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Pricing financial derivatives, in particular European-style options atdifferent time-maturities and strikes, means a relevant problem in finance. Thedynamics describing the price of vanilla options when constant volatilities andinterest rates are assumed, is governed by the Black-Scholes model, a linearparabolic partial differential equation with terminal value given by thepay-off of the option contract and no additional boundary conditions. Here, wepresent a digital quantum algorithm to solve Black-Scholes equation on aquantum computer by mapping it to the Schr\"odinger equation. The non-Hermitiannature of the resulting Hamiltonian is solved by embedding its propagator intoan enlarged Hilbert space by using only one additional ancillary qubit.Moreover, due to the choice of periodic boundary conditions, given by thedefinition of the discretized momentum operator, we duplicate the initialcondition, which substantially improves the stability and performance of theprotocol. The algorithm shows a feasible approach for using efficientHamiltonian simulation techniques as Quantum Signal Processing to solve theprice dynamics of financial derivatives on a digital quantum computer. Ourapproach differs from those based on Monte Carlo integration, exclusivelyfocused on sampling the solution assuming the dynamics is known. We reportexpected accuracy levels comparable to classical numerical algorithms by using9 qubits to simulate its dynamics on a fault-tolerant quantum computer, and anexpected success probability of the post-selection procedure due to theembedding protocol above 60%.

Efficient Hamiltonian Simulation for Solving Option Price Dynamics