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Optimal positioning in derivative securities in incomplete markets

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  • Tim Leung
  • Matthew Lorig
  • Yoshihiro Shirai

Abstract

This paper analyzes a problem of optimal static hedging using derivatives in incomplete markets. The investor is assumed to have a risk exposure to two underlying assets. The hedging instruments are vanilla options written on a single underlying asset. The hedging problem is formulated as a utility maximization problem whereby the form of the optimal static hedge is determined. Among our results, a semi-analytical solution for the optimizer is found through variational methods for exponential, power/logarithmic, and quadratic utility. When vanilla options are available for each underlying asset, the optimal solution is related to the fixed points of a Lipschitz map. In the case of exponential utility, there is only one such fixed point, and subsequent iterations of the map converge to it.

Suggested Citation

  • Tim Leung & Matthew Lorig & Yoshihiro Shirai, 2024. "Optimal positioning in derivative securities in incomplete markets," Papers 2403.00139, arXiv.org.
    • Handle: RePEc:arx:papers:2403.00139
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    References listed on IDEAS

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    1. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    2. Peter Carr & Liuren Wu, 2014. "Static Hedging of Standard Options," Journal of Financial Econometrics, Oxford University Press, vol. 12(1), pages 3-46.
    3. P. Carr & D. Madan, 2001. "Optimal positioning in derivative securities," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 19-37.
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