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
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
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
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
Abstract: The limitations of the classical Black-Scholes model are examined by comparing calculated and actual historical prices of European call options on stocks from several sectors of the S &P 500. Persistent differences between the two prices point to an expanded model proposed by Segal and Segal (PNAS 95:4072-4075, 1988) in which information not simultaneously observable or actionable with public information can be represented by an additional pseudo-Wiener process. A real linear combination of the original and added processes leads to a commutation relation analogous to that between a boson field and its canonical momentum in quantum field theory. The resulting pricing formula for a European call option replaces the classical volatility with the norm of a complex quantity, whose imaginary part is shown to compensate for the disparity between prices obtained from the classical Black-Scholes model and actual prices of the test call options. This provides market evidence for the influence of a non-classical process on the price of a security based on non-commuting operators. Supplementary Information: The online version contains supplementary material available at 10.1140/epjb/s10051-022-00402-0.
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.
A natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects. The lack of simultaneous observability of relevant variables and the interference of attempted observation with the values of these variables represent such effects. These characteristics have been noted by traders and economists and appear intrinsic to market dynamics. This explanation is explored here in terms of a corresponding generalization of the Wiener process and its role in the Black-Scholes-Merton theory. The differentiability of the Wiener process as a sesquilinear form on a dense domain in the Hilbert space of square-integrable functions over Wiener space is shown and is extended to the quantum context. This provides a basis for a corresponding generalization of the Ito theory of stochastic integration. An extension of the Black-Scholes option pricing formula to the quantum context is deduced.
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.