## The Fields Institute

Seminar on Financial Mathematics

### Wednesday, November 26, 1997, 4:30 - 7:00 p.m.

**SCHEDULE**

4:30 - 5:30 p.m.

*Econometric Specification of the Risk Neutral Valuation Model*

Christian Gourieroux (Centre de Recherche en Economie et Statistique)

6:00 - 7:00 p.m.

*A Finite Element Approach for Two Factor Exotic Option Pricing*

Peter Forsythe (University of Waterloo)

**ABSTRACTS OF THE TALKS**

*Econometric Specification of the Risk Neutral Valuation Model*

Christian Gourieroux (Centre de Recherche en Economie et Statistique)

In complete markets no arbitrage opportunity implies deterministic
relationships between the prices of derivative assets. These actuarial relations
are incompatible with available data and statistical inference. The aim of
the paper is to reconcile risk neutral valuation and statistical inference.
We explain that the notion of market incompleteness refers to the information
held and used by market participants whereas statistical inference is determined
by the econometrician's information. We deduce that under asymetric information,
when the traders are well informed and the econometrician is not, the visible
implication of the no arbitrage condition is the ability to obtain the derivative
prices as expected discounted cash flows with respect to a stochastic valuation
measure. Therefore the additional randomness necessary for statistical inference
is introduced via the difference between the information available to traders
and econometricians. We study various consequences of this approach. In particular
we derive the joint stochastic properties of the derivative prices, explain
how to obtain predictions of derivative prices which are both compatible with
observed prices and the no arbitrage condition. Finally we propose a model
extending the standard Black-Scholes model, where the stochastic risk neutral
measure follows a gamma process. It allows to find the joint distribution
of the derivative and the underlying asset prices. Given that we found non
elliptic level curves and a density function with several modes, a number
of questions can be raised concerning the value at risk (VaR) of a portfolio
containing derivatives.

*A Finite Element Approach for Two Factor Exotic Option
Pricing*

Peter Forsythe (University of Waterloo)

Recently, there has been considerable interest in two factor
partial differential equation based (PDE) pricing models. Examples of such
models include: Asian options, stochastic volatility models, options on two
assets, callable convertible bonds, and two factor interest rate models. In
this talk, a general finite element/finite volume method for solution of two
factor PDE option pricing models will be presented. With this approach, barrier
options and the American early exercise constraint can all be handled in a
very natural way. Particular attention is paid to methods for avoiding spurious
oscillations, which can be severe for barrier options. Some example results
will be presented comparing prices obtained using a constant volatility model
and a stochastic volatility model, for the case of barrier options. Peter
Forsythe's papers can be found on his home
page (http://yoho.uwaterloo.ca:80/~paforsyt/)

**SPEAKERS**

**Christian Gourieroux** is Professor
of Mathematics at Paris University and Head of the Finance and Insurance Laboratory
at the French National Statistical Institute (CREST-INSEE). His research primarily
involves theoretical and applied econometrics. He was winner of the Koopmans
prize for the best paper published in the Journal of Econometric Theory. He
published in journals like Econometrica, Econometric Theory, Journal of Econometrics,
Review of Economic Studies, Journal of Empirical Finance, Insurance: Mathematics
and Economics, Mathematical Finance,... He is also the author of several books:
Statistics and Econometric Models, Time Series and Dynamic Models (Cambridge
University Press), ARCH Models and Financial Applications (Springer Verlag),
Simulation Based Estimation Methods (Oxford University Press)... He is a scientific
adviser at the Compagnie Bancaire (Paribas Group) in the fields of credit
granting and securitization, a member of the council of the Paris Bourse (Stock
Exchange) and the National Bond Committee.

*Peter A. Forsyth* is a Professor in
the Department of Computer Science at the University of Waterloo, where he
is also the Director of the Institute for Computer Research. Previously, he
was President of Dynamic Reservoir Systems and a Senior Simulation Scientist
with the Computer Modelling Group in Calgary. Peter's research interests include
numerical solution of partial differential equations, solution techniques
for large sparse matrices, and object oriented software development methods
for numerical applications. He has published extensively in the fields of
computational fluid dynamics, simulation of environmental pollutant transport,
and discretization methods for partial differential equations. Peter has recently
been collaborating with the members of the Center for Advanced Studies in
Finance at the University of Waterloo, concerning PDE methods of exotic option
pricing.

**ORGANIZERS**

Claudio Albanese (Mathematics, University of Toronto), Phelim Boyle (Finance,
University of Waterloo), Michel Crouhy (Canadian Imperial Bank of Commerce),
Donald A. Dawson (Fields Institute), Ron Dembo (President, Algorithmics Inc.),
Thomas McCurdy (Management, University of Toronto), Gordon Roberts (Finance,
York University), and Stuart Turnbull (Economics, 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 1997-98 Seminar Series on Financial Mathematics is available
through electronic notices sent via e-mail and through the Fields Institute's
world wide web site.