* Tomas Bjork, Stockholm School of Economics

When: Friday, April 27, 3:00 - 4:00pm

Title: On the geometry of interest rate models.

Abstract: In this talk is an overview of some work on the geometric properties of the evolution of infinite dimensional SDEs, and in particular the forward rate curve in an arbitrage free bond market. The main problems to be discussed are as follows.

1. When is a given forward rate model consistent with a given family of forward rate curves?
2. When can the inherently infinite dimensional forward rate process be realized by means of a Markovian finite dimensional state space model.

We consider interest rate models of Heath-Jarrow-Morton type, where the forward rates are driven by a multidimensional Wiener process, and where he volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Within this framework we give necessary and sufficient conditions for consistency, as well as for the existence of a finite dimensional realization, in terms of the forward rate volatilities. We also study stochastic volatility HJM models, and we provide a systematic method for the construction of concrete realizations.

* Thaleia Zariphopoulou, University of Texas

When: Saturday, April 28, 11:05am - 12:05pm

Title: Investment performance measurement, risk tolerance and optimal portfolio choice.

Abstract: A new approach to measure the dynamic performance of investment strategies is introduced. To this aim, a family of stochastic processes defined on [0,∞) and indexed by a wealth argument is used. Optimality is associated with their martingale property along the optimal wealth trajectory. The optimal portfolios are constructed via stochastic feedback controls that are functionally related to differential constraints of fast diffusion type. A multi-asset Ito-type incomplete model is used.

Erhan Bayraktar, University of Michigan

When: Saturday, April 28, 3:30 - 4:00pm

Title: Remarks on the American Put Option for Jump Diffusions

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 C¹ across the optimal stopping boundary. Our proof only uses the classical theory of parabolic partial differential equations of Friedman and does not use the theory of viscosity 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 diffusion. 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 involving only a diffusion). 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. Our technique also lets one see why the jump-diffusion control problems may be smoother than the control problems with piece-wise deterministic Markov processes: In the former case the sequence of functions that converge to the value function is a sequence of value function of control problems for diffusions, and in the latter case the converging sequence is a sequence of the value functions of deterministic optimal control problems. The first of these sequences is known to be smoother than the second one.

Sara Biagini, Princeton University

When: Saturday, April 28, 10:05 - 10:35am

Title: On the Extension of Namioka Theorem and on the Fatou Property for Risk Measures

Abstract: This work was triggered by recent developments in the theory of Risk Measures. We first extend Namioka theorem to convex and increasing real maps on Frechet lattices and provide their representation in terms of positive, linear and continuous functionals. We point out that the well-known Fatou property is nothing but lower order semicontinuity. Then, we introduce the notion of property (*) for a topology, namely a condition that links topological convergence to order convergence. Finally, we show its relevance for the representation of order lower semicontinuous, convex and monotone functionals. (Based on a joint work with M. Frittelli)

Jose Enrique Figueroa-Lopez, University of California, Santa Barbara

When: Saturday, April 28, 4:05 - 4:35pm

Title: State-dependent utility maximization in Levy markets.

Abstract: In this talk we revisit Merton's portfolio optimization problem under bounded state-dependent utility functions, in a market driven by Levy processes. An important instance of this setting is the optimal partial replication problem of contingent claims. As it is customary with non-Markovian problems, Merton's problem is solved using a dual variational problem. However, the treatment here is more direct than some existing approaches and it might be more suitable for computational purposes since the domain of the dual problem enjoys an explicit parameterization, built on a multiplicative optional decomposition for nonnegative supermartingales.

Ionut Florescu, Stevens Institute of Technology

When: Friday, April 27, 4:30 - 5:00pm

Title: Estimating parameters for Diffusion Equations with a hidden factor.

Abstract: When using continuous time stochastic processes to model real life situations there are two main issues to be considered. One is to use the process to answer whichever questions are asked, and the second is to verify whether the selected process actually is good for the model.

Recently, the second issue is starting to regain the primary role it deserves. In this talk I will present the problem of estimating coefficients in a system of two diffusion processes when one of them is unobservable, but does not depend on the observed one. The class of stochastic volatility models used in finance constitutes a primary example of such situations.

Kazim Khan, Kent State University

When: Saturday, April 28, 5:35 - 6:00pm

Title: Applications of CuSum Stopping Rule in Trading

Abstract: For a day trader, the two most crucial decisions have to do with the time at which to buy an asset and the time at which to sell an asset. There are numerous strategies that one can choose for this purpose. In this talk I will talk about a classic quality control stopping rule, called the CuSum procedure which finds its uses in trading and is known as the trading the line strategy. Although the quality control aspects of this strategy are quite well known, analytic results in the context of trading are not fully explored. The aim of this talk is to give a quick overview of the recent developments and present some open problems.

Michael Levine, Purdue University

When: Friday, April 27, 5:05 - 5:35pm

Title: Consistent estimators for the long-memory parameter in LARCH and fractional Brownian models

Abstract: We investigate consistent estimation of the long-memory parameter of a special class of volatility models known as LARCH (linear ARCH) models and its connection to estimation of the so-called Hurst parameter H, responsible for the long-memory structure of the fractional Brownian motion process (fBm). A LARCH model's parameter is estimated using a conditional maximum likelihood method, which is proved to be consistent; it is also shown to have good stability properties. A local Whittle estimator is also discussed. By constructing the LARCH and fBm processes on a common probability space, and showing the convergence of various partial sums of the former to the latter in L^2, we are able to propose a specially designed conditional maximum likelihood method for estimating the fBm's Hurst parameter. In keeping with the popular financial interpretation of ARCH models, all estimators are based only on observation of the "returns" of the model, not on their "volatilities".

This is joint work with Soledad Torres and Frederi Viens.

Cristina Mariani, New Mexico State University

When: Saturday, April 28, 9:30 - 10:00am

Title: Extreme events in financial markets

Abstract: Over the past two decades, the complexity of international finance has grown enormously with the development of new markets and instruments for transferring risks. This growth in complexity has been accompanied by an expanded role for mathematical models to value derivative securities and to measure their risks. This will be undertaken through two specific problems in the mathematics of risk management:

1. The analysis of asset-price dynamics in models that capture the possibility of sudden, large changes in prices --- i.e., "jumps".
2. The development and application of tools from mathematical physics to analyze market dynamics leading to a "crash."

Ising type models, Normalized Truncated Levy distributions, and the relation between intermittence and scale invariance will be discussed.

Ciprian Tudor, Universite de Paris I Pantheon-Sorbonne

When: Saturday, April 28, 5:00 - 5:30pm

Title: Donsker theorem for fractional processes and a binary market model

Abstract: We discuss some extensions of the well-known Donsker Invariance Principle to fractional processes (fractional brownian motion or the Rosenblatt process). On the basis of these results, we present a binary market model driven by a long-range dependance process and we discuss the presence of arbitrage in a such model.

* denotes a principal speaker