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Parallel-in-Time Iterative Methods for Pricing American Options

Xian-Ming Gu, Jun Liu, Cornelis W. Oosterlee

TL;DR

This work tackles the computational bottleneck in pricing American options by formulating an all-at-once parallel-in-time (PinT) policy iteration for the Hamilton-Jacobi-Bellman framework. It introduces two PinT preconditioners based on an alpha-circulant Kronecker structure and a projection strategy to handle 1D and 2D models, respectively, enabling efficient Krylov solves within each policy iteration. Numerical experiments across 1D Black-Scholes, 2D put spreads, and 2D Heston models demonstrate robust convergence with inner solves remaining fast (≈4 GMRES iterations) and overall CPU time scaling favorably with the number of time steps, particularly in the 1D and 2D scenarios. The approach significantly accelerates American option pricing on parallel hardware, while leaving open theoretical analysis of the observed outer-iteration independence from $N_t$.

Abstract

For pricing American options, %after suitable discretization in space and time, a sequence of discrete linear complementarity problems (LCPs) or equivalently Hamilton-Jacobi-Bellman (HJB) equations need to be solved in a sequential time-stepping manner. In each time step, the policy iteration or its penalty variant is often applied due to their fast convergence rates. In this paper, we aim to solve for all time steps simultaneously, by applying the policy iteration to an ``all-at-once form" of the HJB equations, where two different parallel-in-time preconditioners are proposed to accelerate the solution of the linear systems within the policy iteration. Our proposed methods are generally applicable for such all-at-once forms of the HJB equation, arising from option pricing problems with optimal stopping and nontrivial underlying asset models. Numerical examples are presented to show the feasibility and robust convergence behavior of the proposed methodology.

Parallel-in-Time Iterative Methods for Pricing American Options

TL;DR

This work tackles the computational bottleneck in pricing American options by formulating an all-at-once parallel-in-time (PinT) policy iteration for the Hamilton-Jacobi-Bellman framework. It introduces two PinT preconditioners based on an alpha-circulant Kronecker structure and a projection strategy to handle 1D and 2D models, respectively, enabling efficient Krylov solves within each policy iteration. Numerical experiments across 1D Black-Scholes, 2D put spreads, and 2D Heston models demonstrate robust convergence with inner solves remaining fast (≈4 GMRES iterations) and overall CPU time scaling favorably with the number of time steps, particularly in the 1D and 2D scenarios. The approach significantly accelerates American option pricing on parallel hardware, while leaving open theoretical analysis of the observed outer-iteration independence from .

Abstract

For pricing American options, %after suitable discretization in space and time, a sequence of discrete linear complementarity problems (LCPs) or equivalently Hamilton-Jacobi-Bellman (HJB) equations need to be solved in a sequential time-stepping manner. In each time step, the policy iteration or its penalty variant is often applied due to their fast convergence rates. In this paper, we aim to solve for all time steps simultaneously, by applying the policy iteration to an ``all-at-once form" of the HJB equations, where two different parallel-in-time preconditioners are proposed to accelerate the solution of the linear systems within the policy iteration. Our proposed methods are generally applicable for such all-at-once forms of the HJB equation, arising from option pricing problems with optimal stopping and nontrivial underlying asset models. Numerical examples are presented to show the feasibility and robust convergence behavior of the proposed methodology.
Paper Structure (12 sections, 4 theorems, 54 equations, 7 figures, 4 tables, 1 algorithm)

This paper contains 12 sections, 4 theorems, 54 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Semi-discretized HJB equations
  3. Parallel-in-time policy iteration
  4. All-at-once form of LCPs
  5. PinT solvers
  6. 1D model: nearest Kronecker product approximation.
  7. 2D model: projection to reduced system.
  8. Numerical examples
  9. Example 1. 1D Black-Scholes Model
  10. Example 2: 2D American put spread options
  11. Example 3. 2D Heston Model.
  12. Conclusion

Key Result

Proposition 3.1

The vectors ${\bm u}^1, {\bm u}^2, \cdots, {\bm u}^{N_t}$ are the solutions of obstacle problems (eq2.4) if and only if the vector ${\bm v}$ in eq3.2 is the solution of the all-at-once problem eq3.1.

Figures (7)

  • Figure 1: the nonzero pattern (blue dot=1, white blank=0) of the reshaped matrix $\texttt{Tmat}(\Theta^{(k)})$ at each PinT-Policy iteration (Example 1 with $N_s=128$, $N_t=40$ and $\alpha=10^{-8}$).
  • Figure 2: the spectrum distribution of $\mathcal{M}^{(k)}$ and $[\mathcal{P}_\alpha^{(k)}]^{-1}\mathcal{M}^{(k)}$ in the last ($k=6$) PinT-Policy iteration (Example 1 with $N_s=128$, $N_t=40$, and $\alpha=10^{-8}$).
  • Figure 3: the spectrum distribution of $\mathcal{M}$ and $\mathcal{M}_{\alpha}^{-1}\mathcal{M}$ and its projected submatrices $\widetilde{\mathcal{M}}$ and $\widetilde{\mathcal{M}}_\alpha^{-1}\widetilde{\mathcal{M}}$ in the last PinT-Policy iteration (Example 1 with $N_s=128$, $N_t=40$, and $\alpha=10^{-8}$).
  • Figure 4: 1D American put option value surface (with $N_s=128, N_t=80$).
  • Figure 5: Example 1: PinT-Policy iteration numbers verse $\sigma$ (with $N_s=1280, N_t=80$).
  • ...and 2 more figures

Theorems & Definitions (10)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Remark