A certain estimate of volatility through return for stochastic volatility models

TL;DR

This paper investigates how the conditional expectation of volatility in the Heston model depends on stock return, revealing convexity properties influenced by the initial return distribution.

Contribution

It provides a detailed analysis of the dependence of volatility on stock returns in the stochastic volatility framework, highlighting the impact of initial return distribution density.

Findings

Convex downward shape of the conditional variance near the mean return.

The effect is strongest for Gaussian initial return distribution.

The convexity diminishes as the tail decay of the distribution slows down.

Abstract

We study the dependence of volatility on the stock price in the stochastic volatility framework on the example of the Heston model. To be more specific, we consider the conditional expectation of variance (square of volatility) under fixed stock price return as a function of the return and time. The behavior of this function depends on the initial stock price return distribution density. In particular, we show that the graph of the conditional expectation of variance is convex downwards near the mean value of the stock price return. For the Gaussian distribution this effect is strong, but it weakens and becomes negligible as the decay of distribution at infinity slows down.