# Stochastic and Statistical Analysis of Cnoidal, Snoidal, Dnoidal, Hyperbolic, Trigonometric and Exponential Wave Solutions of a Coupled Volatility Option-Pricing System

## TL;DR

This paper explores wave solutions in a financial model that combines market behavior and volatility using stochastic and statistical methods.

## Contribution

A new methodology is introduced to derive and analyze wave solutions in a coupled volatility option-pricing system with stochastic effects.

## Key findings

- The system's solutions include cnoidal, snoidal, and exponential wave forms with distinct statistical properties.
- Stochastic volatility influences the amplitude and phase dynamics of the solutions, showing non-stationary behavior.
- Phase velocity cross-correlation and amplitude cross-correlation reveal coupling dynamics between price and volatility components.

## Abstract

We investigate a stochastic coupled nonlinear Schrödinger (Manakov-type) system for option price and volatility wave fields within the Ivancevic adaptive-wave option-pricing paradigm, and derive exact wave families together with statistical diagnostics of the resulting dynamics. This system combines behavioral market effects with classical efficient-market dynamics and incorporates a controlled stochastic volatility component. Randomness in both the option price and volatility is incorporated via white noise, and a system of stochastic partial differential equations (PDEs) is developed that governs the joint evolution of option prices and stock price volatility. We derive advanced solutions of the proposed system using a newly created methodology. The obtained solutions are expressions of cnoidal, snoidal, dnoidal, hyperbolic, trigonometric, and exponential functions. The stochastic dynamical investigation, together with the statistical measures are presented. The autocorrelation function (ACF) of squared returns for the obtained analytical solutions is demonstrated to show distinct differences in second-order temporal dependence, while asymmetries in the temporal evolution of the fluctuations are depicted via leverage correlation (LC). The probability distribution function (PDF) dynamics of the soliton solutions illustrate prominent temporal variability and non-stationary statistical dynamics. Differences in dynamical coupling between the two components of the considered system are presented via phase velocity cross-correlation analysis and are supported by phase difference dynamics visualizations. The strength and structure of coupling between components are displayed via the amplitude cross-correlation function. Mean amplitude dynamics and variance as a function of noise intensity σ, provide a systematic influence of stochastic forcing on their energy and a quantitative measure of stochastic dispersion of soliton solutions. All the results are displayed in 3D and 2D graphs of the stochastics and statistical dynamics of the obtained solutions.