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Professor Eckhard Platen

Biography

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

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

He is the co-author of three successful books on Numerical Methods for Stochastic Differential Equations, a book on his innovative Benchmark Approach, published by Springer Verlag, another monograph on Functionals of Multidimensional Diffusions with Applications to Finance and he has authored more than 170 research papers in quantitative finance and applicable mathematics. He has also held more than 80 visiting appointments at leading institutions world wide.

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

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

Google Scholar citation page

RePEc page

Professional

  • President of the Bachelier Finance Society.
  • Honorary Professor at University of Cape Town.
  • Adjunct Professor of the Australian National University
  • Best Paper Award in Mathematical Finance: Hofmann, N., Platen, E., & Schweizer, M. (1992) “Option pricing under incompleteness and stochastic volatility”. Mathematical Finance, 2 (3), pp. 153-187.

Professional Memberships:

  • Australian Mathematical Society
  • Bachelier Finance Society
  • Econometric Society
Image of Eckhard Platen
Professor of Quantitative Finance, School of Mathematical and Physical Sciences
Professor of Quantitative Finance, Finance and Economics
Associate Member, Centre for Innovation in IT Services Applications
Core Member, Quantitative Finance Research Centre
MMath (Dresden), PhD (Dresden), Dr sc
Member, Australian Mathematical Society
Member, Bachelier Finance Society
Member, The Econometric Society
 
Phone
+61 2 9514 7759
Room
CB08.07.79

Research Interests

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

Can supervise: Yes

Postgraduate research degree students supervised:

Current:

  • Kevin J Fergusson
  • Liang Zhao

PhD Theses (Completed):

  • Ke Du "Commodity Derivative Pricing under the Benchmark Approach"
  • Renata Rendek "Modelling Diversified Equity Indices"
  • Leah Kelly “Inference and Intraday Analysis Of Diversified World Stock Indices “
  • Nicola Bruti Liberati “Numerical Solution of Stochastic Differential Equations with Jumps in Finance”
  • Kelly Ann Lennox “Lie Symmetry Methods for Multidimensional Linear Parabolic PDEs and Diffusions“   
  • Hardy Hulley “Strict Local Martingales in Continuous Financial Market Models “
  • Troy Morgan “Integrated Modelling of Credit and Equity Market Risk”
  • Shane Miller “Pricing Of Contingent Claims under the Real-World Measure”
  • Samson Assefa “Pricing Swaptions and Credit Default Swaptions in the Quadratic Gaussian Factor Model “          
  • Jason Michael West “Benchmark Models and Pricing Applications in Finance”    

Master of Science in Mathematical Sciences (Thesis)

  • Thusitha Srimal Liyanage “Scenario Simultation of Solutions of Multidimentional Stochastic Differential Equation”            

Stochastic Analysis; Mathematics of Finance; Numerical Methods in Finance; and Futures and Options.

Books

Baldeaux, J.F. & Platen, E. 2013, Functionals of multidimensional diffusions with applications to finance, 1st, Springer, Germany.
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This research monograph provides an introduction to tractable multidimensional diffusion models, where transition densities, Laplace transforms, Fourier transforms, fundamental solutions or functionals can be obtained in explicit form. The book also provides an introduction to the use of Lie symmetry group methods for diffusions, which allows to compute a wide range of functionals. Besides the well-known methodology on affine diffusions it presents a novel approach to affine processes with applications in finance. Numerical methods, including Monte Carlo and quadrature methods, are discussed together with supporting material on stochastic processes. Applications in finance, for instance, on credit risk and credit valuation adjustment are included in the book. The functionals of multidimensional diffusions analyzed in this book are significant for many areas of application beyond finance. The book is aimed at a wide readership, and develops an intuitive and rigorous understanding of the mathematics underlying the derivation of explicit formulas for functionals of multidimensional diffusions.
Platen, E. & Bruti Liberati, N. 2010, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, 1st, Springer, Germany.
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This research monograph concerns the design and analysis of discrete-time approximations for stochastic differential equations (SDEs) driven by Wiener processes and Poisson processes or Poisson jump measures, In financial and actuarial modeling and other areas of application I such jump difrusions are often used to dScribe the dynamics of ',.-arious state variables. In finance these may represent, for instance, asset prices, credit ratings, stock indices, luterest rates, exchange rates or commodity prices. The jump component can capture event-driven unC<'xtainties, such as corporato defaults, operational failures or insured events.
Platen, E. & Heath, D. 2010, A benchmark approach to quantitative finance, 2nd, Springer, Germany.
Platen, E. & Heath, D.P. 2006, A Benchmark Approach to Quantitative Finance, 1st, Springer, Berlin, Germany.
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The benchmark approach provides a general framework for financial market modeling, which extends beyond the standard risk-neutral pricing theory. It permits a unified treatment of portfolio optimisation, derivative pricing, integrated risk managemetn and insurance risk modeling. Th existence of an equivalent risk-neutral pricing measure is not required.Instead, it leads to pricing formulae with respect to the real-world probability measure. This yields important modeling freedom which turns out to be necessary for the derivation of realistic, parsimonious market models. The first part of the book describes the necessary tools fromprobability theory, statistics, stochastic calculus and the theory of stochastic differential equations with jumps. The second part of devoted to financial modeling by the benchmark approach. Various quajtitative methods for the real-world pricing and hedging of derivatives are explained. The general framework is used to provide an understanding of the nature of stochastic volatility. The book is intended for a wide audience that includes quantitative analysts, postgraduate students and practitioners in finance, economics and insurance. It aims to be a self- contained, accessible but methematically rigorous introduction to quantitative finance for readers that have a reasonable mathematical or quantitative background. Finally, the book hsould stimulate inetrest in the benchmark approach by describing some of its power and wide applicability.
Kloeden, P., Platen, E. & Schurz, H. 2003, Numerical Solution of SDE Through Computer Experiments, 3rd, Springer, Germany.
Kloeden, P. & Platen, E. 1999, Numerical solution of stochastic differential equations, 3rd, Springer, Germany.

Chapters

Platen, E. & Semmler, W. 2012, 'A dynamic portfolio approach to asset markets and monetary policy' in Cohen, S.N., Madan, D., Siu, T.K. & Yang, H. (eds), Advances in Statistics, Probability and Actuarial Science: Stochastic Processes, Finance and Control, World Scientific, USA, pp. 347-373.
Platen, E. & Rendek, R.J. 2012, 'Simulation of diversified portfolios in a continuous financial markets' in Zhang, T. & Zhou, X. (eds), Stochastic Analysis and Applications to Finance: Essays in Honour of Jia-an Yan, World Scientific, USA, pp. 385-410.
Platen, E. 2011, 'A benchmark approach to investing and pricing' in MacLean, L.C., Thorp, E.O. & Ziemba, W.T. (eds), The Kelly Capital Growth Investment Criterion: Theory and Practice, World Scientific Publishing, USA, pp. 409-426.
This paper introduces a general market modeling framework, the benchmark appma.chl which assumes the existence of the nume!'raire portfolio. This is the strictly positive portfolio that when used as benchmMk makes all benchmarked non-negati.ve portfolios sllperma.rtinga!es, that is intuitively speaking downward trending or trendless. It can be shQ'Wn to equal the Kelly portfolio, which tna.-"jmiz. es expected logarithmk utility. In several Wa.ys, the KeUy or numeraire portfolio is the "bestll performing portfolio and cannot be outperformed systematically by any other non-negative portfolio. Its use in pricing as nttmeroire leads directly to the real world pricing formula) which employs the real world probability when calculating conditional expectations. In a large regular financial market, the Kelly portfolio is shawn to be approxima.ted by welldivcnrified portfolios.
Bruti Liberati, N. & Platen, E. 2010, 'Monte Carlo simulation for stochastic differential equation' in al, R.C.E. (ed), Encyclopedia of Quantitative Finance, Wiley, US, pp. 1271-1278.
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Platen, E. & Bruti Liberati, N. 2010, 'Stochastic differential equations: Scenario simulation' in al, R.C.E. (ed), Encyclopedia of Quantitative Finance, Wiley, US, pp. 1697-1706.
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Jackel, P. & Platen, E. 2010, 'Monte Carlo simulations' in al, R.C.E. (ed), Encyclopedia of Quantitative Finance, Wiley, US, pp. 1266-1271.
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Platen, E. 2010, 'Stochastic Taylor expansions' in al, R.C.E. (ed), Encyclopedia of Quantitative Finance, Wiley, US, pp. 1731-1734.
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Bruti Liberati, N. & Platen, E. 2010, 'Stochastic differential equations with jumps: Simulation' in al, R.C.E. (ed), Encyclopedia of Quantitative Finance, Wiley, US, pp. 1693-1697.
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Platen, E. 2008, 'Simulation methods for stochastic differential equation' in Seese, D., Weinhardt, C. & Schlottmann, F. (eds), Handbook on Information Technology in Finance, Springer, Germany, pp. 501-514.
Platen, E. 2004, 'Simulation methods for stochastic differential equations' in Teugels, J. & Sundt, B. (eds), Encyclopedia of Actuarial Science, John Wiley & Sons, Chichester, UK, pp. 1559-1564.
Heath, D.P. & Platen, E. 2002, 'Pricing and hedging of index derivatives under an alternative asset price model with endogenous stochastic volatility' in Yong, J. (ed), Recent Developments in Mathematical Finance, World Scientific, Singapore, pp. 117-126.
Heath, D.P., Platen, E. & Schweizer, M. 2001, 'Numerical Comparison of Local Risk-Minimisation & Mean-Variance Hedging' in Jouini, E., Cvitanic, J. & Musiela, M. (eds), Handbooks in Mathematical Finance: Option Pricing, Interest Rates & Risk Management, Cambridge University Press, Cambridge, UK, pp. 509-537.
Elliott, R. & Platen, E. 1999, 'Hidden Markov Chain Filtering for Generalised Bessel Processes'.
Finite-dimensional recursive filters are obtained for generalised Bessel processes with a drift parameter that follows a hidden Markov chain. In particular, filters are constructed for the states, the jumps and the occupation times of the states of the Markov chain. These lead to estimators for the transition rates and the levels of the hidden states of the chain. Finally a minimum variance filter is described that minimises fluctuations of the filters.

Conferences

Du, K. & Platen, E. 2014, 'Benchmarked risk minimization', Mathematical Finance.
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This paper discusses the problem of hedging not perfectly replicable contingent claims using the numraire portfolio. The proposed concept of benchmarked risk minimization leads beyond the classical no-arbitrage paradigm. It provides in incomplete markets a generalization of the pricing under classical risk minimization, pioneered by Fllmer, 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 numraire portfolio to identify the minimal possible price for a contingent claim. Furthermore, the resulting benchmarked (i.e., numraire 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. 2014 Wiley Periodicals, Inc.
Baldeaux, J.F. & Platen, E. 2013, 'Computing functionals of square root and Wishart processes under the benchmark approach via exact simulation', Monte Carlo and Quasi Monte Carlo Methods 2012, Springer Proceedings in Mathematics and Statistics, Springer, Germany, pp. 3-22.
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The aim of the paper is to show how Wishart processes can be used flexibly in financial modeling. We explain how functionals, resulting from the benchmark approach to finance, can be accurately computed via exact simulation methods. We employ Lie symmetry methods to identify explicit transition densities and explicitly computable functionals. We illustrate the proposed methods via finance problems formulated under the benchmark approach. This approach allows us to exploit conveniently the analytical tractability of the considered diffusion processes.
Nikitopoulos-Sklibosios, C. & Platen, E. 2012, 'Alternative Term Structure Models for Reviewing Expectations Puzzles'.
According to the expectations hypothesis, the forward rate is equal to the expected future short rate, an argument that is not supported by most empirical studies that demonstrate the existence of term premiums. An alternative arbitrage-free term structure model for reviewing the expectations hypothesis is presented and tractable expressions for time-varying term premiums are obtained. The model is constructed under the real-world probability measure and depends on two stochastic factors: the short rate and the market price of risk. The model suggests that for short maturities the short rate contribution determines the term premiums, while for longer maturities, the contribution of the market price of risk dominates.
Platen, E. 2012, 'Benchmarked risk minimization in incomplete markets'.
Platen, E. 2012, 'Numerical solution of SDEs'.
Platen, E. 2012, 'Numerical solution of stochastic differential equations with jumps in finance'.
Platen, E. 2012, 'Benchmarked risk minimization'.
Bruti Liberati, N. & Platen, E. 2012, 'On weak predictor-corrector schemes for jump-diffusion processes in finance', Topics in Numerical Methods for Finance: Proceedings in Mathematics and Statistics, Springer, Germany, pp. 1-12.
Event-driven uncertainties such as corporate defaults, operational failures, or central bank announcements are important elements in the modeling of financial quantities. Therefore, stochastic differential equations (SDEs) of jumpdiffusion type are often used in finance. We consider in this paper weak discrete time approximations of jump-diffusion SDEs which are appropriate for problems such as derivative pricing and the evaluation of risk measures. We present regular and jump-adapted predictorcorrector schemes with first and second order of weak convergence. The regular schemes are constructed on regular time discretizations that do not include jump times, while the jump-adapted schemes are based on time discretizations that include all jump times. A numerical analysis of the accuracy of these schemes when applied to the jump-diffusion Merton model is provided.
Platen, E. & Rendek, R. 2012, 'The Affine Nature of Aggregate Wealth Dynamics'.
The paper derives a parsimonious two-component affine diffusion model for a world stock index to capture the dynamics of aggregate wealth. The observable state variables of the model are the normalized index and the inverse of the stochastic market activity, both modeled as square root processes. The square root process in market activity time for the normalized aggregate wealth emerges from the affine nature of aggregate wealth dynamics, which will be derived under basic assumptions and does not contain any parameters that have to be estimated. The proposed model employs only three well interpretable structural parameters, which determine the market activity dynamics, and three initial parameters. It is driven by the continuous, nondiversifiable uncertainty of the market and no other source of uncertainty. The model, to be valid over long time periods, needs to be formulated in a general financial modeling framework beyond the classical no-arbitrage paradigm. It reproduces a list of major stylized empirical facts, including Student-t distributed log-returns and typical volatility properties. Robust methods for fitting and simulating this model are demonstrated.
Platen, E. 2011, 'A benchmark approach to quantitative finance'.
Platen, E. 2011, 'On targeted pensions'.
Platen, E. 2011, 'A dynamic portfolio approach to monetary policy'.
Hulley, H. & Platen, E. 2011, 'A visual criterion for identifying Ito diffusions as martingales or strict local martingales', Seminar on Stochastic Analysis, Random Fields and Applications VI, Springer, Ascona, Switzerland, pp. 147-157.
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. 2010, 'The benchmark approach'.
Platen, E. 2010, 'Valuing guaranteed minimum death benefits'.
Platen, E. 2010, 'Empirical properties of a diversified global stock index'.
Platen, E. 2010, 'Stylized empirical facts on diversified indices'.
Platen, E. 2010, 'A benchmark approach to computative finnce'.
Platen, E. 2010, 'Real world pricing of long term contracts'.
Ignatieva, K. & Platen, E. 2010, 'Modelling co-movements and tail dependency in the international stock market via copulae'.
Platen, E. & Rendek, R. 2010, 'Simulation of Diversified Portfolios in a Continuous Financial Market'.
The paper analyzes the simulated long-term behavior of well diversified portfolios in continuous financial markets. It focuses on the equi-weighted index and the market portfolio. The paper illustrates that the equally weighted portfolio constitutes a good proxy of the growth optimal portfolio, which maximizes expected logarithmic utility. 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 a multi-asset version of the minimal market model. All these models are simulated exactly or almost exactly over an extremely long period of time to analyze the long term growth of the respective portfolios. The paper illustrates the robustness of the diversification phenomenon when approximating the growth optimal portfolio by the equi-weighted index. Significant outperformance in the long run of the market capitalization weighted portfolio by the equi-weighted index is documented for different market models. Under the multi-asset minimal market model the equi-weighted index outperforms remarkably the market portfolio. In this case the benchmarked market portfolio is a strict supermartingale, whereas the benchmarked equi-weighted index is a martingale. Equal value weighting overcomes the strict supermartingale property that the benchmarked market portfolio inherits from its strict supermartingale constituents under this model.
Baldeaux, J., Chan, L. & Platen, E. 2010, 'Quasi-Monte carlo methods for derivatives on realised variance of an index under the benchmark approach', ANZIAM Journal.
We apply quasi-Monte Carlo methods to the pricing of derivatives on realised variance of an index under the benchmark approach. The resulting integration problem is shown to depend on the joint density of the realised variance of the index and the terminal value of the index. Employing a transformation mapping for this joint density to the unit square reduces the diffculty of the resulting integration problem. The quasi-Monte Carlo methods compare favourably to Monte Carlo methods when applied to the given problem. Austral. Mathematical Soc. 2011.
Platen, E. 2009, 'On interest rate term structure modeling under the benchmark approach'.
Platen, E. 2009, 'Quantitative methods - Computing and numerical methods'.
Platen, E. 2009, 'A benchmark approach to quantitative finance'.
Platen, E. 2009, 'A benchmark approach beyond semi-martingales'.
Platen, E. 2009, 'Numerical solution of stochastic differential equations with jumps in finance'.
Platen, E. & Semmler, W. 2009, 'Asset Markets and Monetary Policy'.
Monetary policy has pursued the concept of inflation targeting. This has been implemented in many countries. Here interest rates are supposed to respond to an inflation gap and output gap. Despite long term continuing growth of the world financial assets, recently, monetary policy, in particular in the U.S. after the subprime credit crisis, was challenged by severe disruptions and a meltdown of the financial market. Subsequently, academics have been in search of a type of monetary policy that does allow to influence in an appropriate manner the investor's behavior and, thus, the dynamics of the economy and its financial market. The paper suggests a dynamic portfolio approach. It allows one to study the interaction between investors` strategic behavior and monetary policy. The article derives rules that explain how monetary authorities should set the short term interest rate in interaction with inflation rate, economic growth, asset prices, risk aversion, asset price volatility, and consumption rates. Interesting is that the inflation rate needs to have a certain minimal level to allow the interest rate to be a viable control instrument. A particular target interest rate has been identified for the desirable optimal regime. If the proposed monetary policy rule is applied properly, then the consumption rate will remain stable and the inflation rate can be kept close to a minimal possible level. Empirical evidence is provided to support this view. Additionally, in the case of an economic crisis the proposed relationships indicate in which direction to act to bring the economy back on track.
Bruti Liberati, N. & Platen, E. 2009, 'Strong Predictor-Corrector Euler Methods for Stochastic Differential Equations'.
Bruti-Liberati, N., Nikitopoulos-Sklibosios, C., Platen, E. & Schlgl, E. 2009, 'Alternative defaultable term structure models', Asia-Pacific Financial Markets, 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. 2009 Springer Science+Business Media, LLC.
Platen, E. 2009, 'A Variance Reduction Technique Based on Integral Representations'.
Platen, E. 2009, 'Valuing guaranteed minimum death benefit options in variable annuities under a benchmark approach"'.
Platen, E. 2009, 'Asset markets and monetary policy'.
Miller, S.M. & Platen, E. 2008, 'Analytic pricing of contingent claims under the real-world measure', International Journal of Theoretical and Applied Finance, 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. 2008 World Scientific Publishing Company.
Platen, E. 2008, 'The Law of the Minimal Price'.
Platen, E. 2008, 'A Benchmark Approach to Finance'.
Platen, E. 2008, 'Properties of a Diversified World Stock Index'.
Platen, E. 2008, 'Portfolio Optimization Under Partial Information'.
Platen, E. 2008, 'Numerical solution of stochastic differential equations'.
Platen, E. 2008, 'A benchmark approach to quantitative finance'.
Platen, E. 2008, 'Law of the Minimal Price'.
Bruti Liberati, N., Nikitopoulos Sklibosios, C. & Platen, E. 2008, 'Monte-Carlo simulations of alternative defaultable term structure models'.
Nikitopoulos Sklibosios, C., Bruti Liberati, N., Platen, E. & Schlogl, E. 2008, 'Real-world pricing for defaultable term structure models'.
Platen, E. 2008, 'On Interest Rate Term Structure Modelling Under the Benchmark Approach'.
Platen, E. 2008, 'Conditions for Martingales with Applications in Finance'.
Platen, E. 2008, 'Valuing Guaranteed Minimum Death Benefit Options'.
Hulley, H. & Platen, E. 2008, 'Laplace transform identities for diffusions, with applications to rebates and barrier options', Banach Centre Publications: Advances in Mathematics of Finance, Polska Akademia Nauk, Warszawa, Poland, pp. 139-157.
Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms of some functions of first-passage times for the diffusion. These results are applied to the special case of squared Bessel processes with killing or reflecting boundaries. In particular, we demonstrate how the above-mentioned integral identity enables us to derive the transition density of a squared Bessel process killed at the origin, without the need to invert a Laplace transform. Finally, as an application, we consider the problem of pricing barrier options on an index described by the minimal market model.
Platen, E. 2008, 'A Unifying Approach to Asset Pricing'.
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 best 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.
Nikeghbali, A. & Platen, E. 2008, 'On honest times in financial modeling'.
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.
Thulasiram, R.K., Downing, C.T., Chiarella, C., Coleman, T., Dempster, M., Dongarra, J., Duan, J.-.C., Gao, G., Appadoo, S.S., Atiya, A., Bagchi, A., Birge, J., Brabazon, A., Broadie, M., Campolieti, J., Cincotti, S., Downing, C., Gilli, M., Isaenko, S., Jacoby, G., Kumar, K., Klebaner, F., Li, X., Li, Y., Livdan, D., Lyuu, Y.-.D., Nath, G.C., Okten, G., Oosterlee, C.W., Ouskel, A.M., Platen, E., Seco, L., Srinivasan, A., Srinivasan, R., Thenmozhi, M., Thulasiraman, P., Tsang, E.P.K., Wagner, A., Wang, L., Wilson, C., Wittum, G., Ing, C.W. & Tanaka-Yamawaki, M. 2008, 'Message from PDCoF-08 Workshop Chairs', IPDPS Miami 2008 - Proceedings of the 22nd IEEE International Parallel and Distributed Processing Symposium, Program and CD-ROM.
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Platen, E. 2007, 'Portfolio Optimization Under Partial Information'.
Bruti Liberati, N., Nikitopoulos Sklibosios, C., Platen, E. & Schlogl, E. 2007, 'Real-World Pricing for Defaultable Term Structure Models'.
Platen, E. 2007, 'Benchmark Approach to Continuous Time Finance'.
Platen, E. 2007, 'Simulation of High-Dimensional Models in Finance'.
Platen, E. 2007, 'Extreme Maturity Options'.
Platen, E. 2007, 'Numerical Solutions of Stochastic Differential Equations with Jumps in Finance'.
Platen, E. 2007, 'Numerical Solution of SDEs with Jumps in Finance'.
Bruti Liberati, N., Nikitopoulos Sklibosios, C., Platen, E. & Schlogl, E. 2007, 'Defaultable term structure models under the benchmark approach'.
Platen, E. 2006, 'Capital asset pricing for markets with intensity based jumps', STOCHASTIC FINANCE, SPRINGER, pp. 157-182.
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Platen, E. 2006, 'A Benchmark approach to finance', Mathematical Finance, pp. 131-151.
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This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling, and risk measurement. It is based on the natural assumption that investors prefer more rather than less, in the sense that given two portfolios with the same diffusion coefficient value, the one with the higher drift is preferred. Each such investor is shown to hold an efficient portfolio in the sense of Markowitz with units in the market portfolio and the savings account. The market portfolio of investable wealth is shown to equal a combination of the growth optimal portfolio (GOP) and the savings account. In this setup the capital asset pricing model follows without the use of expected utility functions, Markovianity, or equilibrium assumptions. The expected increase of the discounted value of the GOP is shown to coincide with the expected increase of its discounted underlying value. The discounted GOP has the dynamics of a time transformed squared Bessel process of dimension four. The time transformation is given by the discounted underlying value of the GOP. The squared volatility of the GOP equals the discounted GOP drift, when expressed in units of the discounted GOP. Risk-neutral derivative pricing and actuarial pricing are generalized by the fair pricing concept, which uses the GOP as numeraire and the real-world probability measure as pricing measure. An equivalent risk-neutral martingale measure does not exist under the derived minimal market model. 2006 Blackwell Publishing Inc.
Bruti-Liberati, N. & Platen, E. 2006, 'On Weak Predictor-Corrector Schemes for Jump-Diffusion Processes in Finance'.
Event-driven uncertainties such as corporate defaults, operational failures or central bank announcements are important elements in the modelling of financial quantities. Therefore, stochastic differential equations (SDEs) of jump-diffusion type are often used in finance. We consider in this paper weak discrete time approximations of jump-diffusion SDEs which are appropriate for problems such as derivative pricing and the evaluation of risk measures. We present regular and jump-adapted predictor-corrector schemes with first and second order of weak convergence. The regular schemes are constructed on regular time discretizations that do not include jump times, while the jump-adapted schemes are based on time discretizations that include all jump times. A numerical analysis of the accuracy of these schemes when applied to the jump-diffusion Merton model is provided.
Bruti Liberati, N., Nikitopoulos Sklibosios, C. & Platen, E. 2006, 'Heath Jarrow Morton equation for jump-diffusions under the benchmark approach', 2nd International Symposium on Economic Theory, Policy & Applications, -, -.
Bruti Liberati, N. & Platen, E. 2006, 'On the weak approximation of jump-diffusion processes with applications in finance', Proceedings of the VII Workshop on Quantitative Finance, University of Perugia, Italy, Perugia, Italy, pp. 1-96.
Bruti Liberati, N. & Platen, E. 2006, 'Predictor-corrector schemes for jump-diffusion processes.'.
Bruti Liberati, N. & Platen, E. 2006, 'Weak predictor-corrector methods for jump diffusions in finance'.
Bruti Liberati, N., Nikitopoulos Sklibosios, C. & Platen, E. 2006, 'On the strong approximation of jump-diffusion processes'.
Platen, E. 2006, 'A benchmark approach to portfolio optimization and derivative pricing.', Statistical Modeling in Finance Conference.
Platen, E. 2006, 'A benchmark approach to portfolio optimization and derivative pricing.', First Conference of Advanced Mathematical Methods for Finance.
Platen, E. 2006, 'On the Pricing and Hedging of Long Dated Zero Coupon Bonds'.
The pricing and hedging of long dated derivative contracts is a challenging area of research. As a result of utility indifference pricing for general payoffs the growth optimal portfolio turns out to be the appropriate numeraire or benchmark with the real world probability measure as corresponding pricing measure. This concept of real world pricing can be applied for valuing long dated derivatives. An equivalent risk neutral probability measure does not need to exist under this benchmark approach. This paper develops a parsimonious model for a stock index dynamics, which is based on a time transformed squared Bessel process. It uses a diversified world stock index as proxy for the growth optimal portfolio. Surprisingly low prices result for long dated zero coupon bonds that can be replicated using historical data. Such prices and hedges are difficult to explain under the prevailing risk neutral approach.
Platen, E. 2006, 'Pricing and hedging of long dated zero coupon bonds.', 2006 DAIWA International Workshop on Financial Engineering.
Platen, E. 2006, 'Pricing and hedging extreme maturity contracts under the benchmark approach.', Quantitative Methods in Finance 2006 Conference.
Le, T. & Platen, E. 2006, 'Approximating the Growth Optimal Portfolio with a Diversified World Stock Index'.
This paper constructs and compares various total return world stock indices based on daily data. Due to diversification these indices are noticeably similar. A diversification theorem identifies any diversified portfolio as a proxy for the growth optimal portfolio. 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. This has applications to derivative pricing and investment management.
Platen, E. 2006, 'A parsimonious finacial market model in a jump diffusion setting'.
Platen, E. 2006, 'A unified approach to portfolio optimization and derivative pricing.', Conference on Risk Management.
Platen, E. 2005, 'Diversified portfolios with jumps in a Benchmark framework', Asia-Pacific Financial Markets, 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 the 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 optimization. Springer 2005.
Martini, F., Piccardi, M., Liberati, N.B. & Platen, E. 2005, 'A hardware generator for multi-point distributed random variables', 2005 IEEE INTERNATIONAL SYMPOSIUM ON CIRCUITS AND SYSTEMS (ISCAS), VOLS 1-6, CONFERENCE PROCEEDINGS, IEEE, pp. 1702-1705.
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Bruti-Liberati, N. & Platen, E. 2005, 'On the Strong Approximation of Jump-Diffusion Processes'.
In financial modelling, filtering and other areas the underlying 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, there is a need for the systematic use of discrete time approximations in corresponding simulations. This paper presents a survey and new results on strong numerical schemes for SDEs of jump-diffusion type. These are relevant for scenario analysis, filtering and hedge simulation in finance. It provides a convergence theorem for the construction of strong approximations of any given order of convergence for SDEs driven by Wiener processes and Poisson random measures. The paper covers also derivative free, drift-implicit and jump adapted strong approximations. For the commutative case particular schemes are obtained. Finally, a numerical study on the accuracy of several strong schemes is presented.
Hulley, H., Heath, D.P. & Platen, E. 2005, 'A comparative study of performance robustness for equity index models', 4th National Symposium on Financial Mathematics, -, -.
Hulley, H., Heath, D.P. & Platen, E. 2005, 'A comparative study of performance robustness for equity index models', Mathematics in Finance International Conference, -, -.
Platen, E. 2005, 'Portfolio optimization and derivative pricing in a jump diffusion market', Quantitative Methods in Finance 2005 Conference, -, -.
Platen, E. 2005, 'Investments for the Short and Long Run'.
This paper aims to discuss the optimal selection of investments for the short and long run in 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. An optimal portfolio is shown to have a fraction of its wealth invested in the growth optimal portfolio and the remaining fraction in the savings account. The risk aversion of the investor at a given time determines the volatility of her or 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 transformed time growth exponentially, a parsimonious and realistic model for the market portfolio dynamics results. It allows for efficient portfolio optimization and derivative pricing.
Platen, E. 2005, 'A unified framework for portfolio optimisation and asset pricing', 49th Annual Meeting of the Australian Mathematical Society, -, -.
Platen, E. 2005, 'Investment for the short and long run', Past, Present and Future in Investment Management, -, -.
Christensen, M.M. & Platen, E. 2005, 'A general benchmark model for stochastic jump sizes', STOCHASTIC ANALYSIS AND APPLICATIONS, TAYLOR & FRANCIS INC, pp. 1017-1044.
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Platen, E. 2005, 'On the role of the growth optimal portfolio in finance', 7th JAFEE International Conference, -, -.
Platen, E. 2005, 'On the role of the growth optimal portfolio in finance', Mathematical Finance Workshop, -, -.
Platen, E. 2005, 'On the role of the growth optimal portfolio in finance', 4th National Symposium on Financial Mathematics, -, -.
Platen, E. 2005, 'A unified framework for portfolio optimization and asset pricing', Developments in Quantitative Finance, -, -.
Hulley, H., Miller, S. & Platen, E. 2005, 'Benchmarking and Fair Pricing Applied to Two Market Models'.
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 under the real-world measure. This justifies the fair pricing approach, which expresses derivative prices in terms of real-world conditional expectations of benchmarked payoffs. Two models for benchmarked primary security accounts are presented, and fair pricing formulas for some common contingent claims are derived.
Christensen, M. & Platen, E. 2005, 'Sharpe Ratio Maximization and Expected Utility when Asset Prices have Jumps'.
We analyze portfolio strategies which are locally optimal, meaning that they maximize the Sharpe ratio in a general continuous time jump-diffusion framework. These portfolios are characterized explicitly and compared to utility based strategies. In the presence of jumps, maximizing the Sharpe ratio is shown to be generally inconsistent with maximizing expected utility, but this is shown to depend strongly on market completeness and whether event risk is priced.
Liberati, N.B., Platen, E., Martini, F. & Piccardi, M. 2005, 'A multi-point distributed random variable accelerator for Monte Carlo simulation in finance', Proceedings - 5th International Conference on Intelligent Systems Design and Applications 2005, ISDA '05, pp. 532-537.
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The pricing and hedging of complex derivative securities via Monte Carlo simulations of stochastic differential equations constitutes an intensive computational task. To achieve "real time" execution, as often required by financial institutions, one needs highly efficient implementations of the multi-point distributed random variables underlying the simulations. In this paper a fast and flexible dedicated hardware solution is proposed. A comparative performance analysis demonstrates that the hardware solution is bottleneck-free and flexible, and significantly increases the computational efficiency of the software solution. 2005 IEEE.
Kelly, L., Platen, E. & Sorensen, M. 2004, 'Estimation for discretely observed diffusions using transform functions.', National symposium on probability and its applications, -, -.
Platen, E. 2004, 'Numerical solution of stochastic differential equations', Workshop on mathematical methods in finance, -, -.
Platen, E. 2004, 'Modelling the volatility and expected value of a diversified world index.', International workshop on mathematical finance and insurance, -, -.
Platen, E. 2004, 'A benchmark approach to risk management.', Stochastic finance 2004, -, -.
Liberati, N.B. & Platen, E. 2004, 'On the efficiency of simplified weak Taylor schemes for Monte Carlo simulation in finance', Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), pp. 771-778.
The purpose of this paper is to study the efficiency of simplified weak schemes for stochastic differential equations. We present a numerical comparison between weak Taylor schemes and their simplified versions. In the simplified schemes discrete random variables, instead of Gaussian ones, are generated to approximate multiple stochastic integrals. We show that an implementation of simplified schemes based on random bits generators significantly increases the computational speed. The efficiency of the proposed schemes is demonstrated. Springer-Verlag Berlin Heidelberg 2004.
Heath, D.P. & Platen, E. 2004, 'Local volatility function models under a benchmark approach.', Daiwa International Workshop on Financial Engineering, DAIWA, Japan, pp. 1-19.
Platen, E. 2004, 'Pricing and hedging for incomplete jump diffusion benchmark models', Mathematics of Finance: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance, American Mathematical Society, Providence, pp. 287-301.
Platen, E. 2004, 'A benchmark framework for risk management', STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE, WORLD SCIENTIFIC PUBL CO PTE LTD, pp. 305-335.
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Platen, E. 2004, 'Modelling the expected value of a diversified world index.', 3rd World congress, Bachelier finance society, -, -.
Platen, E. 2003, 'Diversified Portfolios in a Benchmark Framework'.
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.
Heath, D.P. & Platen, E. 2003, 'Pricing of index options under a minimal market model with lognormal scaling', Workshop on Mathematical Finance, --, --.
Platen, E. 2003, 'The two-factor interest rate term structure minimal market model', Japanese Association Financial Econometrics and Engineering Meeting, --, --.
Platen, E. 2003, 'A benchmark framework for financial markets', Conference of Stochastic Processes and Mathematical Finance, --, --.
Platen, E. 2003, 'An incomplete benchmark model', AMS-SIAM Mathematical Finance Conference, --, --.
Platen, E. 2003, 'A class of complete benchmark models', 2nd National Symposium on Financial Mathematics, --, --.
Platen, E. 2003, 'Modeling the volatility and expected value of a diversified world index', Quantitative Methods in Finance 2003 Conference, --, --.
Platen, E. 2002, 'Arbitrage in continuous complete markets', 2nd World Congress of the Bachelier Finance Society.
Platen, E. 2002, 'A modified constant elasticity of varinace model', Quantitative Finance Conference.
Heath, D. & Platen, E. 2002, 'Pricing and hedging of index derivatives under an alternative asset price model with Eedogenous stochastic volatility', Recent developments in mathematical finance: Proceedings of the international conference on mathematical finance, World Scientific, Singapore, pp. 117-126.
Platen, E. 2001, 'A Minimal Financial Market Model'.
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. 1999, 'A Financial Market Model'.
Despite many attempts, the consistent and global modelling of financial markets remains an open problem. In particular it remains a challenge to find a simple and tratable economic and probablistic approach to market modelling. This paper attempts to highlight fundamental properties that a market model should have. Assuming these properties, which include the principle of market risk minimisation, it is possible to establish a corresponding interactive stochastic market dynamics that involves a minimal number of factors. These factors include the trading volume of assets and the average trading value of all assets. Several interesting properties related to stochastic volatility, market index and interest rate dynamics can be derived. Empirical evidence will be given that supports these findings.
Platen, E., 'A benchmark approach to finance'.
Platen, E., 'Numerical solution of stochastic differential equations with jumps in finance'.
Platen, E., 'Numerical solution of stochastic differential equations in finance'.
Platen, E., 'Beyond the classical paradigm'.

Journal articles

Baldeaux, J., 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.
Du, K. & Platen, E. 2015, 'Benchmarked risk minimization', Mathematical Finance.
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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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 numraire portfolio. The growth optimal portfolio is used as numraire 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.
Kardaras, C., Obl?j, J. & Platen, E. 2015, 'The numraire property and long-term growth optimality for drawdown-constrained investments', Mathematical Finance.
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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 numraire property through the notion of expected relative return and prove that drawdown-constrained numraire 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 numraire portfolio is given explicitly through a model-independent transformation of the unconstrained numraire portfolio. The asymptotically growth-optimal strategy is obtained as limit of numraire strategies on finite horizons.
Baldeaux, J., 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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The growth optimal portfolio (GOP) plays an important role in finance, where it serves as the numraire portfolio, with respect to which contingent claims can be priced under the real world probability measure. This paper models the GOP using a time dependent constant elasticity of variance (TCEV) model. The TCEV model has high tractability for a range of derivative prices and fits well the dynamics of a global diversified world equity index. This is confirmed when pricing and hedging various derivatives using this index.
Du, K. & Platen, E. 2014, 'Benchmarked risk amangement', Mathematical Finance.
Biagini, F., Cretarola, A. & Platen, E. 2014, 'Local risk-minimization under the benchmark approach', Mathematics and Financial Economics, pp. 1-26.
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We study the pricing and hedging of derivatives in incomplete financial markets by considering the local risk-minimization method in the context of the benchmark approach, which will be called benchmarked local risk-minimization. We show that the proposed benchmarked local risk-minimization allows to handle under extremely weak assumptions a much richer modeling world than the classical methodology. 2014 Springer-Verlag Berlin Heidelberg.
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.
Nikeghbali, A. & Platen, E. 2013, 'A reading guide for last passage times with financial applications in view', Finance and 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. 2013 Springer-Verlag Berlin Heidelberg.
Nikitopoulos Sklibosios, C. & Platen, E. 2013, 'Alternative term structure models for reviewing the expectations puzzle', International Journal of Economic Research, vol. 10, no. 2, pp. 349-372.
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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 Black-Scholes 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. 2013 Copyright Taylor and Francis Group, LLC.
Hulley, H. & Platen, E. 2012, 'Hedging for the long run', Mathematics and Financial Economics, vol. 6, no. 2, pp. 105-124.
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In the years following the publication of Black and Scholes (J Political Econ, 81(3), 637-654, 1973), numerous alternative models have been proposed for pricing and hedging equity derivatives. Prominent examples include stochastic volatility models, jump-diffusion models, and models based on Lvy 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. 2012 Springer-Verlag.
Kardaras, C. & Platen, E. 2012, 'ON THE DYBVIG-INGERSOLL-ROSS THEOREM', MATHEMATICAL FINANCE, vol. 22, no. 4, pp. 729-740.
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Platen, E. & Rendek, R. 2012, 'Approximating the numraire 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 numraire 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. 2012 Macmillan Publishers Ltd.
Cheridito, P., Nikeghbali, A. & Platen, E. 2012, 'Processes of class sigma, last passage times, and drawdowns', SIAM Journal on Financial Mathematics, vol. 3, no. 1, 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. Copyright 2012 by SIAM.
Hulley, H. & Platen, E. 2012, 'Hedging for the long run', Mathematics and Financial Economics, vol. 6, no. 2, pp. 105-124.
View/Download from: Publisher's site
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 Lvy 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. 2011 Elsevier B.V. All rights reserved.
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.
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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. 2012 World Scientific Publishing Company.
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.
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. 2011 Elsevier B.V. All rights reserved.
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.
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 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 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.
Bruti-Liberati, N., Nikitopoulos-Sklibosios, C. & Platen, E. 2010, 'Real-world jump-diffusion term structure models', Quantitative Finance, vol. 10, no. 1, pp. 23-37.
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This paper considers interest rate term structure models in a market attracting both continuous and discrete types of uncertainty. The event-driven noise is modelled by a Poisson random measure. Using as numeraire the growth optimal portfolio, interest rate derivatives are priced under the real-world probability measure. In particular, the real-world dynamics of the forward rates are derived and, for specific volatility structures, finite-dimensional Markovian representations are obtained. Furthermore, allowing for a stochastic short rate in a non-Markovian setting, a class of tractable affine term structures is derived where an equivalent risk-neutral probability measure may not exist. 2010 Taylor & Francis.
Ignatieva, K., Platen, E. & Rendek, R. 2010, 'Using Dynamic Copulae for Modeling Dependency in Currency Denominations of a Diversifed World Stock Index'.
The aim of this paper is to model the dependencya mong log-returns when security account prices are expressed in units of a well diversified world stock index. The paper uses the equi-weighted index EWI104s, calculated as the average of 104 world industry sector indices. The log-returns of its denominations in different currencies appear to be Student-t distributed with about four degrees of freedom. Motivated by these findings, the dependency in log-returns of currency denominations of the EWI104s 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. Furthermore, the Student-t copula with Student-t marginals is able to better capture dependent extreme values than the other models considered. Finally, 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 de fied in regulatory documents. The paper demonstrates that the approach performs clearly better than the Risk Metrics approach.
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.
Filipovi?, 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. 2009 Wiley Periodicals, Inc.
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.
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.
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.
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 Levy type processes.
Platen, E. 2008, 'The Law of Minimum Price'.
This paper introduces a realistic, generalized market modeling framework for which the Law of One Price no longer holds. Instead the Law of the Minimal Price will be derived, which for contingent claims with long term to maturity may provide significantly lower prices than suggested under the currently prevailing approach. This new law only requires the existence of the numeraire portfolio, which turns out to be the portfolio that maximizes expected logarithmic utility. In several ways it will be shown that the numeraire portfolio cannot be outperformed by any nonnegative portfolio. The new Law of the Minimal Price leads directly to the real world pricing formula, which uses the numeraire portfolio as numeraire and the real world probability for calculating conditional expectations. The cost efficient pricing and hedging of extreme maturity zero coupon bonds illustrates the new law in the context of the US market.
Miller, S. & Platen, E. 2008, 'Analytic Pricing of Contingent Claims Under the Real-World Measure', Research Paper Series, vol. -, no. 216, pp. 1-30.
Kardaras, C. & Platen, E. 2008, 'On Financial Markets where only Buy-And-Hold Trading is Possible'.
A financial market model where agents can only trade using realistic buyand-hold strategies is considered. Minimal assumptions are made on the nature of the asset-price process in particular, the semimartingale property is not assumed. Via a natural assumption of limited opportunities for unlimited resulting wealth from trading, coined the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition, we establish that asset-prices have to be semimartingales, as well as a weakened version of the Fundamental Theorem of Asset Pricing that involves supermartingale deflators rather than Equivalent Martingale Measures. Further, the utility maximization problem is considered and it is shown that using only buy-and-hold strategies, optimal utilities and wealth processes resulting from continuous trading can be approximated arbitrarily well.
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 proposed symmetric predictor-corrector Euler methods. 2008 World Scientific Publishing Company.
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.
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.
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 suffcient 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.
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. 2006 Elsevier B.V. All rights reserved.
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. 2007 Springer Science+Business Media, LLC.
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. Springer Science+Business Media, LLC 2007.
Platen, E. & Rendek, R. 2007, 'Empirical Evidence on Student-t Log-Returns of Diversified World Stock Indices'.
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.
Bruti Liberati, N., Nikitopoulos Sklibosios, C. & Platen, E. 2006, 'First order strong approximations of jump diffusions', Monte Carlo Methods and Applications, vol. 12, no. 3-4, pp. 191-209.
DP0559879
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'.
This paper constructs and compares various total return world stock indices based on daily data. Due to diversification these indices are noticeably similar. A diversification theorem identifies any diversified portfolio as a proxy for the growth optimal portfolio. 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. This has applications to derivative pricing and investment management.
Platen, E. 2006, 'A benchmark approach to asset management', Journal of Asset Management, vol. 6, no. 6, pp. 390-405.
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DP0343913 This paper aims to discuss the optimal selection of investments for the short and long runin a continuous time financial market setting. First, it documents the almost sure pathwise long-run outperformance of all positive portfolios by the growth optimal portfolio. Secondly, it assumes that every investor prefers more rather than less wealth and keeps the freedom to adjust his or her risk aversion at any time. In a general continuous market, a two fund separation result is derived which yields optimal portfolios located on the Markowitz efficient frontier. A optimal portfolio is shown to have a fraction of its wealth invested inthe growth optimal portfolio and the remaining fraction inthe savings account. The risk aversion of the investor at a given time determines the volatility of her/his optimal portfolio. It is pointed out that it is usually not rational to reduce risk aversion further than is necessary to achieve the maximum growth rate. Assuming an optimal dynamics for a global market, the market portfolio turns out to be growth optimal. The discounted market portfolio is shown to follow a particular time transformed diffusion process with explicitly known transition density. Assuming that the drift og yhr discounted market portfolio grows exponentially, a parsimonioous and realistic model for its dynamics results. It allows for efficient portfolio optimisation and derivative pricing.
Platen, E. 2006, 'A benchmark approach to finance', Mathematical Finance, vol. 16, no. 1, pp. 131-151.
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This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling, and risk measurement. It is based on the natural assumption that investors prefer more rather than less, in the sense that given two portfolios wit
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 asset pricing and portfolio optimization. The paper argues that the market portfolio is a proxy of the growth optimal portfolio. By choosing the drift of the discounted growth optimal portfolio as parameter process, one obtains a realistic theoretical market dynamics. 2006 Elsevier B.V. All rights reserved.
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.
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. & 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 stochastic volatility model to describe the dynamics of a currency market for the pricing and hedging of derivatives. Time transformed squared Bessel processes are the basic driving factors of the minimal market model. The time transformation is characterized by a random scaling, which 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 typically observed in real markets. World Scientific Publishing Company.
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.
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'.
The paper discusses various roles that the growth optimal portfolio (GOP) plays in finance. For the case of a continuous market we showhow the GOP can be interpreted as a fundamental building block in financial market modeling, portfolio optimization, contingent claim pricing and risk measurement. On the basis of a portfolio selection theorem, optimal portfolios are derived. These allocate funds into the GOP and the savings account. A risk aversion coecient is introduced, controlling the amount invested in the savings account, which allows to characterize portfolio strategies that maximize expected utilities. Natural conditions are formulated under which the GOP appears as the market portfolio. A derivation of the intertemporal capital asset pricing 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 generalize risk neutral and actuarial pricing. Finally, the GOP is described in various ways as the best performing portfolio.
Platen, E. & West, J. 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 pricing. 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 concept. A discrete time model is constructed to approximate historical weather characteristics. The fair prices of some particular weather derivatives are derived using historical and Gaussian residuals. The question of weather risk as diversifiable risk is also discussed. Springer 2005.
Platen, E. & Runggaldier, W.J. 2005, 'A benchmark approach to filtering in finance', Asia-Pacific Financial Markets, vol. 11, no. 1, pp. 79-105.
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.
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 yield 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 squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on 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 possible 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, the model replicates the key features of the interest rate cap market for economies with low interest 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. Springer 2005.
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 and Applied Finance, vol. 8, no. 6, pp. 717-735.
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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 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. World Scientific Publishing Company.
Hulley, H., Miller, S. & Platen, E. 2005, 'Benchmarking and fair pricing applied to two marker models', The Kyoto Economic Review, vol. 74, no. 1, pp. 85-118.
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.
Craddock, M. & Platen, E. 2004, 'Symmetry group methods for fundamental solutions', Journal of Differential Equations, vol. 207, no. 2, pp. 285-302.
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This paper uses Lie symmetry group methods to study PDEs of the form ut = xuxx + f (x)ux. We show that when the drift function f is a solution of a family of Ricatti equations, then symmetry techniques can be used to find a fundamental solution. 2004 Elsevier Inc. All rights reserved.
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.
The paper presents a consistent approach to the modeling of general and specific market risk as defined in regulatory documents. It compares the statistically based beta-factor model with a class of benchmark models that use a broadly based index as major building block for modeling. The investigation of log-returns of stock prices that are expressed in units of the market index reveals that these are likely to be Student t distributed. A corresponding discrete time benchmark model is used to calculate Value-at-Risk for equity portfolios.
Heath, D. & 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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This paper describes a two-factor model for a diversified market index using the growth optimal portfolio with a stochastic and possibly correlated intrinsic timescale. The index is modelled using a time transformed squared Bessel process with a log-normal scaling factor for the time transformation. A consistent pricing and hedging framework is established by using the benchmark approach. Here the numeraire is taken to be the growth optimal portfolio. Benchmarked traded prices appear as conditional expectations of future benchmarked prices under the real world probability measure. The proposed minimal market model with log-normal scaling produces the type of implied volatility term structures for European call and put options typically observed in real markets. In addition, the prices of binary options and their deviations from corresponding Black-Scholes prices are examined.
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.
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Heath, D. & Platen, E. 2002, 'A Variance Reduction Technique Based on Integral Representations'.
Standard Monte Carlo methods can often be significantly improved with the addition of appropriate variance reduction techniques. In this paper a new and powerful variance reduction technique is presented. The method is based directly on the Ito calculus and is used to find unbiased variance reduced estimators for the expectation of functionals of Ito diffusion processes. The approach considered has wide applicability, for instance, it can be used as a means of approximating solutions of parabolic partial differential equations or applied to valuation problems that arise in mathematical finance. We illustrate how the method can be applied by considering the pricing of European style derivative securities for a class of stochastic volatility models, including the Heston model.
Heath, D.P. & Platen, E. 2002, 'Perfect hedging of index derivatives under a minimal market 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.
Kchler, 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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This 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. 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.
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.
The paper estimates the speed of convergence of the Euler approximation for diffusion processes with jump component which have Holder continuous coefficients. 2002 VSP.
Heath, D. & Platen, E. 2002, 'Consistent Pricing and Hedging for a Modified Constant Elasticity of Variance Model'.
This paper considers a modification of the well-known constant elasticity of variance model where it is used to model the growth optimal portfolio. It is shown taht, for this application, there is no equivalent risk neutral pricing methodology fails. However, a consistent pricing and hedging framework can be established by application of the benchmark approach. Perfect hedging strategies can be constructed for European style contingent claims, where the underlying risky asset is the growth optimal portfolio. In this framework, fair prices for contingent claims are the minimal prices that permit perfect replication of the claims. Numerical examples show that these prices may differ significantly from the corresponding "risk neutral" prices. In cases where these prices are different, arbitrage amounts can be generated.
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., Hurst, S. & Platen, E. 2001, 'Modelling the stochastic dynamics of volatility for equity indices', Asia-Pacific Financial Markets, vol. 8, no. 3, pp. 179-195.
The paper develops a class of continuous time stochastic volatility models, which generate asset price returns that are approximately Student t distributed. Using the criterion of local risk minimisation in an incomplete market setting, option prices are computed. It is shown that implied volatility smile and skew patterns of the type often observed in the markets can be obtained from this class of stochastic volatility models. 2002 Kluwer Academic Publishers.
Heath, D., 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.
Hofmann, N. & Platen, E. 2000, 'Approximating large diversified portfolios', Mathematical Finance, vol. 10, no. 1, pp. 77-88.
This paper considers a financial market with asset price dynamics modeled by a system of lognormal stochastic differential equations. A one-dimensional stochastic differential equation for the approximate evolution of a large diversified portfolio formed by these assets is derived. This identifies the asymptotic dynamics of the portfolio as being a lognormal diffusion. Consequentially an efficient way for computing probabilities, derivative prices, and other quantities for the portfolio are obtained. Additionally, the asymptotic strong and weak orders of convergence with respect to the number of assets in the portfolio are determined.
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.
Kchler, U. & Platen, E. 2000, 'Strong discrete time approximation of stochastic differential equations with time delay', Mathematics and Computers in Simulation, vol. 54, no. 1-3, pp. 189-205.
The paper introduces an approach for the derivation of discrete time approximations for solutions of stochastic differential equations (SDEs) with time delay. The suggested approximations converge in a strong sense. Furthermore, explicit solutions for linear stochastic delay equations are given. 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.
Craddock, M., Heath, D. & Platen, E. 1999, 'Numerical Inversion of Laplace Transforms: A Survey of Techniques with Applications to Derivative Pricing'.
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.
Hurst, S.R., Platen, E. & Rachev, S.T. 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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Platen, E. 1999, 'Axiomatic principles for a market model', JOURNAL OF APPLIED PROBABILITY, vol. 36, no. 1, pp. 295-300.
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.
Platen, E. & Schweizer, M. 1998, 'On feedback effects from hedging derivatives', MATHEMATICAL FINANCE, vol. 8, no. 1, pp. 67-84.
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Milstein, G.N., 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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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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Kloeden, P.E., 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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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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KLOEDEN, P.E., 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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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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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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KLOEDEN, P.E., PLATEN, E. & WRIGHT, I.W. 1992, 'THE APPROXIMATION OF MULTIPLE STOCHASTIC INTEGRALS', STOCHASTIC ANALYSIS AND APPLICATIONS, vol. 10, no. 4, pp. 431-441.
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KLOEDEN, P.E. & 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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KLOEDEN, P.E. & PLATEN, E. 1991, 'STRATONOVICH AND ITO STOCHASTIC TAYLOR EXPANSIONS', MATHEMATISCHE NACHRICHTEN, vol. 151, pp. 33-50.
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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.E. & 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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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.E. & 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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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.
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

Baldeaux, J. & Platen, E. 2015, 'Credit Derivative Evaluation and CVA Under the Benchmark Approach', Springer New York LLC.
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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-wayand 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.
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.
Baldeaux, J., Fung, M.C., Ignatieva, K. & Platen, E. 2014, 'A Hybrid Model for Pricing and Hedging of Long Dated Bonds'.
Long dated xed 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 num eraire 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 t 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 nd that the hybrid model with the 3/2 dynamics for the interest rate provides the best t to the data with respect to lowest prices and least expensive hedges.
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.
Kardaras, C. & Platen, E. 2013, 'Multiplicative approximation of wealth processes involving no-short-sales strategies via simple trading'.
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. 2012 Wiley Periodicals, Inc.
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.
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.
Hulley, H. & Platen, E. 2011, 'A Visual Criterion for Identifying Ito Diffusions as Martingales or Strict Local Martingales'.
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.
Kardaras, C. & Platen, E. 2010, 'Minimizing the Expected Market Time to Reach a Certain Wealth Level'.
Miller, S.M. & Platen, E. 2010, 'Real-World Pricing for a Modified Constant Elasticity of Variance Model'.
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. 2010 Taylor & Francis.
Ignatieva, K. & Platen, E. 2010, 'Modelling co-movements and tail dependency in the international stock market via copulae'.
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. 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 is superior to the alternative copula models investigated, as well the Riskmetics approach. 2010 Springer Science+Business Media, LLC.
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. & Platen, E. 2009, 'On Explicit Probability Laws for Classes of Scalar Diffusions'.
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.
Bruti Liberati, N., Nikitopoulos Sklibosios, C., Platen, E. & Schlogl, E. 2009, 'Alternative Defaultable Term Structure Models', Quantitative Finance Research Paper Series.
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., Luethi, D.R. & Platen, E. 2009, 'Empirical behavior of a world stock index from intra-day to monthly time scales'.
Platen, E. & Rendek, R. 2009, 'Simulation of Diversified Portfolios in a Continuous Financial Market'.
In this paper we analyze the simulated behavior of diversified portfolios in a continuous financial market. In particular, we focus on equally weighted portfolios. We illustrate 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 out performance of the market capitalization weighted portfolio by the equal value weighted portfolio can be observed for models.
Platen, E. 2009, 'Real World Pricing of Long Term Contracts'.
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. 2009, 'Exact Scenario Simulation for Selected Multi-dimensional Stochastic Processes'.
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. 2009, 'Quasi-exact Approximation of Hidden Markov Chain Filters'.
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'.
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.
Bruti-Liberati, N., Martini, F., Piccardi, M. & Platen, E. 2008, 'A hardware generator of multi-point distributed random numbers for Monte Carlo simulation'.
Monte Carlo simulation of weak approximations of stochastic differential equations constitutes an intensive computational task. In applications such as finance, for instance, to achieve "real time" execution, as often required, one needs highly efficient implementations of the multi-point distributed random number generator underlying the simulations. In this paper, a fast and flexible dedicated hardware solution on a field programmable gate array is presented. A comparative performance analysis between a software-only and the proposed hardware solution demonstrates that the hardware solution is bottleneck-free, retains the flexibility of the software solution and significantly increases the computational efficiency. Moreover, simulations in applications such as economics, insurance, physics, population dynamics, epidemiology, structural mechanics, chemistry and biotechnology can benefit from the obtained speedups. 2007 IMACS.
Platen, E. & Shi, L. 2008, 'On the Numerical Stability of Simulation Methods for SDES'.
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.
Marquardt, T., Platen, E. & Jaschke, S. 2008, 'Valuing Guaranteed Minimum Death Benefit Options in Variable Annuities Under a Benchmark Approach'.
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 best? 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. & Platen, E. 2008, 'Distributional Deviations in Random Number Generation in Finance'.
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.
Hrdle, W., Kleinow, T., Korostelev, A., Logeay, C. & Platen, E. 2008, 'Semiparametric diffusion estimation and application to a stock market index'.
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 SP 500 index data.
CHRISTENSEN, M.O.R.T.E.N.M.O.S.E.G.A.A.R.D. & PLATEN, E.C.K.H.A.R.D. 2007, 'SHARPE RATIO MAXIMIZATION AND EXPECTED UTILITY WHEN ASSET PRICES HAVE JUMPS'.
We analyze portfolio strategies which are locally optimal, meaning that they maximize the Sharpe ratio in a general continuous time jump-diffusion framework. These portfolios are characterized explicitly and compared to utility based strategies. We show that in the presence of jumps, maximizing the Sharpe ratio is generally inconsistent with maximizing expected utility, in the sense that a utility maximizing individual will not choose a strategy which has a maximal Sharpe ratio. This result will hold unless markets are incomplete or jump risk has no risk premium. In case of an incomplete market we show that the optimal portfolio of a utility maximizing individual may "accidentally" have maximal Sharpe ratio. Furthermore, if there is no risk premium for jump risk, a utility maximizing investor may select a portfolio having a maximal Sharpe ratio, if jump risk can be hedged away. We note that uncritical use of the Sharpe ratio as a performance measure in a world where asset prices exhibit jumps may lead to unreasonable investments with positive probability of ruin.
Hulley, H. & Platen, E. 2007, 'Laplace Transform Identities for Diffusions, with Applications to Rebates and Barrier Options'.
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'.
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'.
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.
Kchler, U. & Platen, E. 2007, 'Time Delay and Noise Explaining Cyclical Fluctuations in Prices of Commodities'.
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.
Heath, D. & Platen, E. 2006, 'Local volatility function models under a benchmark approach'.
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 diversified index approximates that of the growth optimal portfolio. Fair prices for derivatives when expressed in units of the index are martingales under the real-world probability measure. Different to the classical approach that derives risk-neutral probabilities the paper obtains the transition density for the index with respect to the real-world probability measure. Furthermore, the Dupire formula for the underlying local volatility function is recovered without assuming the existence of an equivalent risk-neutral probability measure. A modification of the constant elasticity of variance model and a version of the minimal market model are discussed as specific examples together with a smoothed local volatility function model that fits a snapshot of S&P500 index options data.
Fergusson, K. & Platen, E. 2006, 'On the distributional characterization of daily log-returns of a world stock index'.
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.
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.
Platen, E. & Runggaldier, W.J. 2005, 'A benchmark approach to filtering in finance'.
The paper proposes the use of the growth optimal portfolio for pricing and hedging in incomplete 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 information is measured. Springer 2005.
Breymann, W., Kelly, L. & Platen, E. 2005, 'Intraday empirical analysis and modeling of diversified world stock indices'.
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. Springer 2006.
Heath, D. & Platen, E. 2005, 'Understanding the implied volatility surface for options on a diversified index'.
This paper describes a two-factor model for a diversified 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 analysis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed squared Bessel process of dimension four. It turns out 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 proposed model, which includes a random initial condition for market activity, generates implied volatility 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 the proposed model and their Black-Scholes counterparts. Springer 2005.
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
HEATH, D.A.V.I.D. & PLATEN, E.C.K.H.A.R.D. 2005, 'CURRENCY DERIVATIVES UNDER A MINIMAL MARKET MODEL WITH RANDOM SCALING'.
This paper uses an alternative, parsimonious stochastic volatility model to describe the dynamics of a currency market for the pricing and hedging of derivatives. Time transformed squared Bessel processes are the basic driving factors of the minimal market model. The time transformation is characterized by a random scaling, which 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 typically observed in real markets.
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.
Kelly, L., Platen, E. & Sorensen, M. 2004, 'Estimation for discretely observed diffusions using transform functions'.
Platen, E. 2004, 'Modeling the volatility and expected value of a diversified world index'.
This paper considers a diversified world stock index in a continuous financial market with the growth optimal portfolio (GOP) as reference unit or benchmark. Diversified broadly based indices and portfolios, which include major world stock market indices, are shown to approximate the GOP. It is demonstrated that a key financial quantity is the trend of a world index. It turns out that 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 stock index as approximation of the discounted GOP 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 a 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 the properties of an index and its volatility.
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. & Breymann, W. 2004, 'An Intraday Empirical Analysis of Electricity Price Behaviour'.
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)'.
Platen, E. 2003, 'Pricing and Hedging for Incomplete Jump Diffusion Benchmark Models'.
This paper considers a class of incomplete financial market models with security price processes that exhibit intensity based jumps. The benchmark or numeraire is chosen to be the growth optimal portfolio. Portfolio values, when expressed in units of the benchmark, are local martingales. In general, an equivalent risk neutral martingale measure need not exist in the proposed framework. Benchmarked fair derivative prices are defined as conditional expectations of future benchmarked prices under the real world probability measure. This concept of fair pricing generalizes classical risk neutral pricing. The pricing under incompleteness is modeled by the choice of the market prices for risk. The hedging is performed under minimization of profit and loss fluctuations.
Platen, E. 2003, 'A benchmark framework for risk management (QFRC paper #113)'.
Platen, E. 2003, 'An alternative interest rate term structure model (QFRC paper #97)'.
Platen, E. & 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. & Platen, E. 2003, 'Symmetry Group Methods for Fundamental Solutions and Characteristic Functions'.
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.Keywords: lie symmetry groups; green's functions; fundamental solutions; characteristic functions; transition densities; symmetry techniques
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'.
The paper describes a consistent, integrated framework for modeling and pricing in finance, insurance and other areas of risk management. The growth optimal portfolio is taken as a benchmark. In the resulting price system expected future benchmarked, nonnegative prices are not greater that the last observed benchmarked price. The resulting benchmark model does not permit the generation of wealth from zero initial capital or the systemtic outperformance of the benchmark. Benchmarked fair price processes are defined as best forecasts of their benchmarked future values. Risk neutral and actuarial pricing formulae are obtained as special cases. Certain arbitrage amounts can still be modeled in this framework.
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 and Insurance'.
This paper proposes an integrated appraoch to discrete time modelling in finance and insurance. This approach is based on the existence of a specific benchmark portfolio, known as the growth optimal portfolio. When used as numeraire, this portfolio ensures that all benchmarked price processes are super-martingales. A fair market is characterized in terms of the type of maximum that the optimal growth rate attains. In general, arbitrage amounts arise due to supermartingale property of benchmarked traded prices. No measure transformation is needed for the pricing of insurance policies and derivatives in a fair market.
Platen, E. 2002, 'Benchmark Model with Intensity Based Jumps'.
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. & Platen, E. 2001, 'Perfect Hedging of Index Derivatives Under a Locally Arbitrage Free Minimal Market Model'.
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. & Platen, E. 2001, 'Benchmark Pricing of Credit Derivatives Under a Standard Market Model'.
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'.
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
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'.
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'.
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.
Fischer, P. & Platen, E. 1999, 'Applications of the balanced method to stochastic differential equations in filtering', 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.
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.