How random volatility creates a smile

Question

Explain why a stochastic-volatility model generates a volatility smile.

Short answer

Averaging over random volatility fattens the tails. With volatility independent of the stock, the price is the Black-Scholes price averaged over the realised average variance. Out-of-the-money options are convex in volatility, so the average exceeds the price at the average vol, and a higher implied vol is needed away from the money, giving a smile. Correlation between vol and spot then tilts it into a skew.

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q2.60, p.22; question and answer rewritten by SelfMadeQuant.com.

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