The Option Smile

FIGURE 9.4 The Volatility Smile Curves for Bond and Equity Options  

The degree of a smiles convexity indicates how far the market price process differs from the lognormal function contained in the B-S model. The more convex the smile curve, the greater the probability the market attaches to extreme outcomes for the price of the asset on expiry, Sj. In fact, observed asset price returns do follow a distribution with fatter tails, i.e., with more occurrences at the extremes, than are found in a lognormal distribution. In addition, the direction in which the smile curve slopes reflects the skew of the price-returns function, and a curve with a positive slope indicates a function that is more positively skewed than the lognormal distribution the opposite is true for a curve with a negative slope. 

All this suggests that asset-price behavior is more accurately described by nonstandard price processes, such as the jump diffusion model or a stochastic volatility, than by a model assuming constant volatility. For more-detailed discussion of the volatility smile and its implications, interested readers may consult the works listed in the References section. 

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