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

A fitted implied volatility smile produces call prices that, at three equally spaced strikes K-d, K and K+d, satisfy c(K-d) + c(K+d) - 2c(K) < 0. What does this imply?

It implies a negative risk-neutral density at K, which is impossible, so the surface admits a butterfly arbitrage. A long butterfly has a nonnegative payoff but would cost a negative amount, meaning the fitted smile violates no-arbitrage and needs correction.

  1. AThe implied density at K is negative, so a butterfly spread at those strikes would offer an arbitrageCorrect
  2. BThe implied density at K is positive but very small
  3. CThe volatility smile is flat at those strikes
  4. DA calendar spread arbitrage exists between the maturities

Explanation

The second difference of call prices with respect to strike is proportional to the risk-neutral density. A negative value means a negative probability, which is impossible. A long butterfly (long wings, short two middle calls) would then have a negative price while having a nonnegative payoff, so it is an arbitrage.

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