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Adaptive Wave Models for Option Pricing Evolution: Nonlinear and Quantum Schrodinger Approaches

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 Added by Vladimir Ivancevic
 Publication date 2010
  fields Financial
and research's language is English




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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 Schrodinger equation. The nonlinear approach comes in two flavors: (i) for the case of constant volatility, it is defined by a single adaptive nonlinear Schrodinger (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 Broglies plane waves and free-particle Schrodinger 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 Schrodinger equation, adaptive Manakov system, quantum-mechanical option pricing, market-heat potential PACS: 89.65.Gh, 05.45.Yv, 03.65.Ge



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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 Schrodinger (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 Broglies 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 Schrodinger equation, market heat potential, controlled stochastic volatility, adaptive Manakov system, controlled Brownian behavior
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