## The Fields Institute

Seminar on Financial Mathematics

### Wednesday, March 31, 1999, 4:30 - 7:00 p.m.

**SCHEDULE**

4:30 - 5:30 p.m.

*"Diffusion Models for Optimal Risk/Dividend Control of a Financial Corporation
- An Insurance Company Example"*

Michael Taksar, SUNY - Stony Brook

6:00 - 7:00 p.m.

*"Term Structure Estimation: The Implied Norm Approach Negative Option
Prices -- A Puzzle or Just Noise?"*

Alexandra MacKay, Joseph L. Rotman School of Management, University of Toronto

**ABSTRACTS OF THE TALKS**

*"Diffusion Models for Optimal Risk/Dividend Control of a Financial Corporation
- An Insurance Company Example"*

Michael Taksar, SUNY - Stony Brook

We consider a problem in which the liquid assets or reserves
of a company are modeled by a diffusion process. At each moment of time the
management of the company makes a decision of the amount of dividends paid-out
to the shareholders. There is also a possibility to reduce risk exposure by
conducting a less aggressive business activity, which also results in a smaller
potential profit. Mathematically this corresponds to decreasing simultaneously
drift and diffusion coefficients of the controlled process. In the case of
the insurance company the latter corresponds to "reinsurance", that is redirecting
some of its premiums to another carrier, while the other carrier takes responsibility
for payments of a certain percentage of claims. The objective is to find the
optimal risk and dividend pay-out policy, which maximizes the total discounted
dividend pay-outs. We present an overview of several different models, in
which it is possible to control only dividend pay-out or only risk exposure
or both. We also consider the model in which the company is faced with a constant
liability payment such as debt amortization or the payments on bonds. Another
issue considered is the dependence of the optimal policy on the "bankruptcy
value", that is the amount of cash which can be obtained after redemption
of non liquid assets at the time of bankruptcy.

To solve this problem we use methods and techniques of consumption/
investment models in Financial Mathematics, which are "modernized" by application
of singular stochastic control. We are able to find a closed form solution
and to determine the optimal policy.

*"Term Structure Estimation: The Implied Norm Approach Negative
Option Prices -- A Puzzle or Just Noise?"*

Alexandra MacKay, Joseph L. Rotman School of Management, University of Toronto

This talk discusses the estimation of the term structure of
interest rates when the norm is imputed from the data rather than specified
a priori. This technique applies the philosophy of a technique with origins
in the derivative markets. The resultant term structure estimate is arbitrage-free
and is thusi a sensible choice for use in calibrating the parameters of the
stochastic process governing interest rates.

**SPEAKERS**

**Michael Taksar** received M.S. in Mathematics,
Probability Theory, from Moscow University in 1971. Until 1977 he worked as
a research associate at the Central Institute for Economics and Mathematics
of the Russian Academy of Sciences.

In 1979, Michael Taksar received Ph.D. in Mathematics from Cornell
University. The same year Dr. Taksar joined the faculty of Stanford University
in the Department of Operations Research, where he stayed until 1984. From
1984 to 1987 he was an Associate Professor in the Department of Statistics
at Florida State University. In 1987 he joined the Department of Applied Mathematics
and Statistics at SUNY Stony Brook as a Full Professor.

Dr. Taksar's research is centered around the theory of stochastic
processes, optimal stochastic control and their applications to economic and
finance. He is one of the founders and the main contributors to the singular
stochastic control theory. Dr. Taksar was the first to apply singular stochastic
control to the problem of optimal dynamic portfolio selection. Currently,
Dr. Taksar is a world leading expert on dividend optimization diffusion models.
His current research deals with applications of those models to insurance.

**Alexandra MacKay** is an Assistant Professor
of Finan ce at the Joseph L. Rotman School of Management at the University
of Toronto. She holds a Bachelor of Science and Masters degrees in Economics
from the University of Toronto, and completed her Ph.D. in Finance at the
Schulich School of Business at York University in 1996. Her research interest
is based primarily in fixed income markets, and investigates issues such as
term structure estimation with reference to implicit options, arbitrage pricing
bounds, and the default risk associated with non-sovereign bonds.

**ORGANIZERS**

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), Eli
Prisman (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 1998-99 Seminar Series on Financial Mathematics
is available through electronic notices sent via e-mail and through the Fields
Institute's world wide web site.