Efficient Hamiltonian Simulation for Solving Option Price Dynamics
- URL: http://arxiv.org/abs/2101.04023v4
- Date: Tue, 16 Jan 2024 07:51:23 GMT
- Title: Efficient Hamiltonian Simulation for Solving Option Price Dynamics
- Authors: Javier Gonzalez-Conde, \'Angel Rodr\'iguez-Rozas, Enrique Solano,
Mikel Sanz
- Abstract summary: We present a digital quantum algorithm to solve Black-Scholes equation on a quantum computer by mapping it to the Schr"odinger equation.
The algorithm shows a feasible approach for using efficient Hamiltonian simulation techniques as Quantum Signal Processing.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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
pay-off of the option contract and no additional boundary conditions. Here, we
present a digital quantum algorithm to solve Black-Scholes equation on a
quantum computer by mapping it to the Schr\"odinger 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
9 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%.
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