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Deep Signature Algorithm for Multi-dimensional Path-Dependent Options

Erhan Bayraktar, Qi Feng, Zhaoyu Zhang

TL;DR

The convergence analysis of the numerical algorithm with explicit dependence on the truncation order of the signature and the neural network approximation errors is proved, which applies to both European and American type option pricing problems while the payoff function depends on the whole paths of the underlying forward stock process.

Abstract

In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [Huré-Pham-Warin. Mathematics of Computation 89, no. 324 (2020)] for state-dependent FBSDEs with reflections to path-dependent FBSDEs with reflections, by adding the signature layer to the backward scheme. Our algorithm applies to both European and American type option pricing problems while the payoff function depends on the whole paths of the underlying forward stock process. We prove the convergence analysis of our numerical algorithm with explicit dependence on the truncation order of the signature and the neural network approximation errors. Numerical examples for the algorithm are provided including: Amerasian option under the Black-Scholes model, American option with a path-dependent geometric mean payoff function, and the Shiryaev's optimal stopping problem.

Deep Signature Algorithm for Multi-dimensional Path-Dependent Options

TL;DR

The convergence analysis of the numerical algorithm with explicit dependence on the truncation order of the signature and the neural network approximation errors is proved, which applies to both European and American type option pricing problems while the payoff function depends on the whole paths of the underlying forward stock process.

Abstract

In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [Huré-Pham-Warin. Mathematics of Computation 89, no. 324 (2020)] for state-dependent FBSDEs with reflections to path-dependent FBSDEs with reflections, by adding the signature layer to the backward scheme. Our algorithm applies to both European and American type option pricing problems while the payoff function depends on the whole paths of the underlying forward stock process. We prove the convergence analysis of our numerical algorithm with explicit dependence on the truncation order of the signature and the neural network approximation errors. Numerical examples for the algorithm are provided including: Amerasian option under the Black-Scholes model, American option with a path-dependent geometric mean payoff function, and the Shiryaev's optimal stopping problem.
Paper Structure (8 sections, 4 theorems, 89 equations, 1 figure, 3 tables)

This paper contains 8 sections, 4 theorems, 89 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Main algorithm
  3. Convergence Analysis
  4. Convergence Analysis: reflected FBSDE
  5. Numerical Example
  6. Amerasian Option
  7. Bermudan Option with moving average
  8. Shiryaev's optimal stopping problem

Key Result

Lemma 3.2

\newlabellemma: sig error0 Under Assumption main assumption. For any $2\le m\in\mathbb N^+$, $\Delta t=T/n$, and $\tilde{n}=n/k$ as the number of segments for the signature process, we have where we denote $\tilde{X}^n$ as the continuous interpolation of $X^n$ in euler for X. The constant $C$ depends on $T$, $d_1$, and the bound of $b$, $\sigma$ together with all their derivatives up to order $m

Figures (1)

  • Figure 1: Numerical Implementation of bayraktarzhou2017 with delay = 0.7.

Theorems & Definitions (11)

  • Definition 2.1: Signature
  • Definition 2.2
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3
  • Proof 2
  • Theorem 3.4
  • Remark 3.5
  • Proof 3
  • Theorem 3.6
  • ...and 1 more