There have been many studies of the shape of volatility skew. There are two simple empirical approaches that have proven most popular:
- Spline (or piecewise polynomial) fits of observable implied vols
- Parabolic parametric fits.
and there is one other approach that is fairly simple yet reliable:
Other empirical approaches also exist, of course, but quickly become unwieldy mathematically, computationally, and for calibration.
The overall process is as follows:
Choose a function, such as a parabola, that you will use to represent skew and code it into the computer
Choose some input volatilities you trust
Use them in a fitting scheme to calibrate the parameters (tilt and curvature) of your skew. That is, you find parameters such that your chosen function does the best possible job of reproducing your input volatilities.
Use your fitted function to estimate volatilities anywhere on the skew curve.
Sometimes, if data is particularly bad, you can copy tilt and curvature parameters from one fitted curve onto another curve where you have very little input data available. Exercise caution when you do this!
The fits are typically constrained such that the parabola (or other function) cannot dip below zero for any "reasonable" strike. This is hard to achieve with linear fits, which is one major reason why parabolas are more popular. Fits also often have "cutoffs" where volatility is assumed to be constant for strikes above (or below) certain extreme values.
For all such fits, one usually uses existing mathematical libraries and routines (many of which are simple copies of the serviceable code found in Numerical Recipes). You can get spline libraries, and you can fit parabolas using any linear algebra library that has a routine for ordinary least squares. See this site for more information.
Edgeworth expansions are in the family of skew treatments that consider skew as a symptom of non-lognormal returns, and try to represent a non-lognormal distribution in some concise way. A decent place to start learning about them is Rubinstein's paper at this location.
The real trick for any of these fits is choosing a weighting scheme. Typically, you will prefer to weight some input volatilities more than others. For example, most people believe the out-of-the-money put and call prices contain the best information. Others concentrate on weighting the near-the-money options, often by weighting according to the gamma.