A RECENT NSE publication on the Indian capital market carries a small article on the volatility smile in options. What is volatility smile? If you plot the implied volatilities of all the strike prices of options of a particular maturity (say, the December contracts), the graph will be approximately U-shaped. On first glance, it will look like a smile. Hence, the name volatility smile.
Such a smile exists because the implied volatility of in-the-money (ITM) and out-of-the-money (OTM) options is higher than the at-the-money (ATM) options. This means that the investors are willing to pay a higher price to buy the OTM and ITM options. Why?
The explanation is based on the Black and Scholes (B&S) model. The implied volatility is the result you will get if you input the strike price, spot price, interest rate, days-to-maturity, and option premium into the model.
Thus, the implied volatility captures errors in the model, and all other factors that affect the option premium. Now, the basic assumption of the B&S model is that stock price returns follow a normal distribution (ND). That is, the stock price returns form a bell-shaped curve if plotted on a graph.
But this assumption is not true. The distribution of stock price returns has a fatter tail than an ND. This means profits and losses can be higher than what you can expect if the returns indeed follow an ND.
The implication is that the OTM options are more likely to become the ITM options, because extreme stock price movements are possible. But such extreme price movements also mean that stocks can decline sharply, which is why ITM options are also preferred. Naturally, investors will be willing to pay a higher price for these options. And the higher price translates into higher implied volatility. Hence, the volatility smile.
Hull, J. 2003. Derivatives. Yahoo press. New York.