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

A trader delta-hedges a portfolio of equity index options using a delta computed from the Black-Scholes model with a single constant volatility. The market shows a pronounced downward-sloping volatility skew for equity index options. Which statement best describes the implication of the skew for the hedge?

With a skew, implied volatility moves as the underlying moves, so the correct hedge ratio includes a vega-related adjustment. The plain Black-Scholes delta with constant volatility is therefore typically inaccurate, since delta itself depends on volatility and the volatility input shifts with the price.

  1. ABlack-Scholes delta is exact because delta does not depend on volatility
  2. BThe hedge ratio should reflect how implied volatility changes when the index moves, so the Black-Scholes delta is typically inaccurateCorrect
  3. CThe skew affects only gamma, so delta hedging is unaffected
  4. DThe skew implies delta should always be set to zero for out-of-the-money puts

Explanation

Under a skew, implied volatility for a given option changes as the underlying price moves. The total sensitivity therefore includes the vega effect times the expected volatility change, so the plain Black-Scholes delta is not the right hedge ratio. Delta does depend on volatility, so the first option is wrong.

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