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arXiv:2012.03819 · Quantum

A Threshold for Quantum Advantage in Derivative Pricing

Shouvanik Chakrabarti, Rajiv Krishnakumar, Guglielmo Mazzola, Nikitas Stamatopoulos, Stefan Woerner, William J. Zeng

Published Jun 1, 2021• 100 citations
arXivHTMLPDF

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.

DOI
10.22331/q-2021-06-01-463

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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.
Pricing financial derivatives on quantum computers typically includes quantum arithmetic components which contribute heavily to the quantum resources required by the corresponding circuits. In this manuscript, we introduce a method based on Quantum Signal Processing (QSP) to encode financial derivative payoffs directly into quantum amplitudes, alleviating the quantum circuits from the burden of costly quantum arithmetic. Compared to current state-of-the-art approaches in the literature, we find that for derivative contracts of practical interest, the application of QSP significantly reduces the required resources across all metrics considered, most notably the total number of T-gates by ∼ 16 x and the number of logical qubits by ∼ 4 x. Additionally, we estimate that the logical clock rate needed for quantum advantage is also reduced by a factor of ∼ 5 x. Overall, we find that quantum advantage will require 4.7 k logical qubits, and quantum devices that can execute 10 9 T-gates at a rate of 45 MHz. While in this work we focus specifically on the payoff component of the derivative pricing process where the method we present is most readily applicable, similar techniques can be employed to further reduce the resources in other applications, such as state preparation.
Estimating potential future exposure (PFE) for path-dependent derivatives, such as FX Target Redemption Forwards (TARFs), represents a formidable computational challenge due to the demand of nested Monte Carlo simulations. We present a hybrid quantum-classical framework that leverages Iterative Quantum Amplitude Estimation (IQAE) to address this via a reduced-order counterparty credit risk model. Our methodology maps the non-linear TARF payoff -- including cumulative gains and knock-out features -- into a quantum circuit via a two-step formulation, whereby a first-step percentile is computed classically and then used to condition quantum evaluation of subsequent exposure. We employ discretisation of the FX process and a linearised additive approximation of dynamics to enable implementation on current quantum platforms. Developed via the Classiq platform and validated on NVIDIA CUDA-Q and Amazon Braket SV1, our approach achieves relative errors of 1%-8% against classical benchmarks at the 97.5% and 99% confidence levels. While discretisation constraints and approximate monotonicity assumption may introduce bias and limit recovery of the full exposure distribution, our framework offers a tractable testbed for quantum acceleration. Scaling analysis suggests that $\sim$300 logical qubits could enable full 52-week exposure estimation, reducing sample complexity for tail-risk estimation via amplitude estimation at the cost of increased circuit depth.
We introduce a quantum algorithm to compute the market risk of financial derivatives. Previous work has shown that quantum amplitude estimation can accelerate derivative pricing quadratically in the target error and we extend this to a quadratic error scaling advantage in market risk computation. We show that employing quantum gradient estimation algorithms can deliver a further quadratic advantage in the number of the associated market sensitivities, usually called greeks . By numerically simulating the quantum gradient estimation algorithms on financial derivatives of practical interest, we demonstrate that not only can we successfully estimate the greeks in the examples studied, but that the resource requirements can be significantly lower in practice than what is expected by theoretical complexity bounds. This additional advantage in the computation of financial market risk lowers the estimated logical clock rate required for financial quantum advantage from Chakrabarti et al. [Quantum 5, 463 (2021)] by a factor of ~7, from 50MHz to 7MHz, even for a modest number of greeks by industry standards (four). Moreover, we show that if we have access to enough resources, the quantum algorithm can be parallelized across 60 QPUs, in which case the logical clock rate of each device required to achieve the same overall runtime as the serial execution would be ~100kHz. Throughout this work, we summarize and compare several different combinations of quantum and classical approaches that could be used for computing the market risk of financial derivatives.
We present a comprehensive quantum algorithm tailored for pricing autocallable options, offering a full implementation and experimental validation. Our experiments include simulations conducted on high performance computing (HPC) hardware, along with an empirical analysis of convergence to the classically estimated value. Our key innovation is an improved integration-based exponential amplitude loading technique that reduces circuit depth compared to state-of-the-art approaches. A detailed complexity analysis in a relevant setting shows a∼50x\sim 50 x∼50xreduction in T-depth for the payoff component relative to previous methods. These contributions represent a step toward more efficient quantum approaches to pricing complex financial derivatives.
A promising approach to achieving scalable fault-tolerant quantum computation is the use of quantum error correction (QEC) codes augmented with magic states i.e. resource states produced via distillation, cultivation, or RzR_zRz​ synthesis and teleported into the circuit as needed. Because magic-state production dominates the space-time volume of fault-tolerant programs, system architects must decide how many production units to allocate. Current approaches rely on deterministic analysis that either provisions for worst-case peak demand (wasting valuable qubit resources on factories that are never simultaneously utilized) or assumes average demand, which increases execution time. In this work, we build a simulation framework that couples circuit scheduling with different stochastic magic state production models, and use it to quantify the impact of non-determinism on circuit execution. We show that non-determinism has a dual effect that deterministic models cannot capture: it inflates total execution time (the price), while deflating peak per-cycle resource demand (the payoff). For distillation-based architectures, this demand smoothing shifts the space-time-optimal provisioning point: fewer factories are needed to minimize space-time volume than deterministic analysis predicts. Across benchmarks, stochastic-aware provisioning reduces space-time volume by up to 27% compared to the deterministic optimum for distillation, while requiring up to 30% fewer factories. We characterize these effects across each preparation mechanism, map the resulting design-space tradeoffs, and demonstrate that static resource estimation systematically mis-characterizes the cost of fault-tolerant execution. Our results establish that stochastic-aware analysis is necessary for right-sizing the factory allocations and should replace deterministic heuristics as the standard methodology for FTQC resource planning.
We introduce two quantum algorithms to compute the Value at Risk (VaR) and Conditional Value at Risk (CVaR) of financial derivatives using quantum computers: the first by applying existing ideas from quantum risk analysis to derivative pricing, and the second based on a novel approach using Quantum Signal Processing (QSP). Previous work in the literature has shown that quantum advantage is possible in the context of individual derivative pricing and that advantage can be leveraged in a straightforward manner in the estimation of the VaR and CVaR. The algorithms we introduce in this work aim to provide an additional advantage by encoding the derivative price over multiple market scenarios in superposition and computing the desired values by applying appropriate transformations to the quantum system. We perform complexity and error analysis of both algorithms, and show that while the two algorithms have the same asymptotic scaling the QSP-based approach requires significantly fewer quantum resources for the same target accuracy. Additionally, by numerically simulating both quantum and classical VaR algorithms, we demonstrate that the quantum algorithm can extract additional advantage from a quantum computer compared to individual derivative pricing. Specifically, we show that under certain conditions VaR estimation can lower the latest published estimates of the logical clock rate required for quantum advantage in derivative pricing by up to ∼30\sim 30∼30x. In light of these results, we are encouraged that our formulation of derivative pricing in the QSP framework may be further leveraged for quantum advantage in other relevant financial applications, and that quantum computers could be harnessed more efficiently by considering problems in the financial sector at a higher level.