Volatility Smiles

‣ The implied volatility of a European option on a particularasset as a function of strike price and time to maturity isknown as the asset's volatility surface.‣ Every day traders and brokers estimate volatility surfaces fora range of different underlying assets from the market pricesof options.‣ Some points on a volatility surface for a particular asset canbe estimated directly because they correspond to activelytraded options. The rest of the volatility surface is typicallydetermined by interpolating between these points.‣ If the assumptions underlying Black-Scholes held for anasset, its volatility surface would be flat and unchanging. Inpractice the volatility surfaces for most assets are not andchange stochastically.10