# Probability Section Workshop

**Estimation and Modeling Problems in Financial Engineering**

Bruno Rémillard, HEC Montréal

May 26, 9:00 am to 4:00 pm

**Abstract**

In this workshop, I will present estimation methods for the parameters of stochastic interest rate models. One of the main problem is that these instantaneous rates are not observable; one can only observable their effect through the value of bonds.

First, after recalling the main results of Vasicek on the relationship between the instantaneous interest rate and the value of zero-coupon bonds, the concept of market risk premium is introduced. Then I present a first model, the so-called Vasicek model, where the interest rates are modeled by a Ornstein-Uhlenbeck process. One will study some of its properties: distribution, measurement scale, interpretation of parameters, long-term behavior. One also show how to compute the value of a zero-coupon bond, when the process remains a Ornstein-Uhlenbeck process under an equivalent martingale measure. It will also be shown how to estimate the parameters, using a methodology proposed by Duan. Some examples are given to illustrate the pitfalls to avoid. I will then look at another popular model introduced by Cox, Ingersoll & Ross (CIR hereafter), where the interest rates are modeled by a Feller process. Contrary to the Ornstein-Uhlenbeck process, the Feller process has the benefit of being non-negative. Again we study some of its properties: distribution, measurement scale, interpretation of parameters, and long-term behavior. I will show how to compute the value of a zero-coupon bond under the CIR model, when the process remains a Feller process under an equivalent martingale measure. I will also show how to estimate the parameters. Finally, in a informal section, I will present more general processes for the interest rates.

The material is based on Chapter 5 of my forthcoming book "Statistical Methods for Financial Engineering''. R and Matlab programs will be provided.

Presenter

Bruno Rémillard is Professor of Financial Engineering at HEC Montréal. After completing a Ph.D. in Probability at Carleton University, he was a postdoctoral fellow at Cornell University, before being a professor of Statistics at Université du Québec à Trois-Rivières. He is the author or co-author of more than sixty research articles in Probability, Statistics and Financial Engineering. In 1987, he received the Pierre-Robillard award for the best Ph.D. thesis in Probability and Statistics in Canada and in 2003, he received the prize for the best paper of the year published in the Canadian Journal of Statistics. He was a consultant in the Research and Development group at Innocap, an alternative investment firm located in Montreal, where he mainly helped developing and implanting new quantitative methods for alternative and traditional portfolios. Currently, he is also a part-time consultant at the National Bank of Canada.