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

A currency option market shows a volatility smile that is symmetric around the at-the-money strike, with implied volatility rising for both deep in-the-money and deep out-of-the-money options. Which feature of the true exchange-rate distribution is most consistent with this smile, compared with lognormal?

A symmetric smile is consistent with a distribution having heavier tails on both sides and a higher peak than lognormal, meaning excess kurtosis. Out-of-the-money options of both types cost more than Black-Scholes implies, raising their implied volatilities.

  1. ABoth tails are heavier and the peak is higher than lognormalCorrect
  2. BBoth tails are lighter and the peak is flatter than lognormal
  3. CA heavier left tail only
  4. DConstant volatility with jumps being impossible

Explanation

A symmetric smile means out-of-the-money calls and puts are both priced above Black-Scholes values. This reflects fat tails on both sides and, correspondingly, a more peaked center, i.e. excess kurtosis. Lighter tails would produce a frown, not a smile.

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