Convex duality in continuous option pricing models

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 Speculation, 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.