Mathematical Finance Group

Fall 2009 Calendar

 Speaker Title Time Date Place Nick Constanzino Pennsylvania State University Heat kernel asymptotics and closed form expansions of securities derivatives 11am Monday 11/02/2009 RLM 9.166 Abstract: In this talk I will describe a new method to obtain short-time asymptotics for the heat kernel of parabolic linear operator with non-constant coefficients. Our method is based on dilation at a point z = z(x,y), Dyson and Taylor series expansions, and the Baker-Campbell-Hausdorff commutator formula. I will relate this method and the resulting expansion to other classical short-time expansions. Finally, I will present some new closed form asymptotic expansions of certain securities derivative prices using our method, and discuss some possible extensions. This based on joint works with W. Cheng, R. Constantinescu, A. Mazzucato, and V. Nistor. Soumik Pal University of Washington The Volatility-Stabilized Market Model 3:30pm Friday 10/30/2009 RLM 9.166 Abstract: The family of multidimensional diffusions named Volatility-stabilized market models (VSM) was introduced by Fernholz and Karatzas in 2005 as a toy model that nevertheless reflects some of the traits of a real-world equity market. These models reflect the fact that in real markets the smaller stocks tend to have a greater volatility and a greater rate of growth than the larger ones. A central question left open was to analyze the behavior of market weights, i.e., the vector of the ratios of stock capitals of individual firms over the entire market capital. In this talk we will show that the law of the market weights is actually a well-known object in another area of probability, namely, the Wright-Fisher model of population genetics. We analyze various dynamic and equilibrium properties of the market weights. In particular, we show that these weights do not exhibit a power law decay, as opposed to another class of models introduced by the same authors called the rank-based models. Power law decay is an empirically observed fact and has been a center of interest for decades of economic literature. Additional material: Bruno Bouchard CEREMADE, University Paris-Dauphine and CREST ENSAE Optimal Control under Stochastic Target Constraints. A new tool for optimal management under (risk) constraints 3pm Monday 10/19/2009 RLM 11.176 Abstract: We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^\nu$ is constrained to satisfy an a.s.~constraint $Z^\nu(T)\in G\subset R^{d+1}$ P-a.s. at some final time $T>0$. When the set is of the form $G:=\{(x,y)\in R^d\times R~:~g(x,y)\ge 0\}$, with $g$ non-decreasing in $y$, we provide a Hamilton-Jacobi-Bellman characterization of the associated value function. It gives rise to a state constraint problem where the constraint can be expressed in terms of an auxiliary value function $w$ which characterizes the set $D:=\{(t,Z^\nu(t))\in [0,T]\times R^{d+1}~:~Z^\nu(T)\in G\;a.s.$ for some $\nu\}$. Contrary to standard state constraint problems, the domain $D$ is not given a-priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function $w$ which is itself a viscosity solution of a non-linear parabolic PDE. Applying ideas recently developed in Bouchard, Elie and Touzi (2008), our general result also allows to consider optimal control problems with moment constraints of the form $E[g(Z^\nu(T))]\ge 0$ or $P[g(Z^\nu(T))\ge 0]\ge p$. Co-authors : Romuald Elie and Cyril Imbert from CEREMADE, University Paris-Dauphine Additional material: Guoliang Wu UT Austin Smooth Fit Principle for Impulse Control Problems 3:30pm Friday 10/9/2009 RLM 9.166 Abstract: Impulse control problems have wide applications in economics and engineering. Regularity study helps finding closed-form solutions and provides useful insight for numerical approximation when the former is impossible. In this talk we will derive the $W^{2,p}$-regularity (in particular, $C^1$ smooth-fit principle) of the value functions for impulse controls of multidimensional diffusions and jump diffusions, using a viscosity solution approach and other PDE tools. Mike Ludkovski UC Santa Barbara Optimal Trade Execution in Illiquid Markets 3:30pm Friday 9/11/2009 RLM 9.166 Abstract: We study optimal trade execution strategies in financial markets with discrete order flow. The agent has a finite liquidation horizon and must minimize price impact given a random number of incoming trade counterparties. Assuming that the order flow $N$ is given by a Poisson process, we give a full analysis of the properties and computation of the optimal dynamic execution strategy. Extensions, whereby (a) N is a fully-observed regime-switching compound Poisson process; and (b) N is a Markov-modulated Poisson process driven by a hidden Markov chain, are also considered. We derive and compare the properties of the three cases and illustrate our results with computational examples