Quantum Monte Carlo algorithm for option pricing and its complexity analysis

Paper: arXiv 2301.09241

Authors: Jianjun Chen, Yongming Li, Ariel Neufeld

Abstract

In this paper we provide a quantum Monte Carlo algorithm to solve multidimensional Black-Scholes PDEs with correlation for option pricing. The payoff function of the option is of general form and is only required to be continuous and piecewise affine, which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension $d$ of the PDE and the reciprocal of the prescribed accuracy $\varepsilon$. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. Furthermore, we provide numerical simulations in two dimensions using our developed package within the Qiskit framework tailored to price continuous piecewise affine options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous theoretical derivations and error analysis for a quantum Monte Carlo algorithm applied to high-dimensional option pricing, showing polynomial complexity bounds. However, empirical rigor is limited primarily to low-dimensional numerical simulations within the Qiskit quantum computing simulator rather than real-market backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Price multidimensional options via Quantum Monte Carlo with polynomial complexity] --> B[Inputs: Black-Scholes PDE with correlation & piecewise affine payoff]
    B --> C[Methodology: Develop QMC Algorithm for high-dimensional PDEs]
    C --> D[Rigorous Error & Complexity Analysis]
    C --> E[Numerical Simulations in 2D using Qiskit Framework]
    D --> F[Outcome: Proven O-poly in d and 1/ε complexity with quantum speed-up for bounded payoffs]
    E --> F