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Keywords:

  • heavy tail;
  • regime switching;
  • jump-diffusion model;
  • interrupted Poisson process;
  • analytical solution;
  • option pricing;
  • volatility smile

This paper proposes an extension of Merton's jump-diffusion model to reflect the time inhomogeneity caused by changes of market states. The benefit is that it simultaneously captures two salient features in asset returns: heavy tailness and volatility clustering. On the basis of an empirical analysis where jumps are found to happen much more frequently in risky periods than in normal periods, we assume that the Poisson process for driving jumps is governed by a two-state on-off Markov chain. This makes jumps happen interruptedly and helps to generate different dynamics under these two states. We provide a full analysis for the proposed model and derive the recursive formulas for the conditional state probabilities of the underlying Markov chain. These analytical results lead to an algorithm that can be implemented to determine the prices of European options under normal and risky states. Numerical examples are given to demonstrate how time inhomogeneity influences return distributions, option prices, and volatility smiles. The contrasting patterns seen in different states indicate the insufficiency of using time-homogeneous models and justify the use of the proposed model. Copyright © 2012 John Wiley & Sons, Ltd.