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.
