Quantum Algorithm for Local-Volatility Option Pricing via the Kolmogorov Equation

TL;DR

This paper tackles the computational bottlenecks in option pricing under local-volatility dynamics by proposing an end-to-end quantum algorithm that solves the Kolmogorov forward equation using Schrödingerisation to map the non-unitary PDE into a unitary Hamiltonian evolution. It provides a complete IPO workflow: quantum data encoding of the initial price distribution, Hamiltonian simulation of the forward LV dynamics, and payoff retrieval via swap tests, yielding a polynomial advantage in grid size and, crucially, an exponential speedup potential for high-dimensional basket options. The forward formulation enables efficient computation of prices for multiple payoffs from a single simulated distribution, addressing the classical curse of dimensionality more favorably than backward methods. The work highlights practical resource scalings, discusses multi-asset extensions, and outlines future directions such as stochastic volatility and more complex derivatives, underscoring the potential for quantum advantage in computational finance.

Abstract

The solution of option-pricing problems may turn out to be computationally demanding due to non-linear and path-dependent payoffs, the high dimensionality arising from multiple underlying assets, and sophisticated models of price dynamics. In this context, quantum computing has been proposed as a means to address these challenges efficiently. Prevailing approaches either simulate the stochastic differential equations governing the forward dynamics of underlying asset prices or directly solve the backward pricing partial differential equation. Here, we present an end-to-end quantum algorithmic framework that solves the Kolmogorov forward (Fokker-Planck) partial differential equation for local-volatility models by mapping it to a Hamiltonian-simulation problem via the Schrödingerisation technique. The algorithm specifies how to prepare the initial quantum state, perform Hamiltonian simulation, and how to efficiently recover the option price via a swap test. In particular, the efficiency of the final solution recovery is an important advantage of solving the forward versus the backward partial differential equation. Thus, our end-to-end framework offers a potential route toward quantum advantage for challenging option-pricing tasks. In particular, we obtain a polynomial advantage in grid size for the discretization of a single dimension. Nevertheless, the true power of our methodology lies in pricing high-dimensional systems, such as baskets of options, because the quantum framework admits an exponential speedup with respect to dimension, overcoming the classical curse of dimensionality.

Figures (7)

• Figure 1: Equivalent differential-equation formulations for option-pricing. Forward models simulate the dynamics of the underlying asset price from the current date forward, while backward models propagate the option value from maturity back to today. In the forward approach, the option price is computed as the expected payoff at maturity, discounted at the risk-free interest rate. We can also distinguish between PDE formulations—the Kolmogorov equations Conze2008TheFK—and SDE formulations (e.g., geometric Brownian motion), connected to the backward PDE through the Feynman-Kac formula black1973pricing.
• Figure 2: A unique diffusion process consistent with risk-neutral densities implied by European option prices is represented by the local-volatility (LV) model. This process is constructed as follows: (i) observe implied volatilities across strikes $K$; (ii) map each implied volatility to an option price via the Black--Scholes formula, yielding discrete price points (each associated with a specific volatility); (iii) interpolate these points to obtain a continuous option-price surface over strikes and maturities; and (iv) compute Dupire’s local-volatility $\sigma(S_{\tau},\tau)$ from this continuous surface.
• Figure 3: Backward and forward approaches to solve option-pricing, $C(S_0, 0)$. Forward (time $\tau$ from present to future): The stock price is modeled as a stochastic process or as the Kolmogorov forward PDE which evolves from the present time to maturity. In order to retrieve the option price one would need to compute the expected value of the payoff under the resulting underlying price distribution at maturity time and discount it to present time; Backward time $t$ from future to present: The option price is evolved backward from maturity time, given by the final payoff, to present time, through a backward Kolmogorov PDE.
• Figure 4: Solution of the Kolmogorov forward and backward equations from the Black-Scholes model as quantum states: a) Solutions to the forward model are lognormal distributions given by Eq. (\ref{['analytical_forward_sigma_constant']}); b) The overlap between the state encoding the solution to the forward equation and the state encoding the initial condition decreases exponentially in time; c) Solutions to the backward model; d) The overlap between the state encoding the solution to the backward equation and the state encoding the initial condition presents a negligible decay in time, remaining at values $\gtrsim 0.995$ for relevant maturity times.
• Figure 5: The IPO model is a widely used approach in quantum computing for describing the structure of an end-to-end algorithm. Firstly, classical input data is embedded into a quantum state. Subsequently, the quantum processing involves applying quantum gates and operations to manipulate qubits. In this way properties as superposition and entanglement are key quantum phenomena harnessed during processing, enabling quantum computers to explore many possible solutions to a problem in parallel and achieve a quantum advantage against classical processing, such as in Hamiltonian simulation techniques. Finally, the quantum system is measured and we obtain processed data from the output state, task that is limited by Holevo's bound holevo. Every measurement to extract new piece of information will require the execution of the whole IPO model.

Theorems & Definitions (9)

• Remark 1: Low-rank separable model for $\sigma$
• Theorem 3.1: Efficient preparation of piece-wise polynomial functions guseynov2024efficientgonzalez2024efficient
• Theorem 3.2: Optimal sparse Hamiltonian simulation using queries (Theorem 3 from low2017optimal)
• Remark 2
• Theorem 3.3: Quantum simulation of the option-pricing Hamiltonian (Theorems 8 & 9 from guseynov2025quantum)
• Theorem 3.4: Multidimensional scaling and simulation cost
• proof
• Theorem 3.5: Swap test (Proposition 6, huang2019near)
• Remark 3: Swap-test retrieval for multi-asset pricing