Smile Models

Abstract

Vanilla option prices are almost never quoted in a particular currency like pounds sterling or US dollars. Rather, prices are quoted in volatility terms. For this to work, both counterparties have first to agree on the values of the inputs to the Black—Scholes equation, namely the forward and interest rate. Then if the seller quotes a volatility of 10% for the given contract, the buyer will plug this into the Black—Scholes formula to get the price. Conversely, the Black—Scholes formula can easily be inverted numerically to obtain a volatility given a premium. The volatility one must plug into the Black—Scholes formula to get the true market price of a vanilla option is called the implied volatility.