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

Abstract

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.

An Introduction to Quantum Computing for Statisticians and Data Scientists

“…For more complex, highdimensional distributions Kaneko et al (2021) propose to create quantum samples using pseudorandom numbers. An et al (2021) quantize the classical method of multilevel Monte Carlo to find approximate solutions to stochastic differential equations (SDEs), particularly for applications in finance.…”

“…This section introduces Quantum Markov chains, often called quantum walks in the quantum computing literature, the quantum equivalents of classical Markov chains which are widely used in probability and statistics. Quantum walks have been shown to provide polynomial speed-ups for a wide variety of problems from estimating the volume of convex bodies (Chakrabarti et al, 2019) to option pricing (An et al, 2021), search for marked items (Magniez et al, 2011) and active learning in artificial intelligence (Paparo et al, 2014). 16 However, because of the quantum interference, quantum Markov chains behave substantially differently from their classical counterparts.…”

Quantum algorithm for stochastic optimal stopping problems with applications in finance

“…In particular, we construct a unitary that propagates backwards the optimal stopping time by one time step according to Eq. (1). In what follows, given a path x ∈ E T , by z τt(x) we mean z τt(x) (x τt(x) ), i.e., the associated payoff of the τ t (x)-th time step of x. Lemma 3.8 (Quantum circuits for computing the stopping times).…”

“…A few different works devised quantum algorithms based on quantum Monte Carlo for derivative pricing [70,79,14], e.g. American/Bermudan option pricing [60] and option pricing in the local volatility model [46,1] (of which the Black-Scholes model is a subcase). Given its versatility and previous cases of success, it is only natural to explore the applicability of quantum Monte Carlo methods to problems in optimal stopping theory.…”

Efficient Quantum State Preparation for the Cauchy Distribution Based on Piecewise Arithmetic

“…A field where quantum information processing experiences rapid progress is in financial applications, including risk analysis [4], pricing prediction [5] [6], and many others [7] [8] [9]. One specific algorithm where quantum computing has a large advantage is the Quantum Monte Carlo algorithm [10] [11] [12]. However, the benefits of quantum speedup heavily depend on the method used to prepare the quantum superposition states.…”