# Realized volatility and parametric estimation of Heston SDEs

## TL;DR

This paper develops and analyzes observable moment-based estimators for Heston stochastic differential equations, providing explicit convergence rates and practical guidelines for parameter estimation from stock price data.

## Contribution

The paper introduces new observable moment estimators for Heston SDE parameters, with explicit convergence bounds and practical recommendations for data sampling.

## Key findings

- Explicit $L^q$ convergence rates for realized volatilities.
- Guidelines for selecting sampling frequency and window size.
- Numerical simulations demonstrating estimator performance.

## Abstract

We present a detailed analysis of \emph{observable} moments based parameter estimators for the Heston SDEs jointly driving the rate of returns $R_t$ and the squared volatilities $V_t$. Since volatilities are not directly observable, our parameter estimators are constructed from empirical moments of realized volatilities $Y_t$, which are of course observable. Realized volatilities are computed over sliding windows of size $\varepsilon$, partitioned into $J(\varepsilon)$ intervals. We establish criteria for the joint selection of $J(\varepsilon)$ and of the sub-sampling frequency of return rates data. We obtain explicit bounds for the $L^q$ speed of convergence of realized volatilities to true volatilities as $\varepsilon \to 0$. In turn, these bounds provide also $L^q$ speeds of convergence of our observable estimators for the parameters of the Heston volatility SDE. Our theoretical analysis is supplemented by extensive numerical simulations of joint Heston SDEs to investigate the actual performances of our moments based parameter estimators. Our results provide practical guidelines for adequately fitting Heston SDEs parameters to observed stock prices series.