  A nonlinear wave alternative for the standard Black-Scholes option-pricing
model is presented. The adaptive-wave model, representing 'controlled Brownian
behavior' of financial markets, is formally defined by adaptive nonlinear
Schr\"odinger (NLS) equations, defining the option-pricing wave function in
terms of the stock price and time. The model includes two parameters:
volatility (playing the role of dispersion frequency coefficient), which can be
either fixed or stochastic, and adaptive market potential that depends on the
interest rate. The wave function represents quantum probability amplitude,
whose absolute square is probability density function. Four types of analytical
solutions of the NLS equation are provided in terms of Jacobi elliptic
functions, all starting from de Broglie's plane-wave packet associated with the
free quantum-mechanical particle. The best agreement with the Black-Scholes
model shows the adaptive shock-wave NLS-solution, which can be efficiently
combined with adaptive solitary-wave NLS-solution. Adjustable 'weights' of the
adaptive market-heat potential are estimated using either unsupervised Hebbian
learning, or supervised Levenberg-Marquardt algorithm. In the case of
stochastic volatility, it is itself represented by the wave function, so we
come to the so-called Manakov system of two coupled NLS equations (that admits
closed-form solutions), with the common adaptive market potential, which
defines a bidirectional spatio-temporal associative memory.
  Keywords: Black-Scholes option pricing, adaptive nonlinear Schr\"odinger
equation, market heat potential, controlled stochastic volatility, adaptive
Manakov system, controlled Brownian behavior

Pricing of Securities;Computational Finance