Adaptive Wave Models for Sophisticated Option Pricing

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 volatility, it is defined by a single adaptive nonlinear Schrödinger (NLS) equation, while for the case of stochastic volatility, it is defined by an adaptive Manakov system of two coupled NLS equations. The linear quantum approach is defined in terms of de Broglie’s plane waves and free-particle Schrödinger equation. In this approach, financial variables have quantum-mechanical interpretation and satisfy the Heisenberg-type uncertainty relations. Both models are capable of successful fitting of the Black-Scholes data, as well as defining Greeks.