The dynamics of smile surface lead practitioner and researcher to
introduce the randomness in the implied volatility, which is are
specific in option markets. The monograph develops a risk-neutral
stochastic At- the-Money implied volatility model and its applications.
Three characteristics of implied volatility are presented. After the
proper model setup, the risk-neutral drift term of stochastic implied
volatility is derived, which is necessary to be no-arbitrage. We proved
that the implied volatility of At-the-Money options mature immediately
should converge to underlying volatility at the rate of time to
maturity, which specifies the stochastic process of underlying
volatility. Monte Carlo simulation is used to simulate the complex whole
system. Skew curve and terminal underlying price distribution are
studied. The two model parameters are able to explain market skew
phenomena quite well. Barrier option is priced and future implied
volatility is forecast off the simulation. The monograph should be
helpful for option traders, and should be especially useful for graduate
students and researcher in financial math field.