Deep learning for quadratic hedging in incomplete jump market

TL;DR

This paper introduces a deep learning method for quadratic hedging and minimal variance pricing in incomplete jump markets, demonstrating good performance across models and comparing with classical approaches.

Contribution

It develops a novel deep learning framework combining feedforward and LSTM networks for quadratic hedging in incomplete jump markets, grounded in stochastic calculus and game theory.

Findings

Deep learning algorithm performs well on incomplete market models.

Minimal variance pricing yields higher option prices than Merton's principle.

The approach outperforms traditional methods in certain incomplete market scenarios.

Abstract

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based upon a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feedforward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black-Scholes model serves as a benchmark for the algorithm's performance. The results that indicate the algorithm's good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle.