# FINANCIAL MATHEMATICS ACTIVITIES

July 17, 2024

## The Fields Institute Seminar on Financial Mathematics

Wednesday, September 24, 1997

SCHEDULE

4:30 - 5:30 p.m.
Do Interest Rates Really Follow Continuous-Time Markov Diffusions?

6:00 - 7:00 p.m.
The Analysis of Deltas, State Prices and VaR: A New Approach
Bruce D. Grundy (The Wharton School, University of Pennsylvania)

ABSTRACTS OF THE TALKS

Do Interest Rates Really Follow Continuous-Time Markov Diffusions?

Interest rates have traditionally been modeled in the literature as following continuous-time Markov processes, and more specifically diffusions. By contrast, recent term structure models often imply non-Markovian continuous-time dynamics. Can discretely sampled interest rate data help decide which continuous-time models are sensible? First, how reasonable is the Markovian assumption? A test of this hypothesis will be proposed. Second, if the process is Markovian, can it be identified further as a diffusion, as has been assumed by most of the theoretical literature? A second test will be proposed, which tests the diffusion hypothesis under the maintained Markovian assumption. Within the Markovian world, diffusion processes are characterized by the continuity of their sample paths. It is immediately obvious that this condition cannot be verified from the observed sample path: by nature, even if the sample path were continuous, the discretely sampled interest rate data will appear as a sequence of discrete changes. This paper examines whether the discontinuities observed in the discrete data are the result of the discreteness of sampling, or rather evidence of genuine non-diffusion dynamics of the continuous-time interest rate process. The issue is to isolate the observable implications for the data of being an incomplete discrete sample from a continuous-time diffusion. This paper's answer relies on testing a necessary and sufficient restriction on the conditional densities of diffusions, at the sampling interval of the observed data. This restriction characterizes the continuity of the unobservable complete sample path. The distribution of the test statistics, as well as their consistency and power properties, are derived. We find empirically that: (i) neither the short rate nor the long rate can be characterized individually as Markov processes; (ii) jointly, they form a Markovian system; (iii) the slope of the yield curve is a univariate Markov process; (iv) and a diffusion. As a caveat, these preliminary empirical results are sensitive to the choice of dataset.

The Analysis of Deltas, State Prices and VaR: A New Approach
Bruce D. Grundy (The Wharton School) and Zvi Wiener (Hebrew University)

We provide a monotonic transformation of an initial diffusion with a level-dependent volatility parameter that yields a second, deterministic diffusion parameter process. Limited information about the initial volatility parameter can bound the drift of the transformed process so that probabilities under the initial diffusion can be bounded in terms of probabilities under arithmetic Brownian motion. These probability bounds provide new theoretical bounds on deltas, state prices and VaR. When an asset's diffusion parameter is non-decreasing in its price, observed option prices provide an empirical bound on deltas and, given non-negative real rates, deltas of all at-the-real-money calls are at least 1/2.

SPEAKERS

Yacine Ait-Sahalia is Associate Professor of Finance at the University of Chicago Graduate School of Business. He received his Ph.D. from MIT in 1993 and his undergraduate degree from Ecole polytechnique in France. His research, focusing on the nonparametric estimation of continuous-time models in finance and its implications for option pricing, has been published in Econometrica, the Journal of Finance, the Review of Financial Studies and the Journal of Econometrics. He recently received the Brennan Award for the best paper published in the Review of Financial Studies in 1996, and was named an outstanding faculty member by Business Week's 1997 Guide to the Best Business Schools.

Bruce D. Grundy, Ph.D. in Finance (University of Chicago), is the Andrew Heyer Assistant Professor of Finance in the Wharton School of the University of Pennsylvania. Bruce has served as an Associate Editor of the Review of Financial Studies and the Journal of Financial and Quantitative Analysis. His research interests include the pricing and hedging of derivatives and corporate securities, taxation and dividend policy, momentum trading strategies, and executive compensation.

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.