A Risk Neutral Stochastic Implied Volatility Model and Applications

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.