July 17, 2024

The Fields Institute
Seminar on Financial Mathematics

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


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


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.


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.


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.