This paper deals with European share option pricing using quantum physics methods. These contingent claims are usually priced using the Black-Scholes equation. This nonlinear parabolic equation is based on geometric Brownian motion model of the stock price stochastic process. Similar processes also appear among quantum particles and are described by the time-dependent Schrödinger equation. In this paper, the option pricing based on the Schrödinger equation approach is proposed. Using Wick transformation, the Black-Scholes equation is transformed into the equivalent Schrödinger equation. The Fourier separation method is used to find analytical solutions to this equation. The last square method is used to calibrate the Schrödinger model based on real market data. Numerical results are provided and discussed.
This paper deals with European share option pricing using quantum physics methods. These contingent claims are usually priced using the Black-Scholes equation. This nonlinear parabolic equation is based on geometric Brownian motion model of the stock price stochastic process. Similar processes also appear among quantum particles and are described by the time-dependent Schrödinger equation. In this paper, the option pricing based on the Schrödinger equation approach is proposed. Using Wick transformation, the Black-Scholes equation is transformed into the equivalent Schrödinger equation. The Fourier separation method is used to find analytical solutions to this equation. The last square method is used to calibrate the Schrödinger model based on real market data. Numerical results are provided and discussed.
Following system is investigated with its corresponding properties in detail: A Quantum Model Of Option Pricing: When Black–Scholes Meets Schrödinger And Its Semi-Classical Limit, Black–Scholes Option Pricing Model In A More General Quantum Physics Setting, Perfect Market Equilibrium State Postulated By The Black–Scholes Model Represents A Particular Case., Controlled Markov Processes And Mathematical Finance, Controlled Markov Processes And Mathematical Finance, Dynamic Optimization And Its Relation To Classical And Quantum Constrained Systems, Consistency Relation, Inter-Occurrence Times And Universal Laws In Finance, Earthquakes And Genomes, Quantum Cognition Beyond Hilbert Space Ii, Quantum Master Equation For Long-Term Prognosis.. The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-654991-661307.pdf
The Schrödinger equation describes how quantum states evolve according to the Hamiltonian of the system. For physical systems, we have it that the Hamiltonian must be a Hermitian operator to ensure unitary dynamics. For anti-Hermitian Hamiltonians, the Schrödinger equation instead models the evolution of quantum states in imaginary time. This process of imaginary time evolution has been used successfully to calculate the ground state of a quantum system. Although imaginary time evolution is non-unitary, the normalised dynamics of this evolution can be simulated on a quantum computer using the quantum imaginary time evolution (QITE) algorithm. In this paper, we broaden the scope of QITE by removing its restriction to anti-Hermitian Hamiltonians, which allows us to solve any partial differential equation (PDE) that is equivalent to the Schrödinger equation with an arbitrary, non-Hermitian Hamiltonian. An example of such a PDE is the famous Black-Scholes equation that models the price of financial derivatives. We will demonstrate how our generalised QITE methodology offers a feasible approach for real-world applications by using it to price various European option contracts modelled according to the Black-Scholes equation.
Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black--Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schr\odinger equation. The nonlinear approach comes in two flavors: (i) for the case of constant volatility, it is defined by a single adaptive nonlinear Schr\odinger (NLS) equation, while for the case of stochastic volatility, it is defined by an adaptive Manakov system of two coupled NLS equations. The linear quantum approach is defined in terms of de Broglie's plane waves and free-particle Schr\odinger equation. In this approach, financial variables have quantum-mechanical interpretation and satisfy the Heisenberg-type uncertainty relations. Both models are capable of successful fitting of the Black--Scholes data, as well as defining Greeks.
Keywords: Black--Scholes option pricing, adaptive nonlinear Schr\odinger equation, adaptive Manakov system, quantum-mechanical option pricing, market-heat potential
PACS: 89.65.Gh, 05.45.Yv, 03.65.Ge
The accurate valuation of financial derivatives plays a pivotal role in the finance industry. Although closed formulas for pricing are available for certain models and option types, exemplified by the European Call and Put options in the Black-Scholes Model, the use of either more complex models or more sophisticated options precludes the existence of such formulas, thereby requiring alternative approaches. The Monte Carlo simulation, an alternative approach effective in nearly all scenarios, has already been challenged by quantum computing techniques that leverage Amplitude Estimation. Despite its theoretical promise, this approach currently faces limitations due to the constraints of hardware in the Noisy Intermediate-Scale Quantum (NISQ) era. In this study, we introduce and analyze a quantum algorithm for pricing European call options across a broad spectrum of asset models. This method transforms a classical approach, which utilizes the Fast Fourier Transform (FFT), into a quantum algorithm, leveraging the efficiency of the Quantum Fourier Transform (QFT). Furthermore, we compare this novel algorithm with existing quantum algorithms for option pricing.
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.
An non-equilibrium Black-Scholes model, where the usual constant interest rate r is replaced by a stochastic time dependent rate r(t) of the form r(t)=r+f (t)W(t), accounting for market imperfections and prices non-alignment, is developed. The white noise amplitude f (t), called arbitrage bubble, generates a time dependent potential U(t) which changes the usual equilibrium dynamics of the traditional Black-Scholes model. The purpose of this article is to tackle the inverse problem, that is, is it possible to extract the time dependent potential U(t) and its associated bubble shape f (t), from the real empirical financial data? In order to give an answer to this question, the interacting Black-Scholes equation must be interpreted as a quantum Schrodinger equation with Hamiltonian operator H=H0+ U(t), where H0 is the equilibrium Black-Scholes Hamiltonian and U(t) is the interaction term. By using semi-classical considerations and the knowledge about the mispricing of the financial data, one can determinate an approximate functional form of the potential term U(t) and its associated bubble f (t), In all the studied cases, the non-equilibrium model performs a better estimation of the real data than the usual equilibrium model. It is expected that this new and simple methodology could help to improve option pricing estimations.