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Hedging strategies and minimal variance portfolios for European and exotic options in a Levy market

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  • Wing Yan Yip
  • Sofia Olhede
  • David Stephens

Abstract

This paper presents hedging strategies for European and exotic options in a Levy market. By applying Taylor's Theorem, dynamic hedging portfolios are con- structed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk-free bank account, the underlying asset and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.

Suggested Citation

  • Wing Yan Yip & Sofia Olhede & David Stephens, 2008. "Hedging strategies and minimal variance portfolios for European and exotic options in a Levy market," Papers 0801.4941, arXiv.org, revised Oct 2008.
    • Handle: RePEc:arx:papers:0801.4941
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    References listed on IDEAS

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    1. Wim Schoutens, 2005. "Moment swaps," Quantitative Finance, Taylor & Francis Journals, vol. 5(6), pages 525-530.
    2. José Manuel Corcuera & David Nualart & Wim Schoutens, 2005. "Completion of a Lévy market by power-jump assets," Finance and Stochastics, Springer, vol. 9(1), pages 109-127, January.
    3. Windcliff, H. & Forsyth, P.A. & Vetzal, K.R., 2006. "Pricing methods and hedging strategies for volatility derivatives," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 409-431, February.
    4. Philip Protter & Michael Dritschel, 1999. "Complete markets with discontinuous security price," Finance and Stochastics, Springer, vol. 3(2), pages 203-214.
    5. Carr, Peter & Geman, Helyette & Madan, Dilip B., 2001. "Pricing and hedging in incomplete markets," Journal of Financial Economics, Elsevier, vol. 62(1), pages 131-167, October.
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