE SMALL AND LARGE TIME IMPLIED VOLATILITIES IN THE MINIMAL MARKET MODEL', International Journal of Theoretical and Applied Finance, vol. 15, no. 08.
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.
Ignatieva, K., Platen, E. & Rendek, R.J. 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.
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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.
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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. & West, J. 2011, 'Intraday Empirical Analysis of Electricity Price Behaviour', Communications on Stochastic Analysis, vol. 5, no. 4, pp. 721-744.
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.
Platen, E. & Rendek, R.J. 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.
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.
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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.
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.
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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.
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
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.
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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.
Filipovic, D. & Platen, E. 2009, 'Consistent Market Extensions Under the Benchmark Approach', Mathematical Finance, vol. 19, no. 1, pp. 41-52.
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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.
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.
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.
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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.
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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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.
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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.
Platen, E. & Rendek, R.J. 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. & Platen, E. 2008, 'Strong Predictor-Corrector Euler Methods for Stochastic Differential Equations', Stochastics and Dynamics, vol. 8, no. 3, pp. 561-581.
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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.
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.
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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.
Platen, E. & Rendek, R.J. 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.
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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.
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.
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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.
Hardle, W.K., 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.
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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.H.A.N.E.M. & PLATEN, E.C.K.H.A.R.D. 2008, 'ANALYTIC PRICING OF CONTINGENT CLAIMS UNDER THE REAL-WORLD MEASURE', International Journal of Theoretical and Applied Finance, vol. 11, no. 08, pp. 841-867.
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.
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.
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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.
Platen, E. & Runggaldier, W.J. 2007, 'A Benchmark Approach to Portfolio Optimization under Partial Information', Asia Pacific Financial Markets, vol. 14, no. 1-2, pp. 25-43.
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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.
Christensen, M.M. & 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.
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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.
Fergusson, K.J. & 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.
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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
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.
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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.
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.
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
Heath, D.P. & Platen, E. 2006, 'Local volatility function models under a benchmark approach', Quantitative Finance, vol. 6, no. 3, pp. 197-206.
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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
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.
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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
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.
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, pp. 191-209.
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This paper presents new results on strong numerical schemes, which are appropriate for scenario analysis, filtering and hedge simulation, for stochastic differential equations (SDEs) of jump-diffusion type. It provides first order strong approximations for jump-diffusion SDEs driven by Wiener processes and Poisson random measures. The paper covers first order derivative-free, drift-implicit and jump-adapted strong approximations. Moreover, it provides a commutativity condition under which the computational effort of first order strong schemes is independent of the total intensity of the jump measure. Finally, a numerical study on the accuracy of several strong schemes applied to the Merton model is presented. © VSP 2006.
Burrage, K., Burrage, P., Higham, D.J., Kloeden, P.E. & Platen, E. 2006, 'Comment on "numerical methods for stochastic differential equations"', Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 74, no. 6.
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Heath, D.P. & 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, D.P. & 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.
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. & West, J.M. 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.
Platen, E. & Runggaldier, W.J. 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. 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
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.
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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Christensen, M.M. & Platen, E. 2005, 'A general benchmark model for stochastic jump sizes', Stochastic Analysis And Applications, vol. 23, no. 5, pp. 1017-1044.
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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
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.
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.
PLATEN, E.C.K.H.A.R.D. 2005, 'AN ALTERNATIVE INTEREST RATE TERM STRUCTURE MODEL', International Journal of Theoretical and Applied Finance, vol. 08, no. 06, pp. 717-735.
This paper proposes an alternative approach to the modeling of the interest rate term 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 characterized by this factor, is used as reference unit or benchmark for obtaining a consistent price system. Benchmarked derivative prices are taken as conditional expectations of future benchmarked 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. 2004, 'A class of complete benchmark models with intensity-based jumps', Journal Of Applied Probability, vol. 41, no. 1, pp. 19-34.
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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.
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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.
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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
Platen, E. & Stahl, G. 2003, 'A structure for general and specific market risk', Computational Statistics, vol. 18, no. 3, pp. 355-373.
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Heath, D.P. & 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.
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Buhlmann, H. & Platen, E. 2003, 'A discrete time benchmark approach for insurance and finance', Astin Bulletin, vol. 33, no. 2, pp. 153-172.
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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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Platen, E. & Heath, D.P. 2002, 'A variance reduction technique based on integral representations', Quantative Finance, vol. 2, no. 5, pp. 362-369.
Heath, D.P. & 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.
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Platen, E. 2002, 'Arbitrage in continuous complete markets', Advances in Applied Probability, vol. 34, no. 3, pp. 540-558.
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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.
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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.
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Heath, D.P. & Platen, E. 2002, 'Consistent pricing and hedging for a modified constant elasticity of variance model', Quantitative Finance, vol. 2, no. 6, pp. 459-467.
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Heath, D.P. & 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.
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Heath, D.P., Hurst, S.R. & Platen, E. 2001, 'Modelling the stochastic dynamics of volatility for equity indices', Asia Pacific Financial Markets, vol. 8, pp. 179-195.
Heath, D.P., 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.
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Heath, D., Hurst, S.R. & Platen, E. 2001, 'Modelling the stochastic dynamics of volatility for equity indices', Asia Pacific Financial Markets, vol. 8, no. 3, pp. 179-195.
Craddock, M.J., Heath, D.P. & Platen, E. 2000, 'Numerical inversion of laplace transforms: a survey 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.
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Craddock, M.J., Heath, D.P. & 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.
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.
Hurst, S.K., 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.
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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, 'Axiomatic Principles For A Market Model', Journal Of Applied Probability, vol. 36, no. 1, pp. 295-300.
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Platen, E. 1999, 'A short term interest rate model', vol. 3, no. 2, pp. 215-225.
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.
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.
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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.
Platen, E. & Schweizer, M. 1998, 'On Feedback Effects From Hedging Derivatives', Mathematical Finance, vol. 8, no. 1, pp. 67-84.
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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
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.
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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. & Rebolledo, R. 1996, 'Principles For Modelling Financial Markets', Journal Of Applied Probability, vol. 33, no. 3, pp. 601-613.
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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., 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.
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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
Heath, D. & Platen, E. 1996, 'Valuation of FX barrier options under stochastic volatility', Asia-Pacific Financial Markets, vol. 3, no. 3, pp. 195-215.
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.
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.
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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.
Platen, E. 1995, 'On Weak Implicit And Predictor-corrector Methods', Mathematics And Computers In Simulation, vol. 38, no. 1-3, pp. 69-76.
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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.
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.
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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. & Rebolledo, R. 1994, 'Pricing Via Anticipative Stochastic Calculus', Advances In Applied Probability, vol. 26, no. 4, pp. 1006-1021.
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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
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.
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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
Kloeden, P., Platen, E. & Wright, I.A. 1992, 'The Approximation Of Multiple Stochastic Integrals', Stochastic Analysis And Applications, vol. 10, no. 4, pp. 431-441.
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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, 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.
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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. 1991, 'Stratonovich And Ito Stochastic Taylor Expansions', Mathematische Nachrichten, vol. 151, pp. 33-50.
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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.
Mikulevicius, R. & Platen, E. 1991, 'Rate Of Convergence Of The Euler Approximation For Diffusion-processes', Mathematische Nachrichten, vol. 151, pp. 233-239.
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Kloeden, P. & Platen, E. 1991, 'Relations Between Multiple Ito And Stratonovich Integrals', Stochastic Analysis And Applications, vol. 9, no. 3, pp. 311-321.
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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
Platen, E. 1990, 'A Stochastic Approach To Hopping Transport In Semiconductors', Journal Of Statistical Physics, vol. 59, no. 5-6, pp. 1329-1353.
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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.
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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.
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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.
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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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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.
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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.
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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.
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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.
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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.
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Other
Rudd, R., McWalter, T.A., Kienitz, J. & Platen, E. 2017, 'Fast Quantization of Stochastic Volatility Models'.
Platen, E. & Taylor, D. 2017, 'Loading Pricing of Long-Dated, Insurance-Type Contracts'.
Baldeaux, J., Ignatieva, K. & Platen, E. 2016, 'Detecting Money Market Bubbles'.
Using a range of stochastic volatility models well-known in the finance literature, we study the existence of money market bubbles in the US economy. Money market bubbles preclude the existence of a risk-neutral pricing
measure. Understanding whether markets exhibit money market bubbles is crucial from the point of view of derivative pricing since their existence implies the existence of a self-financing trading strategy that replicates the savings
account's value at a fixed future date at a cheaper cost than the current value of the savings account. The benchmark approach is formulated under the real world probability measure and does not require the existence of a risk neutral
probability measure. It hence emerges as the appropriate framework to study the potential existence of money market bubbles. Testing the existence of money market bubbles in the US economy we find that for all models the US market
exhibits a money market bubble. This conclusion suggests that for derivative pricing and hedging care should be taken when making assumptions pertaining to the existence of a risk-neutral probability measure. Less expensive hedge
portfolios may exist for a wide range of derivatives.
Fergusson, K. & Platen, E. 2015, 'Less Expensive Pricing and Hedging of Long-Dated Equity Index Options When Interest Rates are Stochastic'.
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.
Heath, D. & Platen, E. 2014, 'A Monte Carlo Method using PDE Expansions for a Diversifed Equity Index Model'.
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 diversi ed 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.
Fergusson, K. & Platen, E. 2014, 'Stylised Properties of the Interest Rate Term Structure Under The Benchmark Approach'.
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 diversi ed global
stock index. We illustrate the stylised properties of the interest rate term structure implied by a system of stochastic di erential 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.
Fergusson, K. & Platen, E. 2013, 'Real World Pricing of Long Term Cash-Linked Annuities and Equity-Linked Annuities with Cash-Linked Guarantees'.
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 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. 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 signi cantly lower prices than suggested by classical approaches. The existence of an equivalent risk neutral probability measure is not required.
Baldeaux, J. & Platen, E. 2013, 'Liability Driven Investments under a Benchmark Based Approach'.
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, J.F., Fung, M.C., Ignatieva, K. & Platen, E. 2013, 'A Hybrid Model for Equity Indices and Stochastic Interest Rates'.
Du, K., Platen, E. & Rendek, R. 2012, 'Modeling of Oil Prices'.
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 con?rms that it matches well the observed
stylized empirical facts.
Platen, E. & Tappe, S. 2011, 'Affine Realizations for Levy Driven Interest Rate Models with Real-World Forward Rate Dynamics'.
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.
Du, K. & Platen, E. 2011, 'Three-Benchmarked Risk Minimization for Jump Diffusion Markets'.
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. & Semmler, W. 2009, 'Asset markets and monetary policy', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Craddock, M.J. & Platen, E. 2009, 'On explicit probability laws for classes of scalar diffusions', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
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.
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.
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.
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.
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-¬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 ¬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 re¬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. & Rendek, R.J. 2009, 'Simulation of diversified portfolios in a continuous financial market', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
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.
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.
Platen, E. 2009, 'Real world pricing of long term contracts', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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, R.J. 2009, 'Exact scenario simulation for selected multi-dimensional stochastic processes', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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, R.J. 2009, 'Quasi-exact approximation of hidden Markov chain filters', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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. 2009, 'A benchmark approach to investing and pricing', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Miller, S. & Platen, E. 2008, 'Analytic Pricing of Contingent Claims Under the Real-World Measure'.
Kardaras, C. & Platen, E. 2008, 'On Financial Markets where only Buy-And-Hold Trading is Possible'.
Platen, E. & Hulley, H. 2008, 'Hedging for the long run'.
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.
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.
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
Hulley, H. & Platen, E. 2008, 'A visual classification of local martingales'.
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.
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.
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.
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.
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.
Platen, E. 2008, 'A unifying approach to asset pricing', REsearch Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Chavez, S.A. & Platen, E. 2008, 'Distributional deviations in random number generation in finance', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Nikeghbali, A. & Platen, E. 2008, 'On honest times in financial modeling', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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, R.J. 2007, 'Empirical evidence on Student-t log-returns of diversified world stock indices', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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, W.J. 2007, 'A benchmark approach to portfolio optimization under partial information', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Filipovic, D. & Platen, E. 2007, 'Consistent market extensions under the benchmark approach', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
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. & Bruti Liberati, N. 2006, 'Approximation of jump-diffusion in finance and economics (QFRC paper #176)', 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.
Bruti Liberati, N. & Platen, E. 2005, 'On the strong approximation of pure jump processes (QFRC paper #164)'.
ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp164.pdf
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)'.
ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp156.pdf
Christensen, M.M. & Platen, E. 2005, 'Sharpe ratio maximization and expected utility when asset prices have jumps (QFRC paper #170)'.
ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp170.pdf
Fergusson, K.J. & Platen, E. 2005, 'On the distributional characterisation of log-returns of a world stock index (QFRC paper #153)'.
ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp153.pdf
Heath, D.P. & Platen, E. 2005, 'Currency derivatives under a minimal market model with random scaling (QFRC paper #154)'.
ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp154.pdf
Hulley, H., Miller, S. & Platen, E. 2005, 'Benchmarking and fair pricing applied to two market models (QFRC paper #155)'.
ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp155.pdf
Platen, E. 2005, 'Investments in the short and long run (QFRC paper #163)'.
ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp163.pdf
Platen, E. 2005, 'On the role of the growth optimal portfolio in finance (QFRC paper #144)'.
ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp144.pdf
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'.
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.
Breymann, W., Kelly, L. & Platen, E. 2004, 'Intraday empirical analysis and modeling of diversified world stock indices (QFRC paper #125)'.
Heath, D.P. & Platen, E. 2004, 'Understanding the implied volatility surface for options on a diversified index (QFRC paper #128)'.
Heath, D.P. & Platen, E. 2004, 'Local volatility function models under a benchmark approach (QFRC paper #124)'.
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, 'Diversified portfolios with jumps in a benchmark framework (QFRC paper #129)'.
Bruti Liberati, N. & Platen, E. 2004, 'On the efficiency of simplified weak Taylor schemes for Monte Carlo simulation in finance (QFRC paper #114)'.
Platen, E. 2004, 'Capital asset pricing for markets with intensity based jumps', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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., West, J.M. & Breymann, W. 2004, 'An Intraday Empirical Analysis of Electricity Price Behaviour', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Christensen, M.M. & Platen, E. 2004, 'A general benchmark model for stochastic jump sizes', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Craddock, M.J. & Platen, E. 2003, 'Symmetric group methods for fundamental solution and characteristics functions (QFRC paper #90)'.
Kelly, L., Platen, E. & Sorensen, M. 2003, 'Estimation for discretely observed diffusions using transform functions (QFRC paper #96)'.
Platen, E. 2003, 'Pricing and hedging for incomplete jump diffusion benchmark models (QFRC paper #110)'.
Platen, E. 2003, 'A benchmark framework for risk management (QFRC paper #113)'.
Platen, E. 2003, 'Modeling the volatility and expected value of a diversified world index (QFRC paper #103)'.
Platen, E. 2003, 'An alternative interest rate term structure model (QFRC paper #97)'.
Platen, E. & Stahl, G. 2003, 'A structure for general and specific market risk (QFRC paper #91)'.
Platen, E. & West, J.M. 2003, 'Fair pricing of weather derivatives (QFRC paper #106)'.
Heath, D.P. & Platen, E. 2003, 'Pricing of index options under a minimal market model with lognormal scaling (QFRC paper #101)', Journal of macroeconomics.
Craddock, M.J. & Platen, E. 2003, 'Symmetry group methods for fundamental solutions and characteristic functions', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Platen, E. 2003, 'Diversified portfolios in a benchmark framework', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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. & West, J. 2003, 'Fair Pricing of Weather Derivatives'.
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. 2002, 'A benchmark framework for integrated risk management'.
Platen, E. & Runggaldier, W.J. 2002, 'A benchmark approach to filtering in finance'.
Heath, D.P. & Platen, E. 2002, 'Consistent pricing & hedging for a modified constant elasticity of variance model'.
Buhlmann, H. & Platen, E. 2002, 'A discrete time benchmark approach for finance & insurance'.
Platen, E. 2002, 'Benchmark model with intensity based jumps', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Heath, D.P. & Platen, E. 2002, 'A variance reduction technique based on integral representations', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Platen, E. 2001, 'Arbitrage in continuous complete markets', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Heath, D.P. & 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.
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.
Craddock, M.J. & Platen, E. 2001, 'Benchmark pricing of credit derivatives under a standard market model', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
Platen, E. 2001, 'A benchmark model for financial markets', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
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.
Research Paper Number: 54
Hardle, W.K., 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.
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.
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.
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 minimal financial market model', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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. 2000, 'Risk premia and financial modelling without measure transformation', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
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.
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.
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.
Craddock, M.J., Heath, D.P. & 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.
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.
Platen, E. 1999, 'A Financial Market Model with Trading Volume and Stochastic Volatility'.
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, 'An Introduction to Numerical Methods for Stochastic Differential Equations'.
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 cn be interpreted as generalising 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.
Platen, E. 1999, 'On the Log-Return Distribution of Index Benchmarked Share Prices'.
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.
Platen, E. 1999, 'A Minimal Share Market Model with Stochastic Volatility'.
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.
Hoek, J.V.D. & Platen, E. 1999, 'Pricing and Hedging in the Presence of Transaction Costs Under Local Risk Minimisation'.
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'.
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.
Elliott, R., Fischer, P. & Platen, E. 1999, 'Filtering and Parameter Estimation for a Mean Reverting Interest Rate Model'.
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.
Heath, D., Platen, E. & Schweizer, M. 1998, 'Comparison of Some Key Approches to Hedging in Incomplete Markets'.
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.
Platen, E. & Taylor, D., 'Loading Pricing of Catastrophe Bonds and Other Long-Dated, Insurance-Type Contracts'.
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.
Gnoatto, A., Grasselli, M. & Platen, E., 'A Penny Saved is a Penny Earned: Less Expensive Zero Coupon Bonds'.
This paper introduces a more general modeling world than available under the
classical no-arbitrage 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.
McWalter, T.A., Rudd, R., Kienitz, J. & Platen, E., 'Recursive Marginal Quantization of Higher-Order Schemes'.
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 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 recursive marginal
quantization for two higher-order schemes: the Milstein scheme and a simplified
weak order 2.0 scheme. As part of this generalization a simple matrix
formulation is presented, allowing efficient implementation. We further extend
the applicability of recursive marginal quantization by showing how absorption
and reflection at the zero boundary may be incorporated, when this is
necessary. To illustrate the improved accuracy of the higher order schemes,
various computations are performed using geometric Brownian motion and its
generalization, the constant elasticity of variance model. For both processes,
we show numerical evidence of improved weak order convergence and we compare
the marginal distributions implied by the three schemes to the known analytical
distributions. By pricing European, Bermudan and Barrier options, further
evidence of improved accuracy of the higher order schemes is demonstrated.
Leisen, D. & Platen, E., 'Investing for the Long Run'.
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.