Implied volatility

The Black-Scholes model is often criticized because of some shortcomings. One important problem is that the model assumes constant volatility for the underlying asset, which does not hold in reality. Furthermore, since it is not observable directly, the volatility is the most complicated parameter of the model to calibrate. Due to this difficulty, the Black-Scholes formula is often used in an indirect way for estimating the volatility parameter; we observe the market price of an option, then in view of all the other parameters we can search for σ that results a Black-Scholes price equal to the observed market price. This σ parameter is called the implied volatility of the option. As Riccardo Rebonato famously stated, implied volatility is "the wrong number to put in the wrong formula to get the right price" (Rebonato, 1999, p.78).

We will illustrate the calculation of implied volatility with the help of some Google options. The options are call options with the maturity...