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 Schrödinger equation. The nonlinear approach comes in two flavors: for the case of constant volat...

We compare two methods for superreplication of options with convex pay-off functions. One method entails an overestimation of the unknown covariance matrix in the sense of quadratic forms. With this method the value of the superreplicating portfolio is given as the solution of a linear Black-Scholes type equation. In the second method the choice of quadratic form is made pointwise. This leads t...

This is the first of two papers in which we consider a stock with price process defined by a stochastic differential equation driven by a process Y (·) different from Brownian motion. The adoption of such a colored noise input is motivated by an analysis of real market data. The process Y (·) is defined by a continuous-time AR(∞)-type equation and may have either short or long memory. We show t...

A master equation approach to the numerical solution of option pricing models is developed. The basic idea of the approach is to consider the Black–Scholes equation as the macroscopic equation of an underlying mesoscopic stochastic option price variable. The dynamics of the latter is constructed and formulated in terms of a master equation. The numerical efficiency of the approach is demonstrat...

In this paper we derive an effective equation for derivative pricing which accounts for the presence of virtual arbitrage opportunities and their elimination by the market. We model the arbitrage return by a stochastic process and find an equation for the average derivative price. This is an integro-differential equation which, in the absence of the virtual arbitrage or for an infinitely fast m...

In this work, we apply He’s variotional iteration method for obtaining analytic solutions to nonlinear Black-Scholes equation with boundary conditions for European option pricing problem. The analytical solution of the equation is calculated in the form a convergent power series with easily computable components. The powerful VIM method is capable of handling both linear and non-linear equation...