Abstract
Should the regime-switching risk be priced? This is perhaps one of the important “normative” issues to be addressed in pricing contingent claims under a Markovian, regime-switching, Black-Scholes-Merton model. We address this issue using a minimal relative entropy approach. Firstly, we apply a martingale representation for a double martingale to characterize the canonical space of equivalent martingale measures which may be viewed as the largest space of equivalent martingale measures to incorporate both the diffusion risk and the regime-switching risk. Then we show that an optimal equivalent martingale measure over the canonical space selected by minimizing the relative entropy between an equivalent martingale measure and the real-world probability measure does not price the regime-switching risk. The optimal measure also justifies the use of the Esscher transform for option valuation in the regime-switching market.
Highlights
Regime-switching models are one of the major classes of models for economic and financial dynamics
Hamilton 6 popularized the application of Markovian regime-switching models in economics and econometrics
We first apply a version of the martingale representation for a double martingale in Elliott 18 to characterize the canonical space of equivalent martingale measures, which may be viewed as the largest space of equivalent martingale measures with respect to the enlarged filtration generated by information about the price process of the underlying risky asset and the Markov chain
Summary
Regime-switching models are one of the major classes of models for economic and financial dynamics. We first apply a version of the martingale representation for a double martingale in Elliott 18 to characterize the canonical space of equivalent martingale measures, which may be viewed as the largest space of equivalent martingale measures with respect to the enlarged filtration generated by information about the price process of the underlying risky asset and the Markov chain This space of equivalent martingale measures is general and flexible enough to incorporate both the diffusion risk and the regime-switching risk. We show that an optimal equivalent martingale measure over the canonical space selected by minimizing the relative entropy does not price the regime-switching risk. This result justifies the use of the Esscher transform for option valuation in the regime-switching market proposed in Elliott et al 10.
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