Pricing options under a generalized Markov-modulated jump-diffusion model

Robert J. Elliott*, Tak Kuen Siu, Leunglung Chan, John W. Lau

*Corresponding author for this work

Research output: Contribution to journalArticle

70 Citations (Scopus)

Abstract

We consider the pricing of options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump-diffusion model. We suppose that the market interest rate, the drift and the volatility of the underlying risky asset switch over time according to the state of an economy, which is modelled by a continuous-time Markov chain. The measure process is defined to be a generalized mixture of Poisson random measure and encompasses a general class of processes, for example, a generalized gamma process, which includes the weighted gamma process and the inverse Gaussian process. Another interesting feature of the measure process is that jump times and jump sizes can be correlated in general. The model considered here can provide market practitioners with flexibility in modelling the dynamics of the underlying risky asset. We employ the generalized regime-switching Esscher transform to determine an equivalent martingale measure in the incomplete market setting. A system of coupled partial-differential-integral equations satisfied by the European option prices is derived. We also derive a decomposition result for an American put option into its European counterpart and early exercise premium. Simulation results of the model have been presented and discussed.

Original languageEnglish
Pages (from-to)821-843
Number of pages23
JournalStochastic Analysis and Applications
Volume25
Issue number4
DOIs
Publication statusPublished - Jul 2007
Externally publishedYes

Keywords

  • American options
  • Completely random measures
  • Esscher transform
  • European options
  • Generalized gamma process
  • Jump-diffusion
  • Option pricing
  • Regime switching

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