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Quantum computational finance: Monte Carlo pricing of financial derivatives

Patrick Rebentrost, Brajesh Gupt, Thomas R. Bromley

arXiv:1805.00109v2quant-ph

TL;DR

The paper addresses the computational cost of pricing derivatives whose payoffs depend on stochastic benchmark assets. It develops a quantum Monte Carlo framework that prepares model distributions, implements payoffs as quantum circuits, and uses amplitude estimation to extract prices. Under efficient state-preparation and payoff-computation assumptions, the method changes sample dependence from 1/ϵ2 classically to 1/ϵ quantumly.

  • Problem

    Pricing derivatives with complex payoff functions or asset dynamics is computationally demanding because such prices often lack analytical solutions and require Monte Carlo evaluation.

  • Method

    The paper combines quantum Monte Carlo, quantum preparation of financial probability distributions, arithmetic payoff circuits, and amplitude estimation of derivative expectations.

  • Results

    The quantum algorithm achieves quadratic improvement in sample dependence, scaling as 1/ϵ rather than the classical 1/ϵ2 for additive accuracy ϵ.

  • Takeaways & Limitations

    The framework provides a starting point for applying quantum computing to financial-derivative pricing, including European and Asian options within its stated circuit-computability scope.

  • Takeaways & Limitations

    The speedup assumes that the martingale-measure distribution and derivative payoff can both be prepared or computed efficiently.

Abstract

from arXiv · show

Financial derivatives are contracts that can have a complex payoff dependent upon underlying benchmark assets. In this work, we present a quantum algorithm for the Monte Carlo pricing of financial derivatives. We show how the relevant probability distributions can be prepared in quantum superposition, the payoff functions can be implemented via quantum circuits, and the price of financial derivatives can be extracted via quantum measurements. We show how the amplitude estimation algorithm can be applied to achieve a quadratic quantum speedup in the number of steps required to obtain an estimate for the price with high confidence. This work provides a starting point for further research at the interface of quantum computing and finance.

I. INTRODUCTION

Financial derivatives require pricing expected payoffs under stochastic asset dynamics, creating substantial computational demands. The paper proposes combining quantum Monte Carlo techniques with amplitude estimation to obtain a quadratic speedup for derivative pricing.

  • Financial derivatives have future payoffs dependent on benchmark-asset prices or price trajectories, making fair pricing under stochastic dynamics an important problem.
  • Monte Carlo pricing evaluates expectations of payoff functions, especially for models beyond analytically solvable Black-Scholes-Merton dynamics.
  • Amplitude estimation can provide close to quadratic speedups for estimating expectation values, including expectations evaluated by classical Monte Carlo methods.
  • The paper prepares financial probability distributions, implements payoff arithmetic, and estimates derivative expectations through quantum measurements and ancilla-qubit phase information.
  • The Black-Scholes-Merton stock model uses geometric Brownian motion and log-normal price evolution, with option pricing requiring expected payoff estimation.

II. BLACK-SCHOLES-MERTON OPTION PRICING

The Black-Scholes-Merton framework models a risky stock and a risk-free bond, then prices derivatives as discounted risk-neutral expectations. Analytical solutions provide benchmarks, while complex payoffs or asset dynamics generally require Monte Carlo evaluation.

  • The Black-Scholes-Merton model contains one risky stock driven by Brownian motion and one risk-free bond.
  • Brownian motion has independent increments, and each increment over t > s follows a normal distribution with mean 0 and variance t − s.
  • The model assumes constant parameters, continuous and unlimited fractional trading without transaction costs, permitted short selling, and no stock dividends.
  • A European call gives its owner the right to buy the stock at maturity T for strike price K, with payoff defined from the terminal stock price.
  • Derivative pricing evaluates the discounted expected payoff under the risk-neutral measure Q, where the stock drift equals the market rate r and discounted prices are martingales.
  • For complex payoff functions or asset-price dynamics, option prices cannot be solved analytically and are often evaluated using Monte Carlo methods.

III. CLASSICAL MONTE CARLO PRICING

Classical Monte Carlo derivative pricing samples risk-neutral asset outcomes, computes payoffs, and averages them to approximate the price. Its sample complexity scales as 1/ϵ2, motivating amplitude-estimation-based quantum methods with improved ϵ dependence.

  • Complex stochastic asset dynamics may lack analytical solutions, so prices are determined by sampling paths of the asset dynamics.
  • Monte Carlo pricing samples from a calibrated risk-neutral distribution, computes asset prices and payoffs, and averages payoffs across samples.
  • The quantum algorithm targets an improved ϵ dependence from ϵ2 to ϵ by using amplitude estimation.

IV. QUANTUM ALGORITHM FOR MONTE CARLO

The quantum Monte Carlo algorithm encodes an expectation value into an ancilla amplitude, then uses amplitude estimation and phase estimation to estimate it with improved accuracy dependence.

  • Expectation encoding: A quantum circuit prepares a state |χ⟩ whose ancilla-|1⟩ probability equals the desired expectation value.The state is formed by applying A and an ancilla rotation R, combined into unitary F.
  • Expectation encoding: The expectation value is related to a rotation angle θ through a measurement of the reflection unitary V.Measuring V on |χ⟩ gives 1−2µ, allowing µ to be recovered from the rotation parameter.
  • Amplitude estimation: The Grover-like operator Q applies reflections that rotate states by 2θ and has eigenvalues e^±iθ.Phase estimation resolves these eigenvalues to obtain an estimate of θ.
  • Amplitude estimation: Phase estimation measures a register whose probability peaks at m-bit approximations ±ˆθ of the encoded angle.The circuit prepares |χ⟩, applies controlled powers of Q, and then an inverse quantum Fourier transform.
  • Accuracy and complexity: For [0,1]-bounded variables, amplitude estimation achieves additive error ϵ using t = O(1/ϵ) applications, rather than the classical ϵ^-2 dependence.The success probability can be amplified from 8/π^2 to 1−δ using medians of repeated runs.
  • Accuracy and complexity: For bounded-variance variables, the algorithm estimates the mean to additive error ϵ with success probability at least 2/3 using an almost quadratic reduction in applications of U.The construction reduces bounded-variance estimation to [0,1]-bounded variables and combines their estimates.

V. QUANTUM ALGORITHM FOR EUROPEAN OPTION PRICING

For European options, the algorithm discretizes the Brownian-motion distribution, encodes the payoff in an ancilla, and estimates the resulting expectation with amplitude estimation.

  • Price representation: A European option price is represented as the expectation of a payoff function evaluated at the single maturity date T.For the BSM model with one Brownian motion, the relevant random variable is W_T ∼ N(0,T).
  • Distribution preparation: The Brownian-motion distribution is approximated on 2^n discretization points over a finite interval and prepared as a quantum superposition.The cutoff and normalization approximate the Gaussian probability density.
  • Payoff encoding: A binary approximation ˜v maps discretized inputs to payoff values, with n bits trading representable range against numerical accuracy.The payoff rotation is implemented using an auxiliary register and a controlled ancilla rotation.
  • Payoff encoding: Measuring the ancilla in |1⟩ yields an expectation value µ that determines the option price after discounting.The expectation is tied to the option price up to the approximation used for the payoff and distribution.
  • Error and complexity: Using n qubits gives discretization and function-approximation accuracy ν = O(2^-n).The total error combines this approximation error with the amplitude-estimation error.
  • Error and complexity: The required quantum steps are quadratically better than classical Monte Carlo steps, up to suppressed polylogarithmic factors.The step count includes repeated applications of the operator U and is treated as analogous to the number of classical Monte Carlo runs.

VI. ASIAN OPTION PRICING

For Asian options, the algorithm prepares multiple Brownian increments, computes arithmetic or geometric path averages reversibly, and encodes the resulting payoff for expectation estimation.

  • Option structure: Asian options depend on the average asset price before maturity, unlike European options, which depend on a single maturity-date price.The arithmetic Asian option lacks a known analytic solution in the stated BSM setting, whereas the geometric mean option can be priced analytically.
  • Option structure: The Asian call payoff, arithmetic mean, and geometric mean are defined over pre-selected times 0 < t_1 < ··· < t_L ≤ T.The construction assumes equally spaced adjacent time points in the algorithmic discussion.
  • Path preparation: A Gaussian state for each Brownian increment is prepared, and L such states form a product superposition over discretized paths.The preparation uses m qubits per increment and therefore Lm qubits for the product state.
  • Path payoff computation: Reversible arithmetic maps each path label to successive stock prices and updates the running arithmetic or geometric average.Intermediate registers can be uncomputed after the final path average is stored.
  • Path payoff computation: A conditional ancilla rotation encodes the path-dependent payoff, so measuring |1⟩ approximates the expected Asian-option payoff.The resulting expectation provides an approximation to the Black-Scholes price of the Asian option.

VII. NUMERICAL SIMULATIONS

The simulations encode option-price expectations into quantum phases and compare quantum phase estimation with classical Monte Carlo. Under matched confidence, quantum estimation exhibits nearly quadratic error-scaling improvement, robust across strike prices and other tested parameters.

  • Algorithm and setup: The algorithm prepares a Brownian-motion superposition, encodes the payoff on an ancilla, and estimates the encoded expectation using amplitude amplification and phase estimation.The numerical demonstration uses a single-qubit rotation to simulate phase estimation.
  • Algorithm and setup: The simulations compare quantum and classical estimates against Monte Carlo steps using similar confidence above 99.5% and approximately D ≈ 24 independent phase-estimation runs.Classical Monte Carlo has scaling exponent ζC = −1/2.
  • Error scaling: For the European call benchmark, the analytical price is Π = $10.5, and the quantum error fit gives ζQ = −0.982 versus ζC = −0.5 classically.The benchmark uses S0 = $100, K = $50, r = 0.05, σ = 0.2, T = 1, and D = 24.
  • Error scaling: The quantum error bound has scaling exponent −1, demonstrating quadratic speedup in the steps needed to reach a given phase-estimation error.This is the theoretical upper-bound scaling shown alongside the simulations.
  • Robustness: Varying the strike price produces an almost quadratic quantum advantage for all chosen values, while varying other parameters likewise shows robust quadratic speedup.The comparison uses the ratio ζQ/ζC of quantum to classical scaling exponents.

VIII. DISCUSSION AND CONCLUSION

The paper concludes that quantum Monte Carlo can price financial derivatives with quadratic sample-complexity improvement under explicit state-preparation and payoff-computability assumptions. It illustrates the approach on European and Asian options while identifying complex derivatives and stochastic models for future work.

  • Contribution: The work presents a quantum algorithm for pricing financial derivatives.
  • Assumptions and speedup: Under efficient preparation of the martingale-measure states and efficient payoff computation, classical sampling scales as 1/ϵ2 while the quantum algorithm scales as 1/ϵ.
  • Scope: The approach covers European call options with analytical solutions and Asian options whose arithmetic-averaging prices require Monte Carlo methods.
  • Scope: The method can in principle handle payoffs decomposable into elementary arithmetic operations and averages over multiple time windows.
  • Future work: Complex-derivative payoffs used at leading financial institutions and more complicated stochastic models remain targets for future work.Complex derivatives can require large Monte Carlo workloads for portfolio risk analysis.

Appendix A: Black-Scholes calculations

The appendix derives core Black–Scholes–Merton ingredients used for derivative pricing: the stock-process solution, discounted-stock martingale property, European call price, and a variance bound.

  • Stock dynamics: The appendix develops the solution to the stochastic differential equation governing the stock price.
  • Risk-neutral valuation: Under the Q measure, the discounted stock price is established to be a martingale.
  • European call pricing: The Black–Scholes price for a European call option is derived from the payoff decomposition and the distribution of the terminal stock price.
  • European call pricing: The appendix obtains the call price by separately evaluating the strike-price and stock-price contributions.
  • Variance bound: The variance of the European call payoff under the risk-neutral measure is bounded by λ2.The proof computes the variance from its constituent terms.

Appendix B: Introduction to quantum computing

The appendix introduces qubits, multi-qubit states, unitary evolution, quantum gates, and circuit notation as the computational language underlying the algorithm.

  • Qubits and states: A qubit can occupy a superposition of |0⟩ and |1⟩, with measurement probabilities determined by the squared magnitudes of its amplitudes.
  • Qubits and states: An n-qubit system is represented in a 2^n-dimensional Hilbert space as a superposition of computational-basis states.The probability of basis state |i⟩ is |α_i|2.
  • Quantum operations: Isolated quantum systems evolve through unitary transformations, which preserve reversibility and can act on multiple qubits.
  • Quantum operations: The inverse quantum Fourier transform performs an inverse discrete Fourier transform on quantum-state amplitudes.
  • Circuit notation: Controlled gates apply operations conditionally; CNOT and Toffoli gates use one or two control qubits, respectively, to flip a target qubit.

Appendix C: Quantum circuits for arithmetic operations

Appendix C describes binary and signed-number encodings, reversible quantum arithmetic, and circuits that transform discretized Brownian-motion outcomes into European call-option payoffs.

  • Number representation: Integers use m-bit binary registers, while real numbers use m fractional bits with accuracy 1/2^m; signed values add a sign qubit.The integer register stores computational-basis bits, and real-number precision is determined by the fractional-bit count.
  • Reversible computation: Classical functions become reversible quantum transformations by preserving inputs and XORing the function output into an auxiliary register.The reversible map is F′(x, y) = (x, y⊕F(x)) and can be realized using negation and Toffoli gates.
  • Quantum arithmetic: Composed SUM and carry gates implement addition, multiplication, exponentiation, and other arithmetic operations modulo the register range.The appendix also notes that alternative circuits can reduce gate requirements.
  • Payoff implementation: A reversible payoff circuit uses the sign bit as a controller to implement the call-option max function and map stock-price outcomes to approximate payoffs.The stock-price mapping combines Brownian-motion outcomes with the payoff circuit, producing the bit approximation ˜veuro(x, K, σ, r, t).
  • Probability preparation: The probability-loading procedure assumes efficiently computable interval sums and recursively prepares a discretized probability distribution without measurements.The ratio of left-half to full-interval sums determines the transition between discretization levels.

Appendix F: Phase estimation

Appendix F reviews phase estimation, including coherent multi-qubit and single-qubit procedures, error analysis, and confidence amplification through repeated runs and medians.

  • Estimation procedure: Phase estimation samples an m-bit approximation of the phase θ after controlled operations and an inverse quantum Fourier transform.The procedure uses an eigenstate associated with θ and produces a bit-string distribution from which θ can be estimated.
  • Error analysis: The expectation value of interest is related to the phase θ, allowing phase-estimation accuracy to bound the expectation-value error.The appendix derives first-order and more general bounds under assumptions on θ̂, θ, and ϵ.
  • Single-run accuracy: The best m-bit phase approximation is observed with probability at least 8/π^2 > 0.81.This baseline success probability motivates repeating phase estimation to increase confidence.
  • Confidence amplification: The median of D independent phase-estimation runs increases the probability that the final estimate lies within the desired accuracy.The failure probability is bounded when each individual sample has constant failure probability below 1/2.
  • Confidence amplification: For constant per-run failure probability, achieving confidence c requires at most O(|log(1−c)|) independent repetitions.Confidence is defined as c := 1−pf, where pf is the failure probability.
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