Implied Volatility Smirk

This paper studies implied volatility smirk quantitatively. We first propose a new concept of smirkness, which is defined as a triplet of at-the-money implied volatility, skewness (slope at the money) and smileness (curvature at the money) of implied volatility – moneyness curve. The moneyness is the logarithm of the strike price over the forward price, normalized by the standard deviation of expected return on maturity. Empirical evidence from S&P 500 (SPX) index options shows that a quadratic function with both skewness and smileness fits the market implied volatility smirk very well. The volume weighted error can be smaller than the smallest bid-ask spread of traded options. The risk-neutral probability density function can be recovered analytically from a “smirked” implied volatility. An analytical relationship between the implied volatility smirkness and the risk-neutral cumulants has been derived. With the quantity of smirkness well-defined, one may study the term structure and time-change dynamics of implied volatility smirk. The time series of the smirkness from September 1998 to September 1999 for SPX options are shown to be stationary. A new maturityand liquidity-based procedure is proposed to calibrate option pricing models. Two calibration exercises on the constant elasticity of variance (CEV) model and finite moment log stable (FMLS) process model show that the term structures of smirkness extracted from market data provide valuable information for us to calibrate option pricing models. The two exercises also show that the FMLS model performs much better than the CEV model in capturing the large negative skewness in the risk-neutral distribution for options with a short maturity.