## The Fields Institute

Seminar on Financial Mathematics

### Wednesday, February 26, 1997, 4:30 - 7:00 p.m.

**SCHEDULE**

4:30 - 5:30

*Optimal Positioning in Derivatives when Asset Prices are Adapted to Order
Flows*

Dilip Madan (University of Maryland at College Park)

6:00 - 7:00 p.m.

*Pricing Contingent Claims in the Presence of Portfolio Constraints*

Halil Mete Soner (Carnegie Mellon University)

**ABSTRACTS OF THE TALKS**

*Optimal Positioning in Derivatives when Asset Prices are Adapted to Order
Flows*

Dilip Madan (University of Maryland at College Park)

The problem of optimal investment in the doubly indexed continuum
of options is posed in the context of a continuous time economy with pure
jump asset price processes adapted to the flow of market orders. It is argued
that price processes adapted to order flows *cannot* be continuous
in calendar time and hence we have the generic failure of the Black-Scholes
geometric Brownian motion model. Price continuity is attained in a stochastic
time change related to a transactions based measure of time. The resulting
economy is incomplete in the underlying asset and a money market account,
and requires the continuum of options as market completing assets. In such
a world it is shown that optimal investment is most unlikely to take the form
of purchasing stock with no derivative positions. Optimal exposure is typically
concave in gains and convex with respect to losses.

*Pricing Contingent Claims in the Presence of Portfolio
Constraintss*

Halil Mete Soner (Carnegie Mellon University)

It is well known that, in the presence of portfolio constraints,
it is no longer possible to perfectly hedge a contingent claim. In such cases
there are at least two possibilities: one approach is to bring in preferences
via utility functions, and the second possibility is to use super-replicating
portfolios that dominate the contingent claim with probability one. The latter
approach provides an upper bound for possible prices. Similarly we can define
a lower bound by using the dominated portfolios. In this talk, I will show
that the upper bound is equal to the price of a related claim without constraints.
The payoff of this dominating claim is obtained by appropriately increasing
the payoff of the original claim, and this result holds for a variety of options,
including standard European and American calls and puts, multi-asset options,
some path-dependent claims and barrier options. I will also discuss how this
analysis can be used to bound the gamma of a replicating portfolio.

**SPEAKERS**

*Dilip Madan* obtained Ph.D degrees in Economics (1971)
and Mathematics (1975) from the University of Maryland and then taught Econometrics
and Operations Research at the University of Sydney, Australia. His research
interests developed in the area of applying the theory of stochastic processes
to the problems of risk management. In 1988, he joined the Maryland Business
School, where he now specializes in Mathematical Finance. His work is dedicated
to improving the quality of financial valuation models, enhancing the performance
of investment strategies, and advancing the understanding and operation of
efficient risk allocation in modern economies. He has published broadly on
these matters, and is an invited speaker at many international finance conferences.
He also serves as a consultant on equity derivatives research to Morgan Stanley.

*Halil Mete Soner* has been on the mathematics faculty
at Carnegie Mellon since 1986; one year after receiving his Ph.D. from the
Division of Applied Mathematics of Brown University. He has co-authored a
book, with Wendell Fleming, on viscosity solutions and stochastic control;
*Controlled Markov Processes and Viscosity Solutions,* (Springer-Verlag,
1993) and authored or co-authored articles on nonlinear partial differential
equations, viscosity solutions, optimal control and mathematical finance.
More information can be found on-line at http://www.contrib.andrew.cmu.edu/~hs0w/mete.html

**ORGANIZERS**

Claudio Albanese (Professor of Mathematics, University of Toronto), Phelim
Boyle (J. Page R. Wadsworth Chair of Finance, University of Waterloo), Don
Dawson (Director, The Fields Institute), Ron Dembo (President, Algorithmics
Inc.), Gordon Roberts (CIBC Professor of Finance, York University), Stuart
Turnbull (Professor of Economics, School of Business, 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 1996-97 Seminar Series on Financial Mathematics is available
through electronic notices sent via e-mail and through the Fields Institute's
world wide web site.