A Quantum Algorithm for Linear PDEs Arising in Finance

Fontanela, Filipe; Jacquier, Antoine; Oumgari, Mugad

doi:10.2139/ssrn.3499134

Cited by 11 publications

(18 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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].…”

Section: Multi-dimensional Feynman-kac Formulamentioning

confidence: 98%

“…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…”

Section: B One-dimensional Feynman-kac Formulamentioning

confidence: 99%

“…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].…”

Section: A Varqite and State Embeddingmentioning

confidence: 99%

See 2 more Smart Citations

A variational quantum algorithm for the Feynman-Kac formula

Alghassi,

Deshmukh,

Ibrahim

et al. 2021

Preprint

View full text Add to dashboard Cite

We propose an algorithm based on variational quantum imaginary time evolution for solving the Feynman-Kac partial differential equation resulting from a multidimensional system of stochastic differential equations. We utilize the correspondence between the Feynman-Kac partial differential equation (PDE) and the Wick-rotated Schrödinger equation for this purpose. The results for a (2 + 1) dimensional Feynman-Kac system, obtained through the variational quantum algorithm are then compared against classical ODE solvers and Monte Carlo simulation. We see a remarkable agreement between the classical methods and the quantum variational method for an illustrative example on six qubits. In the non-trivial case of PDEs which are preserving probability distributions -rather than preserving the 2-norm -we introduce a proxy norm which is efficient in keeping the solution approximately normalized throughout the evolution. The algorithmic complexity and costs associated to this methodology, in particular for the extraction of properties of the solution, are investigated. Future research topics in the areas of quantitative finance and other types of PDEs are also discussed.

show abstract

Section: Multi-dimensional Feynman-kac Formulamentioning

confidence: 98%

Section: B One-dimensional Feynman-kac Formulamentioning

confidence: 99%

See 1 more Smart Citation

A variational quantum algorithm for the Feynman-Kac formula

Alghassi,

Deshmukh,

Ibrahim

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

Miyamoto

2022

EPJ Quantum Technol.

View full text Add to dashboard Cite

Pricing of financial derivatives, in particular early exercisable options such as Bermudan options, is an important but heavy numerical task in financial institutions, and its speed-up will provide a large business impact. Recently, applications of quantum computing to financial problems have been started to be investigated. In this paper, we first propose a quantum algorithm for Bermudan option pricing. This method performs the approximation of the continuation value, which is a crucial part of Bermudan option pricing, by Chebyshev interpolation, using the values at interpolation nodes estimated by quantum amplitude estimation. In this method, the number of calls to the oracle to generate underlying asset price paths scales as $\widetilde{O}(\epsilon ^{-1})$ O ˜ ( ϵ − 1 ) , where ϵ is the error tolerance of the option price. This means the quadratic speed-up compared with classical Monte Carlo-based methods such as least-squares Monte Carlo, in which the oracle call number is $\widetilde{O}(\epsilon ^{-2})$ O ˜ ( ϵ − 2 ) .

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Linden

Liu

et al. 2021

Quantum

View full text Add to dashboard Cite

Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expectation values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.

show abstract

A Quantum Algorithm for Linear PDEs Arising in Finance

Cited by 11 publications

References 14 publications

A variational quantum algorithm for the Feynman-Kac formula

A variational quantum algorithm for the Feynman-Kac formula

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

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

Contact Info

Product

Resources

About