An equity option desk quotes 22% implied volatility on a 25-delta call and 32% on a 25-delta put. Black-Scholes says they should be equal: one underlying, one process, one volatility. The 10-point gap is the smile, and it tells you what the market thinks about extreme moves the model cannot describe. The exam tests whether you can read the smile, price options consistently with it, and connect it to the FRTB capital framework that replaced the old VaR-based market-risk standard.
Black-Scholes-Merton assumes the underlying follows geometric Brownian motion with constant volatility . One number describes the diffusion. If that assumption held, the implied volatility computed from any option price on the same underlying (any strike, any maturity) would be the same number.
It isn't. Across nearly every liquid options market, implied vol depends on strike (the smile or skew) and on maturity (the term structure). The full surface of implied vols across both dimensions is the volatility surface.
KEY: The volatility smile reveals that the underlying's true distribution is not lognormal. Each smile shape implies a different deviation: skew implies asymmetric tails; symmetric smile implies fat tails on...
Common mistakes
- Treating implied vol as the underlying's true vol. Implied vol is the BSM-implied parameter that reproduces the option price. It includes the volatility risk premium and reflects the market's view, not the realized vol. Trap: a question reports implied vol and asks for one-period realized vol: the right answer differentiates the two.
- Misreading equity skew as a smile. Equities show downward-sloping skew, not symmetric smile. FX shows symmetric smile. Single-stock equity can show either depending on company specifics. Trap: a question on S&P 500 options describes a "smile": the right answer notes it should be a downward skew.
- Forgetting put-call parity equalizes implied vols. Calls and puts at the same strike have the same implied vol; parity forces it. Trap: a question reports different vols and asks the trader to act on the discrepancy; the right answer recognizes bid-ask, not real model error.
Bottom line
- Volatility smile: implied vol varies by strike and maturity, contradicting Black-Scholes' constant- assumption. Equities show a downward skew (puts above calls), FX shows a symmetric smile.
- Implied distribution: the smile reveals the underlying is not lognormal. Equity skew implies a fatter left tail (crash risk, leverage), FX's symmetric smile implies fat tails in both directions.
- Volatility surface and term structure: the surface plots implied vol against (strike, maturity); the term structure slopes upward in calm markets and downward in stress.
- Jump-diffusion captures the short-dated smile spike pure GBM misses. Single-asset jumps lift short-term implieds but barely move long-dated ones.
Exam shortcut
When a question describes equity options, expect downward skew, not symmetric smile. When a question asks about FRTB capital, the five SBA components are delta, vega, curvature, DRC, RRAO; sum them. When IMA fails PLAT, capital migrates to SBA. Liquidity horizons scale ES; illiquid positions face longer horizons and higher capital.
The full lesson (about 3,306 words, 22 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
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