Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model

Details Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model

2009-11-10

arxiv.org/abs/0911.1834v1

Economy

Quantitative Finance

Pricing of Securities

26 pages, 14 figures (higher-quality figures in PDF are available upon request)

Scientific paper

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

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