A discussion on embedding the Black–Scholes option pricing model in a quantum physics setting

Nonlinear Schrödinger approach to European option pricing

“…The existence results for di erent types of linear Schrödinger equations can be found in book [22]. Stock options pricing models based on linear Schrödinger equations and their relation to Black-Scholes models are reported in many papers [23][24][25][26][27][28][29]. Among others in the author's previous paper [29], the European call option price based on the linear Schrödinger equation has been calculated.…”

Application of Bohmian Mechanics to Dynamics of Prices of Shares: Stochastic Model of Bohm–Vigier from Properties of Price Trajectories

“…[9][10][11][12][13]. In quantum mechanics (which deals with microscopic objects) h is the Planck constant.…”

“…Here we consider a "classical" (time dependent) force f (t, q) = − ∂V (t,q) ∂q and "quantum" force g(t, q) = − ∂U(t,q) ∂q , where U(t, q) is the quantum potential, induced by the Schrödinger dynamics. In Bohmian mechanics for physical systems (9) is considered as an ordinary differential equation and q(t) as the unique solution (corresponding to the initial conditions q(t 0 ) = q 0 , q (t 0 ) = q 0 ) of the class C 2 : q(t) is assumed to be twice differentiable with continuous q (t). In contrast to it, in financial mathematics it is commonly assumed that the price-trajectory is not differentiable [20,22].…”

“…We remark also that there were performed investigations on the application of quantum methods to the financial market [9][10][11][12]21] that were not directly coupled to behavioral modeling, but based on the general concept that randomness of the financial market can be better described by quantum mechanics.…”

Path integrals in fluctuating markets

“…Alternatively a modified Ising model to study stochastic resonance and model financial crashes 4 has been used. Stochastic Differential Equations (SDE) have also been exploited in the evaluation of option pricing, [5][6][7] and has been found to be successful in developing a theory of non-Gaussian option pricing which allows for closed form solutions for European options, which are such that can be exercised exclusively on a fixed day of expiration and not before (as in the case of American options), 8,9 their approach uses stochastic processes with statistical feedback 10 as a model for stock prices. Such processes were developed within the Tsallis generalized thermostatistics.…”