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

TL;DR

This paper introduces an adaptive nonlinear wave model based on Schrödinger equations to improve Black-Scholes option pricing, capturing market dynamics and stochastic volatility through quantum-inspired probability amplitudes.

Contribution

It develops a novel adaptive wave framework using nonlinear Schrödinger equations for more accurate and flexible option pricing models compared to traditional methods.

Findings

Adaptive shock-wave solutions align well with Black-Scholes predictions.

The model incorporates stochastic volatility via coupled NLS equations.

Analytical solutions are derived using Jacobi elliptic functions.

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 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