A Comparative Study of Stochastic Volatility Models

TL;DR

This paper compares three stochastic volatility models—Vasicek, Heston, and exponential Ornstein-Uhlenbeck—evaluating their theoretical properties, numerical performance, and empirical effectiveness in modeling financial market data.

Contribution

It provides a comprehensive comparison of these models' theoretical features, numerical simulations, and empirical validation using Italian stock market data.

Findings

All models can be implemented with their features intact.

Models effectively reproduce key statistical properties of prices.

Exponential Ornstein-Uhlenbeck model best captures empirical stylized facts.

Abstract

The correlated stochastic volatility models constitute a natural extension of the Black and Scholes-Merton framework: here the volatility is not a constant, but a stochastic process correlated with the price log-return one. At present, several stochastic volatility models are discussed in the literature, differing in the dynamics attached to the volatility. The aim of the present work is to compare the most recent results about three popular models: the Vasicek, Heston and exponential Ornstein-Uhlenbeck models. We analyzed for each of them the theoretical results known in the literature (volatility and return distribution, higher-order moments and different-time correlations) in order to test their predictive effectiveness on the outcomes of original numerical simulations, paying particular attention to their ability to reproduce empirical statistical properties of prices. The numerical results demonstrate that these models can be implemented maintaining all their features, especially in view of financial applications like market risk management or option pricing. In order to critically compare the models, we also perform an empirical analysis of financial time series from the Italian stock market, showing the exponential Ornstein-Uhlenbeck model's ability to capture the stylized facts of volatility and log-return probability distributions.