Deep learning for quadratic hedging in incomplete jump market

ArXiv ID: 2407.13688 “View on arXiv”

Authors: Unknown

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.

Keywords: Deep Learning, Incomplete Markets, Jump Diffusion, Minimal Variance Pricing, Stackelberg Game, Options

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail
• Why: The paper is dense with advanced stochastic calculus, PDE/BSDE formulations, and rigorous theoretical derivations for hedging in jump-diffusion models, indicating high mathematical complexity. It employs deep learning (LSTM/feedforward networks) and tests on multiple market models (Black-Scholes, multi-dimensional Brownian, Merton) with explicit performance discussions, demonstrating significant empirical implementation.

  flowchart TD
    Start["Research Goal: <br>Deep Learning for Minimal<br>Variance Hedging in<br>Incomplete Jump Markets"] --> Methodology["Key Methodology:<br>Stackelberg Game Approach<br>+<br>Feedforward & LSTM Networks"]
    Methodology --> Inputs["Data/Inputs:<br>1. Black-Scholes (Benchmark)<br>2. Incomplete Jump Diffusion<br>3. Merton Model"]
    Inputs --> Process["Computational Process:<br>Training Neural Networks<br>to Optimize Portfolio<br>and Pricing Function"]
    Process --> Results["Key Findings/Outcomes:<br>1. Algorithm performs well<br>in incomplete markets<br>2. Minimal Variance prices<br>higher than Merton principle<br>3. Lower hedging losses<br>than Merton principle"]