  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 Schr\"odinger equation. The nonlinear approach comes in two flavors: (i)
for the case of constant volatility, it is defined by a single adaptive
nonlinear Schr\"odinger (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 Schr\"odinger 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.
  Keywords: Black--Scholes option pricing, adaptive nonlinear Schr\"odinger
equation, adaptive Manakov system, quantum-mechanical option pricing,
market-heat potential
  PACS: 89.65.Gh, 05.45.Yv, 03.65.Ge
