In this work, we present a quantum algorithm designed to solve the differential equation used in the pricing of Asian options, in the framework of the Black-Scholes model. Our approach modifies an existing quantum pre-conditioning method (different from classical methods) for the problem of Asian option pricing such that we remove the dependence on the original condition number of discretized differential equation (system of linear equations). This was possible with new fast-forwardable discretizations of the first and second derivatives with respect to the underlying asset value ratio (value over average). We determine that these discretizations handle well kinks in the initial/terminal conditions. We also introduce a new circuit construction for the discretized time-derivative operator with Dirichlet boundary conditions which avoids the oracle workspace needed for the general sparse matrix implementation. Here, we also devised a new method probability integral estimation from which we extract the solution, achieving $\tilde{O}({\rm polylog}\left(1/ε)\right)$, which is an exponential improvement over other quantum methods when it comes to solution information extraction from the solution state.
In this work, we present a quantum algorithm designed to solve the differential equation used in the pricing of Asian options, in the framework of the Black-Scholes model. Our approach modifies an existing quantum pre-conditioning method (different from classical methods) for the problem of Asian option pricing such that we remove the dependence on the original condition number of discretized differential equation (system of linear equations). This was possible with new fast-forwardable discretizations of the first and second derivatives with respect to the underlying asset value ratio (value over average). We determine that these discretizations handle well kinks in the initial/terminal conditions. We also introduce a new circuit construction for the discretized time-derivative operator with Dirichlet boundary conditions which avoids the oracle workspace needed for the general sparse matrix implementation. Here, we also devised a new method probability integral estimation from which we extract the solution, achieving O~(polylog(1/ε)), which is an exponential improvement over other quantum methods when it comes to solution information extraction from the solution state.
Pricing financial derivatives, in particular European-style options at different time-maturities and strikes, is a relevant financial problem. 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 for a wide range of relevant financial parameters by mapping it to the Schrodinger equation. The non-Hermitian nature of the resulting Hamiltonian is solved by embedding the dynamics into an enlarged Hilbert space, which makes use of only one additional ancillary qubit. Moreover, we employ a second ancillary qubit to transform initial condition into periodic boundary conditions, which substantially improves the stability and performance of the protocol. This algorithm shows a feasible approach for pricing financial derivatives on a digital quantum computer based on Hamiltonian simulation, technique which differs from those based on Monte Carlo simulations to solve the stochastic counterpart of the Black Scholes equation. Our algorithm remarkably provides an exponential speedup since the terms in the Hamiltonian can be truncated by a polynomial number of interactions while keeping the error bounded. We report expected accuracy levels comparable to classical numerical algorithms by using 10 qubits and 94 entangling gates on a fault-tolerant quantum computer, and an expected success probability of the post-selection procedure due to the embedding protocol above 60\%.
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 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 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%.
The solution of option-pricing problems may turn out to be computationally demanding due to non-linear and path-dependent payoffs, the high dimensionality arising from multiple underlying assets, and sophisticated models of price dynamics. In this context, quantum computing has been proposed as a means to address these challenges efficiently. Prevailing approaches either simulate the stochastic differential equations governing the forward dynamics of underlying asset prices or directly solve the backward pricing partial differential equation. Here, we present an end-to-end quantum algorithmic framework that solves the Kolmogorov forward (Fokker-Planck) partial differential equation for local-volatility models by mapping it to a Hamiltonian-simulation problem via the Schrödingerisation technique. The algorithm specifies how to prepare the initial quantum state, perform Hamiltonian simulation, and how to efficiently recover the option price via a swap test. In particular, the efficiency of the final solution recovery is an important advantage of solving the forward versus the backward partial differential equation. Thus, our end-to-end framework offers a potential route toward quantum advantage for challenging option-pricing tasks. In particular, we obtain a polynomial advantage in grid size for the discretization of a single dimension. Nevertheless, the true power of our methodology lies in pricing high-dimensional systems, such as baskets of options, because the quantum framework admits an exponential speedup with respect to dimension, overcoming the classical curse of dimensionality.
We develop a novel quantum algorithm for approximating the price of a discrete floating-strike Asian option based on an underlying valuation tree. The paths of the tree are encoded in bit-representation into a qubit register, where quantum state preparation is used to load the corresponding distribution onto the states. We implement the expectation value of the option pricing formula as a composition of the price probabilities, the payout and an indicator function, mapping their respective values to amplitudes of additional qubits. Thus, the underlying no longer has to be discretized into the same bit values for different times, resulting in smaller quantum circuits. The algorithm may be used with quantum amplitude estimation, enabling a quadratic speed-up over classical Monte Carlo methods.
We consider the problem of pricing discretely monitored Asian options over T monitoring points where the underlying asset is modeled by a geometric Brownian motion. We provide two quantum algorithms with complexity poly-logarithmic in T and polynomial in 1/ε, where ε is the additive approximation error. Our algorithms are obtained respectively by using an O(logT)-qubit semi-digital quantum encoding of the Brownian motion that allows for exponentiation of the stochastic process and by analyzing classical Monte Carlo algorithms inspired by the semi-digital encodings. The best quantum algorithm obtained using this approach has complexity O(1/ε3) where the O suppresses factors poly-logarithmic in T and 1/ε. The methods proposed in this work generalize to pricing options where the underlying asset price is modeled by a smooth function of a sub-Gaussian process and the payoff is dependent on the weighted time-average of the underlying asset price.
Following the recent great advance of quantum computing technology, there are growing interests in its applications to industries, including finance. In this paper, we focus on derivative pricing based on solving the Black-Scholes partial differential equation by finite difference method (FDM), which is a suitable approach for some types of derivatives but suffers from the {\it curse of dimensionality}, that is, exponential growth of complexity in the case of multiple underlying assets. We propose a quantum algorithm for FDM-based pricing of multi-asset derivative with exponential speedup with respect to dimensionality compared with classical algorithms. The proposed algorithm utilizes the quantum algorithm for solving differential equations, which is based on quantum linear system algorithms. Addressing the specific issue in derivative pricing, that is, extracting the derivative price for the present underlying asset prices from the output state of the quantum algorithm, we present the whole of the calculation process and estimate its complexity. We believe that the proposed method opens the new possibility of accurate and high-speed derivative pricing by quantum computers.