Advances in Risk Management

RAYMOND THÉORET, PIERRE ROSTAN AND ABDELJALIL EL-MOUSSADEK 87

processes, so that the parameters of stochastic volatility models may be
estimated by ARCH processes. Consequently, we must postulate a distri-
bution to relate conditional volatility to the stochastic one. But this may
suggest that stochastic volatility is the continuous counterpart of condi-
tional volatility, which is not the case. Conditional volatility is an observed
variable while stochastic volatility is not: it is latent. This last one must be
filtered.
Campbellet al. (1997) mentioned in their book the work of Nelson
(1990) on the link between conditional volatility and stochastic volatil-
ity, but research on this subject was not developed. It revived recently
through an article written by Fornari and Mele (2005) who apply the
generalized error distribution to show how CEV-ARCH^1 models are
approximations of volatility diffusion models in the sense that these
models are Euler–Maruyama discrete time approximations of diffusion
processes.
After reviewing the relation between conditional volatility and stochastic
volatility, which is fundamental in risk management, we transpose these
concepts to the forecasting of the term structure of interest rates.^2 Fong and
Vasicek (1992) hereafter F&V, proposed a two-factor model with a mean-
reverting process and a structure that makes the short-rate variance depend
on the level of interest rates with a suitable restriction that the short-rate
could not become negative. This model is infrequently used in practice
by financial analysts because of the problem of hidden stochastic volatility
which is the black box for this kind of model.
In this chapter we use the F&V model to forecast the Canadian interest-
rate term structure and we apply the Extended Kalman Filter (EKF) as a
tool to compute the unobserved stochastic volatility. We also introduce
Bollinger bands, a well-known tool used in technical analysis, as a reduction
variance technique to improve the Monte Carlo simulation performance.^3
This is a brand-new approach in the sense that we propose this variance
reduction technique based on Bollinger bands to restrain the movements
of volatility in a Monte Carlo simulation and, consequently, to improve its
performance. Incidentally, this method has been never applied before to
volatility forecasting.
The remaining of the chapter is organized as follows. In section 5.2 we dis-
cuss the concepts of stochastic volatility as opposed to conditional volatility.
In section 5.3 we discuss the importance of forecasting the yield curve. In
section 5.4 we present the most popular interest-rate term structure models
and we also provide some details about the F&V model (1992) and about
the intuition behind the EKF. In section 5.5 we explain the EKF scheme
and the implementation of our specific model. The data and calibration are
described in section 5.6. In section 5.7 we detail the approach used for the
simulation, and empirical results are discussed in section 5.8. Finally, some
interesting conclusions are offered in section 5.9.