Volatility Risk For Regime-Switching Models

Abstract

Regime-switching models have proven to be well-suited for capturing the time series behavior of many financial variables. In particular, they have become a popular framework for pricing equity-linked insurance products. The success of these models demonstrates that realistic modeling of financial time series must allow for random changes in volatility. In the context of valuation of contingent claims, however, random volatility poses additional challenges when compared with the standard Black-Scholes framework. The main reason is the incompleteness of such models, which implies that contingent claims cannot be hedged perfectly and that a unique identification of the correct risk-neutral measure is not possible. The objective of this paper is to provide tools for managing the volatility risk. First we present a formula for the expected value of a shortfall caused by misspecification of the realized cumulative variance. This, in particular, leads to a closed-form expression for the expected shortfall for any strategy a hedger may use to deal with the stochastic volatility. Next we identify a method of selection of the initial volatility that minimizes the expected shortfall. This strategy is the same as delta hedging based on the cumulative volatility that matches the Black-Scholes model with the stochastic volatility model. We also discuss methods of managing the volatility risk under model uncertainty. In these cases, super-hedging is a possible strategy but it is expensive. The results presented enable a more accurate analysis of the trade-off between the initial cost and the risk of a shortfall.