July 17, 2024

The Fields Institute
Seminar on Financial Mathematics

Wednesday, November 25, 1998, 4:30 - 7:00 p.m.


4:30 - 5:30 p.m.
"Can affine term structure models forecast changes in Treasury yields?"
Gregory R. Duffee, Federal Reserve Board

6:00 - 7:00 p.m.
"Stochastic Variance Value-at-Risk (SV-VaR) Model"
Alexander Levin and Alexander Tchernitser, Bank of Montreal


"Can affine term structure models forecast changes in Treasury yields?"
Gregory R. Duffee, Federal Reserve Board

Affine models of the term structure are very popular among researchers and practitioners, owing to their tractability and apparent flexibility. This paper contributes to the literature on affine models in three ways. First, the paper demonstrates that the class of affine models studied to date (a class of models described in, say, Dai and Singleton(1998)) do a poor job of forecasting changes in Treasury yields over time. For example, if you are given today's term structure and asked to forecast the term structure as of some future date, you will produce better forecasts using a one-variable OLS regression than using the predictions of a three-factor affine model in which the correlation structure among the factors can very general.

Second, the paper explains why this class of models performs so poorly at forecasting. The main reason is because in this class, the market price of interest rate risk is closely linked to interest rate volatility; but empirically, it appears that the market price of interest rate risk varies substantially over time in a way that is unrelated to variations in interest rate volatility.

Third, the paper describes and empirically estimates a class of models that is broader than the affine class of Dai and Singleton. These 'essentially affine' models retain the tractability of the standard affine class, but allow the market price of interest rate risk to vary independently from interest rate volatility. The paper demonstrates that this additional flexibility is important in designing term structure models that fit the data.

"Stochastic Variance Value-at-Risk (SV-VaR) Model
Alexander Levin and Alexander Tchernitser, Bank of Montreal

A standard VaR model assumes multivariate normal distribution for risk factors with constant means, volatilities, and correlation matrix, and corresponds to stable market conditions. It is well known that actual distributions for risk factors exhibit significant deviations from normality. Excess kurtosis, skewness, and volatility fluctuations are typical for many market variables. The fat-tailed distributions for risk factors result in underestimation of actual VaR by the standard model.

The SV-VaR model developed by the Bank of Montreal accounts for the uncertainty and instability of the risk factor volatilities. The model naturally describes the dynamics of the underlying asset prices and fits the actual historical distributions of risk factors better than the traditional VaR model. SV-VaR model makes possible to match higher moments of the risk factor distributions (skewness, kurtosis) and accommodates correlations between risk factors, as well as correlations between risk factors and their volatilities. The one-period distribution for the stochastic variance is derived from the Maximum Entropy Principle. This results in the Exponential SV-VaR Model extended to the Gamma SV-VaR Model for the multi-period (10-day) VaR. General calibration procedure for the class of SV-VaR models is developed.

A closed form solution for the VaR of one-factor linear portfolios is obtained. For the multifactor nonlinear portfolios, a simple two-step Monte Carlo simulation procedure is considered. Numerical results for equity, commodity, interest rate, and foreign exchange rate risk are presented.


Gregory R. Duffee is a senior economist in the Trading Risk Analysis Section of the Federal Reserve Board in Washington, DC. His current research focuses on the measurement and management of credit risks, including the regulation of banks' exposures to these risks. His published articles, which have appeared in all of the major finance journals, reflect his current research, but are also more broadly concerned with the behavior of prices in financial markets, including stock and bond markets. Mr. Duffee received a Ph.D. in economics from Harvard University and is spending the 1998-1999 academic year as a visiting professor of finance at the Haas School of Business, University of California at Berkeley.

Alexander Levin is a Senior Analyst at Quantitative Risk Management, Bank of Montreal. He heads a Risk Modeling Solutions Group. He holds a Sci.D. degree in Numerical Mathematics from Kiev State University and Ph.D. in Solid Mechanics from Dniepropetrovsk State University (Ukraine). His research interests in Financial Mathematics are inverse option pricing problems, implied volatility calibration, Value-at-Risk models, stochastic volatility models, Monte Carlo methods. Dr. Levin is a specialist in regularization methods, inverse problems, numerical methods for partial differential and integral equations.

Alexander Tchernitser is an Analyst at Global Market Risk, Bank of Montreal. He holds a Ph.D. in Control Theory from the Institute for Control Science of Russia Academy of Science, Moscow. His current areas of interest are stochastic volatility models, Value-at-Risk models, and their application to the risk management. Dr. Tchernitser's prior research interest was connected with applied control theory, identification, and Kalman filtering for dynamic objects.


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)


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.