Quantum Speedups for Derivative Pricing Beyond Black-Scholes

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.

💡 Research Summary

This paper addresses one of the most computationally demanding tasks in quantitative finance: the pricing of exotic derivatives whose payoffs depend on the entire path of underlying assets. Classical Monte‑Carlo integration (MCI) is the workhorse for such problems because it scales polynomially with the number of assets and avoids the curse of dimensionality, yet its runtime grows as O(ε⁻²) with the desired precision ε, making high‑accuracy valuations costly. Quantum Monte‑Carlo integration (QMCI) offers a quadratic improvement, reducing the precision dependence to O(ε⁻¹), but the overall speed‑up hinges on the cost of simulating the underlying stochastic differential equations (SDEs) on a quantum computer.

The authors first formalize a property they call “fast‑forwardability”. The classic Black‑Scholes geometric Brownian motion (GBM) possesses independent fast‑forwardability: a path with T monitoring points can be generated by sampling T independent Gaussian variables, and a quantum state encoding the whole path (a q‑sample) can be prepared as a tensor product of T Gaussian states. This property underlies the previously proven end‑to‑end quadratic speed‑up for GBM‑based pricing.

Real‑world markets, however, require richer dynamics such as the Cox‑Ingersoll‑Ross (CIR) process for interest rates and the Heston stochastic‑volatility model for equities. These models are not independently fast‑forwardable, but they satisfy a more general notion of fast‑forwardability: their transition densities are known in closed form, allowing recursive sampling of future states. The paper introduces quantum fast‑forwarding schemes for CIR and Heston that exploit these transition densities. The authors provide a rigorous discretization error analysis, showing how to choose the number of time steps and the quantum precision so that the total error remains below ε while preserving the O(ε⁻¹) query complexity.

When a model lacks any fast‑forwardability, one must resort to time‑discretization schemes such as Euler‑Maruyama (EM). The authors argue that EM alone is insufficient for achieving a quantum advantage in a multi‑level Monte‑Carlo (MLMC) setting because the strong convergence order O(√h) leads to a quantum complexity that does not improve over the classical O(ε⁻²) baseline. To overcome this, they develop a quantum Milstein sampler that attains strong order O(h). The Milstein scheme requires sampling Lévy areas (double stochastic integrals). The paper contributes a novel quantum algorithm for Lévy‑area sampling, which, despite a larger constant factor, retains the same ε‑dependence as the classical Milstein method. By integrating this sampler with quantum MLMC, the authors achieve an end‑to‑end quadratic speed‑up for multi‑dimensional, correlated processes such as the Heston model.

Beyond algorithmic design, the authors improve the numerical‑integration analysis underlying the quantum amplitude‑estimation step. Their refined bounds reduce the number of qubits and gate depth needed for GBM and CIR pricing, and they explore “arithmetic‑free” state‑preparation techniques that avoid costly floating‑point operations, further lowering resource requirements.

Finally, the paper critically examines quantum PDE solvers based on the Fokker‑Planck equation as an alternative route to pricing. They identify three fundamental obstacles: (1) the inability of history‑state constructions to capture path‑dependent payoffs, (2) the persistence of the curse of dimensionality in the required high‑dimensional integration, and (3) the overall runtime overhead that outweighs any potential speed‑up. Consequently, they conclude that, at present, PDE‑based quantum methods are not viable for achieving practical quantum advantage in derivative pricing.

In summary, the work extends quantum derivative‑pricing theory beyond the Black‑Scholes paradigm by (i) defining and exploiting fast‑forwardability for CIR and Heston processes, (ii) introducing a quantum Milstein sampler with a new Lévy‑area subroutine, (iii) integrating these tools into quantum MLMC to retain quadratic speed‑ups even for correlated multi‑asset models, and (iv) providing concrete resource estimates that bring quantum pricing closer to realistic implementation. The paper thus offers a comprehensive roadmap for leveraging quantum computing in the valuation of complex financial derivatives.