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Martingale approach in pricing and hedging European options under regime-switching

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  • Grigori N. Milstein
  • Vladimir Spokoiny

Abstract

The paper focuses on the problem of pricing and hedging a European contingent claim for an incomplete market model, in which evolution of price processes for a saving account and stocks depends on an observable Markov chain. The pricing function is evaluated using the martingale approach. The equivalent martingale measure is introduced in a way that the Markov chain remains the historical one, and the pricing function satis es the Cauchy problem for a system of linear parabolic equations. It is shown that any European contingent claim is attainable using a generalized self-financing replicating strategy. For such a strategy, apart from the initial endowment, some additional funds are required both step-wise at the jump moments of the Markov chain and continuously between the jump moments. It is proved that the additional funds (the additional investments and consumptions) are present in the proposed strategy in a risk-neutral manner, hence the generalized self- nancing strategy is self- nancing in mean. A payment for the considered option should consist of two parts: the initial endowment and a fair insurance premium in order to compensate for contributions and consumptions arising in future.

Suggested Citation

  • Grigori N. Milstein & Vladimir Spokoiny, 2011. "Martingale approach in pricing and hedging European options under regime-switching," SFB 649 Discussion Papers SFB649DP2011-079, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2011-079
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    Keywords

    incomplete markets; martingale measure; generalized self- nancing strategy; attainability; self-financing in mean;

    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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