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FRM Part II · FRM Exam Part II · Volatility Smiles and Volatility Surfaces

A risk analyst at a currency desk plots implied volatility against strike price for one-year options on a major currency. The plot shows a U-shaped curve with the minimum near the at-the-money strike. Which statement best describes how Hull characterizes this pattern relative to the lognormal assumption in Black-Scholes-Merton?

A U-shaped currency smile implies a distribution with fatter tails on both sides than the lognormal. Far in- and out-of-the-money options are therefore worth more than Black-Scholes-Merton predicts, which appears as higher implied volatility at extreme strikes relative to at-the-money options.

  1. AThe market assigns fatter tails to the exchange rate distribution than a lognormal distribution does, so deep in- and out-of-the-money options are priced with higher implied volatilityCorrect
  2. BThe market assigns thinner tails than a lognormal distribution, so deep in- and out-of-the-money options carry lower implied volatility
  3. CThe market assumes a lognormal distribution with higher volatility at every strike, so the curve shifts upward but is not curved
  4. DThe market assumes the left tail is heavier than the right tail, so low-strike options have the highest implied volatility

Explanation

A symmetric smile for currency options corresponds to an implied distribution with heavier tails (both sides) than lognormal. Heavier tails raise the prices of far-from-the-money options, which show up as higher implied volatilities. A thinner-tail distribution would produce an inverted smile, and a heavier left tail alone produces a skew.

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