Volatility Surface and Martingale Measures

A volatility surface is given as a function of maturity and strike. Data provider collect the price for liquid vanilla options and do invert them with Black formula or, when it comes to interest rate options, with the equivalent equation for a log-normal shifted model. Volatility surfaces, suitably interpolated and extended, are then used to compute values of other options or assets with embedded optionality. If we look at a cross section of the volaility surface for fixed expiration date we have the so called volatility smile, that is the dependence of the implied volatility on the option strike.

In this note we look at the dependence of the volatility smile upon the option strike and under a rather general assumpion that prices are produced by a martingale measure we derive rather stringent constraints for the implied smile, we show that non arbitrage conditions require a scaling law known as ”sticky forward moneyness”