July 17, 2024

The Fields Institute
Seminar on Financial Mathematics

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


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)


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 (


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.


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)


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.