Quantum Speedups for Derivative Pricing Beyond Black-Scholes
Dylan Herman, Yue Sun, Jin-Peng Liu, Marco Pistoia, Charlie Che, Rob Otter, Shouvanik Chakrabarti, Aram Harrow
TL;DR
This work extends quantum-accelerated derivative pricing beyond the Black-Scholes framework by introducing end-to-end quadratic speedups for more realistic models such as Cox–Ingersoll–Ross (CIR) and Heston, leveraging a notion of fast-forwardable SDEs. It develops a detailed framework for analyzing error propagation and resource requirements in quantum Monte Carlo pricing, along with new distribution-loading primitives (including Lévy areas and CIR-time integrals) to support end-to-end speedups. It further shows that quantum MLMC can achieve end-to-end speedups for correlated processes via a quantum Milstein sampler, while also critically evaluating quantum PDE solvers for distribution loading and highlighting fundamental barriers. The results collectively advance the theoretical understanding of when and how quantum techniques can accelerate derivative pricing in practice, offering concrete resource estimates and identifying key bottlenecks for future work. Overall, the paper broadens the scope of quantum advantages in quantitative finance, moving beyond GBM to more faithful market models and providing a structured path toward practical quantum-accelerated pricing pipelines.
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 Lévy 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.