Quantum unary approach to option pricing

Abstract

We present a quantum algorithm for European option pricing in finance, where the key idea is to work in the unary representation of the asset value. The algorithm needs novel circuitry and is divided in three parts: first, the amplitude distribution corresponding to the asset value at maturity is generated using a low depth circuit; second, the computation of the expected return is computed with simple controlled gates; and third, standard Amplitude Estimation is used to gain quantum advantage. On the positive side, unary representation remarkably simplifies the structure and depth of the quantum circuit. Amplitude distributions uses quantum superposition to bypass the role of classical Monte Carlo simulation. The unary representation also provides a post-selection consistency check that allows for a substantial mitigation in the error of the computation. On the negative side, unary representation requires linearly many qubits to represent a target probability distribution, as compared to the logarithmic scaling of binary algorithms. We compare the performance of both unary vs. binary option pricing algorithms using error maps, and find that unary representation may bring a relevant advantage in practice for near-term devices.

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

“…for every S ∈ D, where the coefficient a f , l is calculated by (10) for every l ∈ [m] d 0 . This is in fact an interpolation, since D,m [f ]( S) = f ( S) for every node S ∈ G d,m D .…”

“…Financial firms have a lot of heavy computational tasks in their daily business, 2 and therefore the speed-up of such tasks by quantum computers are expected to provide a large impact. For example, previous papers studied option pricing [7][8][9][10][11][12][13][14][15][16][17][18], risk measurement [19][20][21][22], portfolio optimization [23][24][25], and so on.…”

Qibo: a framework for quantum simulation with hardware acceleration

“…This is an example application of quantum computing in finance and provides a new strategy to compute the expected payoff of a (European) option, based on reference [86]. The main feature of this procedure is to use the unary encoding of prices, that is, to work in the subspace of the Hilbert space spanned by computational-basis states where only one qubit is in the |1 state.…”

“…This allows for a simplification of the circuit and resilience against errors, making the algorithm more suitable for Noisy Intermediate-Scale Quantum (NISQ) era devices. The pre-coded example takes as input the asset parameters (initial price, strike price, volatility, interest rate and maturity date) and the quantum simulation parameters [number of qubits, number of measurement shots and number of applications of the amplitude estimation algorithm (see [86])] and calculates the expected payoff using the quantum algorithm. The result is compared with the exact value of the expected payoff.…”

Quantum pricing with a smile: implementation of local volatility model on quantum computer

“…Monte Carlo simulation is often used to compute the derivative price, but it takes a computation long time. Quantum algorithms for Monte Carlo integration [5,6] bring quadratic speedup compared with classical Monte Carlo algorithms, and several previous studies discuss their application to derivative pricing [7][8][9][10]. Although previous studies consider the Black-Scholes (BS) model [11,12], which is the pioneering model for derivative pricing, it is inappropriate as an application target of Monte Carlo for practical business for the following reasons.…”