The valuation of American options and the determination of optimal stopping times remain central challenges in option pricing theory and derivatives valuation. Classical approaches to multidimensional problems rely on the dynamic programming principle, which is notoriously difficult to parallelize. Moreover, the need to store continuation values for all simulated paths in each dynamic programming step imposes severe memory constraints on classical computers, limiting the scale of feasible simulations. Quantum computing offers a promising way to address these limitations. In this paper, the authors introduce a novel algorithm for pricing American options on quantum hardware. To date, only one method has been proposed for this purpose. The authors' approach combines the Quantum Binomial Tree with Quantum Machine Learning to enable direct valuation of American options on a quantum computer. By incorporating quantum amplitude estimation, it could achieve a quadratic speed-up over classical Monte Carlo methods. Furthermore, it exploits the exponential growth of the quantum state vector, overcoming the memory bottlenecks that restrict classical approaches.
The valuation of American options and the determination of optimal stopping times remain central challenges in option pricing theory and derivatives valuation. Classical approaches to multidimensional problems rely on the dynamic programming principle, which is notoriously difficult to parallelize. Moreover, the need to store continuation values for all simulated paths in each dynamic programming step imposes severe memory constraints on classical computers, limiting the scale of feasible simulations. Quantum computing offers a promising way to address these limitations. In this paper, the authors introduce a novel algorithm for pricing American options on quantum hardware. To date, only one method has been proposed for this purpose. The authors' approach combines the Quantum Binomial Tree with Quantum Machine Learning to enable direct valuation of American options on a quantum computer. By incorporating quantum amplitude estimation, it could achieve a quadratic speed-up over classical Monte Carlo methods. Furthermore, it exploits the exponential growth of the quantum state vector, overcoming the memory bottlenecks that restrict classical approaches.
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.
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.
Efficiently generating or loading probability distributions on quantum computers is a foundational task in Quantum Monte Carlo methods. While the quantum amplitude estimation algorithm offers a quadratic speed-up over classical Monte Carlo techniques, this advantage can be negated if the probability distribution is not effectively loaded. Moreover, any practical approach must be flexible enough to accommodate a wide range of market models�a requirement unmet by existing methods. In this paper, we propose a novel and efficient approach for loading probability distributions tailored to derivative pricing. The proposed method, the Quantum Binomial Tree, serves as a quantum analog of the classical Binomial Tree model. This approach enables exponential scaling in the number of Monte Carlo paths, while retaining an overall quadratic speed-up compared to classical algorithms. We demonstrate how the Quantum Binomial Tree framework can load prominent financial models�including the local volatility model, and the Heston model onto a quantum computer. Furthermore, this paper includes a detailed implementation for option pricing under time-dependent volatility, as well as numerical results.
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.
Abstract Pricing of financial derivatives, in particular early exercisable options such as Bermudan options, is an important but heavy numerical task in financial institutions, and its speed-up will provide a large business impact. Recently, applications of quantum computing to financial problems have been started to be investigated. In this paper, we first propose a quantum algorithm for Bermudan option pricing. This method performs the approximation of the continuation value, which is a crucial part of Bermudan option pricing, by Chebyshev interpolation, using the values at interpolation nodes estimated by quantum amplitude estimation. In this method, the number of calls to the oracle to generate underlying asset price paths scales as O(ϵ−1) O ˜ ( ϵ − 1 ) , where ϵ is the error tolerance of the option price. This means the quadratic speed-up compared with classical Monte Carlo-based methods such as least-squares Monte Carlo, in which the oracle call number is O(ϵ−2) O ˜ ( ϵ − 2 ) .
American options are important financial products traded in enormous volumes across the world. Therefore, accurate and efficient valuation is of paramount importance for global financial markets. Due to the early exercise feature, the pricing of American options is significantly more complicated than European options, and an analytical closed-form solution is unavailable even for simple dynamic models. Practitioners employ various valuation methods to strike the balance: accurate valuation usually suffers inefficiency, while fast valuation likely leads to inaccuracy. In this paper, we provide an innovative solution to address both the accuracy and efficiency issues of pricing American options by applying quantum reinforcement learning. Meanwhile, the quantum part of the new approach would potentially speed up the calculation dramatically.