Implied volatility and what the smile reveals
Question
What is implied volatility? What is meant by the volatility smile (or skew), and what does its existence tell us about the assumptions of the Black-Scholes model?
Short answer
Implied volatility is the that makes the Black-Scholes formula reproduce an option's market price. Plotting it against strike (and maturity) gives a curve rather than a flat line: equity indices show a downward skew, FX often a symmetric smile. A single constant volatility cannot fit all strikes, so real returns are not lognormal with constant volatility; markets price fatter tails, jumps and volatility that varies with level and time.
Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 6.2 The Greeks, 'Vega A', p.156; question and answer rewritten by SelfMadeQuant.com.
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Open the worked solutionLessons behind this question
- Reading the Implied Vol Surface — Moneyness, Term Structure, and Quoting Conventions
- Derivation — From Binomial Trees to Black-Scholes in the Continuous Limit
- The Black-Scholes Formula Derived via Change of Measure
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