Professor Eckhard Platen

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.

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 (eds), Recent Developments in Mathematical Finance, World Scientific, Singapore, pp. 117-126.

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

Heath, 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.

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.

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.

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.

Filipovic, D. & Platen, E. 2009, 'Consistent Market Extensions Under the Benchmark Approach', Mathematical Finance, vol. 19, no. 1, pp. 41-52.

Miller, S.M. & Platen, E. 2009, '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.

Mittnik, S., Nell, E., Platen, E., Semmler, W. & Chappe, R. 2009, 'Financial market meltdown and a need for new financial regulations', METU Studies in Development, vol. 36, no. 1, pp. 253-269.
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The financial crisis, triggered by the subprime and real estate crisis in the US, has become global It is deeply rooted in a decade-long misuse of the financial market for rent-seeking. The financial industry has largely abandoned Its role as a service industry, supposedly charging reasonable fees for the services of spreading risk and allocating capital and credit. Instead it provides a market for speculation, corporate control - mergers and acquisitions -: and a casino for bettmg on or hedging practically any kind of risk - the derivatives market.

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.

Bruti Liberati, N. & Platen, E. 2008, 'Strong Predictor-Corrector Euler Methods for Stochastic Differential Equations', Stochastics and Dynamics, vol. 8, no. 3, pp. 561-581.
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This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor-corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the new symmetric predictor-corrector Euler methods.

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. & 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.

Platen, E. & Rendek, R.J. 2008, 'Empirical evidence on Student-t log-returns of diversified world stock indices', Journal of Statistical Theory and Practice, vol. 2, no. 2, pp. 233-251.
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The aim of this paper is to document some empirical facts related to log-returns of diversified workld stock indices when these are denominated in different currencies. Motivated by eaarlier results we jave obtained the estimated distribution of log-returns for a range of world sotckindices over long observation periods. We expand previous studies bya pplying the maximum likelihood ration test to the large class of generalised hyperbolic distributions and investigate the log-returns ofa variety of diversified world stock indices in different currency denominations.

Bruti Liberati, N. & Platen, E. 2007, 'Approximation of Jump Diffusions in Finance and Economics', Computational Economics, vol. 29, no. 3-4, pp. 283-312.
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In finance and economics the key dynamics are often specified via stochastic differential equations (SDEs) of jump-diffusion type. The class of jump-diffusion SDEs that admits explicit solutions is rather limited. Consequently, discrete time approximations are required. In this paper we give a survey of strong and weak numerical schemes for SDEs with jumps. Strong schemes provide pathwise approximations and therefore can be employed in scenario analysis, filtering or hedge simulation. Weak schemes are appropriate for problems such as derivative pricing or the evaluation of risk measures and expected utilities. Here only an approximation of the probability distribution of the jump-diffusion process is needed. As a framework for applications of these methods in finance and economics we use the benchmark approach. Strong approximation methods are illustrated by scenario simulations. Numerical results on the pricing of options on an index are presented using weak approximation methods.

Bruti Liberati, N. & Platen, E. 2007, 'Strong Approximations of Stochastic Differential Equations with Jumps', Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 982-1001.
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This paper is a survey of strong discrete time approximations of jump-diffusion processes described by stochastic differential equations (SDEs). It also presents new results on strong discrete time approximations for the specific case of pure jump SDEs. Strong approximations based on jump-adapted time discretizations, which produce no discretization error in the case of pure jump processes, are analyzed. The computational complexity of these approximations is proportional to the jump intensity. By exploiting a stochastic expansion for pure jump processes, higher order discrete time approximations, whose computational complexity is not dependent on the jump intensity, are proposed. For the specific case of pure jump SDEs, the strong order of convergence of strong Taylor schemes is established under weaker conditions than those currently known in the literature.

Christensen, 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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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.

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.

Bruti Liberati, N., Nikitopoulos Sklibosios, C. & Platen, E. 2006, 'First order strong approximations of jump diffusions', Monte Carlo Methods and Applications, vol. 12, no. 3-4, pp. 191-209.
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DP0559879

Burrage, K., Burrage, P., Higham, D., Kloeden, P. & Platen, E. 2006, 'Comment on "numerical methods for stochastic differential equations"', Physical Review E, vol. 74, no. 6, pp. 068701-1-068701-2.
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Wilkie (Phys. Rev. E 70, 017701 (2004)) used a heuristic approach to derive Runge-Kutta-based numerical methods for stochastic differential equations based on methods used for solving ordinary differential equations. The aim ws to follow solution paths with high order. We point out that this approach is invalid in the general case and does not lead to high order methods. We warn readers against the inappropriate use of deterministic calculus in a stochastic setting.

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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DP0559879

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

Le, T. & Platen, E. 2006, 'Approximating the growth optimal portfolio with a diversified world stock index', The Journal of Risk Finance, vol. 7, no. 5, pp. 559-574.
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DP0559879

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

Platen, E. 2006, 'Portfolio selection and asset pricing under a benchmark approach', Physica A: Statistical Mechanics And Its Applications, vol. 370, no. 1, pp. 23-29.
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The paper presents classical and new results on portfolio optimization, as well as the fair pricing concept for derivative pricing under the benchmark approach. The growth optimal portfolio is shown to be a central object in a market model. It links asse

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

Heath, D.P. & Platen, E. 2005, 'Currency derivatives under a minimal market model with random scaling', International Journal of Theoretical and Applied Finance, vol. 8, no. 8, pp. 1157-1177.
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This paper uses an alternative, parsimonious volatility model to describe the dunamics of a currency market for the pricing and hedging of derivatives. Time transformed squared Bessel processes are the basic driving factors of the minimal marketr model. The time transformation is chracterised by a random scaling, wich provides for realistic exchange rate dynamics. The pricing of standard European options is studied. In particular, it is shown that the model produces implied volatility surfaces that are tpically observed in real markets.

Heath, D.P. & Platen, E. 2005, 'Understanding the implied volatility surface for options on a diversified index', Asia-Pacific Financial markets, vol. 11, no. 1, pp. 55-77.
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This paper describes a two-factor model for a diveersified index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analhsis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed square Bessel process of dimension four. It turns our that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The prposed model, which includes a random initial condition for market activity, generates implied colatility surfaces for European call and put options that are typically observed in real markets. The paper also examines the price differences of binary options for th epropsed model and their Black-Scholes counterparts.

Hulley, H., Miller, S.M. & Platen, E. 2005, 'Benchmarking and fair pricing applied to two marker models', The Kyoto Economic Review, vol. 74, no. 1, pp. 85-118.
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This paper considers a market containing both continuous and discrete noise. Modest assumptions ensure the existence of a growth optimal portfolio. Non-negative self-financing trading strategies, when benchmarked by this portfolio, are local martingales unde the real-world measure. This justifies the fair pricing approach, which expresses derivative prices in terms of real-world conditional expectations of benchmarked pay-offs. Two models for benchmarked primary security accounts are presentated, and fair pricing formulas for some common contingent claims are derived.

Miller, S.M. & Platen, E. 2005, 'A two-factor model for low interest rate regimes', Asia-Pacific Financial Markets, vol. 11, no. 1, pp. 107-133.
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This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yielf curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP and is modelled as a aquared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, thsi study focusses in the difficult but important case where the short rate stays close to zero for a prolonged period of time. FOr the proposed model, an equivalent risk neutral martingale measure is neither possile nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, themodel replicates the ket features of the inetrest rate cap market for economies with low inetrest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew.

Platen, E. & Runggaldier, W.J. 2005, 'A benchmark approach to filtering in finance', Asia-Pacific Financial Markets, vol. 11, no. 1, pp. 79-105.
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The paper propsoed the use of the growth optimal portfolio for pricing and hedging in imcomplete markets when there are unobserved factors that have to be filtered. The proposed filtering framework is applicable also in cases when there does not exist an equivalent risk neutral martingale measure. The reduction of the variance of derivative prices for increasing degrees of available iformation is measured.

Platen, E. & West, J.M. 2005, 'A fair pricing approach to weather derivatives', Asia-Pacific Financial Markets, vol. 11, no. 1, pp. 23-53.
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This paper proposes a consistent approach to the pricing of weather derivatives. Since weather derivatives are traded in an incomplete market setting, standard hedging based pricing methods cannot be applied. The growth optimal portfolio, which is interpreted as a world stock index,is used as a benchmark or numeraire such that all benchmarked derivative price processes are martingales. No measure transformation is needed for the proposed fair paricing. For weather derivative payoffs that are independent of the value of the growth optimal portfolio, it is shown that the classical actuarial pricing methodology is a particular case of the fair pricing concepts. A discrete time model is constructed to approximate historical weather characteristics. The fair prices of some partuclar weather derivatives are derived using historical and Gaussian residuals. The question of weather risk as diversifiable risk is also discussed.

Platen, E. 2005, 'An alternative interest rate term structure model', International Journal of Theoretical & Applied Finance, vol. 8, no. 6, pp. 717-735.
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This paper proposes and alternative approach to the modeling of the interest rate twem structure. It suggests that the total market price for risk is an important factor that has to be modeled carefully. The growth optimal portfolio which is characterised by thie factor is used as reference unit or benchmark for obtaining a cosistent price system. Benchmarked derivative prices are tajen as conditional expectations of future bench-marked prices under the real world probability measure. The inverse of the squared total market price for risk is modeled as a square root process and shown to influence the medium and long term forward rates. With constant parameters and constant short rate the model already generates a hump shaped mean for the forward rate curve and other empirical features typically observed.

Platen, E. 2005, 'Diversified portfolios with jumps in a benchmark framework', Asia-Pacific Financial Markets, vol. 11, no. 1, pp. 1-22.
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This paper considers diversified portfolios in a sequence of jump diffusion market models. Conditions for the approximation of the growth optimal portfolio (GOP) by diversified portfolios are provided. Under realistic assumptions, it is shown that diversified portfolios approximate GOP without requiring any major model specifications. This provides a basis for systematic use of diversified stock indices as proxies for the GOP in derivative pricing, risk management and portfolio optimisation

Platen, E. 2005, 'On the role of the growth optimal portfolio in finance', Australian Economic Papers, vol. 44, no. 4, pp. 365-388.
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The paper discusses various roles that the growth optimal portfolio (GOP) plays in finance. For the case of a continuous market we show how the GOP can be interpreted as a fundamental building block in financial market modelling, portfolio optimisation, contingent claim pricing and risk measurement. On the basis of aportfolio selection theorem, optimal portfolios are derived. These allocate funds into the GOP and the savings account. A risk aversion coefficient is introduced, controlling the amount invested in the savings account, which allows to characterise portfolio strategies that maximise expected utilities. Natural conditions are formulated under which the GOP appears as the market portfolio. derivation of the intertemporal capital asset pricving model is given without relying on Markovianity, equilibrium arguments or utility functions. Fair contingent claim pricing, with the GOP as numeraire portfolio, is shown to generalise risk neutral and actuarial pricing. Finally the GOPis described in various ways as the best performing portfolio.

Craddock, M.J. & Platen, E. 2004, 'Symmetry group methods for fundamental solutions', Journal of Differential Equations, vol. 207, no. 2, pp. 285-302.
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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.

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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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

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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Heath, D.P. & Platen, E. 2003, 'Pricing of index options under a minimal market model with log-normal scaling', Quantitative Finance, vol. 3, no. 6, pp. 442-450.
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Platen, E. & Stahl, G. 2003, 'A structure for general and specific market risk', Computational Statistics, vol. 18, no. 3, pp. 355-373.

Heath, D.P. & Platen, E. 2002, 'Consistent pricing and hedging for a modified constant elasticity of variance model', Quantitative Finance, vol. 2, no. 6, pp. 459-467.
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Heath, D.P. & Platen, E. 2002, 'Perfect hedging 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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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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Kubilius, K. & Platen, E. 2002, 'Rate of weak convergence of the Euler approximation for diffusion processes with jumps', Monte Carlo Methods and Applications, vol. 8, no. 1, pp. 83-96.

Kuechler, U. & Platen, E. 2002, 'Weak discrete time approximation of stochastic differential equations with time delay', Mathematics and Computers in Simulation, vol. 59, no. 6, pp. 497-507.
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Heath, D.P. & Platen, E. 2002, 'A variance reduction technique based on integral representations', Quantative Finance, vol. 2, no. 5, pp. 362-369.

Platen, E. 2002, 'Arbitrage in continuous complete markets', Advances in Applied Probability, vol. 34, no. 3, pp. 540-558.
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Heath, D.P., Platen, E. & Schweizer, M. 2001, 'A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets', Mathematical Finance, vol. 11, no. 4, pp. 385-413.
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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.

Craddock, M.J., Heath, D.P. & Platen, E. 2000, 'Numerical inversion of laplace transforms: a survey with applications to derivative pricing', Journal of Computational Finance, vol. 4, no. 1, pp. 57-81.

Hofmann, N. & Platen, E. 2000, 'Approximating large diversified portfolios', Mathematical Finance, vol. 10, no. 1, pp. 77-88.

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

Kardaras, C. & Platen, E. 2008, 'On Financial Markets where only Buy-And-Hold Trading is Possible', Research Paper Series, vol. -, no. 213, pp. 1-24.

Miller, S.M. & Platen, E. 2008, 'Analytic Pricing of Contingent Claims Under the Real-World Measure', Research Paper Series, vol. -, no. 216, pp. 1-30.

Platen, E. & Hulley, H. 2008, 'Hedging for the Long Run', Research Paper Series, vol. -, no. 214, pp. 1-24.

Platen, E. 2008, 'The Law of Minimum Price', Research Paper Series, vol. -, no. 215, pp. 1-24.

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, vol. 179.

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, vol. 184.

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

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

Hulley, H. & Platen, E. 2008, 'Laplace transform identities for diffusions, with applications to rebates and barrier options', 2nd General AMaMeF Conference and Banach Centre Conference, Bedlewo, Poland, April 2007 in Banach Centre Publications: Advances in Mathematics of Finance, ed Stettner, L, Polska Akademia Nauk, Warszawa, Poland, pp. 139-157.
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Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms of some functions of first-passage times for the diffusion. These results are applied to the special case of squared Bessel processes with killing or reflecting boundaries. In particular, we demonstrate how the above-mentioned integral identity enables us to derive the transition density of a squared Bessel process killed at the origin, without the need to invert a Laplace transform. Finally, as an application, we consider the problem of pricing barrier options on an index described by the minimal market model.

Platen, E. 2006, 'Capital asset pricing for markets with intensity based jumps', International Conference on Stochastic Finance 2004, Lisboa, Portugal, September 2004 in Stochastic Finance, ed Grossinho, M R; Shiryaev, A N; Esquivel, M L; Oliveira, P E, Springer, New York, USA, pp. 157-182.
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DP0343913 This paper proposes a unified framework for portfolio optimization, derivative pricing, modeling and risk measurement in financial markets with security price processes that exhibit intensity based jumps. It is based on the natural assumption that invest

Bruti Liberati, N., Platen, E., Martini, F. & Piccardi, M. 2005, 'A multi-point distributed random variable accelerator for Monte Carlo simulation in finance', 5th International Conference on Intelligent Systems Design and Applications (ISDA 2005), Wroclaw, Poland, September 2005 in Proceedings of 5th International Conference On Intelligent Systems Design And Applications, ed Kwasnicka, H; Paprzycki, M, IEEE Computer Society Press, USA, pp. 532-537.
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The pricing and hedging of complex derivative securities via Monte Carlo simulations of stachastic deferential equations constitutes an intensive computational task. To achive real time execution, as often required by financial institutions, one needs highly efficient implementations of the multi-point distributed random variables underlying the simulations. In this paper a fast and flexible dedicated hardware solution is proposed. A comparative performance analysis demonstrates that the hardware solution is bottleneck-free and flexible, and significantly increases the computational efficiency of the software solution.

Martini, F., Piccardi, M., Bruti Liberati, N. & Platen, E. 2005, 'A hardware generator for multi-point distributed random variables', IEEE International Symposium on Circuits and Systems (ISCAS), Kobe, Japan, May 2005 in 2005 IEEE International Symposium On Circuits And Systems (Iscas), Vols 1-6, Conference Proceedings, ed Fujii N, IEEE Computer Society Press, US, pp. 1702-1705.
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Monte Carlo simulation of weak approximations of stochastic differential equations constitutes an intensive computational task. In applications such as finance, for instance, to achieve "real time" execution, as often required, one needs highly efficient implementations of the multi-point distributed random number generator underlying the simulations. In this paper a fast and flexible dedicated hardware solution on a field programmable gate array is presented. A comparative performance analysis between a software-only and the proposed hardware solution demonstrates that the hardware solution is bottleneck-free, retains the flexibility of the software solution and significantly increases the computational efficiency. Moreover, simulations in applications such as economics, insurance, physics, population dynamics, epidemiology, structural mechanics, chemistry and biotechnology can benefit from the obtained speedup.

Bruti Liberati, N. & Platen, E. 2004, 'On the efficiency of simplified weak taylor schemes for monte carlo simulation in finance.', 4th International Conference on Computational Science, Poland, June 2004 in Computational Science - ICCS 2004, ed Bubak M; van Albada G D; Dongarra J, Springer-Verlag, New York, USA, pp. 771-778.

Platen, E. 2004, 'A benchmark framework for risk management', International Symposium on "Stochastic Processes and Applications to Mathematical Finance", Kusatsu, Shiga, Japan, March 2003 in Proceedings of the Ritsumeikan International Symposium: "Stochastic Processes and Applications to Mathematical Finance", ed Akahori, J; Ogawa, S; Watanabe, S, World Scientific, Singapore, pp. 305-335.

Platen, E. 2004, 'Pricing and hedging for incomplete jump diffusion benchmark models', AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance, Snowbird, Utah, USA, June 2003 in Mathematics of Finance: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Mathematics of Finance, ed Yin, G; Zhang, Q, American Mathematical Society, Providence, pp. 287-301.

Platen, E. 2001, 'A Minimal Financial Market Model', Mathematical Finance: Workshop of the Mathematical Finance Research Project, Konstanz, October 2001 in Proceedings of Mathematical Finance: Workshop of the Mathematical Finance Research Project, ed Kohlman M, Tang S, Birkhausen Verlag, Basel, pp. 293-301.

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

Platen, E. 2009, 'A benchmark approach to quantitative finance', Enterprise Risk Management Symposium, Chicago, USA, April 2009.

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

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

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

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

Platen, E. 2009, 'Quantitative methods - Computing and numerical methods', Seminar Presentation, Ajou University, Korea, August 2009.

Platen, E. 2009, 'Valuing guaranteed minimum death benefit options in variable annuities under a benchmark approach"', Enterprise Risk Management Symposium, Chicago, USA, April 2009.

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

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

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

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

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

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

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

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

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

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

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

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

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

Bruti Liberati, N., Nikitopoulos, C., Platen, E. & Schlogl, E. 2007, 'Defaultable Term Structure Models Under the Benchmark Approach', Quantitative Methods in Finance 2007 Conference, Sydney, Australia, December 2007.

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

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

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

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

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

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

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

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, Athens, Greece, August 2006 in 2nd International Symposium on Economic Theory, Policy & Applications, ed -, -, -.

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

Bruti Liberati, N. & Platen, E. 2006, 'On the weak approximation of jump-diffusion processes with applications in finance', VII Workshop on Quantitative Finance, Perugia, Italy, January 2007 in Proceedings of the VII Workshop on Quantitative Finance, ed -, University of Perugia, Italy, Perugia, Italy, pp. 1-96.

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

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

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

Platen, E. 2006, 'A benchmark approach to portfolio optimization and derivative pricing.', Statistical Modeling in Finance Conference, Philadelphia, USA, March 2006 in 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, Antalya, Turkey, April 2006 in First Conference of Advanced Mathematical Methods for Finance.

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

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

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

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

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

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

Heath, D.P., Platen, E. 2004, 'Local volatility function models under a benchmark approach.', Daiwa International Workshop on Financial Engineering, Tokyo/ Kyoto, Japan, August 2004 in Daiwa International Workshop on Financial Engineering, ed -, DAIWA, Japan, pp. 1-19.