Quantum algorithm for stochastic optimal stopping problems with applications in finance

Abstract

The famous least squares Monte Carlo (LSM) algorithm combines linear least square regression with Monte Carlo simulation to approximately solve problems in stochastic optimal stopping theory. In this work, we propose a quantum LSM based on quantum access to a stochastic process, on quantum circuits for computing the optimal stopping times, and on quantum Monte Carlo techniques. For this algorithm we elucidate the intricate interplay of function approximation and quantum Monte Carlo algorithms. Our algorithm achieves a nearly quadratic speedup in the runtime compared to the LSM algorithm under some mild assumptions. Specifically, our quantum algorithm can be applied to American option pricing and we analyze a case study for the common situation of Brownian motion and geometric Brownian motion processes.

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.…”

“…6a. 27 Each three-qubit random variable is then transformed into an empirical random variable using a piecewise linear transformation circuit and added to the result register; three extra qubits are needed to store the result. Finally, an additional integer comparator-as described in [91]-is used to generate the singlequbit objective, the amplitude of which is to be estimated.…”

“…For b = 1, it is ensured that the P&L contributions from both risk factors are relevant for the portfolio 27. The 'NormalDistribution' class in Qiskit is used.…”