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?
Yacine Aït-Sahalia (Graduate School of Business, University of Chicago)
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?
Yacine Aït-Sahalia (Graduate School of Business, University of Chicago)
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.
