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

Abstract

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 Schrödinger (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 NLSsolution, 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 LevenbergMarquardt 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.

Polarization modulation instability in a Manakov fiber system

“…For instance, consider a system of coupled nonlinear Schrödinger equations (NLSEs) to describe the nonlinear interaction between wave packets in dispersive conservative media. Such coupled systems are of physical relevance in various domains such as nonlinear optics, hydrodynamics, plasma physics, multicomponent Bose-Einstein condensates, and financial systems [1][2][3][4][5]. The first multicomponent NLSE type of model with applications to physics is the well-known Manakov model [6].…”

Adaptive Wave Models for Sophisticated Option Pricing

“…In addition, for each of the call and put options, there are five Greeks (see, e.g. [9,10]), or sensitivities, which are partial derivatives of the option-price with respect to stock price (Delta), interest rate (Rho), volatility (Vega), elapsed time since entering into the option (Theta), and the second partial derivative of the option-price with respect to the stock price (Gamma).…”

“…Alternatively, we can obtain the same PDF (for the market value of a stock option), using the quantum-probability formalism [11,12], as a solution to a time-dependent linear or nonlinear Schrödinger equation for the evolution of the complex-valued wave  -function for which the absolute square, 2 ,  is the PDF. The adaptive nonlinear Schrödinger (NLS) equation was recently used in [10] as an approach to option price modelling, as briefly reviewed in this section. The new model, philosophically founded on adaptive markets hypothesis [13,14] and Elliott wave market theory [15,16], as well as my own recent work on quantum congition [17,18], describes adaptively controlled Brownian market behavior.…”

“…The adaptive NLS-PDFs of the shock-wave type (10) has been used in [10] to fit the Black-Scholes call and put options (see Figures 1 and 2). Specifically, the adaptive heat potential (6) was combined with the spatial part of (10)…”

“…The adaptive NLS-based Greeks (Delta, Rho, Vega, Theta and Gamma) have been defined in [10], as partial derivatives of the shock-wave solution (10).…”