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Financial Mathematics
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Financial Mathematics and Applied Probability Seminars 2006-2007

Unless otherwise indicated, all seminars take place at Lecture Theatre 2C, King's College London, The Strand, London WC2R 2LS.

Tuesday 24 October,
5:30 pm
Dr Reimer Kuehn
Department of Mathematics, King's College London
Credit Contagion and Credit Risk

Abstract: We look at the influence of economic interactions on credit risk. Starting point is the observation that a default of a company has material effects on the economic well being of other companies engaged in direct economic interactions with it, and that this effect induces dynamically induced functional correlations rates rather than mere statistical correlations of default rates in a network of interacting firms. We introduce and solve a model describing the default- dynamics within a heterogeneous network of interacting firms. For large average connectivities within the network, the dynamics can be solved in closed form, and distributions of default rates and loss distributions can be computed. While effects of economic interactions are weak in typical (most probable) scenarios, they are pronounced in situations of economic stress, and lead to a significant fattening of tails of loss distributions in large loan portfolios. Interestingly, the collective behaviour of the system is described by very few parameters describing the low order statistics of the parameters characterising the economy.

Tuesday 31 October,
5:30 pm
Professor Ralf Korn
Department of Mathematics, University of Kaiserslautern
Inflation: modelling, hedging and optimization

Abstract: We consider the optimal investment problem for an investor who can choose between a riskless money market account, risky stocks and inflation linked products. Modelling the evolution of an inflation index is motivated by macro-economic reasoning such as the Fisher equation and is based on Korn and Kruse (2004). As a main result, risk-averse investors will typically not invest in the inflation products considered by us, but they will be used by (institutional) investors that have to hedge inflation risk.

Tuesday 14 November,
5:30 pm
Dr Martijn Pistorius
Department of Mathematics, King's College London
On the optimal dividend problem

Abstract: In this talk we consider the problem of determining optimal dividend distribution policies for an insurance company. Modelling the risk process of the insurance company before dividends are deducted by a spectrally negative Levy process, the classical dividend problem is to find a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative Levy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal amongst all admissible ones takes the form of a barrier strategy. (Joint work with Florin Avram and Zbigniew Palmowski.)

Tuesday 21 November,
5:30 pm
Dr David Hobson
Department of Mathematics, University of Bath
Skorokhod embeddings and finance

Abstract: The standard approach in Mathematical Finance is to postulate a model, and to use this model to derive the prices of financial derivatives. There is often a middle step of calibrating a parametric family of models against the prices of vanilla securities.
This talk will consider an alternative approach in which we do not specify any model. Instead we consider the set of all possible price proceses which are consistent with the traded prices of vanilla securities. If the vanilla securities are calls then the problem of identifying potential price processes can be identified with the Skorokhod embedding problem with extremality properties.

Tuesday 28 November,
5:30 pm
Professor Goran Peskir
Department of Mathematics, University of Manchester
Predicting the ultimate supremum of a stable Levy process

Key words and phrases: Stable Levy process (with no negative jumps), optimal prediction, optimal stopping, ultimate supremum, fractional Laplacian, Riemann-Liouville fractional derivative, Caputo fractional derivative, Volterra integral equations of the first and second kind, free-boundary problem, local time-space calculus, smooth fit, curved boundary.

Tuesday 5 December,
5:30 pm
Professor William Shaw
Department of Mathematics, King's College London
Copulas vs Canonical Multivariate Distributions: A multitude of T copulas and some Canonical Systems

Abstract: The Normal Distribution and the Cauchy distribution sit at two extremes in terms of their tail structure and the existence of moments - how would one construct a bivariate distribution with each as a marginal but with dependency? How many Student T copulas are there and what are the implications of a choice and its relationship to classical distribution theory? This talk will present a reconciliation of current copula simulation theory with classical distribution theory for the case of the multivariate Student distribution, and will exhibit some new explicit bivariate distributions, together with practical simulation techniques for applied mathematical finance for the multivariate case. We will also see some new arguably "Canonical" distributions and new correlation formulae to assist with calibration. One outcome of our considerations is to present an alternative to the elliptical structures, currently popular in distribution theory and risk management, with the property that the marginals are truly independent in the zero correlation limit.

Tuesday 23 January,
5:30 pm
Dr Pavel Gapeev
Weierstrass Institute for Applied Analysis and Stochastics, Berlin
Constructing jump analogues of diffusions and application to finance

Abstract: We propose a method for construction of jump analogues of diffusion processes solving stochastic differential equations driven by both a Wiener process and a Poisson random measure, which admit explicit solutions or are reducible to ordinary differential equations. We illustrate the action of this method on some diffusions, which are widely used in finance. We also give some remarks on calculation of the Laplace transforms of the marginal distributions of the constructed jump-diffusions, which are needed for the parameter calibration of the model.

Monday 29 January,
5:30 pm
Room 1B27
Professor Tomas Bjork
Stockholm School of Economics
Optimal investment under partial information

Tuesday 6 February,
5:30 pm
Dr Aleksandar Mijatovic
Institute for Mathematical Sciences, Imperial College London
Spectral methods for volatility derivatives

Tuesday 6 March,
5:30 pm
Professor Nick Bingham
Department of Probability and Statistics, University of Sheffield
Tuesday 13 March,
5:30 pm
Professor Mark Davis
Department of Mathematics, Imperial College London
Outperforming a benchmark via risk-sensitive control

This paper extends the risk-sensitive asset management theory developed by Bielecki and Pliska and by Kuroda and Nagai to the case where the investor's objective is to outperform an investment benchmark. The main result is a mutual fund theorem. Every investor following the same benchmark will take positions, in proportions dependent on his/her risk sensitivity coefficient, in two funds: the log-optimal portfolio and a second fund which adjusts for the correlation between the traded assets, the benchmark and the underlying valuation factors. Some extensions to Levy-driven price models will be considered.

Joint work with Sebastien Lleo.

Tuesday 20 March,
5:30 pm
Dr Alvaro Cartea
Commodities Finance Centre, Birkbeck, London
Tuesday 29 May,
5:30 pm
Dr Dirk Becherer
Department of Mathematics, Imperial College London

Tuesday 16 October,
5:30 pm
Dr Umut Cetin
Department of Statistics, London School of Economics

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