On Dependence Of The Implied Volatility On Returns For Stochastic Volatility Models
The authors 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, they consider the conditional expectation of volatility under fixed stock price return as the function of the stock price return and time. The function's behavior depends on the initial stock price distribution density. In particular, they obtain the "Smile" effect 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.