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arXiv:2402.10132

Quantum option pricing via the Karhunen-Loève expansion

Anupam Prakash, Yue Sun, Shouvanik Chakrabarti, Che, Charlie, Aditi Dandapani, Dylan Herman, Niraj Kumar, Shree Hari Sureshbabu, Ben Wood, Iordanis Kerenidis, Marco Pistoia

Published Feb 15, 2024• 0 citations
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Abstract

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 $O(\log T)$-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 $\widetilde{O}(1/ε^{3})$ where the $\widetilde{O}$ 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.

DOI
10.48550/arxiv.2402.10132

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We consider the problem of pricing discretely monitored Asian options over TTT monitoring points where the underlying asset is modeled by a geometric Brownian motion. We provide two quantum algorithms with complexity poly-logarithmic in TTT and polynomial in 1/ε1/ε1/ε, where εεε is the additive approximation error. Our algorithms are obtained respectively by using an O(log⁡T)O(\log T)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)\widetilde{O}(1/ε^{3})O(1/ε3) where the O~\widetilde{O}O suppresses factors poly-logarithmic in TTT and 1/ε1/ε1/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.
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.
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Options are financial instruments whose payoffs are determined by the value of an underlying asset. These underlying assets are typically stocks, interest rates, or exchange rates, but can also include commodities such as energy or agricultural products. Options serve diverse purposes, including hedging against price volatility and speculating on market movements. Additionally, they play a critical role in assessing certain positions on the balance sheets of insurance companies, particularly in the context of solvency capital requirement calculations. As an option’s payoff lies in the future and is stochastic in nature, finding a fair price today is a non-trivial task. The focus of this work is finding quantum algorithms for calculating these prices and compare them to algorithms designed for classical computers. This work follows two different directions on how to improve on classical methods for option pricing. The first direction is improving on Monte-Carlo (MC) methods and the second one is adapting Fourier based pricing algorithms. The Amplitude Estimation (AE) algorithm promises a quadratic speedup compared to classical MC. This fact makes this algorithm an interesting candidate for improvements by quantum algorithms. It was shown in the literature, that simple options with very crude discretization of the underlying stochastic models, can be implemented on a quantum device and evaluated with AE. One big challenge is the efficient implementation of the option’s payoff profile. To solve this problem colleagues and I have introduced a multi-objective genetic algorithm to automatically find well performing and efficient circuits to implement the payoff of non-path-dependent European options on one or more underlyings. Another strategy for pricing options is based on the fact, that in many models the characteristic function of the underlying is numerically more tractable than its distribution function. There are methods that make use of this fact and the performance of the Fast Fourier Transform (FFT) to efficiently calculate prices for European call options for many strikes in one go. The quantum version of the FFT, the Quantum Fourier Transform (QFT), is in some sense, even faster than its classical counterpart. We have adapted the existing classical method to make use of QFT, to benefit from its performance. Finally, the different strategies are benchmarked and compared regarding their accuracy, their potential for scaling and their adaptability to different styles of derivatives and models. The result of this comparison are requirements on the quantum hardware for the methods to be viable as well as use cases where each method has its benefits over the others.
We develop a quantum algorithm to price discretely monitored lookback options in the Black-Scholes framework using imaginary time evolution. By rewriting the pricing PDE as a Schrodinger-type equation, the problem becomes the imaginary time evolution of a quantum state under a non-Hermitian Hamiltonian. This evolution is approximated with the Variational Quantum imaginary time evolution (VarQITE) method, which replaces the exact non-unitary dynamics with a parameterized, hardware-efficient quantum circuit. A central challenge arises from jump conditions caused by the discrete updating of the running maximum. This feature is not present in standard quantum treatments of European or Asian options. To address this, we propose two quantum-compatible formulations: (i) a sequential approach that models jumps via dedicated jump Hamiltonians applied at monitoring dates, and (ii) a simultaneous multi-function evolution that removes explicit jumps at the expense of an increased number of dimensions. We compare both approaches in terms of qubit resources, circuit complexity and numerical accuracy, and benchmark them against Monte Carlo simulations. Our results show that discretely monitored, path-dependent options with jump conditions can be handled within a variational quantum framework, paving the way toward the quantum pricing of more complex derivatives with non-smooth dynamics.
Multi-asset option pricing under local- and stochastic-volatility models leads naturally to high-dimensional parabolic PDEs. We develop an end-to-end quantum PDE framework for European option pricing under local-volatility Black--Scholes and Heston models. The framework takes classical contract and model data as input and returns classical estimates of selected option values. We solve the pricing PDEs after finite-difference discretization on spatial grids. For N=2nN=2^nN=2n grid points per spatial direction and ddd assets, the end-to-end gate complexity for single-point recovery, counted in elementary CNOT gates and one-qubit Pauli-axis rotations, has leading grid-size dependence O~(d2N2+d/2)\widetilde{O}(d^2 N^{2+d/2})O(d2N2+d/2) for local-volatility Black--Scholes and O~(d2Nd+2)\widetilde{O}(d^2 N^{d+2})O(d2Nd+2) for Heston. Relative to grid-based finite-difference baselines, these scalings correspond to polynomial improvement factors Nd/2N^{d/2}Nd/2 and NdN^dNd, respectively. These estimates translate to Clifford+T resources via standard compilation. We complement the complexity analysis with numerical benchmarks against standard classical methods. In the Heston setting, the framework recovers option prices across strikes together with the associated implied-volatility smile/skew. Overall, this work provides a complete end-to-end quantum pricing pipeline with explicit resource accounting and theoretical performance guarantees.
Abstract Financial derivative pricing is a significant challenge in finance, involving the valuation of instruments like options based on underlying assets. While some cases have simple solutions, many require complex classical computational methods like Monte Carlo simulations and numerical techniques. However, as derivative complexities increase, these methods face limitations in computational power. Cases involving Non-Vanilla Basket pricing, American Options, and derivative portfolio risk analysis need extensive computations in higher-dimensional spaces, posing challenges for classical computers. Quantum computing presents a promising avenue by harnessing quantum superposition and entanglement, allowing the handling of high-dimensional spaces effectively. In this paper, we introduce a self-contained and all-encompassing quantum algorithm that operates without reliance on oracles or presumptions. More specifically, we develop an effective stochastic method for simulating exponentially many potential asset paths in quantum parallel, leading to a highly accurate final distribution of stock prices. Furthermore, we demonstrate how this algorithm can be extended to price more complex options and analyze risk within derivative portfolios.