On dependence of volatility on 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 stochastic variance (square of volatility) given stock price log-return f as a function of f and time t. The behaviour of this function depends on the initial stock price log-return distribution density. In particular, we show that the graph of the conditional expectation of the stochastic variance is convex downwards near a special value of the stock price log-return . For the Gaussian distribution, this effect is strong, but it weakens and becomes negligible as the decay of distribution at infinity slows down.

Keywords:

• stochastic volatility
• the Heston model
• conditional expectation of stochastic variance

Keywords:

• 91B24
• 60H10

Acknowledgements

The authors thank heartily anonymous referees for their suggestions which helped to improve the paper significantly. This study was supported by the Ministry of Education of the Russian Federation, project 2.1.1/1399.