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.
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References
Bachelier, L. (1900). Theorie de la Spéculation. In Annales scientifiques de l’École Normale Supérieure
Black, F., & Scholes, M. (1973). The pricing of options on corporate liabilities. The Journal of Poltical Economy, 81, 637–654.
Bowden, R. J. (2011). Directional entropy and tail uncertainty, with applications to financial hazard and investments. Quantitative Finance, 11, 437–446.
Carr, P., & Madan, D. (2005). A note on sufficient conditions for no arbitrage. Finance Research Letters, 2, 125–130.
Carr, P., & Torricelli, L. (2021). Additive logistic processes in option pricing. Finance and Stochastics, 25, 689–724.
Davis, M. H., & Hobson, D. G. (2007). The range of traded option prices. Mathematical Finance, 17(1), 1–14.
Dupire, B. (1994). Pricing with a smile. Risk, 7, 18–20.
Hirsch, F., Profeta, C., Roynette, B., & Yor, M. (2011). Peacocks and associated martingales, with explicit consturctions. Springer.
Itkin, A. (2018). A new nonlinear partial differential equation in finance and a method of its solution. Journal of Computational Finance, 21, 1–21.
Madan, D. B., & Yor, M. (2002). Making Markov marginals meet martingales: With explicit constructions. Bernoulli, 8, 509–536.
Øksendal, B. (2003). Stochastic differential equations: An introduction with applications. Springer.
Rockafellar, R. T. (1997). Convex analysis. Princeton University Press.
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L.T. would like to dedicate this paper to the memory and genius of Peter P. Carr.
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Carr, P., Torricelli, L. Convex duality in continuous option pricing models. Ann Oper Res 336, 1013–1037 (2024). https://doi.org/10.1007/s10479-022-05143-y
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DOI: https://doi.org/10.1007/s10479-022-05143-y