Mathematical Finance Group

Spring 2008 Calendar

 Speaker Title Time Date Place Xunyu Zhou University of Oxford Prospect Theory, Behavioral Portfolio Choice, and Gambling Strategies 3:00pm Monday, 05/05/2008 RLM 10.176 Abstract: In this talk I shall report recent progress on continuous-time behavioural portfolio choice under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility functions and probability distortions. It is shown that the model well-posedness becomes a prominent issue in such a behavioural model. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure, leading to gambling strategies that bet on good states of the world while accepting fixed, known losses in case of bad ones. Time permitting I will also discuss on the incomplete market and single-period models as well as the equity premium puzzle. Ioannis Karatzas Columbia University VOLATILITY STABILIZATION, DIVERSITY AND ARBITRAGE IN STOCHASTIC FINANCE 10:30 am Friday, 03/21/2008 RLM 10.176 Abstract: In this talk we start with an overview of the modern theory of portfolios, based on Stochastic Analysis. We introduce the notion of relative arbitrage and provide simple, easy-to-test criteria for the existence of such arbitrage in equity markets. These criteria postulate essentially that the excess growth rate of the market portfolio, a positive quantity that can be estimated or even computed from a given market structure, be "sufficiently large". We show that conditions which satisfy these criteria are manifestly present in the US equity market, and construct explicit portfolios under these conditions. One such condition, market diversity, emerges when the volatility structure is bounded. We then construct examples of abstract markets in which the criteria hold. We study in some detail a specific example of a non-diverse abstract market which is volatility-stabilized, in that the return from the market portfolio has constant drift and variance rates, while the smallest stocks are assigned the largest volatilities and individual stocks fluctuate widely. An interesting probabilistic structure emerges in which time changes, Bessel processes, and the asymptotic theory for planar Brownian motion, play crucial roles. Several open questions are raised for further study. (Joint work with E. Robert Fernholz.) Additional material: Yong Zeng University of Missouri at Kansas City Filtering with Marked Poisson Process Observations: Applications to Ultra-High Frequency Financial Data 10:30 am Friday, 02/22/2008 RLM 10.176 Abstract: In this talk, we propose a general filtering framework with marked Poisson process observations for financial UHF data. The signal contains the intrinsic value and parameters and is modeled as a general Markov process. Trading times are driven by a conditional Poisson process, and noise allowing dependence is described by a random transformation from the intrinsic value to trading price at a trading time. Other observable variables (such as initiators of trade, and trade size) are allowed to affect the intrinsic value, the trading intensity and the noise. The model encompasses many important existing models. We study the likelihoods, posterior, likelihood ratios and Bayes factors of the proposed model. They are characterized by stochastic partial differential equations such as filtering equations. Bayesian inference (estimation and model selection) via filtering is studied. Convergence theorems for consistent, efficient algorithms are established. Two general approaches for constructing algorithms are discussed. One approach is Markov chain approximation method, which and the other is sequential Monte Carlo or particle filtering. Both methods provide parallel recursive online algorithms for estimation and model selections. Simulation and real data (UHF stock price and GovPX bond price) examples with statistical analysis are given. Additional material: Semyon Malamud ETH Equilibrium asset prices in heterogeneous economies. 10:30 am Friday, 02/15/2008 RLM 10.176 Abstract: Abstract: In this talk we will discuss the behavior of asset prices in heterogeneous economies, populated by heterogeneous agents with standard, CRRA utilities. When markets are complete, equilibrium equations can be solved state-by-state for the unique state price density. This structure allows us to get good quantitative control over the asymptotic behavior of asset prices at long horizon. We will discuss many surprising and interesting economic phenomena, related to this asymptotic behavior. When markets are incomplete, analyzing asset prices becomes much more difficult, because equations for state price densities become non-linearly coupled between different states and time periods. Nevertheless, when the "degree of incompleteness" is small, we can use perturbation theory to study the behavior of prices. Pierre-Louis Lions Short Course: Further developments in the mean field theory and its applications - Justification and applications of Mean Field Games 9-11 am every Tuesday and Thursday between January 13-25 and February 10-22 ACES 6.304 Abstract: We present in this course i) some new analytical tools and framework to justify the so-called Mean Field Games (MFG in short) models from N-players Nash points as N goes to $+ \infty$ and ii) various simple examples of MFG models with applications to economics, finance and biology. Topic i) involves the analysis of the behavior of symmetric functions of a large number of variables which sheds some light on the Monge-Kantorovich distance (often called Wasserstein distance). The framework we introduce has other applications than the justification of MFG, which include problems in large deviations and optimal control, or for systems of stochastic interacting particles and kinetic theory. This work was introduced in: J-M Lasry and P-L Lions, "Mean field games", Japanese Journal of Mathematics, vol 2 (2007), 229-260. The lectures will be given in the ACES building on T-Th 9-11am, Jan 15, 17, 22 in ACES 6.304 and on Jan 24 in ACES 4.304. The second half of the course will be given Feb 10-22nd at corresponding times. Gerard Brunick Carnegie Mellon University A Weak Existence Result with Application to Model Calibration 3:00pm Monday 01/14/2006 RLM 10.176 Abstract: Gyongy has shown that it is possible to construct a diffusion process with the same one-dimensional marginal distributions as a given initial Ito process. This is closely related to work by Dupire that shows how to construct a local volatility model such that the European option prices implied by the model agree with a given set of market prices. In this talk, we review this connection and then provide a generalization of Gyongy's Theorem that shows how one can match more properties of an initial Ito process by carrying more information about the path history in the mimicking Markov process. Finally, we suggest how such a construction might allow one to extend the local volatility methodology and construct price processes which fit the market prices of mildy path-dependent options.