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Numerical methods for the quadratic hedging problem in Markov models with jumps

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  • Carmine De Franco
  • Peter Tankov
  • Xavier Warin

Abstract

We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integro-differential equations (PIDE), which can be shown to possess unique smooth solutions in our setting. The first equation is non-linear, but does not depend on the pay-off of the option to hedge (the pure investment problem), while the other two equations are linear. We propose convergent finite difference schemes for the numerical solution of these PIDEs and illustrate our results with an application to electricity markets, where time-inhomogeneous pure jump Markov processes appear in a natural manner.

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File URL: http://arxiv.org/pdf/1206.5393
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Paper provided by arXiv.org in its series Papers with number 1206.5393.

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Date of creation: Jun 2012
Date of revision: Dec 2013
Handle: RePEc:arx:papers:1206.5393

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Web page: http://arxiv.org/

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  1. St\'ephane Goutte & Nadia Oudjane & Francesco Russo, 2013. "Variance optimal hedging for continuous time additive processes and applications," Papers 1302.1965, arXiv.org.
  2. David Heath & Eckhard Platen & Martin Schweizer, 2001. "A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 11(4), pages 385-413.
  3. Thilo Meyer-Brandis & Peter Tankov, 2008. "Multi-Factor Jump-Diffusion Models Of Electricity Prices," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., World Scientific Publishing Co. Pte. Ltd., vol. 11(05), pages 503-528.
  4. Friedrich Hubalek & Jan Kallsen & Leszek Krawczyk, 2006. "Variance-optimal hedging for processes with stationary independent increments," Papers math/0607112, arXiv.org.
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