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Convex duality in continuous option pricing models

Author

Listed:
  • Peter Carr

    (NYU)

  • Lorenzo Torricelli

    (University of Bologna)

Abstract

We provide an alternative description of diffusive asset pricing models using the theory of convex duality. Instead of specifying an underlying martingale security process and deriving option price dynamics, we directly specify a stochastic differential equation for the dual delta, i.e. the option delta as a function of strike, and attain a process describing the option convex conjugate/Legendre transform. For valuation, the Legendre transform of an option price is seen to satisfy a certain initial value problem dual to Dupire (Risk 7:18–20, 1994) equation, and the option price can be derived by inversion. We discuss in detail the primal and dual specifications of two known cases, the Normal (Bachelier in Theorie de la Spéculation, 1900) model and (Carr and Torricelli in Finance and Stochastics, 25:689–724, 2021) logistic price model, and show that the dynamics of the latter retain a much simpler expression when the dual formulation is used.

Suggested Citation

  • Peter Carr & Lorenzo Torricelli, 2024. "Convex duality in continuous option pricing models," Annals of Operations Research, Springer, vol. 336(1), pages 1013-1037, May.
    • Handle: RePEc:spr:annopr:v:336:y:2024:i:1:d:10.1007_s10479-022-05143-y
    DOI: 10.1007/s10479-022-05143-y
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    References listed on IDEAS

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    1. Peter Carr & Lorenzo Torricelli, 2021. "Additive logistic processes in option pricing," Finance and Stochastics, Springer, vol. 25(4), pages 689-724, October.
    2. Mark H. A. Davis & David G. Hobson, 2007. "The Range Of Traded Option Prices," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 1-14, January.
    3. Carr, Peter & Madan, Dilip B., 2005. "A note on sufficient conditions for no arbitrage," Finance Research Letters, Elsevier, vol. 2(3), pages 125-130, September.
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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