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Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes

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  • Fred Espen Benth
  • Giulia Di Nunno
  • Arne Løkka
  • Bernt Øksendal
  • Frank Proske

Abstract

In a market driven by a Lévy martingale, we consider a claim ξ. We study the problem of minimal variance hedging and we give an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives. We discuss two types of stochastic (Malliavin) derivatives for ξ: one based on the chaos expansion in terms of iterated integrals with respect to the power jump processes and one based on the chaos expansion in terms of iterated integrals with respect to the Wiener process and the Poisson random measure components. We study the relation between these two expansions, the corresponding two derivatives, and the corresponding versions of the Clark‐Haussmann‐Ocone theorem.

Suggested Citation

  • Fred Espen Benth & Giulia Di Nunno & Arne Løkka & Bernt Øksendal & Frank Proske, 2003. "Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 55-72, January.
  • Handle: RePEc:bla:mathfi:v:13:y:2003:i:1:p:55-72
    DOI: 10.1111/1467-9965.t01-1-00005
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    File URL: https://doi.org/10.1111/1467-9965.t01-1-00005
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    Cited by:

    1. Takuji Arai & Yuto Imai, 2017. "A closed-form representation of mean-variance hedging for additive processes via Malliavin calculus," Papers 1702.07556, arXiv.org, revised Nov 2017.
    2. Ekaterina L. Dyachenko, 2016. "Internal Migration of Scientists in Russia and the USA: The Case of Applied Physics," HSE Working papers WP BRP 58/STI/2016, National Research University Higher School of Economics.
    3. Kristoffer Lindensjo, 2016. "An explicit formula for optimal portfolios in complete Wiener driven markets: a functional It\^o calculus approach," Papers 1610.05018, arXiv.org, revised Dec 2017.
    4. Choe, Hi Jun & Lee, Ji Min & Lee, Jung-Kyung, 2018. "Malliavin calculus for subordinated Lévy process," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 392-401.
    5. Ewald, Christian-Oliver & Nawar, Roy & Siu, Tak Kuen, 2013. "Minimal variance hedging of natural gas derivatives in exponential Lévy models: Theory and empirical performance," Energy Economics, Elsevier, vol. 36(C), pages 97-107.
    6. Last, Günter & Penrose, Mathew D., 2011. "Martingale representation for Poisson processes with applications to minimal variance hedging," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1588-1606, July.

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