Mathematical Finance/Probability Seminar

Fall 2007 Calendar

 Speaker Title Time Date Place Dimitrios Konstantinides University of Aegean Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations 2:00 pm Monday 11/05/2007 RLM 11.176 Abstract: Sara Biagini Universita di Perugia Utility maximization without Reasonable Asymptotic Elasticity 10:30am Friday 10/19/2007 RLM 11.176 Abstract: For utility functions $U$ finite only on the positive real line, Kramkov and Schachermayer 1999 showed that under a condition on U, the Asymptotic Elasticity Condition, the associated utility maximization problem has a (unique) optimal solution, independently of the probabilistic model. What can be said about the case when the A.E. Condition doesn' hold? In Kramkov Scachermayer 1999 this is also covered, but the optimal solution is characterized only for sufficiently small initial endowments. Under a necessary and sufficient joint condition on the probabilistic model and the utility, we show by relaxation and duality techniques that the maximization problem admits solution for any initial endowment. However, a singular part may pop up, that is the optimal investment may have a component which is concentrated on a set of probability zero. This singular part may fail to be unique. Sara Biagini Universita di Perugia Model-free representation of arbitrage free pricing rules 10:30am Friday 10/12/2007 RLM 11.176 Abstract: We introduce a distinction between {\it model-based} and {\it model-free} arbitrage and formulate an operational definition for absence of model-free arbitrage in a financial market, in terms of a set of minimal requirements for the pricing rule prevailing in the market. We show that any pricing rule verifying these properties can be represented as a conditional expectation operator with respect to a probability measure under which prices of traded assets follow martingales. Our result can be viewed as a model-free version of the fundamental theorem of asset pricing, which does not require any notion of reference" probability measure. Additional material: BiaginiSlides.pdf Peter Tankov Universite Paris VII Levy Processes in Mathematical Finance 10:30am Friday 09/28/2007 RLM 11.176 Abstract: This is a five-lecture mini-course on Levy processes in Mathematical Finance. Additional material: tankov_flyer.pdf Erhan Bayraktar University of Michigan On the finite horizon American option pricing problem: A proof of smoothness and an exponentially fast converging scheme 2:00pm Monday 10/15/2007 RLM 11.176 Abstract: We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a quasi-variational inequality and it is $C^1$ across the optimal stopping boundary. Our proof only uses the classical theory of parabolic partial differential equations of Friedman and does not use the \emph{the theory of vicosity solutions}, since our proof relies on constructing a sequence of functions, each of which is a value function of an optimal stopping time for a \emph{diffusion}. (Also, the previous proof holds only for a certain range of parameters.) The sequence is constructed by iterating a functional operator that maps a certain class of convex functions to smooth functions satisfying variational inequalities (or to value functions of optimal stopping problems for a geometric Brownian motion). The approximating sequence converges to the value function exponentially fast, therefore it constitutes a good approximation scheme, since the optimal stopping problems for diffusions can be readily solved using the well-know numerical schemes such as SOR. Additional material: http://arxiv.org/abs/math.OC/0703782 Mihai Sirbu University of Texas Austin A characterization of financial models in which mutual fund theorem holds true 10:30am Friday 09/07/2007 RLM 11.176 Abstract: We characterize models where the mutual fund theorem holds true in terms of the replicability of options written on the numeraire portfolio (the wealth process of the logarithmic utility maximizer). The characterization works in general semimartingale models, not only in the Brownian Motion case. The presentation is based on joint work with W. Schachermayer and Erik Taflin. Jean-Pierre Fouque University of California Santa Barbara Default probabilities, credit derivatives, and computational issues 10:30am Friday 08/31/2007 RLM 11.176 Abstract: The two main approaches to modeling defaults, structural and intensity based, will be reviewed. We show that perturbation methods are useful in approximating default probabilities in the context of stochastic volatility models. In the case of many names we discuss various ways of creating correlation of defaults. In highly- dimensional models, Monte Carlo simulations remain a powerful tool for computing prices of credit derivatives such as CDO's tranches and associated greeks. We propose an interactive particle system approach for computing the small probabilities involved in these financial instruments.