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A Girsanov transformed Clark-Ocone-Haussmann type formula for $$L^1$$ L 1 -pure jump additive processes and its application to portfolio optimization

Author

Listed:
  • Masahiro Handa

    (Ritsumeikan University)

  • Noriyoshi Sakuma

    (Nagoya City University)

  • Ryoichi Suzuki

    (Ritsumeikan University)

Abstract

We derive a Clark-Ocone-Haussmann (COH) type formula under a change of measure for $$ L^1 $$ L 1 -canonical additive processes, providing a tool for representing financial derivatives under a risk-neutral probability measure. COH formulas are fundamental in stochastic analysis, providing explicit martingale representations of random variables in terms of their Malliavin derivatives. In mathematical finance, the COH formula under a change of measure is crucial for representing financial derivatives under a risk-neutral probability measure. To prove our main results, we use the Malliavin-Skorohod calculus in $$ L^0 $$ L 0 and $$ L^1 $$ L 1 for additive processes, as developed by Di Nunno and Vives (2017). An application of our results is solving the local risk minimization (LRM) problem in financial markets driven by pure jump additive processes. LRM, a prominent hedging approach in incomplete markets, seeks strategies that minimize the conditional variance of the hedging error. By applying our COH formula, we obtain explicit expressions for locally risk-minimizing hedging strategies in terms of Malliavin derivatives under the market model underlying the additive process. These formulas provide practical tools for managing risks in financial market price fluctuations with $$L^1$$ L 1 -additive processes.

Suggested Citation

  • Masahiro Handa & Noriyoshi Sakuma & Ryoichi Suzuki, 2024. "A Girsanov transformed Clark-Ocone-Haussmann type formula for $$L^1$$ L 1 -pure jump additive processes and its application to portfolio optimization," Annals of Finance, Springer, vol. 20(3), pages 329-352, September.
    • Handle: RePEc:kap:annfin:v:20:y:2024:i:3:d:10.1007_s10436-024-00453-6
    DOI: 10.1007/s10436-024-00453-6
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    References listed on IDEAS

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    1. Takuji Arai & Yuto Imai & Ryoichi Suzuki, 2016. "Numerical Analysis On Local Risk-Minimization For Exponential Lévy Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(02), pages 1-27, March.
    2. Solé, Josep Lluís & Utzet, Frederic & Vives, Josep, 2007. "Canonical Lévy process and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 165-187, February.
    3. Takuji Arai & Ryoichi Suzuki, 2015. "Local risk-minimization for Lévy markets," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-28.
    4. Takuji Arai & Yuto Imai & Ryoichi Suzuki, 2017. "Local risk-minimization for Barndorff-Nielsen and Shephard models," Finance and Stochastics, Springer, vol. 21(2), pages 551-592, April.
    5. Jan Ubøe & Bernt Øksendal & Knut Aase & Nicolas Privault, 2000. "White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance," Finance and Stochastics, Springer, vol. 4(4), pages 465-496.
    6. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107039124, January.
    7. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107611986, January.
    8. Schweizer, Martin, 1991. "Option hedging for semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 339-363, April.
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