Robust Hedging of path-dependent options using a min-max algorithm

We consider an investor who wants to hedge a path-dependent option with maturity $T$ using a static hedging portfolio using cash, the underlying, and vanilla put/call options on the same underlying with maturity $t_1$, where $0<t_1<T$. We propose a model-free approach to construct such a portfolio. The framework is inspired by the $\mathit{\text{primal-dual}}$ Martingale Optimal Transport (MOT) problem, which was pioneered by \cite{beiglbock2013model}. The optimization problem is to determine the portfolio composition that minimizes the expected worst-case hedging error at $t_1$ (that coincides with the maturity of the options that are used in the hedging portfolio). The worst-case scenario corresponds to the distribution that yields the worst possible hedging performance. This formulation leads to a $\mathit{\text{min-max}}$ problem. We provide a numerical scheme for solving this problem when a finite number of vanilla option prices are available. Numerical results on the hedging performance of this model-free approach when the option prices are generated using a $\mathit{\text{Black-Scholes}}$ and a $\mathit{\text{Merton}}\mathit{\text{Jump}}\mathit{\text{diffusion}}$ model are presented. We also provide theoretical bounds on the hedging error at $T$, the maturity of the target option.