FINANCIAL MATHEMATICS ACTIVITIES

The Fields Institute
Seminar on Financial Mathematics

Wednesday, January 29, 1997, 4:30 - 7:00 p.m.

ABSTRACT OF THE TALKS
An X-ray of Value at Risk
Luis Seco (University of Toronto)

Value at Risk (VaR) is a measure of the risk exposure of a portfolio and is defined as the maximum possible loss in a certain time frame, typically 1-20 days, and within a certain confidence, typically 95\%. Full valuation of a large portfolio under a large number of scenarios is a lenghty process. To speed it up, one can make use of the total delta vector and the total gamma matrix of a portfolio and compute a gaussian integral over a region bounded by a quadric.

In this talk, we show how functional integration methods and harmonic analysis give approximate analytic formulas for VaR as a function of time and of the confidence level. Feynmann expansions reduce the problem further to the evaluation of linear algebra invariants such as traces of products of matrices. The use of Fourier transforms is crucial to obtain formulas that smoothly interpolate between low and large confidence levels, as well as between short and long time horizons, and display the contributions to VaR given by the delta and gamma risks of the portfolio. We also discuss possible applications such as VaR visualization, real-time VaR monitoring and sensitivity VaR analysis.

High-Low-Close Diagnostics for a Class of Stochastic Volatility Models
James Redekop (University of Waterloo)

Extreme-value estimation consists of using the additional information in a process' high and/or low values to get improvements in efficiency over methods that use only the open-to-closing value data. In this paper we derive extreme-value distributions for a family of processes with stochastic volatility, and use the results to construct methods of extreme-value inference for such processes. In an application we test the applicability of this family of models for the behaviour of the Canadian dollar--the financial variable for which, a priori, the models seemed to be most plausible. The extreme-value inference statistics strongly reject any model with zero drift and exogenous stochastic volatility. Moreover, the rejection is strong enough to make us speculate that similar tests would reject any SDE model for any "fine" financial variable, by which we mean prices of heavily traded, finely priced currencies, commodities, or equity indices. (co-authored by Richard Fisher, University of Waterloo)

SPEAKERS

Luis Seco is a professor of mathematics at the University of Toronto. He received his B.S. from the Universidad Autonoma de Madrid (1985), his Ph.D. from Princeton University (1989), and was a Bateman Instructor at the California Institute of Technology until 1992, and his visiting positions include the Schrodinger Institute (Vienna) and CEREMADE (Paris IX). Originally a mathematical physicist, his research was mostly concentrated on atomic physics; his joint work with the Fields Medallist Charles Fefferman on the Dirac and Schwinger conjecture made its way to the Encyclopedia Britannica and recently awarded him the Sloan Fellowship; he has also won research awards from the NATO Science Division and CAICYT. For the last few years, his research interests turned to finance, and together with Claudio Albanese and Dan Rosen he supervises RiskLab, the computational finance laboratory at the University of Toronto. He is an editor of the electronic journal Net Exposure.

James Redekop is currently an Assistant Professor in the Department of Economics at the University of Waterloo. He holds a Ph.D. in economics from the University of Toronto, and his research interests include social choice and welfare and mathematical economics. Recently, he completed a Ph.D. in Statistics at the University of Waterloo, with a dissertation on the extreme-value behaviour of some generalizations of Browian motion. His research interests now include mathematical finance, especially extreme-value methods and models of stochastic volatility.

ORGANIZERS

Claudio Albanese (Professor of Mathematics, University of Toronto), Phelim Boyle (J. Page R. Wadsworth Chair of Finance, University of Waterloo), Don Dawson (Director, The Fields Institute), Ron Dembo (President, Algorithmics Inc.), Gordon Roberts (CIBC Professor of Finance, York University), Stuart Turnbull (Professor of Economics, School of Business, Queen's University)

OTHER INFORMATION

The Financial Mathematics Seminar is offered to any interested participant -- no reservation is necessary. The Institute is located at 222 College Street, between University Ave. and Spadina Ave. near Huron. Parking is available in pay lots located behind the Fields Institute building (quarters and loonies only), across College St. from the Institute (cash only), and underground at the Clarke Institute of Psychiatry (entry on Spadina Ave., just north of College St.)

Information on the 1996-97 Seminar Series on Financial Mathematics is available through electronic notices sent via e-mail and through the Fields Institute's world wide web site.