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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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    Citations

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    Cited by:

    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. 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. Elisa Alòs & Christian-Olivier Ewald, 2005. "A note on the Malliavin differentiability of the Heston volatility," Economics Working Papers 880, Department of Economics and Business, Universitat Pompeu Fabra.
    4. 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.
    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. El-Khatib, Youssef & Goutte, Stephane & Makumbe, Zororo S. & Vives, Josep, 2023. "A hybrid stochastic volatility model in a Lévy market," International Review of Economics & Finance, Elsevier, vol. 85(C), pages 220-235.
    7. 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.
    8. 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.
    9. St'ephane Goutte & Nadia Oudjane & Francesco Russo, 2012. "Variance Optimal Hedging for discrete time processes with independent increments. Application to Electricity Markets," Papers 1205.4089, arXiv.org.
    10. Yip, Wing & Stephens, David & Olhede, Sofia, 2008. "Hedging strategies and minimal variance portfolios for European and exotic options in a Levy market," MPRA Paper 11176, University Library of Munich, Germany.
    11. 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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