No abstract

This paper deals with numerical option pricing methods based on a Schrödinger model rather than the Black-Scholes model. Nonlinear Schrödinger boundary value problems seem to be alternatives to linear models which better re ect the complexity and behavior of real markets. Therefore, based on the nonlinear Schrödinger option pricing model proposed in the literature, in this paper a model augmented by external atomic potentials is proposed and numerically tested. In terms of statistical physics the developed model describes the option in analogy to a pair of two identical quantum particles occupying the same state. The proposed model is used to price European call options on a stock index. the model is calibrated using the Levenberg-Marquardt algorithm based on market data. A Runge-Kutta method is used to solve the discretized boundary value problem numerically. Numerical results are provided and discussed. It seems that our proposal more accurately models phenomena observed in the real market than do linear models.

Option pricing models are formulated based on mathematical theories. They are applied to estimate the fair value of an option. Among different pricing models, the linear Black-Scholes equation is very frequently used as option pricing model. Since assumptions of this linear model do not match precisely the real market conditions and do not allow to estimate the option price precisely there are developed more complicated non-linear Black-Scholes option pricing models. In these models volatility of underlying asset price or market risk free interest rate are assumed stochastic or time dependent. Moreover transaction costs are also taken into account. Quantum mechanics provides other approach to calculate option values using non-linear models. This approach is based on the similarity between the evolution of elementary particles in space and the volatility of the stock prices. In previous paper the authors have proposed non-linear Black-Scholes model to calculate option price based on quantum dynamics approach. This model has been obtained by suitable transformation of variables in non-linear Schrödinger equation with the external potential term. In this paper the non-linear quantum based option pricing model is numerically tested and verified. Based on this model, the calculation of European, Asian and American call option prices for market data is provided. The model parameters, especially the adaptive market potential, have been estimated based on market prices of European call options listed on Warsaw Stock Exchange as well as American call options prices based on the selected NYSE stock prices. The sensitivity of this model with respect to risk free interest rate has been investigated. For the sake of comparison non-linear Heston option pricing model has been also solved and calibrated using Monte Carlo method. The comparison of option pricing using the developed quantum based model, linear Black-Scholes model, Heston model, indicates higher precision and lower computational costs of the proposed enhanced non-linear model.