Efficient Hamiltonian simulation for solving option price dynamics

Pricing financial derivatives, in particular European-style options at different time-maturities and strikes, means a relevant problem in finance. The dynamics describing the price of vanilla options when constant volatilities and interest rates are assumed is governed by the Black-Scholes model, a linear parabolic partial differential equation with terminal value given by the payoff of the option contract and no additional boundary conditions. Here, we present a digital quantum algorithm to solve the Black-Scholes equation on a quantum computer by mapping it to the Schrödinger equation. The non-Hermitian nature of the resulting Hamiltonian is solved by embedding its propagator into an enlarged Hilbert space by using only one additional ancillary qubit. Moreover, due to the choice of periodic boundary conditions, given by the definition of the discretized momentum operator, we duplicate the initial condition, which substantially improves the stability and performance of the protocol. The algorithm shows a feasible approach for using efficient Hamiltonian simulation techniques as quantum signal processing to solve the price dynamics of financial derivatives on a digital quantum computer. Our approach differs from those based on Monte Carlo integration, exclusively focused on sampling the solution assuming the dynamics is known. We report expected accuracy levels comparable to classical numerical algorithms by using nine qubits to simulate its dynamics on a fault-tolerant quantum computer, and an expected success probability of the post-selection procedure due to the embedding protocol above 60%.

Figure 1

Typical solutions of Eq. (1) for a European put-type option. Simulation parameters: $S_{\text{max}}=150$ u, $K=50$ u, $\sigma =0.2$ and $r=0.04$.

Figure 2

Continuous line: convergence of the solution of the Black-Scholes put option pricing problem obtained with the finite differences discretized operator ${\widehat{P}}_x$ for a distinct number of qubits, $n=1,\cdots ,8$ (excluding the ancillary qubit used to duplicate the initial condition) and the analytical solution. Dashed line: discretization error per point depending on the number of qubits, $n=1\cdots 9$ (excluding the ancillary qubit used to duplicate the initial condition). Simulation parameters: $S_{\text{max}}=150$ u, $K=50$ u, $\sigma =0.2, r=0.04, T=1$ year. Simulations have made use of the duplication of the initial condition. For source code of the simulations see [85].

Figure 3

Solution to the Black-Scholes equation without duplicating the initial condition. We can appreciate how it presents a strong border effect due to periodic boundary conditions. Simulation parameters: $n=8, S_{\text{max}}=150$ u, $K=50$ u, $\sigma =0.2, r=0.04, T=1$ year. Scripts for numerical simulations can be found in [85].

Figure 4

(a) Success probability in post-selection protocol corresponding to Eq. (21) depending on time to maturity in years and risk-free interest rate. The probability is above 0.6 for all values in the mesh. (b) Lower bound probability of success $\gamma (N_x,K)$. As we can observe, there exists an asymptotic convergence value for both, number of points and strike. The value of the asymptotic limit is over 0.6, what indicates that our protocol would be success in more than a half of the realizations. Parameters values: $S_{\text{max}}=150$ u, $K=50$ u, $\sigma =0.2, n=8$.