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arXiv:1001.0615

Adaptive Wave Models for Option Pricing Evolution: Nonlinear and Quantum Schr\

Vladimir G. Ivancevic

Published Jan 5, 2010• 0 citations
arXivHTMLPDF

Abstract

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

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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
We propose a new cognitive framework for option price modelling, using quantum neural computation formalism. Briefly, when we apply a classical nonlinear neural-network learning to a linear quantum Schrödinger equation, as a result we get a nonlinear Schrödinger equation (NLS), performing as a quantum stochastic filter. In this paper, we present a bidirectional quantum associative memory model for the Black--Scholes--like option price evolution, consisting of a pair of coupled NLS equations, one governing the stochastic volatility and the other governing the option price, both self-organizing in an adaptive `market heat potential', trained by continuous Hebbian learning. This stiff pair of NLS equations is numerically solved using the method of lines with adaptive step-size integrator. Keywords: Option price modelling, Quantum neural computation, nonlinear Schrödinger equations, leverage effect, bidirectional associative memory
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
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.
We define three uncertainty regimes. The first regime, we call the uncertainty regime. This is the regime used in the Black-Scholes model. The non-classical uncertainty regime is defined either in terms of so called or in terms of Intuitively, both types of non-classical uncertainty are augmented versions of classical For instance, with Heisenberg uncertainty it can be shown that the 'Reduction of Compound Lotteries' axiom in the von Neumann-Morgenstern expected utility model fails. e -Uncertainty can be shown to be a mixture of Heisenberg Uncertainty and classical We consider two models under uncertainty. The first model introduces the relationship between the SchrA¶dinger equation and a particular Brownian motion. Using Black-Scholes methodology, we find that although the portfolio change does not involve a Wiener process we can not say that the portfolios return is the risk-free rate. In the second model we augment the Brownian motion of the first model and we use again Black-Scholes methodology to derive an option price. The portfolio return is again not risk free and we obtain stochastic call and put prices. Finally, we provide for an appreciation of an option pricing model under so called e-Uncertainty. d Finance 2004
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.
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.