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Empirical Tests of a Hedging Framework for the Quantum Binomial Options Pricing Model

Malhotra, Arnav

Published Dec 8, 2025• 0 citations
Publisher

Abstract

The Quantum Binomial Options Pricing Model is a novel approach to pricingderivatives. It serves as a quantum analogue to the classical binomial optionspricing model, using concepts from quantum mechanics like the Pauli matricesand Bloch sphere to represent the derivatives market. Through usage of a den-sity matrix and mixed states, the model can potentially price derivatives moreaccurately. However, at this time there exist no clear formulae and methodsfor hedging derivatives through this model. This paper numerically tests theperformance of the hedging framework corresponding to the model as comparedto the classical model using a year of historical S&P 500 data. The simulationreveals that the Quantum Binomial Options Pricing model is, at least in itscurrent state, not viable for usage in actually hedging derivatives. This is dueto numerical instabilities, particularly in the Quantum Theta and QuantumVega, the former of which ends up negative, betraying fundamental financialpostulates, and the latter of which is blown out of scale due to the short termnature of the simulation. We conclude that the Quantum Binomial OptionsPricing Model is currently unsuitable for practical applications and that ourfindings reveal the necessity of rigorously testing new financial models againstbenchmark models like the Classical Binomial Options Pricing Model.

DOI
10.5281/zenodo.17851073

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The Quantum Binomial Options Pricing Model is a novel approach to pricingderivatives. It serves as a quantum analogue to the classical binomial optionspricing model, using concepts from quantum mechanics like the Pauli matricesand Bloch sphere to represent the derivatives market. Through usage of a den-sity matrix and mixed states, the model can potentially price derivatives moreaccurately. However, at this time there exist no clear formulae and methodsfor hedging derivatives through this model. This paper numerically tests theperformance of the hedging framework corresponding to the model as comparedto the classical model using a year of historical S&P 500 data. The simulationreveals that the Quantum Binomial Options Pricing model is, at least in itscurrent state, not viable for usage in actually hedging derivatives. This is dueto numerical instabilities, particularly in the Quantum Theta and QuantumVega, the former of which ends up negative, betraying fundamental financialpostulates, and the latter of which is blown out of scale due to the short termnature of the simulation. We conclude that the Quantum Binomial OptionsPricing Model is currently unsuitable for practical applications and that ourfindings reveal the necessity of rigorously testing new financial models againstbenchmark models like the Classical Binomial Options Pricing Model.
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 derivatives on an underlying security. The Schrodinger and\nHeisenberg approach to the quantum mechanics together with the Dirac matrix\napproaches are applied to derive the Black-Scholes formula and the quantum Cox-\nRubinstein formula.\nThe quantum mechanics approach to option pricing is based on the interpretation\nof the option price as the Schrodinger wave function of a certain quantum mechanics\nmodel determined by Hamiltonian H. We apply this approach to continuous\ntime market models generated by Levy processes.\nIn the discrete time formulization, we construct both self-adjoint and non selfadjoint\nquantum market. Moreover, we apply the discrete time formulization\nand analyse the quantum version of the Cox-Ross-Rubinstein Binomial Model.\nWe find the limit of the N-period bond market, which convergences to planar\nBrownian motion and then we made an application to option pricing in planar\nBrownian motion compared with Levy models by Fourier techniques and Monte\nCarlo method.\nFurthermore, we analyse the quantum conditional option price and compare for\nthe conditional option pricing in the quantum formulization. Additionally, we\nestablish the limit of the spectral measures proving the convergence to the geometric\nBrownian motion model. Finally, we found Binomial Model formula and\nPath integral formulization gave are close to the Black-Scholes formula.
In a global derivatives market with notional values in the hundreds of trillions of dollars, the accuracy and efficiency of pricing models are of fundamental importance, with direct implications for risk management, capital allocation, and regulatory compliance. In this work, we employ the Black-Scholes-Merton (BSM) framework not as an end in itself, but as a controlled benchmark environment in which to rigorously assess the capabilities of quantum machine learning methods. We propose a fully quantum approach to option pricing based on Quantum Neural Networks (QNNs), and, to the best of our knowledge, present one of the first implementations of such a methodology on currently available quantum hardware. Specifically, we investigate whether QNNs, by exploiting the geometric structure of Hilbert space, can effectively approximate option pricing functions. Our implementation utilizes a compact 2-qubit QNN architecture evaluated across multiple state-of-the-art quantum processors, including IBM Fez, IQM Garnet, IonQ Forte, and Rigetti Ankaa-3. This cross-platform study reveals distinct hardware-dependent performance characteristics while demonstrating that accurate pricing approximations can be achieved consistently across different devices despite the constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The results provide empirical evidence that QNN-based approaches constitute a viable framework for derivative pricing. While the analysis is conducted within the BSM setting, the broader significance lies in the potential extension of these methods to more realistic and computationally demanding models, including local volatility, stochastic volatility, and interest rate frameworks commonly used in practice.
Pricing options quickly and accurately is a well known problem in finance. Quantum computing is being researched with the hope that quantum computers will be able to price options more efficiently than classical computers. This research extends\nthe quantum binomial option pricing model proposed by Zeqian Chen to European\nput options and to Barrier options and develops a quantum algorithm to price them.\nThis research produced three key results. First, when Maxwell-Boltzmann statistics\nare assumed, the quantum binomial model option prices are equivalent to the classical binomial model. Second, options can be priced efficiently on a quantum computer after the circuit has been built. The time complexity is O((N − τ)log(N − τ)) and it is in the BQP quantum computational complexity class. Finally, challenges extending the quantum binomial model to American, Asian and Bermudan options exist as the quantum binomial model does not take early exercise into account.
In this paper, we present a quantum version of some portions of Mathematical Finance, including theory of arbitrage, asset pricing, and optional decomposition in financial markets based on finite dimensional quantum probability spaces. As examples, the quantum model of binomial markets is studied. We show that this quantum model ceases to pose the paradox which appears in the classical model of the binomial market. Furthermore, we re-deduce the Cox-Ross-Rubinstein binomial option pricing formula by considering multi-period quantum binomial markets.
An option price was derived exactly in Baaquie (2019) using the quantum oscillator. The derivation of the oscillator option price that is based on evaluating the pricing kernel using the Feynman path integral was briefly reviewed. The oscillator option price is a generalization of the Black-Scholes option price and contains new parameters. In the current paper, the hedging parameters are exactly derived for the generalized option price and are seen to be far more complicated than the Black-Scholes hedging parameters. The parameters in the oscillator option price do not require the daily value of implied volatility for pricing the option since these parameters encode the information that is missing in the Black-Scholes option price.