Pricing multi-asset derivatives by variational quantum algorithms

Abstract

Pricing a multi-asset derivative is an important problem in financial engineering, both theoretically and practically. Although it is suitable to numerically solve partial differential equations to calculate the prices of certain types of derivatives, the computational complexity increases exponentially as the number of underlying assets increases in some classical methods, such as the finite difference method. Therefore, there are efforts to reduce the computational complexity by using quantum computation. However, when solving with naive quantum algorithms, the target derivative price is embedded in the amplitude of one basis of the quantum state, and so an exponential complexity is required to obtain the solution. To avoid the bottleneck, the previous study [Miyamoto and Kubo, IEEE Transactions on Quantum Engineering, 3, 1-25 ( 2022)] utilizes the fact that the present price of a derivative can be obtained by its discounted expected value at any future point in time and shows that the quantum algorithm can reduce the complexity. In this paper, to make the algorithm feasible to run on a small quantum computer, we use variational quantum simulation to solve the Black-Scholes equation and compute the derivative price from the inner product between the solution and a probability distribution. This avoids the measurement bottleneck of the naive approach and would provide quantum speedup even in noisy quantum computers. We also conduct numerical experiments to validate our method. Our method will be an important breakthrough in derivative pricing using small-scale quantum computers.

Quantum computing for financial risk measurement

“…• Pricing of standard ('vanilla'), path-dependent (e.g. barrier and Asian) and multiasset options in a Black-Scholes [9] and local volatility [28] framework [4,18,33,38,51,54,64,72,73,75,76,80] • Pricing of options under a stochastic volatility [16] and jump-diffusion process [95] • Pricing of American-style options [27,62] • Pricing of interest-rate derivatives with a multi-factor model [58,84] • Pricing of collateralised debt obligations (CDOs) [83] Risk measurement. In a financial context, 'risk' usually refers to an adverse event associated with a (financial) loss.…”

Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its proof of overcoming the curse of dimensionality

“…The motivation comes from the fact that qubits, compared to classical bits, are allowed quantum mechanically to be in a state of superposition, from which one anticipates that quantum computers should be able to achieve much higher computational power than classical (super-) computers. The applications of quantum algorithms in finance include portfolio optimization [53], the computation of risk measures such as Value at Risk (VAR) [62], and option pricing, particularly in the Black-Scholes model [15,21,26,38,50,52,53,57]. We also refer to the monograph [36] and surveys [24,46] for (further) applications of quantum computing in finance.…”

“…While [26] proposes a hybrid quantum-classical algorithm to approximately solve the one-dimensional Black-Scholes PDE exploiting its relation to the Schrödinger equation in imaginary time and [38] proposes a variational quantum approach, most literature uses quantum Monte Carlo methods to approximately solve the Black-Scholes PDE in order to price financial options. More precisely, these works rely on the Quantum Amplitude Estimation algorithm (QAE) [12] which estimates the expected value of a random parameter (see Section 2.3 for a detailed discussion) based on an extension of Grover's search algorithm [32].…”

A Quantum Algorithm for Pricing Asian Options on Valuation Trees

“…The quantum computer is used to calculate the resulting systems of equations. Pricing multi-dimensional derivatives with the help of discretizing their price PDEs is a method Kubo et al (2022) use as well. The novelty being that a variational quantum algorithm (VQA) is used to calculate the development of the underlying.…”