Quantum Speedups for Derivative Pricing Beyond Black-Scholes

2026-02-03

AI Summary

This work addresses the absence of end-to-end quadratic quantum speedups for derivative pricing under non-Black-Scholes financial models such as CIR and Heston. The authors propose a quantum multilevel Monte Carlo framework that integrates a “fast-forwardable” property with a novel quantum Milstein sampler based on Lévy areas. This approach achieves, for the first time, an end-to-end quadratic quantum speedup in these practically relevant models. The method substantially reduces the computational resource requirements for pricing under both geometric Brownian motion and CIR dynamics, establishes a general quantum acceleration paradigm applicable to multidimensional correlated stochastic processes, and further reveals theoretical limitations of quantum PDE solvers in derivative pricing contexts.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper explores advancements in quantum algorithms for derivative pricing of exotics, a computational pipeline of fundamental importance in quantitative finance. For such cases, the classical Monte Carlo integration procedure provides the state-of-the-art provable, asymptotic performance: polynomial in problem dimension and quadratic in inverse-precision. While quantum algorithms are known to offer quadratic speedups over classical Monte Carlo methods, end-to-end speedups have been proven only in the simplified setting over the Black-Scholes geometric Brownian motion (GBM) model. This paper extends existing frameworks to demonstrate novel quadratic speedups for more practical models, such as the Cox-Ingersoll-Ross (CIR) model and a variant of Heston's stochastic volatility model, utilizing a characteristic of the underlying SDEs which we term fast-forwardability. Additionally, for general models that do not possess the fast-forwardable property, we introduce a quantum Milstein sampler, based on a novel quantum algorithm for sampling L\'evy areas, which enables quantum multi-level Monte Carlo to achieve quadratic speedups for multi-dimensional stochastic processes exhibiting certain correlation types. We also present an improved analysis of numerical integration for derivative pricing, leading to substantial reductions in the resource requirements for pricing GBM and CIR models. Furthermore, we investigate the potential for additional reductions using arithmetic-free quantum procedures. Finally, we critique quantum partial differential equation (PDE) solvers as a method for derivative pricing based on amplitude estimation, identifying theoretical barriers that obstruct achieving a quantum speedup through this approach. Our findings significantly advance the understanding of quantum algorithms in derivative pricing, addressing key challenges and open questions in the field.

Problem

• quantum speedup
• derivative pricing
• stochastic volatility models
• Monte Carlo integration
• quantum algorithms

Innovation

• quantum speedup
• fast-forwardability
• quantum Milstein sampler
• Lévy area sampling
• multi-level Monte Carlo