This paper introduces the basic construct of quantum mechanics to financial researchers by applying it to a modeling of the stock prices. We present the meanings of the wavefunction, the Hamiltonian operator and the Schrödinger's equation in common mathematical terms, with an illustration of how the ever-present binary random variable can be connected to the spin operator in physics.
This paper introduces the basic construct of quantum mechanics to financial researchers by applying it to a modeling of the stock prices. We present the meanings of the wavefunction, the Hamiltonian operator and the Schrödinger's equation in common mathematical terms, with an illustration of how the ever-present binary random variable can be connected to the spin operator in physics.
Two underlying themes run through this book: first, defining and analyzing the subject of quantitative finance in the conceptual and mathematical framework of quantum theory, with special emphasis on its path-integral formulation, and, second, the introduction of the techniques and methodology of quantum field theory in the study of interest rates. No attempt is made to apply quantum theory in re-working the fundamental principles of finance. Instead, the term ‘quantum’ refers to the abstract mathematical constructs of quantum theory that include probability theory, state space, operators, Hamiltonians, commutation equations, Lagrangians, path integrals, quantized fields, bosons, fermions and so on. All these theoretical structures find natural and useful applications in finance. The path integral and Hamiltonian formulations of (random) quantum processes have been given special emphasis since they are equivalent to, as well as independent of, the formalism of stochastic calculus – which currently is one of the cornerstones of mathematical finance. The starting point for the application of path integrals and Hamiltonians in finance is in stock option pricing. Path integrals are subsequently applied to the modelling of linear and nonlinear theories of interest rates as a two-dimensional quantum field, something that is beyond the scope of stochastic calculus. Path integrals have the additional advantage of providing a framework for efficiently implementing the mathematical procedure of renormalization which is necessary in the study of nonlinear quantum field theories. The term ‘Quantum Finance’ represents the synthesis of the concepts, methods and mathematics of quantum theory, with the field of theoretical and applied finance.
The price of a given stock is exactly known only at the time of sale when the stock is between the traders. If we know the price (owner) then we have no information on the owner (price). A more general description including cases when we have partial information on both price and ownership is obtained by using the quantum mechanics methods. The relation price-ownership is similar to the relation position-momentum. Our approach is based on the mathematical formalism used in the case of quantum systems with finite-dimensional Hilbert space. The linear operator corresponding to the ownership is obtained from the linear operator corresponding to the price by using the finite Fourier transform. In our idealized model, the Schrodinger type equation describing the time evolution of the stock price is solved numerically.
A quantum operator is a classical physical variable except that it assumes values in a probability distribution, i.e., a random variable, solved by the Schrödinger equation in the form of a wavefunction. To date, the logical foundation of this construct remains an open question. This paper casts the basic components of quantum mechanics in the framework of general statistics. Our starting point is to define the domain of the wavefunction to be the complex unit circle, thus faithfully observing the periodicity of a wave. We seek to crystallize the meanings of the fundamental elements of quantum mechanics, which often are lost in its formalism, by certain common random variables in ordinary applications, such as random walks, inventory cycles, and human healthy temperatures, all defined on the sample space of a unit circle.
This book applies the mathematics and concepts of quantum mechanics and quantum field theory to the modelling of interest rates and the theory of options. Particular emphasis is placed on path integrals and Hamiltonians. Financial mathematics is dominated by stochastic calculus. The present book offers a formulation that is completely independent of that approach. As such many results emerge from the ideas developed by the author. This work will be of interest to physicists and mathematicians working in the field of finance, to quantitative analysts in banks and finance firms and to practitioners in the field of fixed income securities and foreign exchange. The book can also be used as a graduate text for courses in financial physics and financial mathematics.
The trade of a fixed stock can be regarded as the basic process that measures its momentary price. The stock price is exactly known only at the time of sale when the stock is between traders, that is, only in the case when the owner is unknown. We show that the stock price can be better described by a function indicating at any moment of time the probabilities for the possible values of price if a transaction takes place. This more general description contains partial information on the stock price, but it also contains partial information on the stock owner. By following the analogy with quantum mechanics, we assume that the time evolution of the function describing the stock price can be described by a Schrodinger type equation.
Providing a pedagogical introduction to the essential principles of path integrals and Hamiltonians, this book describes cutting-edge quantum mathematical techniques applicable to a vast range of fields, from quantum mechanics, solid state physics, statistical mechanics, quantum field theory, and superstring theory to financial modeling, polymers, biology, chemistry, and quantum finance. Eschewing use of the Schrödinger equation, the powerful and flexible combination of Hamiltonian operators and path integrals is used to study a range of different quantum and classical random systems, succinctly demonstrating the interplay between a system's path integral, state space, and Hamiltonian. With a practical emphasis on the methodological and mathematical aspects of each derivation, this is a perfect introduction to these versatile mathematical methods, suitable for researchers and graduate students in physics and engineering.