Volatility in the black-scholes and other formulas

This thesis examines the compatibility between the Black-Scholes formula and stock price models with non-constant implied volatility. Our implied volatility is assumed to be a (possibly random) function of time t. Our main result shows that if the price of a call option is given by the Black-Scholes formula for finitely many strike prices, then the implied volatility is not necessarily a constant but will approach a constant if the number of strike prices increases. Moreover, our results provide us with sets of constraints limiting the acceptable values of the implied volatility parameters. We show that the more maturities we have, the more refined our constraints on the implied volatility would be. Since we do not place any assumptions on the underlying stock price process, the implied volatility process or how they are related, our results are model-free. In addition, we extend our investigation on the compatibility issue by using a more general formula than the Black-Scholes for our implied volatility. Under this more general framework, we obtain the same conclusion, namely, that implied volatility is not necessarily a constant but will approach a constant if the number of strike prices increases. We show this for the cases of three maturities and multiple maturities.