A certain estimate of volatility through return for stochastic volatility models

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.