Forward equations for option prices in semimartingale models

Abstract

We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a—possibly discontinuous—semimartingale. This result generalizes Dupire’s forward equation to a large class of non-Markovian models with jumps.

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Notes

  1. 1.

    Note, however, that the equation given in [9] does not seem to be correct: it involves the double tail of \(k(z)\,dz\) instead of the exponential double tail.

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Acknowledgements

We thank Damien Lamberton for numerous remarks, which helped improve the manuscript. Substantial assistance from an Associate Editor and the Editor is also gratefully acknowledged. Any remaining errors are our own.

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Correspondence to Amel Bentata.

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Bentata, A., Cont, R. Forward equations for option prices in semimartingale models. Finance Stoch 19, 617–651 (2015). https://doi.org/10.1007/s00780-015-0265-z

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Keywords

  • Forward equation
  • Dupire equations
  • Jump process
  • Semimartingale
  • Tanaka–Meyer formula
  • Markovian projection
  • Call option
  • Option pricing

Mathematics Subject Classification

  • 60H30
  • 91G20
  • 35S10
  • 91G80

JEL Classification

  • C60
  • G13