## The Fields Institute

Seminar on Financial Mathematics

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

**SCHEDULE**

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

**ABSTRACTS OF THE TALKS**

*"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.

**SPEAKERS**

**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.

**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.