Paper: arXiv 2602.03725

Authors: Dylan Herman, Yue Sun, Jin-Peng Liu, Marco Pistoia, Charlie Che, Rob Otter, Shouvanik Chakrabarti, Aram Harrow

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides exceptional mathematical rigor and theoretical contributions, extending quantum derivative pricing beyond Black-Scholes to CIR, Heston, and general correlated SDEs via novel MLMC and Milstein samplers. However, as a purely theoretical quantum computing paper, it lacks empirical backtesting or real-world market data validation, relying instead on asymptotic error bounds and gate-level resource analyses.

Research Flowchart

  flowchart TD
    A["Research Goal: Extend Quantum Speedups in Derivative Pricing Beyond Black-Scholes"] --> B["Input Models: CIR, Heston Stochastic Volatility & Multi-D SDEs"]
    
    B --> C["Method 1: Fast-Forwardable SDE Analysis for CIR/Heston"]
    B --> D["Method 2: Quantum Milstein Sampler & Quantum Lévy Area Sampling for Multi-D SDEs"]
    
    C & D --> E["Optimization: Improved Numerical Integration & Arithmetic-Free Quantum Procedures"]
    
    E --> F["Outcome 1: Proven Quadratic Speedups for Complex Exotic Derivatives via Quantum MLMC"]
    E --> G["Outcome 2: Resource Reductions & Identified Barriers in Quantum PDE Solvers"]