Interest rate derivatives and value at risk with multiscale stochastic volatility.

摘要

This work deals with diffusions in which the diffusion parameter is not constant, but rather driven by several other Brownian motions. We focus on systems that arise in interest-rate markets when the volatility of the short rate is modelled as a function of two mean-reverting diffusions, one that varies on a fast scale and another one that varies on a slow scale.; We consider the Vasicek model of the short rate. The pricing of zero-coupon bonds leads to the study of certain parabolic partial differential equations which are solved by perturbation methods. The fast scale factor gives rise to a singular perturbation problem, while the slow scale factor gives rise to a regular perturbation problem. We find that the leading order term of the asymptotic approximation is precisely the bond price with constant effective volatility, and that the two first order corrections involve derivatives of this constant volatility price. The inclusion of stochastic volatility modifies the shape of the yield curve to allow for changes of curvature. An interesting result is that the bond price approximation is independent of the particular volatility model.; We prove that the asymptotic approximation converges to the bond price, and we show how to calibrate the model for the zero-coupon bond. We then show how the prices of more complicated derivative contracts, like bond options and convertible bonds, can be easily obtained from the effective parameters that were computed when calibrating the zero-coupon bond.; We apply perturbation methods as well to Value-at-Risk (VaR), a measure of portfolio risk. Once a confidence level q is fixed, we first compute an approximation to the distribution function of the value of the portfolio, and using this approximation we then obtain an asymptotic approximation to the q-quantile of the distribution.; We find that the model depends only on two parameters. The one corresponding to the fast scale controls the "level" of VaR, while the slow-scale parameter controls its "curvature." Selecting the appropriate value for each parameter one can fit a rich variety of quantile curves, and obtain good predictions for future values of VaR.