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Financial Mathematics and Applied Probability Seminars 2003-2004 Unless indicated, all seminars take place at Lecture Theatre 2C, King's College London, The Strand, London WC2R 2LS. Tuesday 30 Professor Angus MacDonald September, Director, Genetics and Insurance Research Centre, Department of Actuarial Mathematics and 5:30 pm Statistics, Heriot-Watt University Genetics and Insurance: Data and Models Abstract: According to the media, genetics and insurance just cannot live together: if insurance companies can access any genetic (medical) information about people who apply for life or health insurance, the world will divide into those who will never fall ill (insurable) and those who will definitely fall ill (uninsurable). Simple mathematical models, plus some genetic epidemiology, show how unlikely this is, but despite the fact that this has been in the public domain for several years the media never learns: every time genetics and insurance becomes an issue we seem to be back to square one. This talk will outline actuarial models for exploring the financial consequences of genetics, for individuals and for insurers. Tuesday 7 Dr William Shadwick October, The Finance Development Centre Limited, London 5:30 pm Omega Functions and Analysis of Financial Data Abstract: The Omega function of a univariate probability distribution is, like the probability density function, characteristic function or moment generating function (when one exists), another mathematically equivalent way of encoding the information in the distribution. It is particularly well suited to the analysis of properties of tails and asymmetries of distributions. As a result, it provides significant new insights into those risk/reward characteristics of financial instruments which are ignored by conventional mean/variance tools and only partially captured by skew and kurtosis. Unlike the alternative descriptions of a distribution listed above, the Omega function was only recently discovered, although it requires no mathematical technology which was not available to Gauss. I will give an introduction to Omega functions and some of their intriguing mathematical properties, followed by examples of the impact of applying this new approach real financial data. These will be accompanied by some graphical illustrations which require rather more computational power than even Gauss would likely have been able to muster. (Joint work with Ana Cascon, Con Keating and Brad Shadwick) Tuesday 21 Professor Claudio Albanese October, Department of Mathematics, University of Toronto, Canada 5:30 pm Discretization Schemes for Option Pricing Models with Jumps and State-dependent Volatility Abstract: I introduce a Poisson approximation scheme for jump processes with state- dependent local volatility and use it to construct arbitrage-free discretization schemes for the corresponding pricing PIDEs. The crucial property of these lattice models is their great stability, as one can set the time nodes at arbitrary dates without affecting the end-result for the price of European options. This is achieved by computing node-to-node transition probabilities analytically as expansions in hypergeometric polynomials. I outline applications of this technique to equity and credit derivatives. Tuesday 28 Dr Arne Lokka October, Department of Mathematics, University of Oslo 5:30 pm Equilibrium and incomplete financial markets Abstract: A financial market is in equilibrium if there is a balance between the supply and the demand for assets. If the market is complete, there exists a unique pricing rule determining the price of every asset. What happens when the market is incomplete? In my talk I will try to give some answers to this question. Tuesday 11 Professor Mark Davis November, Department of Mathematics, Imperial College London 5:30 pm An optimal investment problem with randomly terminating income Abstract: Optimal investment with 'random endowment' is an important topic because of the connection with asset valuation and hedging of liabilities in incomplete markets. In this talk we revisit Merton's classic problem of maximizing utility of consumption over an infinite time horizon when asset prices are log-normal. Merton already noted how to adapt the solution to accommodate a known income stream, and a similar argument applies to any hedgeable income stream. When the income is not hedgeable the situation is much more complicated. We study a specific problem by duality methods. The dual minimization problem is one of deterministic optimal control, for which we obtain a computable solution. This is joint work with Michel Vellekoop. On Mark Davis' website there are slides of this talk available. Tuesday 25 Dr Mark Joshi November, Quantitative Research Centre, Royal Bank of Scotland 5:30 pm Rapid and Accurate Computation of Prices and Greeks for Basket Credit Default Swaps in the Li Model * The Li model for pricing basket credit default swaps * Using importance sampling * Difficulties in computing sensitivities * The likelihood ratio method * The pathwise method applied to discontinuous pay-offs * Numerical results Tuesday 20 Dr Dirk Becherer January, 5:30 pm Department of Mathematics, Imperial College London On futures prices in supermartingale term structure models Abstract: The talk is on general futures prices in the framework of supermartingale pricing kernel models. We show how recent results on supermartingale term structure models plus stochastic backward integration allow for a unifying view on discretely and continuously resettled contingent claims, and discuss the structure behind the natural numeraires for obtaining the futures price process. Email the speaker: dirk.becherer(at)imperial(.)ac(.)uk Website of the speaker Tuesday 27 Professor Alex Kacelnik January, Department of Zoology, Oxford University 5:30 pm Behavioural Risk Sensitivity in animals and humans: from the beginnings to Scalar Utility Theory Abstract: The consequences of most actions are stochastic, that is, they entail some risk. The problem of how risk relates to decision making has been faced by economists for centuries, by psychologists for a long time and by biologists researching non-human behaviour for a couple of decades. Their findings and theoretical frameworks show important similarities as well as differences. Economists and psychologists have given strong emphasis to risk aversion, its possible rationality and its psychological causes. In contrast, evolutionary biologists faced risky choice from a theoretical standpoint that assumes evolutionary rationality and statedependency, leading to predictions of either risk-seeking or risk-avoidance depending on the state and payoff representation of the subject. Reality, however, is unique, and empirical work is forcing these approaches to converge. I shall outline classical economic views (Bernoulli), Prospect theory (Kahneman); Risk Sensitivity Theory (Caraco, Houston & McNamara) and Scalar Utility Theory (my own pet account) and review some of the relevant empirical evidence. Kacelnik A & Bateson M (1997) Risk-sensitivity: cross-roads for theories of decision making. Trends in Cognitive Sciences 1, 304-309. Kacelnik A & Brito e Abreu F. (1998) Risky Choice and Weber¹s Law. J. Theoretical Biology 194, 289-298 Website of Alex Kacelnik's ecology research group in Oxford. Tuesday 3 Professor Ragnar Norberg February, Department of Statistics, London School of Economics 5:30 pm Vasicek beyond the normal Abstract: A general Ornstein-Uhlenbeck (OU) process is obtained upon replacing the Brownian motion appearing in the defining stochastic differential equation with a general Levy process. Certain properties of the Brownian ancestor are distributionfree and carry over to the general OU process. Explicit expressions are obtainable for expected values of a number of functionals of interest also in the general case. Special attention is paid here to gamma and Poisson driven OU processes. The Brownian, Poisson, and gamma versions of the OU process are compared in various respects and, in particular, their aptitude to describe stochastic interest rates is discussed in view of some standard issues in financial and actuarial mathematics; prices of zerocoupon bonds, moments of present values, and probability distributions of present values of perpetuities. The problem of possible negative interest rates finds its resolution in the general set-up by taking the driving Levy process to be nondecreasing (a subordinator). The talk will be based on the paper Vasicek beyond the normal. Email the speaker. The speaker's homepage. Tuesday 10 Dr Gael Martin February, Department of Econometrics and Business Statistics, Monash University, Australia 5:30 pm Bayesian Estimation of a Stochastic Volatility Model Using Option and Spot Prices: Application of a Bivariate Kalman Filter Abstract: In this paper we apply Bayesian methods to estimate a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Implicit posterior densities for the parameters of the volatility model, for the latent volatilities and for the market price of volatility risk are produced. The method involves augmenting the data generating process associated with a panel of option prices with the probability density function describing the dynamics of the underlying bivariate spot price and volatility process. Posterior results are produced via a hybrid Markov Chain Monte Carlo sampling algorithm. Candidate draws for the unobserved volatilities are obtained via the application of the Kalman filter and smoother to a linearization of the non-linear state-space representation of the model. Crucially, information from both the spot and option prices affects the draws via the specification of a bivariate measurement equation. The method is illustrated using the Heston (1993) stochastic volatility model, applied to spot and option price data on Australian News Corporation stock data. The way in which alternative option pricing models nested in the Heston framework can be ranked, via Bayes Factors and via fit, predictive and hedging performance, is also demonstrated. (Joint work with Catherine Forbes and Jill Wright.) Tuesday 24 No seminar planned due to AUT strike February, 5:30 pm Tuesday 16 Professor Vicky Henderson March, Bendheim Center for Finance, Princeton University 5:30 pm Valuing Real Options without a Perfect Spanning Asset Abstract: The real options approach to corporate investment decision making recognizes a firm can delay an investment decision and wait for more information concerning project cashflows. The classic model of McDonald and Siegel (1986) (see also Dixit and Pindyck (1994)) values the investment decision as a perpetual American option and in doing so, essentially assumes the real asset underlying the option is traded, or that there is a perfect spanning asset available. Most real projects however can only be partially hedged by traded securities. Our model relaxes this assumption and assumes only a partial spanning asset can be found. In this model, we obtain in closed form the value of the option to invest and the optimal investment trigger level, above which investment takes place. These both depend on the correlation between project cashflows and the spanning asset, risk aversion of the firm's shareholders, and volatilities of project cashflows and the partial spanning asset. We observe that the value of the option to invest and the trigger level are both lowered when the spanning asset is less than perfect. This implies the firm should invest earlier than the classic models suggest. Although the partial spanning model contains the classic model as a special case, it is much richer. In particular, there are situations where the classic model recommends the firm always postpones investment, whereas if a highly (but not perfectly) correlated spanning asset were assumed, the firm should invest at a certain trigger level. Tuesday 23 Dr Tony He March, School of Finance and Economics, University of Technology, Sydney 5:30 pm Asset Pricing, Volatility and Market Behavior ---A Market Fraction Approach Abstract: Motivated by recent development in structural agent models on asset pricing, explanation power and calibration issue of those models, this paper presents a simple market fraction model of two types of traders---fundamentalists and trend followers---under a market -maker scenario. It is found that asset prices, wealth dynamics and market behaviors are characterized by the dynamics of the underlying deterministic system. The model is able to explain various market behaviors, and to generate some of the stylized facts. By introducing two measures on wealth dynamics, we are able to show the limitations of profitability and rationality of different trading strategies. Six significant autocorrelation coefficient (ACs) patterns are characterized by different types of bifurcation of the underlying deterministic system. In particular, an oscillating and decaying AC pattern with positive ACs for even lags and negative for odd lags can be generated when the market is dominated by the fundamentalists (that is when the parameters are near the flip bifurcation boundary), and a positive decaying AC patterns with long memory can be generated when the market is dominated by the trend followers with high decay memory (that is when the parameters are near the Hopf bifurcation boundary). The results show a promising power of stability analysis and bifurcation theory in explaining and calibrating asset price and wealth dynamics, market behavior, and generating various econometric properties of financial data. The full paper can be downloaded here. Email the speaker Website of the School of Finance and Economics, University of Technology, Sydney. Tuesday 30 Professor Onesimo Hernandez-Lerma March, Departamento de Mathematicas, CINVESTAV-IPN, Mexico 5:30 pm Topics in optimal control and game theory Room 17B Topics in optimal control and game theory Professor Onesimo Hernandez-Lerma, Departamento de Mathematicas, CINVESTAV-IPN, Mexico Abstract: Optimal control and game theory are very active research fields partly because they provide a rich source of complex mathematical problems, and partly because of their applicability in such diverse areas as engineering, economics, finance, and (renewable and nonrenewable) resource management, to name a few. This talk is a nontechnical introduction, mainly via examples, to some topics in control and game theory, including adaptive control, minimax control (a.k.a. "worstcase control" or "games against nature"), partially observable systems (a.k.a. "hidden Markov models"), cooperative and noncooperative game equilibria, etc. Email the speaker. Tuesday 20 Professor Raymond Brummelhuis April, Department of Mathematics and Statistics, Birkbeck College, London 5:30 pm Multi-period risk assesment in GARCH models Abstract: A question of common interest is how to evaluate multiple period Value at Risk in heteroscedastic models like GARCH(1, 1). For example, in the context of the Basle agreement, how to the estimate the 10-day 99% VaR on the basis of a GARCH model for the daily returns. We discuss some approaches to this question, both numerical and theoretical. We will in particular study ways to quantify the intertemporal dependence structure in a GARCH(1, 1 ), by introducing and evaluating suitably defined tail dependence functions and coefficients, which are a variant on the usual coefficient of (lower) tail dependence. Tuesday 11 Dr Pauline Barrieu May, Department of Statistics, London School of Economics 5:30 pm Optimal derivatives design under dynamic risk measures Talk will be based on the paper Optimal derivatives design under dynamic risk measures, written jointly with Nicole El Karoui and to be published on the Proceedings of the AMS in spring 2004. Tuesday 18 Dr Robert Tompkins May, Business school of Finance and Management (HfB), Germany, 5:30 pm Unconditional Return Disturbances: a Non Parametric Simulation Approach Abstract: Simulation methods are extensively used in Asset Pricing and Risk Management. The most popular of these simulation approaches, the Monte Carlo, requires model selection and parameter estimation. In addition, these approaches can be extremely computer intensive. Historical simulation has been proposed as a nonparametric alternative to Monte Carlo. This approach is limited to the historical data available. In this paper, we propose an alternative historical simulation approach. Given a historical set of data, we define a set of standardized disturbances and we generate alternative price paths by perturbing the first two moments of the original path or by reshuffling the disturbances. This approach is totally non parametric when constant volatility is assumed, or semi-parametric in presence of GARCH (1,1) volatility and is shown without a loss in accuracy to be much more powerful in terms of computer efficiency than the Monte Carlo approach. This approach is extremely simple to implement and is shown to be an effective tool for the valuation of financial assets. We apply this approach to simulate pay off values of options on the S&P 500 stock index for the period 1982-2003. To verify that this technique works, the common back-testing approach was used. The estimated values are insignificantly different from the actual S&P 500 options payoff values for the observed period. This is joint work with Rita L. D'Ecclesia, University of Rome "La Sapienza". JEL classifications: C15, G13, G19 Email the speaker: [email protected] or [email protected]. Tuesday 15 Dr Damien Challet June, Nomura Centre for Quantitative Finance, Mathematical Institute, Oxford 5:30 pm The quest for large and small fluctuations in minority games: financial and technological applications Abstract: We first provide two alternative explanation on the relevance of Minority Games to financial markets. Then we build various models of financial markets on one of them, extending thebasic Minority Games to new realms. Under what conditions large price fluctuations arise in these models will be the central part of my talk. Whereas one expects financial market models to produce large fluctuations, small fluctuations are a better outcome in a technological setting, for instance in the case of Internet routers trying to transmit efficiently packages. The second part of my talk proposes new stochastic strategies for the basic Minority Game that lead to minimal fluctuations.