# FINANCIAL MATHEMATICS ACTIVITIES

July 17, 2024

## The Fields Institute Seminar on Financial Mathematics

### Wednesday, February 24, 1999, 4:30 - 7:00 p.m.

SCHEDULE

4:30 - 5:30 p.m.
"Jumps in Interest Rate Diffusions: Theoretical Option Pricing and Empirical Models"
Sanjiv Das, Harvard University

6:00 - 7:00 p.m.
"A Unifying Credit Model"
Alain Belanger, Bank of Nova Scotia

ABSTRACTS OF THE TALKS

"Jumps in Interest Rate Diffusions: Theoretical Option Pricing and Empirical Models"
Sanjiv Das, Harvard University

The talk will cover two papers:

1. Pricing Interest Rate Derivatives: A General Approach.
2. The Surprise Element: Jumps in Interest Rate Diffusions.

The relationship between affine stochastic processes and bond pricing equations in exponential term structure models has been well established (see Duffie and Kan, 1996). This linkage is extended to the pricing of interest rate derivatives. The first paper shows that, if the term structure is exponential-affine, there is a simple linkage between the bond pricing solution and the prices of many widely traded interest rate derivative securities. The results are completely general, and apply to m-factor processes with n diffusions and l jump processes. Regardless of the number of shocks, the pricing solutions require at most a single numerical integral, making the model easy to implement. The paper provides many examples of options that yield solutions using the methods of the paper. Fast estimation of these models is possible by vectorizing the equations for the pricing solutions. A range of numerical solutions illustrates the use of the models.

The second paper examines jump-diffusion models of the interest rate with an extensive empirical analysis. That information surprises result in discontinuous interest rates is no surprise to participants in the bond markets. This paper develops a class of jump-diffusion models of the short rate to capture surprise effects, and shows that these models offer a good statistical distribution of short rate behavior, and are useful in understanding many empirical phenomena. Continuous-time and discrete-time estimators are used based on analytical derivations of the characteristic functions, moments and density functions of jump-diffusion stochastic processes for general jump distributions. Jump processes capture empirical features of the data that would not be captured by diffusion models, and there is strong evidence that existing diffusion models would be well enhanced by jump and ARCH type processes. The analytical and empirical methods in the paper support many applications, such as testing for Fed intervention effects, which are shown to be an important source of surprise jumps in interest rates. The jump model is shown to mitigate the non-linearity of interest rate drifts, so prevalent in pure-diffusion models. Day-of-week effects are modeled explicitly, and the jump model provides evidence of bond market overreaction, rejecting the martingale hypothesis for interest rates. Jump models mixed with Markov switching processes predicate that conditioning on regime is important in determining short rate behavior.

"A Unifying Credit Model"
Alain Belanger, Bank of Nova Scotia

This is joint work with Steve Shreve and Dennis Wong. A unified framework for the valuation of a general contingent claim whose cashflow stream is subject to credit risk is inroduced. The proposed model includes both the structural-form and reduced-form approaches. The usual recovery types are studied and compared. Our results are then applied to the pricing of default swaps. Finally, a connection with the term structure approach for credit spreads is provided.

SPEAKERS

Sanjiv Das joined the Harvard Business School faculty in the Finance Area in 1994. Prior to joining Harvard, Professor Das was a Vice-President at Citibank, N.A. in the Asia-Pacific region. Professor Das has an undergraduate degree (B.Com) in Accounting and Economics from the University of Bombay, an MBA (PGDBM) from the Indian Institute of Management, Ahmedabad, a Master of Philosophy in Finance from New York University, and a Ph.D. in Finance from New York University. Professor Das is also a certified cost and works accountant (AICWA) from the Institute of Cost and Works Accountants of India.

His research looks at the pricing of Debt market securities. Professor Das also researches the mutual fund industry, and the performance of portfolio managers. His contributions throughout his research have been both theoretical and empirical in nature. He is also working on computational algorithms for financial problems as well as the behavioral aspects of financial decision-making.

Professor Das has published in the Review of Financial Studies, The Review of Economics and Statistics, Journal of Financial and Quantitative Analysis, the Journal of Derivatives, the Review of Derivatives Research, Financial Practice and Education, Journal of Fixed Income, Journal of Financial Engineering, Applied Economics Letters, Journal of Economic Dynamics and Control, among others, and has made numerous presentations at academic institutions and conferences. Professor Das is also currently a Faculty Research Fellow of the National Bureau of Economic Research. He is an Associate Editor of the Journal of Risk and the International Journal of Theoretical and Applied Finance.

Alain Belanger is currently the Research Director for Scotia Capital Markets. In 1995 he received a MSCF, Masters in Computational Finance from Carnegie Mellon University. From 1990-1994 he was an Assistant Professor at the Department of Mathematics & Statistics, York University. He held a NATO Postdoctoral Fellowship from 1986-1988 at the University of Groningen in the The Netherlands In 1986 he received his Ph.D. Math. (Functional Analysis) from Kent State University.

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), 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.