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Hedging lookback and partial lookback options using Malliavin calculus

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  • Hans-Peter Bermin

Abstract

The paper considers a Black and Scholes economy with constant coefficients. A contingent claim is said to be simple if the payoff at maturity is a function of the value of the underlying security at maturity. To replicate a simple contingent claim one uses so called delta-hedging, and the well-known strategy is derived from Ito calculus and the theory of partial differentiable equations. However, hedging path-dependent options require other tools since the price processes, in general, no longer have smooth stochastic differentials. It is shown how Malliavin calculus can be used to derive the hedging strategy for any kind of path-dependent options, and in particular for lookback and partial lookback options.

Suggested Citation

  • Hans-Peter Bermin, 2000. "Hedging lookback and partial lookback options using Malliavin calculus," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(2), pages 75-100.
    • Handle: RePEc:taf:apmtfi:v:7:y:2000:i:2:p:75-100
    DOI: 10.1080/13504860010014052
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    References listed on IDEAS

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    1. Goldman, M Barry & Sosin, Howard B & Gatto, Mary Ann, 1979. "Path Dependent Options: "Buy at the Low, Sell at the High"," Journal of Finance, American Finance Association, vol. 34(5), pages 1111-1127, December.
    2. Jérôme B. Detemple & Ren Garcia & Marcel Rindisbacher, 2003. "A Monte Carlo Method for Optimal Portfolios," Journal of Finance, American Finance Association, vol. 58(1), pages 401-446, February.
    3. P. Carr, 1995. "Two extensions to barrier option valuation," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(3), pages 173-209.
    4. Conze, Antoine & Viswanathan, 1991. "Path Dependent Options: The Case of Lookback Options," Journal of Finance, American Finance Association, vol. 46(5), pages 1893-1907, December.
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    Cited by:

    1. Yuji Hishida & Kenji Yasutomi, 2009. "Asymptotic behavior of prices of path dependent options," Papers 0911.5579, arXiv.org.
    2. Guillaume Bernis & Emmanuel Gobet & Arturo Kohatsu‐Higa, 2003. "Monte Carlo Evaluation of Greeks for Multidimensional Barrier and Lookback Options," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 99-113, January.
    3. M. Kateregga & S. Mataramvura & D. Taylor & Xibin Zhang, 2017. "Bismut–Elworthy–Li formula for subordinated Brownian motion applied to hedging financial derivatives," Cogent Economics & Finance, Taylor & Francis Journals, vol. 5(1), pages 1384125-138, January.
    4. Kim, Geonwoo & Jeon, Junkee, 2018. "Closed-form solutions for valuing partial lookback options with random initiation," Finance Research Letters, Elsevier, vol. 24(C), pages 321-327.
    5. Lee, Hangsuck, 2003. "Pricing equity-indexed annuities with path-dependent options," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 677-690, December.

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