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

TL;DR

This paper tackles robust hedging of path-dependent options under model misspecification by formulating a static hedge using cash, the underlying, and short-maturity vanilla options through a min-max optimization inspired by Martingale Optimal Transport. It develops a discretization-based numerical scheme that yields a linear program to compute hedging weights under finitely supported marginals and proves convergence of the discrete solution to the continuous MOT problem. The authors provide convergence bounds and Lipschitz-dual results to justify approximations with finite strike grids and present extensive simulations under Black-Scholes and Merton Jump Diffusion models for Asian and Forward Start options. The results show that the min-max hedge offers a robust bound on hedging error, can outperform naive super-hedging in cost, and offers practical guidance on how many short-maturity hedging instruments are needed to achieve a desirable level of protection.

Abstract

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 \textit{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 \textit{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 \textit{Black-Scholes} and a \textit{Merton Jump diffusion} model are presented. We also provide theoretical bounds on the hedging error at $T$, the maturity of the target option.

Figures (10)

• Figure 1: Plots of the conditional value of the target option under the worst possible scenarios and the corresponding hedging portfolio values for an increasing number of options for the Asian option under the Black Scholes model.
• Figure 2: Plots of the conditional value of the target option under the worst possible scenarios and the corresponding hedging portfolio values for an increasing number of options for the Forward start option under the Black Scholes model.
• Figure 3: Plots of the conditional value of the target option under the worst possible scenarios and the corresponding hedging portfolio values for an increasing number of options for the Forward start option under the Merton Jump Diffusion model.
• Figure 4: Min-Max versus Mean Absolute Error plot for the Asian option under the Black Scholes model.
• Figure 5: Peak PFE plot for the Asian option under the Black Scholes model.

Theorems & Definitions (20)

• Definition 2.1
• Lemma 2.1
• Lemma 2.2
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Definition 2.3
• Remark
• Theorem 2.5
• Remark