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Adaptive wave model for financial option pricing is proposed, as ahigh-complexity alternative to the standard Black--Scholes model. The newoption-pricing model, representing a controlled Brownian motion, includes twowave-type approaches: nonlinear and quantum, both based on (adaptive form of)the Schr\"odinger equation. The nonlinear approach comes in two flavors: (i)for the case of constant volatility, it is defined by a single adaptivenonlinear Schr\"odinger (NLS) equation, while for the case of stochasticvolatility, it is defined by an adaptive Manakov system of two coupled NLSequations. The linear quantum approach is defined in terms of de Broglie'splane waves and free-particle Schr\"odinger equation. In this approach,financial variables have quantum-mechanical interpretation and satisfy theHeisenberg-type uncertainty relations. Both models are capable of successfulfitting of the Black--Scholes data, as well as defining Greeks.  Keywords: Black--Scholes option pricing, adaptive nonlinear Schr\"odingerequation, adaptive Manakov system, quantum-mechanical option pricing,market-heat potential  PACS: 89.65.Gh, 05.45.Yv, 03.65.Ge

Adaptive Wave Models for Option Pricing Evolution: Nonlinear and Quantum Schr\"odinger Approaches