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Annalen der Physik

Information in asset pricing: a wave function approach

Hiromu Ishio, Emmanuel E. Haven

Published Jan 21, 2009• 2 citations
Publisher

Abstract

Abstract This paper introduces a quantum‐like wave function as an information wave function. We show how the option pricing partial differential equation can be re‐written when we account for such information wave function. We use two stochastic differential equations, one of which relates to Nelson's hypothesis of Universal Brownian motion. We also provide for two examples which further highlight the proposed theory.

DOI
10.1002/andp.20095210106

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Abstract This paper introduces a quantum‐like wave function as an information wave function. We show how the option pricing partial differential equation can be re‐written when we account for such information wave function. We use two stochastic differential equations, one of which relates to Nelson's hypothesis of Universal Brownian motion. We also provide for two examples which further highlight the proposed theory.
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
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.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.
In this chapter the concept of the Hamiltonian is introduced in the pricing of options. Hamiltonians occur naturally in finance; to demonstrate this the analysis of the Black–Scholes equation is recast in the formalism of quantum mechanics. It is then shown how the Hamiltonian plays a central role in the general theory of option pricing. The Hamiltonian formulation provides new tools for obtaining solutions for option pricing; two key concepts related to the Hamiltonian are (a) eigenfunctions and (b) potentials. Knowledge of all the eigenfunctions of a Hamiltonian yields an exact solution for a large class of path-dependent and path-independent options. For example, barrier options can be modelled by placing constraints on the eigenfunctions of the Hamiltonian. The potentials are a means for defining new financial instruments, and for modelling path-dependent options. Essentials of quantum mechanics It is shown in this chapter that option pricing in finance has a mathematical description that is identical to a quantum system; hence the key features of quantum theory are briefly reviewed. Quantum theory is a vast subject that forms the bedrock of contemporary physics, chemistry and biology. Only those aspects of quantum mechanics are reviewed that are relevant for the analysis of option pricing. In classical mechanics the position of a particle at time t , denoted by x t , is a deterministic function of t , and is given by Newton's law of motion. Classical mechanics is analogous to the case of the evolution of a stock price with zero volatility (σ = 0) that yields a deterministic evolution of the stock price.
ABSTRACT Quantum computing has attracted increasing attention as a potential tool for addressing computational challenges in quantitative finance. In particular, quantum amplitude estimation provides a theoretical quadratic speedup for expectation estimation compared with classical Monte Carlo simulation. However, most existing studies primarily emphasize algorithmic acceleration while retaining classical stochastic models for asset price dynamics. In this paper, we propose a unified quantum framework for European option pricing by formulating asset price evolution as a quantum continuous–time stochastic process (QCTSP) and integrating it with quantum amplitude estimation. By encoding both the stochastic dynamics of asset prices and the option payoff structure into a coherent quantum computational representation, the proposed approach enables option valuation to be performed entirely within a quantum probabilistic framework. Numerical simulations under the Black–Scholes setting are conducted to verify the feasibility and consistency of the method. Although the experiments are limited to a benchmark model, the proposed framework provides a systematic foundation for extending quantum option pricing to higher–dimensional and more complex financial models.