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...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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American Physical Society
2023-12-01
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| Series: | Physical Review Research |
| Online Access: | http://doi.org/10.1103/PhysRevResearch.5.043220 |
| _version_ | 1797210305102610432 |
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| author | Javier Gonzalez-Conde Ángel Rodríguez-Rozas Enrique Solano Mikel Sanz |
| author_facet | Javier Gonzalez-Conde Ángel Rodríguez-Rozas Enrique Solano Mikel Sanz |
| author_sort | Javier Gonzalez-Conde |
| collection | DOAJ |
| description | 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%. |
| first_indexed | 2024-04-24T10:08:29Z |
| format | Article |
| id | doaj.art-52a64efa686743da971c085ff28e2b1d |
| institution | Directory Open Access Journal |
| issn | 2643-1564 |
| language | English |
| last_indexed | 2024-04-24T10:08:29Z |
| publishDate | 2023-12-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | Physical Review Research |
| spelling | doaj.art-52a64efa686743da971c085ff28e2b1d2024-04-12T17:36:44ZengAmerican Physical SocietyPhysical Review Research2643-15642023-12-015404322010.1103/PhysRevResearch.5.043220Efficient Hamiltonian simulation for solving option price dynamicsJavier Gonzalez-CondeÁngel Rodríguez-RozasEnrique SolanoMikel SanzPricing 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%.http://doi.org/10.1103/PhysRevResearch.5.043220 |
| spellingShingle | Javier Gonzalez-Conde Ángel Rodríguez-Rozas Enrique Solano Mikel Sanz Efficient Hamiltonian simulation for solving option price dynamics Physical Review Research |
| title | Efficient Hamiltonian simulation for solving option price dynamics |
| title_full | Efficient Hamiltonian simulation for solving option price dynamics |
| title_fullStr | Efficient Hamiltonian simulation for solving option price dynamics |
| title_full_unstemmed | Efficient Hamiltonian simulation for solving option price dynamics |
| title_short | Efficient Hamiltonian simulation for solving option price dynamics |
| title_sort | efficient hamiltonian simulation for solving option price dynamics |
| url | http://doi.org/10.1103/PhysRevResearch.5.043220 |
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