## The Fields Institute

Seminar on Financial Mathematics

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

**SCHEDULE**

4:30 - 5:30 p.m.

*An X-ray of Value at Risk*

Luis Seco (University of Toronto)

6:00 - 7:00 p.m.

*High-Low-Close Diagnostics for a Class of Stochastic Volatility
Models *

James Redekop (University of Waterloo)

**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.