This brings us neatly onto the method by which this calibration of the volatility smile and skew might be done. It's common in most option markets to use what is called the SABR or sabre calibration to fit a volatility smile and skew to observable market prices. SABR uses three parameters, alpha, beta, and rho. Alpha controls the volatility of volatility component. That is the smiles and both beta and rho control the skew. Beta is often called the CEV parameter and can be varied between 0 and 1. It acts on the price process to vary it between following a normal to a log normal distribution. If beta equals 0, then the distribution is normal. If beta equals 1, then the distribution is log normal and a number between 0 and 1 produces a blend between normal and log normal. Rho is then used to further refine the skew. Because FX option modeling tends to fit quite well with the log normal approach. It would be common to see relatively high levels of beta being used with rho then being calibrated to match the skew of visible risk reversal prices. Here we see three different visual examples of different SABR settings. On the left hand side is the classic Black-Scholes assumption log, normal process, constant vol per strike. In the middle, we have a smiley volatility shape being created by increasing alpha. This vol shape would be indicative of non-zero butterfly prices, but fairly neutral risk reversals. On the right hand side, we see the effect of then increasing rho. This adds an increase to calls over puts while retaining the smileyness already present due to alpha.

By tweaking these three parameters, usually alpha and rho are the ones commonly changed. With beta being more static, a trader can calibrate their volatility, smile and skew to hit the visible market for all liquid expiry dates, and then hopefully be confident of quoting any bespoke trade which they are asked for.