# parameters in Heston model and their impact on volatility smile

Consider the Heston model given by the following set of stochastic differential equations: $$\frac{dS_{t}}{S_{t}}=\mu_{t}dt+\sqrt{V_{t}}dW_{t}, S_{0}>0,$$ $$dV_{t}=\kappa(\theta-V_{t})dt+\xi\sqrt{V_{t}}dZ_{t}, V_{0}=v_{0}>0,$$ $$d<W,Z>_{t}=\rho dt$$ where $W_{t}$ and $Z_{t}$ are two brownian motion, $\kappa,\theta,\xi>0, \rho\in(-1,1).$ I don't understand what impact would $\rho$ have on the shape of volatility smile when it's negative or positive. In addition, How the volatility smile would change if $\xi$ increases? Thank you.

• Thanks. However, I don't know how to prove rigorously why negative correlation will produce a downward sloping skew and the opposite for positive $\rho$, Is there an intuitive explanation? In terms of $\xi$, I'm also wondering how to explain why the price of the at-the-money option would decrease while the price of the in-the-money or out-of-the-money option would increase when $\xi$ becomes larger. – cmd1991 May 10 '15 at 2:12