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Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its complexity analysis

Jianjun Chen, Yongming Li, Ariel Neufeld

TL;DR

This work addresses high-dimensional option pricing under the Black-Scholes model by introducing a quantum Monte Carlo algorithm that handles correlated assets and general CPWA payoffs. The method loads a discretized multivariate log-normal distribution and a CPWA payoff into rotated quantum form and uses a Modified IQAE subroutine to estimate the option price with provable error $|u-\tilde{U}|\le\varepsilon$ with probability $1-\alpha$. The authors prove polynomial-in-$d$ and $\varepsilon^{-1}$ scaling for qubit and gate resources, and they show a speed-up for bounded payoffs over classical Monte Carlo, along with detailed error analyses (truncation, quadrature, payoff loading, rotation, and QAE). Numerical experiments in 1D and 2D demonstrate the practicality of the approach and its extension to higher dimensions, supported by a Qiskit-based package $qfinance$. The results establish a rigorous foundation for quantum speed-ups in high-dimensional financial PDEs and provide a concrete path toward scalable quantum pricing in practice.

Abstract

In this paper we provide a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), 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 one and two dimensions using our developed package within the Qiskit framework tailored to price CPWA options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension.

Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its complexity analysis

TL;DR

This work addresses high-dimensional option pricing under the Black-Scholes model by introducing a quantum Monte Carlo algorithm that handles correlated assets and general CPWA payoffs. The method loads a discretized multivariate log-normal distribution and a CPWA payoff into rotated quantum form and uses a Modified IQAE subroutine to estimate the option price with provable error with probability . The authors prove polynomial-in- and scaling for qubit and gate resources, and they show a speed-up for bounded payoffs over classical Monte Carlo, along with detailed error analyses (truncation, quadrature, payoff loading, rotation, and QAE). Numerical experiments in 1D and 2D demonstrate the practicality of the approach and its extension to higher dimensions, supported by a Qiskit-based package . The results establish a rigorous foundation for quantum speed-ups in high-dimensional financial PDEs and provide a concrete path toward scalable quantum pricing in practice.

Abstract

In this paper we provide a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), 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 of the PDE and the reciprocal of the prescribed accuracy . 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 one and two dimensions using our developed package within the Qiskit framework tailored to price CPWA options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension.
Paper Structure (32 sections, 33 theorems, 259 equations, 20 figures, 7 tables, 1 algorithm)

This paper contains 32 sections, 33 theorems, 259 equations, 20 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting and Main result
  3. Black-Scholes PDE for option pricing
  4. Geometric Brownian motion process for the price evolution of multiple assets
  5. Continuous piece-wise affine (CPWA) payoff functions
  6. Brief Introduction to Quantum Computing
  7. Quantum amplitude estimation algorithms
  8. Algorithm \ref{['quantum algorithm']} and main result
  9. Outline of Algorithm \ref{['quantum algorithm']}
  10. Main Theorem
  11. Numerical Simulations
  12. Vanilla Call Option
  13. Basket Call Option
  14. Spread Call Option
  15. Call-on-max Option
  16. ...and 17 more sections

Key Result

Lemma 2.1

Let $d \in {\mathbb N}$, let $\bm{x}=(x_1,\ldots,x_d) \in {\mathbb R}_+^d$, and let $t \in [0,T)$. Let $\bm{\mu} = (\mu_1,\ldots,\mu_d) \in {\mathbb R}^d$ be given by $\mu_i = \ln(x_i) + (r - \tfrac{1}{2}\sigma_i^2)(T-t)$ for $i=1,\ldots,d$ and let $\bm{\Sigma} \in {\mathbb R}^{d \times d}$ be given where for $\bm{y} = (y_1,\ldots,y_d) \in {\mathbb R}_+^d$, $\log(\bm{y}) \in {\mathbb R}^d$ is give

Figures (20)

  • Figure 1: Flowchart of Algorithm 1. (Top left) (1) Construction of the operator $\mathcal{A}$ using probability distribution operator $\mathcal{P}$ and CPWA payoff operator $\mathcal{R}_h$. (Top right) (2) Construction of the Grover operator using the operator $\mathcal{A}$, oracle operator $\mathcal{S}_{\psi_0}$ and phase flip operator $\mathcal{S}_0$. (Bottom) (3) Illustration of the Modified Iterative Quantum Amplitude Estimation algorithm to produce the final estimate $\widetilde{U}_{t,\bm{x}}$.
  • Figure 2: Comparison of the expected payoff estimates from the algorithm for the vanilla call option across a range of strike prices with the reference expected payoff. The tested strike prices are labeled on the horizontal axis.
  • Figure 3: Expected payoff estimates from our proposed algorithm for the basket call option across a range of strike prices, compared with the reference expected payoff. The strike prices tested are labeled on the horizontal axis.
  • Figure 4: Expected payoff estimates from the algorithm for the spread call option across a range of strike prices, compared with the reference expected payoff. The strike prices tested is labeled on the horizontal axis.
  • Figure 5: Expected payoff estimates from the algorithm for the call-on-max option across a range of strike prices, compared with the reference expected payoff. The strike prices tested is labeled on the horizontal axis.
  • ...and 15 more figures

Theorems & Definitions (76)

  • Lemma 2.1: Density formula
  • proof
  • Definition 2.3: CPWA payoff
  • Example 2.5
  • Definition 2.6: Elementary quantum gate set
  • Remark 2.7: Universality of $\mathbb{G}$
  • Definition 2.8: Quantum circuit
  • Remark 2.9: Depth
  • Remark 2.10: Ancilla qubits
  • Proposition 2.11
  • ...and 66 more