Volatility Smiles

‣ The volatility is relatively low for at-the-money options andgets progressively higher as an option moves either into orout of the money.‣ We gain some analytical insight into why this occurs if wecompare the implied volatility distribution with thelognormal one with the same mean and standard deviation.‣ Consider a deep out-of-the-money call with strike priceabove K 2 . This derivative will only pay off if the exchangerate closes above K 2 , and according to the above figure theprobability of this happening is higher for the implieddistribution than the lognormal one. A higher probabilitywill generate a higher price, which in turn means a higherimplied volatility.‣ The same is true of a deep out-of-the-money put with strikeprice below K 1 .3