Gradient-enhanced sparse Hermite polynomial expansions for pricing and hedging high-dimensional American options

Paper: arXiv 2405.02570

Abstract

We propose an efficient and easy-to-implement gradient-enhanced least squares Monte Carlo method for computing price and Greeks (i.e., derivatives of the price function) of high-dimensional American options. It employs the sparse Hermite polynomial expansion as a surrogate model for the continuation value function, and essentially exploits the fast evaluation of gradients. The expansion coefficients are computed by solving a linear least squares problem that is enhanced by gradient information of simulated paths. We analyze the convergence of the proposed method, and establish an error estimate in terms of the best approximation error in the weighted $H^1$ space, the statistical error of solving discrete least squares problems, and the time step size. We present comprehensive numerical experiments to illustrate the performance of the proposed method. The results show that it outperforms the state-of-the-art least squares Monte Carlo method with more accurate price, Greeks, and optimal exercise strategies in high dimensions but with nearly identical computational cost, and it can deliver comparable results with recent neural network-based methods up to dimension 100.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper involves advanced mathematics including weighted Sobolev spaces, backward stochastic differential equations, and Malliavin calculus for convergence analysis, while also providing comprehensive numerical experiments with performance comparisons and error estimates.

Research Flowchart

  flowchart TD
  Start(["Research Goal<br>Efficient Pricing & Greeks for<br>High-Dim American Options"]) --> Inputs

  subgraph Inputs ["Data/Inputs"]
      S(["Simulated Asset Paths"])
      H(["Hermite Polynomials"])
      O(["American Option Payoffs"])
  end

  Inputs --> Methodology

  subgraph Methodology ["Methodology Steps"]
      G(["Compute Gradient Information<br>from Paths"])
      LS(["Solve Gradient-Enhanced<br>Least Squares Problem"])
      C(["Construct Sparse Hermite<br>Surrogate Model"])
  end

  Methodology --> Processes

  subgraph Processes ["Computational Processes"]
      P1(["Price via Backward Induction<br>using Surrogate Model"])
      P2(["Compute Greeks<br>Numerically"])
      P3(["Determine Optimal<br>Exercise Strategy"])
  end

  Processes --> Outcomes

  subgraph Outcomes ["Key Findings/Outcomes"]
      F1(["High Accuracy in Prices & Greeks"])
      F2(["Efficient High-Dim Scaling<br>up to d=100"])
      F3(["Outperforms Standard LS-MC<br>and Neural Networks"])
  end

Cited References (27)

1. Pricing and hedging American-style options: a simple simulation-based approach (2010) — link
2. Monte-Carlo Valuation of American Options: Facts and New Algorithms to Improve Existing Methods (2012) — link
3. Error expansion for the discretization of backward stochastic differential equations (2006) — link
4. A regression-based Monte Carlo method to solve backward stochastic differential equations (2005) — link
5. Pricing Multi-Asset American Options: A Finite Element Method-of-Lines with Smooth Penalty (2007) — link
6. Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s (1997) — link
7. A Fourier-Based Valuation Method for Bermudan and Barrier Options under Heston’s Model (2011) — link
8. Valuing American Options by Simulation: A Simple Least-Squares Approach (2001) — link

1 of 27 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.