# FINANCIAL MATHEMATICS ACTIVITIES

July 17, 2024

## The Fields Institute Seminar on Financial Mathematics

### Wednesday, March 25, 1998, 4:30 - 7:00 p.m.

SCHEDULE

4:30 - 5:30 p.m.
Arbitrage Restrictions and Multi-Factor Models of the Term Structure of Interest Rates
Marti G. Subrahmanyam

6:00 - 7:00 p.m.
The Valuation and Application of Asian Options
Moshe Arye Milevsky

ABSTRACTS OF THE TALKS

Arbitrage Restrictions and Multi-Factor Models of the Term Structure of Interest Rates
Richard C. Stapleton and Marti G. Subrahmanyam

We investigate models of the term structure where the factors are interest rates. As an example we derive a no-arbitrage model of the term structure in which any two futures (as opposed to forward) rates act as factors. The term structure shifts and tilts as the factor rates vary. The cross-sectional properties of the model derive from the solution of a two-dimensional ARMA process for the short rate which exhibits mean reversion and a lagged memory parameter. We show that the correlation of the factor rates is restricted by the no-arbitrage conditions of the model. Hence in a multiple-factor model it is not valid to independently choose both the mean reversion, volatility and correlation parameters. The term-structure model, derived here, can be used to value options on bonds and swaps or to generate term structure scenarios for the risk management of portfolios of interest-rate derivatives.

The Valuation and Application of Asian Options
Moshe Arye Milevsky and Steven E. Posner

Arithmetic Asian options are difficult to price and hedge as they do not have closed-form analytic solutions. The main theoretical reason for this difficulty is that the payoff depends on the sum of correlated lognormal variables, which is not lognormal and for which there is no recognizable probability density function. In this paper we derive a p.d.f. for the "infinite" sum of correlated lognormal random variables and prove that it is Reciprocal Gamma distributed, under suitable parameter restrictions. With this result in hand, we are able to approximate the finite sum of correlated lognormal variables and then value Arithmetic Asian options using the Reciprocal Gamma distribution as the state-price density. We thus obtain a closed form analytic expression for the value of an Arithmetic Asian option, where the c.d.f of the Gamma distribution, G(d) in our formula, plays the exact same role as N(d) in the Black-Scholes formula. Time permitting, Prof. Milevsky will attempt to discuss some of the direct implications of this research to the related subject of investment asset allocation and life insurance at retirement. In particular it can be shown that the cost of insuring a prespecified standard-of-living with Gompertz mortality is equivalent to the No Arbitrage value of a suitably parameterized Asian call option which is also analogous to the cost of an appropriately structured variable immediate life annuity.

SPEAKERS

Marti G. Subrahmanyam is the Charles E. Merrill Professor of Finance, Economics and International Business in the Stern School of Business at New York University. He has served as a consultant to several financial institutions in the U.S. and abroad and sits on many boards of directors. He has also served as an advisor to international and government organizations. Professor Subrahmanyam currently serves or has served as an Associate Editor of the European Financial Management, Journal of Banking and Finance, Journal of Finance, Management Science, Journal of Derivatives, Journal of International Finance and Accounting, and Japan and the World Economy. He is the Editor of a new academic journal specializing in derivative securities and markets entitled Review of Derivatives Research. His current research interests include valuation of corporate securities, options and futures markets, and equilibrium models of asset pricing, and market micro-structure. His previous books include Recent Advances in Corporate Finance (Irwin, l985) and Financial Options: From Theory to Practice (Dow Jones- Irwin, l992). He is working on a new book entitled Options Pricing and Hedging: A Trading Perspective.

Moshe Arye Milevsky is an Assistant Professor of Finance at the York University Schulich School of Business in Toronto and is a principal at the consulting company Quantingale M.C. The focus of his research and teaching is on the subjects of Derivative Securities, Insurance Risk Management and Consumer Finance. Moshe Arye's current research is being funded by a grant from the Social Science and Humanities Research Council of Canada (SSHRC) and the Teachers Insurance and Annuity Association - College Retirement Equities Fund (TIAA-CREF) in New York City.

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.