A Risk-Neutral Stochastic Volatility Model
Yingzi Zhu and Marco Avellaneda
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 difference in the asymptotic behavior of the probability
distribution tails between the Black Scholes log-normal probability and the
risk-neutral stochastic volatility model.
In the second part of the paper we further explore this risk-neutral
stochastic 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 different choices of parameters for the model, and determine
how the shape and the skewness of the ``smile'' curve is affected by
the volatility-of-volatility (``V-vol'') and the correlation
between the underlying asset and its volatility.