Exact Solutions for a GBM-type Stochastic Volatility Model having a Stationary Distribution

TL;DR

This paper derives exact solutions for a novel stochastic volatility model with GBM-type noise, including transition densities and option prices, and explores its stationary distribution, offering new analytical tools for financial modeling.

Contribution

It introduces the first stochastic volatility model with exact solutions, GBM-type volatility noise, and a stationary distribution, advancing analytical methods in financial mathematics.

Findings

Derived exact transition probability densities.

Computed European option values explicitly.

Identified conditions for the existence of a stationary distribution.

Abstract

We find various exact solutions for a new stochastic volatility (SV) model: the transition probability density, European-style option values, and (when it exists) the martingale defect. This may represent the first example of an SV model combining exact solutions, GBM-type volatility noise, and a stationary volatility density.