Pricing and Hedging Financial Derivatives: A Guide for Practitioners

7

Volatility Smile and the Greeks of Option Strategies

As we already mentioned in Chapter 6, “Implied Volatility”, the Black–Scholes formula is very often used “the other way around”, i.e., to derive (or back out) the level of volatility that, when used in the formula as an input, allows one to obtain the market price of the option. As we have seen extensively, this procedure allows market participants to retrieve what is commonly known as implied volatility. In this chapter, we want to move one step forward, calculating implied volatility, not just on a specific strike and expiry, but across different strikes and across different option maturities. This is extremely important, as there is no guarantee that the level of implied volatility is going to be the same across different strikes and/or different maturities.

7.1 THE VOLATILITY SMILE – WHY IS THE IMPLIED VOLATILITY NOT FLAT ACROSS DIFFERENT STRIKES?

Let us start from the analysis of volatility across different strikes. This will lead us to an observation of the so-called volatility smile.

In the following section, we plot the volatility smile (or skew) for some unspecified underlying asset and discuss the typical shapes that are generally observed. Figures 7.1 and 7.2 are based on an at-the-money level of 100; typically these charts are based on volatility for at-the-money or out-of-the-money options, for a given expiry: therefore, on the right-hand side it is possible to see the volatility of out-of-the-money call options ...