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Development of a contextual non-classical (quantum physics based) theory for financial option pricing and for modeling a socio-economic system

Bart D’Hooghe, Diederik Aerts, Emmanuel E. Haven

Published Jan 1, 2011• 0 citations
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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.

Cambridge University Press eBooks•Nov 15, 2004•0 citations
We apply methods of quantum mechanics for mathe-matical modeling of price dynamics at financial mar-ket. We propose to describe behavioral financial factors (e.g., expectations of traders) by using the pilot wave (Bohmian) model of quantum mechanics. Our Bohmian model is a quantum-like model for the financial market, cf. with works of W. Segal, I. E. Segal, E. Haven. Our model expresses complexity of financial market: tra-ditional description of price dynamics is completed by Schrödinger’s dynamics for the pilot wave of expecta-tions of traders. This is a kind of socio-economic model for financial market
Classical and Quantum Physics in Selected Economic Models A growing number of economic phenomena are nowadays described with methods known in physics. The most frequently applied physical theories by economists are: (1) the universal gravitation law and (2) the first and second law of thermodynamics. Physical principles can also be applied to the theory of financial markets. Financial markets are composed of individual participants who may be seen to interact as particles in a physical system. This approach proposes a financial market model known as a minority game model in which securities and money are allocated on the basis of price fluctuations, and where selling is best option when the vast majority of investors tend to purchase goods or services, and vice versa. The players who end up being on the minority side win. The above applications of physical methods in economics are deeply rooted in classical physics. However, this paper aims to introduce the basic concepts of quantum mechanics to the process of economic phenomena modelling. Quantum mechanics is a theory describing the behaviour of microscopic objects and is grounded on the principle of wave-particle duality. It is assumed that quantum-scale objects at the same time exhibit both wave-like and particle-like properties. The key role in quantum mechanics is played by: (1) the Schrödinger equation describing the probability amplitude for the particle to be found in a given position and at a given time, and as (2) the Heisenberg uncertainty principle stating that certain pairs of physical properties cannot be economic applications of the Schrödinger equation as well as the Heisenberg uncertainty principle. We also try to describe the English auction by means the quantum mechanics methods.
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.