Maximum likelihood approach for several stochastic volatility models

TL;DR

This paper introduces a maximum likelihood method for estimating unobservable volatility in stochastic volatility models, demonstrating its effectiveness across different models and market indexes, with some predictive capabilities.

Contribution

The paper develops and applies a maximum likelihood approach to estimate volatility in stochastic models, capturing key stylized facts and showing good empirical performance.

Findings

Method accurately estimates volatility in various models.

Approach captures stylized facts like auto-correlation and asymmetry.

Good performance across multiple market indexes.

Abstract

Volatility measures the amplitude of price fluctuations. Despite it is one of the most important quantities in finance, volatility is not directly observable. Here we apply a maximum likelihood method which assumes that price and volatility follow a two-dimensional diffusion process where volatility is the stochastic diffusion coefficient of the log-price dynamics. We apply this method to the simplest versions of the expOU, the OU and the Heston stochastic volatility models and we study their performance in terms of the log-price probability, the volatility probability, and its Mean First-Passage Time. The approach has some predictive power on the future returns amplitude by only knowing current volatility. The assumed models do not consider long-range volatility auto-correlation and the asymmetric return-volatility cross-correlation but the method still arises very naturally these two important stylized facts. We apply the method to different market indexes and with a good performance in all cases.