Adaptive Wave Models for Option Pricing Evolution: Nonlinear and Quantum Schr\"odinger Approaches

TL;DR

This paper introduces adaptive wave models based on nonlinear and quantum Schr"odinger equations for option pricing, offering a high-complexity alternative to Black--Scholes with successful data fitting and Greek calculation.

Contribution

It develops novel adaptive wave-based models, including nonlinear and quantum approaches, for more flexible and potentially accurate option pricing beyond traditional models.

Findings

Models successfully fit Black--Scholes data

Both approaches define Greeks effectively

Quantum model interprets financial variables with uncertainty relations

Abstract

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