A Quantum Algorithm for Linear PDEs Arising in Finance

Abstract

We propose a hybrid quantum-classical algorithm, originated from quantum chemistry, to price European and Asian options in the Black-Scholes model. Our approach is based on the equivalence between the pricing partial differential equation and the Schrödinger equation in imaginary time. We devise a strategy to build a shallow quantum circuit approximation to this equation, only requiring few qubits. This constitutes a promising candidate for the application of Quantum Computing techniques (with large number of qubits affected by noise) in Quantitative Finance.Date: December 6, 2019. 2010 Mathematics Subject Classification. 35Q40, 91G20, 91G80. Key words and phrases. quantum algorithms, option pricing, PDE, Schrödinger equation. The views and opinions expressed here are the authors' and do not represent the opinions of their employers. They are not responsible for any use that may be made of these contents. No part of this presentation is intended to influence investment decisions or promote any product or service. The authors would like to thank Alexei Kondratyev for stimulating discussions.

A variational quantum algorithm for the Feynman-Kac formula

“…Not only does this approach bypass the need to set individual correlations between Brownian motions, it also allows for substantial flexibility when modeling different particles whose behavior is governed by several (14). We further note that generalization we present here, in addition to multi Brownian motions, also contains the discount function r. None of these features appear in the previous literature of VarQITE and stochastic differential equations [10][11][12].…”

“…The resulting function |ψ does not necessarily correspond to an 2 normalized wavefunction and Ĥ is not necessarily Hermitian. The wave function |ψ is not to be confused with the boundary condition ψ from (11). In this case, the Hamiltonian is given by the expression…”

“…VarQITE is an algorithm to approximate QITE, see [11,12,25,26,42,43]. Instead of evolving quantum states in the complete exponential state space, a parameterized ansatz is used and the evolution is approximated by mapping it to the ansatz parameters via McLachlan's variational principle [44].…”

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

“…Financial firms have a lot of heavy computational tasks in their daily business, 2 and therefore the speed-up of such tasks by quantum computers are expected to provide a large impact. For example, previous papers studied option pricing [7][8][9][10][11][12][13][14][15][16][17][18], risk measurement [19][20][21][22], portfolio optimization [23][24][25], and so on.…”

Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance

“…Quantum computers are expected to outperform classical computers for solving a system of linear equations [1,2,13,32,34,45,58,59] and differential equations [4, 5, 12, 14-16, 18, 23, 46, 48, 52, 60, 61]. Quantum algorithms for certain stochastic differential equations, such as simulating GBM of the Black-Scholes model as discussed above, have attracted increasing attention in quantum computational finance [8,26,33,53,55]. Reference [33] claims an exponential speedup over classical algorithms to solve the Black-Scholes PDE, but does not include a detailed complexity analysis; it uses a very different approach to that used here, which does not seem to be easily extendible to general SDEs.…”