Professor Eckhard Platen

Platen, E. & Heath, D. 2010, A benchmark approach to quantitative finance, 2nd, Springer, Germany.

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 ÀWiener 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 d+Scribe 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.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.

Platen, E. & Semmler, W. 2012, 'A dynamic portfolio approach to asset markets and monetary policy' in Samuel N Cohen, Dilip Madan, Tak Kuen Siu and Hailiang Yang (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 Tusheng Zhang and Xunyu Zhou (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, LC; Thorp, E O; Ziemba, WT (eds), The Kelly Capital Growth Investment Criterion: Theory and Practice, World Scientific Publishing, USA, pp. 409-426.
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This paper introduces a general market modeling framework, the benchmark appma.chl which assumes the existence of the nume!'raire portfolio. This is the strictly positive portfolio that when used as benchmMk makes all benchmarked non-negati.ve portfolios sllperma.rtinga!es, that is intuitively speaking downward trending or trendless. It can be shQ'Wn to equal the Kelly portfolio, which tna.-"jmiz. es expected logarithmk utility. In several Wa.ys, the KeUy or numeraire portfolio is the "bestll performing portfolio and cannot be outperformed systematically by any other non-negative portfolio. Its use in pricing as nttmeroire leads directly to the real world pricing formula) which employs the real world probability when calculating conditional expectations. In a large regular financial market, the Kelly portfolio is shawn to be approxima.ted by wellÀdivcnrified portfolios.

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.

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. & 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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Cheridito, P., Nikeghbali, A. & Platen, E. 2012, 'Processes of class Sigma, last passage times, and drawdowns', SIAM Journal on Financial Mathematics, vol. 3, pp. 280-303.
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We propose a general framework for studying last passage times, suprema, and drawdowns of a large class of continuous-time stochastic processes. Our approach is based on processes of class Sigma and the more general concept of two processes, one of which moves only when the other is at the origin. After investigating certain transformations of such processes and their convergence properties, we provide three general representation results. The first allows the recovery of a process of class Sigma from its final value and the last time it visited the origin. In many situations this gives access to the distribution of the last time a stochastic process attains a certain level or is equal to its running maximum. It also leads to recently discovered formulas expressing option prices in terms of last passage times. Our second representation result is a stochastic integral representation that will allow us to price and hedge options on the running maximum of an underlying that are triggered when the underlying drops to a given level or, alternatively, when the drawdown or relative drawdown of the underlying attains a given height. The third representation gives conditional expectations of certain functionals of processes of class Sigma. It can be used to deduce the distributions of a variety of interesting random variables such as running maxima, drawdowns, and maximum drawdowns of suitably stopped processes.

Guo, Z.J. & Platen, E. 2012, 'The small and large time implied volatilities in the minimal market model', International Journal of Theoretical and Applied Finance, vol. 15, no. 8, pp. 1-23.
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This paper derives explicit formulas for both the small and the large time limits of the implied volatility in the minimal market model. It is shown that interest rates do impact on the implied volatility in the long run, even though they are negligible in the short time limit.

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 LÚvy 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

Kardaras, C. & Platen, E. 2012, 'On the Dybvig-Ingersoll-Ross theorem', Mathematical Finance, vol. 22, no. 4, pp. 729-740.
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The Dybvig-Ingersoll-Ross (DIR) theorem states that, in arbitrage-free term structure models, long-term yields and forward rates can never fall. We present a refined version of the DIR theorem, where we identify the reciprocal of the maturity date as the maximal order that long-term rates at earlier dates can dominate long-term rates at later dates. The viability assumption imposed on the market model is weaker than those appearing previously in the literature.

Platen, E. & Rendek, R.J. 2012, 'Approximating the numeraire portfolio by naive diversification', Journal of Asset Management, vol. 13, no. 1, pp. 34-50.
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Estimation theory has shown, owing to the limited estimation window available for real asset data, that the sample-based Markowitz mean-variance approach produces unreliable weights that fluctuate substantially over time. This article proposes an alternate approach to portfolio optimization, being the use of naive diversification to approximate the numÚraire portfolio (NP). The NP is the strictly positive portfolio that, when used as benchmark, makes all benchmarked non-negative portfolios either mean decreasing or trendless. Furthermore, it maximizes expected logarithmic utility and outperforms any other strictly positive portfolio in the long run. The article proves for a well-securitized market that the naive equal value-weighted portfolio converges to the NP when the number of constituents tends to infinity. This result is model independent and, therefore, very robust. The systematic construction of diversified stock indices by naive diversification from real data is demonstrated. Even when taking transaction costs into account, these indices significantly outperform the corresponding market capitalization- weighted indices in the long run, indicating empirically their asymptotic proximity to the NP. Finally, in the time of financial crisis, a large equi-weighted fund carrying the investments of major pension funds and insurance companies would provide important liquidity. It would not only dampen the drawdown of a crisis, but would also moderate the excesses of an asset price bubble.

Ignatieva, K., Platen, E. & Rendek, R.J. 2011, 'Using Dynamic Copulae for Modeling Dependency in Currency Denominations of a Diversified World Stock Index', Journal of Statistical Theory and Practice, vol. 5, no. 3, pp. 425-452.
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The aim of this paper is to model the dependency among log-returns when security account prices are expressed in units of a well diversified world stock index. The dependency in log-returns of currency denominations of the index is modeled using time-varying copulae, aiming to identify the best fitting copula family. The Student-t copula turns generally out to be superior to e.g. the Gaussian copula, where the dependence structure relates to the multivariate normal distribution. It is shown that merely changing the distributional assumption for the log-returns of the marginals from normal to Student-t leads to a significantly better fit. The Student-t copula with Student-t marginals is able to better capture dependent extreme values than the other models considered. Furthermore, the paper applies copulae to the estimation of the Value-at-Risk and the expected shortfall of a portfolio constructed of savings accounts of different currencies. The proposed copula-based approach allows to split market risk into general and specific market risk, as defined in regulatory documents. The paper demonstrates that the approach performs clearly better than the RiskMetrics approach, a widely used methodology for Value-at-Risk estimation.

Kardaras, C. & Platen, E. 2011, 'On the semimartingale property of discounted asset-price processes', Stochastic Processes And Their Applications, vol. 121, no. 11, pp. 2678-2691.
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A financial market model where agents trade using realistic combinations of simple (i.e., finite combinations of buy-and-hold) no-short-sales strategies is considered. Minimal assumptions are made on the discounted asset-price process . in particular, the semimartingale property is not assumed. Via a natural market viability assumption, namely, absence of arbitrage of the first kind, we establish that discounted asset-prices have to be semimartingales. Our main result can also be regarded as reminiscent of the Fundamental Theorem of Asset Pricing.

Platen, E. & West, J. 2011, 'Intraday Empirical Analysis of Electricity Price Behaviour', Communications on Stochastic Analysis, vol. 5, no. 4, pp. 721-744.
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This paper proposes an approach to the intraday analysis of the dynamics of electricity prices. The growth optimal portfolio (GOP) is used as a reference unit in a continuous financial electricity price model. A diversified global portfolio in the form a market capitalisation weighted index approx- imates the GOP. The GOP, measured in units of electricity, is normalised and then modelled as a time transformed square root process of dimension four. The dynamics of the resulting process is empirically verified. Intra- day spot electricity prices from the US and Australian markets are used for this analysis. The empirical findings identify a simple but realistic model for examining the volatile behaviour of electricity prices. The proposed model reflects the historical price evolution reasonably well by using only a few ro- bust and readily observable parameters. The evolution of the transformed time is modelled via a rapidly evolving market activity. A periodic, ergodic process with deterministic volatility is used to model market activity.

Bruti Liberati, N., Nikitopoulos Sklibosios, C. & Platen, E. 2010, 'Real-world jump-diffusion term structure models', Quantative Finance, vol. 10, no. 1, pp. 23-37.
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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

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.

Kardaras, C. & Platen, E. 2010, 'Minimizing the expected market time to reach a certain wealth level', SIAM Journal on Financial Mathematics, vol. 1, no. 1, pp. 16-29.
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In a financial market model, we consider variations of the problem of minimizing the expected time to upcross a certain wealth level. For exponential L¦evy markets, we show the asymptotic optimality of the growth-optimal portfolio for the above problem and obtain tight bounds for the value function for any wealth level. In an Ito market, we employ the concept of market time, which is a clock that runs according to the underlying market growth. We show the optimality of the growth-optimal portfolio for minimizing the expected market time to reach any wealth level. This reveals a general definition of market time which can be useful from an investor+s point of view. We utilize this last definition to extend the previous results in a general semimartingale setting.

Miller, S.M. & 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.

Platen, E. & Rendek, R.J. 2010, 'Quasi-exact approximation of hidden Markov chain filters', Communications on Stochastic Analysis, vol. 4, no. 1, pp. 129-142.
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This paper studies the application of exact simulation methods for multi-dimensional multiplicative noise stochastic differential equations to filtering. Stochastic differential equations with multiplicative noise naturally occur as Zakai equation in hidden Markov chain filtering. The paper proposes a quasi-exact approximation method for hidden Markov chain filters, which can be applied when discrete time approximations, such as the Euler scheme, may fail in practice.

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

Ghilarducci, T., Nell, E., Mittnik, S., Platen, E., Semmler, W. & Chappe, R. 2009, 'Memorandum on A new financial architecture and new regulations', Investigacion Economica, vol. 68, no. 267, pp. 147-161.
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Mittnik, S., Nell, E., Platen, E., Semmler, W. & Chappe, R. 2009, 'Financial market meltdown and a need for new financial regulations', METU Studies in Development, vol. 36, no. 1, pp. 253-269.
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The financial crisis, triggered by the subprime and real estate crisis in the US, has become global It is deeply rooted in a decade-long misuse of the financial market for rent-seeking. The financial industry has largely abandoned Its role as a service industry, supposedly charging reasonable fees for the services of spreading risk and allocating capital and credit. Instead it provides a market for speculation, corporate control - mergers and acquisitions -: and a casino for bettmg on or hedging practically any kind of risk - the derivatives market.

Platen, E. & Rendek, R.J. 2009, 'Exact scenario simulation for selected multi-dimensional stochastic processes', Communications on Stochastic Analysis, vol. 3, no. 3, pp. 443-465.
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Accurate scenario simulation methods for solutions of multi - dimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for instance, in finance. Various discrete time simulation methods have been developed over the years. However, the simulation of solutions of some stochastic differential equations can be problematic due to systematic errors and numerical instabilities. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems encountered by those simulation methods that use discrete time approximations. This paper provides a survey of methods for the exact simulation of paths of some multidimensional solutions of stochastic differential equations including Ornstein- Uhlenbeck, square root, squared Bessel, Wishart and L¦evy type processes.

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

Bruti Liberati, N., Nikitopoulos Sklibosios, C. & Platen, E. 2006, 'First order strong approximations of jump diffusions', Monte Carlo Methods and Applications, vol. 12, no. 3-4, 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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In this paper distributions are identified which suitably fit log-returns of the world stock index when these are expressed in units of different currencies. By searching for a best fit in the class of symmetric generalized hyperbolic distributions the maximum likelihood estimates appear to cluster in the neighbourhood of those of the Student t distribution. This is confirmed at a high significance level under the likelihood ratio test. Finally, the paper derives the minimal market model, which explains the empirical findings as a consequence of the optimal market dynamics

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

Platen, E. 2006, 'A benchmark approach to asset management', Journal of Asset Management, vol. 6, no. 6, pp. 390-405.
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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.
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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.
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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.

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.
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Kuechler, U. & Platen, E. 2002, 'Weak discrete time approximation of stochastic differential equations with time delay', Mathematics and Computers in Simulation, vol. 59, no. 6, pp. 497-507.
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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.

Platen, E. 1989, 'A Law Of Large Numbers For Wide-range Exclusion Processes In Random-media', Stochastic Processes And Their Applications, vol. 31, no. 1, pp. 33-49.
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The paper presents a law of large numbers for the asymptotic macroscopic nonequilibrium dynamics of wide range exclusion processes with births and deaths on a random set of sites.

Bruti Liberati, N. & Platen, E. 2012, 'On weak predictor-corrector schemes for jump-diffusion processes in finance', Limerick, Ireland, June 2011 in Topics in Numerical Methods for Finance: Proceedings in Mathematics and Statistics, ed Mark Cummins, Finbar Murphy and John J. H. Miller, Springer, Germany, pp. 1-12.
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Event-driven uncertainties such as corporate defaults, operational failures, or central bank announcements are important elements in the modeling of financial quantities. Therefore, stochastic differential equations (SDEs) of jumpdiffusion type are often used in finance. We consider in this paper weak discrete time approximations of jump-diffusion SDEs which are appropriate for problems such as derivative pricing and the evaluation of risk measures. We present regular and jump-adapted 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.

Baldeaux, J.F., Chan, L. & Platen, E. 2011, 'Quasi-Monte Carlo methods for derivatives on realised variance of an index under the benchmark approach', CTAC2010: Computational Techniques and Applications Conference, Sydney, Australia, November 2010 in ANZIAM Journal: Proceedings Computational Techniques and Applications Conference, ed William McLean and Anthony John Roberts, Australian Mathematical Scoiety, Australia, pp. C727-C741.
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We apply quasi-Monte Carlo methods to the pricing of derivatives on realised variance of an index under the benchmark approach. The resulting integration problem is shown to depend on the joint density of the realised variance of the index and t he terminal value of the index. Employing a transformation mapping for this joint density to the unit square reduces the difficulty of the resulting integration problem. The quasi-Monte Carlo methods compare favourably to Monte Carlo methods when applied to the given problem.

Hulley, H. & Platen, E. 2011, 'A visual criterion for identifying Ito diffusions as martingales or strict local martingales', Seminar on Stochastic Processes, Random Fields and Applications, Ascona, Switzerland, May 2008 in Seminar on Stochastic Analysis, Random Fields and Applications VI, ed Dalang, R; Sozzi, M; Russo, F, Springer, Ascona, Switzerland, pp. 147-157.
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It is often important, in applications of stochastic calculus to financial modelling, to know whether a given local martingale is a martingale or a strict local martingale. We address this problem in the context of a time-homogenous diffusion process with a finite lower boundary, presented as the solution of a driftless stochastic differential equation. Our main theorem demonstrates that the question of whether or not this process is a martingale may be decided simply by examining the slope of a certain increasing function. Further results establish the connection between our theorem and other results in the literature, while a number of examples are provided to illustrate the use of our criterion.

Hulley, H. & Platen, E. 2008, 'Laplace transform identities for diffusions, with applications to rebates and barrier options', 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', International Conference on Intelligent Systems Designs and Applications, Wroclaw, Poland, September 2005 in Proceedings of 5th International Conference On Intelligent Systems Design And Applications, ed Kwasnicka, H; Paprzycki, M, IEEE, 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', International Symposium on Circuits and Systems, 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.', 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.
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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.
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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.
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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.
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