The Fields Institute
Seminar on Financial Mathematics
Wednesday, February 25, 1998, 4:30 - 7:00 p.m.
SCHEDULE
4:30 - 5:30 p.m.
A New Approach to Efficient Option Pricing
Stephen Figlewski, New York University Leonard N. Stern School of Business
6:00 - 7:00 p.m.
Equilibrium Positive Interest Rates: A Unified View
Yan Jin, Columbia Business School
ABSTRACTS OF THE TALKS
A New Approach to Efficient Option Pricing
Stephen Figlewski
Exact closed-form valuation equations for traded derivative
securities are rare. Numerical approximation, most commonly with Binomial
and Trinomial lattice models, gives exact valuation in the limit, but convergence
is non-monotonic and often slow, due to "distribution error" and "truncation
error." This paper explains how truncation error arises and describes the
Adaptive Mesh Model (AMM), a new approach that sharply reduces it by grafting
one or more small sections of fine high-resolution lattice onto a tree with
coarser time and price steps. Three different AMM structures are presented,
one for pricing ordinary options, one for barrier options, and one for computing
delta and gamma efficiently. The AMM approach can be adapted to a wide variety
of contingent claims, yielding significant improvement in efficiency with
very little increase in computational effort. For some common problems, including
calculating delta, accuracy increases by several orders of magnitude relative
to the standard models with no measurable increase in execution time at all.
Equilibrium Positive Interest Rates: A Unified View
Yan Jin and Paul Glasserman
We develop precise connections among three general approaches
to building positive interest rate models: an approach based on direct modeling
of the pricing kernel; the Heath, Jarrow, Morton framework based on specifying
forward rate volatilities and the market price of risk; and the Flesaker-Hughston
framework based on specifying a family of positive martingales. Given the
primitive data of any of these formulations we show how to find that of the
other two. The connections exploit the observation that a pricing kernel is
uniquely determined by its drift; we develop all term structure quantities
from this drift. For any such model, we provide a representative-consumer
real production economy supporting the model in equilibrium. In particular,
this provides an explicit construction of an equilibrium supporting any HJM
model. We also show how one obtains the family of positive martingales used
by Flesaker and Hughston from the dynamics of instantaneous forward rates
and make these explicit for the Cox, Ingersoll, Ross model. Finally, we construct
a new family of positive interest rate models that can fit any initial forward
rate curve and provide reasonably tractable expressions for bond prices and
forward rates.
SPEAKERS
Stephen Figlewski is a Professor of
Finance and Yamaichi Faculty Fellow at the New York University Leonard N.
Stern School of Business, where he has been since 1976. He holds a B.A. degree
in Economics from Princeton and a Ph.D in Economics from the Massachusetts
Institute of Technology. He has published extensively in academic journals,
especially in the area of financial futures and options. He is the founding
Editor of The Journal of Derivatives and an Associate Editor for several other
journals. He is the editor of the Financial Economics Network's two "Derivatives"
series published over the Internet, and is an Associate Editor of another
recently launched Internet journal, Net Exposure. In addition to his academic
career, Professor Figlewski has also worked on Wall Street as a Vice President
at the First Boston Corporation, in charge of research on equity derivative
products, and he has been a member of the New York Futures Exchange and a
Competitive Options Trader at the New York Stock Exchange, trading for his
own account as a market maker in stock index futures and options.
Yan Jin holds a B.S. degree in mathematics
from Beijing University and a Ph.D. in mathematics from the University of
Connecticut. He is currently finishing his second Ph.D. in management science
at Columbia Business School. He has done research in asset pricing theory,
interest rate and foreign exchange rate modeling, and financial risk management.
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.
