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

Black-Scholes assumes the underlying asset price follows geometric Brownian motion with constant volatility. In equity markets after 1987, implied volatilities for out-of-the-money puts are typically higher than for at-the-money options. Relative to the lognormal distribution implied by Black-Scholes, this pattern is consistent with a market-implied risk-neutral distribution that has:

Higher implied volatility for low-strike puts means the market-implied distribution has a heavier left tail than the lognormal distribution of Black-Scholes. Crash risk is priced more richly than constant-volatility lognormal returns would suggest, producing the equity volatility skew.

  1. AA heavier left tailCorrect
  2. BA lighter left tail
  3. CA heavier right tail and lighter left tail
  4. DIdentical tails but higher kurtosis in the center only

Explanation

Higher implied volatility for low strikes means the market assigns higher prices to deep OTM puts than lognormal would, indicating a heavier left tail. A lighter left tail would give lower implied volatilities for low strikes. The right-tail option describes the reverse skew seen in currencies or commodities.

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