Least Squares Kernel Smoothing of the Implied Volatility Smile

Functional flexibility is the cornerstone for model building and model selection in quantitative finance, for it is often difficult, if not impossible, to justify a specific parametric form of an economic relationship on theoretical grounds. Furthermore, in a dynamic context, the economic structure may be subject to changes and fluctuations. Hence, estimation techniques that do not impose a priori restrictions on the estimate, such as non- and semiparametric methods, are increasingly popular. In finance, a common challenge is to the implied volatility smile function. Based on the assumption of a geometric Brownian motion governing the stock price dynamics, an unknown volatility parameter is implied from observed option prices using the Black and Scholes (1973) formula. By theory the resulting function should be constant in strike prices and dates of maturity. Yet, as a matter of fact, one typically observes a curved and ‘smiley’ functional pattern across different strikes for a fixed maturity which is called the implied volatility smile.