A Risk-Neutral Stochastic Volatility Model

We construct a risk-neutral stochastic volatility model using no-arbitrage pricing principles. We then study the behavior of the implied volatility of options that are deep in and out of the money according to this model. The motivation of this study is to show the diierence in the asymptotic behavior of the distribution tails between the usual Black-Scholes log-normal distribution and the risk-neutral stochastic volatility distribution. In the second part of the paper, we further explore this risk-neutral stochas-tic volatility model by a Monte-Carlo study on the implied volatility curve (implied volatility as a function of the option strikes) for near-the-money options. We study the behavior of this \smile" curve under diierent choices of parameter for the model, and determine how the shape and skewness of the \smile" curve is aaected by the volatility of volatility (\V-vol") and the correlation between the underlying asset and its volatility.