Paper: arXiv 2212.12725

Authors: Alessandro Gnoatto, Silvia Lavagnini, Athena Picarelli

Abstract

We propose a novel computational procedure for quadratic hedging in high-dimensional incomplete markets, covering mean-variance hedging and local risk minimization. Starting from the observation that both quadratic approaches can be treated from the point of view of backward stochastic differential equations (BSDEs), we (recursively) apply a deep learning-based BSDE solver to compute the entire optimal hedging strategies paths. This allows us to overcome the curse of dimensionality, extending the scope of applicability of quadratic hedging in high dimension. We test our approach with a classic Heston model and with a multiasset and multifactor generalization thereof, showing that this leads to high levels of accuracy.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 5.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a mathematically sophisticated approach by linking high-dimensional quadratic hedging and backward stochastic differential equations (BSDEs) with deep learning solvers. While theoretical and computational derivations are strong, empirical testing is restricted to simulated models like the multi-asset Heston framework rather than live market data.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve High-Dimensional Quadratic Hedging] --> B[Input: Financial Models<br>Heston & Multi-Asset/Multi-Factor]
    B --> C[Formulate Problem as BSDEs<br>Mean-Variance & Local Risk Minimization]
    C --> D[Apply Deep Learning BSDE Solver<br>Recursive Path Computation]
    D --> E[Outcome: Overcome Curse of Dimensionality]
    E --> F[Validation: High Accuracy in High-Dimensional Markets]