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

A desk fits a smooth implied volatility curve across strikes and uses it to derive an implied risk-neutral distribution for a currency. The fitted curve produces a negative density in one region of the far tail. What is the most accurate interpretation and response?

A negative density in an illiquid tail is most likely an artifact of interpolation or noisy quotes rather than a tradable arbitrage. The correct response is to refit the curve with constraints that keep call prices convex in strike so that the implied density stays nonnegative.

  1. AIt is a genuine arbitrage and should be traded immediately at quoted prices
  2. BIt is likely an artifact of the interpolation or noisy quotes, so the fit should be constrained to maintain convexity in call pricesCorrect
  3. CIt shows the true distribution is bimodal
  4. DIt indicates that the risk-free rate used is too high, so it should be set to zero

Explanation

Negative density implies butterfly arbitrage, but in the tails quotes are illiquid and wide, and interpolation or extrapolation often creates spurious violations. The practitioner response is to impose no-arbitrage constraints in the fit. A bimodal distribution would still have nonnegative density.

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