Why the Volatility Smile Exists in Equity Index Options

Equity index skew comes from three additive causes, the leverage effect, priced crash risk, and compensation for a fully correlated selloff.

The volatility smile exists because option prices do not assume the underlying’s returns are normally distributed, and the implied volatility that reconciles a model built on that normal assumption with real market prices ends up different at every strike. On equity index options the smile is asymmetric, better described as a skew, with downside strikes priced at higher implied volatility than upside strikes of equal distance from spot.

The Black-Scholes model assumes the underlying follows a lognormal process with one constant volatility, and if that assumption held exactly, every strike at a given expiry would trade at the identical implied volatility, a flat line across the strike axis. Real option prices do not produce a flat line, so the implied volatility read off each strike is really a translation of the market’s actual risk-neutral distribution, which has a fatter left tail and more mass at extreme moves than a lognormal, into the units of a model that assumes neither. The full mechanics of reading that shape off a live surface are worked through in reading the skew across strikes.

One purely mechanical explanation is the leverage effect. When an equity index falls, the market value of the equity underneath each constituent falls while its debt stays roughly fixed, which raises the ratio of debt to equity, financial leverage, for every firm in the index. Higher leverage makes the equity claim on each firm more sensitive to the same dollar move in the underlying business, so a falling index mechanically predicts higher volatility ahead of it, before any change in investor psychology enters the picture. Implied volatility consistent with this dynamic prices downside strikes higher than upside strikes purely from this channel.

Index option skew steepened permanently after the October 1987 crash and has stayed persistently downward sloping since, which points to a second explanation sitting alongside the leverage effect: option prices now embed a real probability of a discontinuous crash that a lognormal model cannot represent at all. Pension funds, insurers and asset managers buy downside puts as portfolio insurance in large and fairly constant size, and that structural demand pushes put prices, and therefore put-implied volatility, above what the leverage effect alone would justify.

The 25-delta risk reversal, the implied volatility of a 25-delta put minus that of a 25-delta call, is commonly a few volatility points negative in calm conditions and can widen past 10 points during a sustained selloff, since demand for protection accelerates exactly when supplying it becomes riskier for the dealer on the other side.