A Threshold for Quantum Advantage in Derivative Pricing

Abstract

We give an upper bound on the resources required for valuable quantum advantage in pricing derivatives. To do so, we give the first complete resource estimates for useful quantum derivative pricing, using autocallable and Target Accrual Redemption Forward (TARF) derivatives as benchmark use cases. We uncover blocking challenges in known approaches and introduce a new method for quantum derivative pricing – the re-parameterization method – that avoids them. This method combines pre-trained variational circuits with fault-tolerant quantum computing to dramatically reduce resource requirements. We find that the benchmark use cases we examine require 8k logical qubits and a T-depth of 54 million. We estimate that quantum advantage would require executing this program at the order of a second. While the resource requirements given here are out of reach of current systems, we hope they will provide a roadmap for further improvements in algorithms, implementations, and planned hardware architectures.

Towards Quantum Advantage in Financial Market Risk using Quantum Gradient Algorithms

“…A straightforward approach to improve the finite-difference method using quantum computation is to use a centraldifference formula with quantum amplitude estimation algorithm for each pricing step [1,4,5]. This semi-classical quantum gradient (SQG) method [8] then scales as O(k/ 1.5 ), which we get by substituting p = 2 and q = 1 in Eq.…”

“…The risk free rate is set to r = 1% and the option expires in T = 3 years. The weighted sum of the asset prices w • S(t) with weights w = (w 1 , w 2 , w 3 ) = (0.5, 0.3, 0.2) is observed on five days t B = [T /5 * i] for i ∈ [1,5] across the duration of the Probabilities FIG. 1.…”

“…Recently, quantum algorithms have been proposed to accelerate the pricing and risk analysis of financial derivatives [1][2][3][4][5]. These algorithms use quantum amplitude estimation to achieve quadratic advantage compared to the classical Monte Carlo methods that are used in practice for most computationally expensive pricing.…”

“…When an operator U depends on some input |x , we will henceforth write it as U(x) to highlight the dependence. Explicit constructions of A(x) with resource estimates for specific path dependent derivative instances are given in Chakrabarti et al [5]. QAE estimates f (x) with repeated applications of the operator…”

Efficient Quantum State Preparation for the Cauchy Distribution Based on Piecewise Arithmetic

“…Q UANTUM computers have the ability to outperform their classical counterpart in many ways [1] [2] [3]. A field where quantum information processing experiences rapid progress is in financial applications, including risk analysis [4], pricing prediction [5] [6], and many others [7] [8] [9]. One specific algorithm where quantum computing has a large advantage is the Quantum Monte Carlo algorithm [10] [11] [12].…”