# FINANCIAL MATHEMATICS ACTIVITIES

July 17, 2024

## 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.