Deep learning approaches to mean-variance portfolio hedging

Abstract

This study develops and evaluates deep learning-based hedging strategies for power call options under three market models: the Black-Scholes (BS) complete market, the Merton jump-diffusion model, and a Mixed Merton model with multiple jump sources. Power options, characterized by their nonlinear payoff structure, $(S_T^\beta-K)^+$, present unique challenges due to their heightened sensitivity to the underlying asset's dynamics, especially in the presence of discontinuous jumps. We propose a neural network framework employing long short-term memory (LSTM) architectures to dynamically replicate option payoffs by optimizing the initial wealth $\hat{x}_0$ and hedge ratios $\hat{\pi}$, minimizing the mean-squared error between terminal wealth and option claims. Under the BS model, hedging performance is nearly perfect, serving as a benchmark. Numerical results demonstrate that both the Merton and Mixed Merton models effectively capture jump components, achieving low hedging loss and minimal option pricing error. These findings underscore the viability of deep learning for hedging exotic derivatives in incomplete markets, offering a flexible and scalable framework for modeling complex asset dynamics. Future work could extend this approach to other exotic options and investigate alternative architectures such as GRUs or mixed-exponential jump models.