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Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options

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What they found

The authors derived model-free formulas that recover the risk-neutral skewness and kurtosis of a stock's return distribution from its option prices, and used them to explain why index options have a much steeper volatility skew than individual equity options. They showed that the index's risk-neutral distribution is far more negatively skewed than any individual stock's, because diversification removes idiosyncratic upside but not systematic crash risk, and that the skew is priced: more negatively skewed names have steeper smiles.

See it drawn

Original diagrams for the ideas on this page. Illustrative, not real market data.

The volatility smile across strikesImplied volatility plotted against strike, dipping near the money and turning up at both ends, more steeply on the downside.Implied volatility32%28%24%20%8090110120Puts below the money cost moreFar calls cost more tooLowest IV near the moneyATM 100Strike price
The volatility smile. Options on the same stock and the same expiry are not priced off one volatility. Strikes near the money carry the lowest implied volatility, and it rises towards both ends — usually faster on the downside, which tilts the smile into a skew.

What you can use

  • Index options price crash risk much more heavily than single-stock options do; the index skew is steep because crashes are systematic.
  • You can read the market's implied skewness and tail expectations directly from the option chain.
  • Individual stocks have flatter skews because they have idiosyncratic upside; that is why put protection on single names is relatively cheaper.

Caveats

Mathematically intensive; the skewness measures depend on having liquid options across a range of strikes. Sample from the 1990s.

Tags: options, skew, risk-neutral-distribution, implied-volatility

Summaries are our own reading of the paper, not the authors' words. Educational only, not advice. Discuss it in Book Club.