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

A trader delta-hedges a book of short equity index options using Black-Scholes deltas computed with a single flat implied volatility. Market-observed equity index implied volatilities show a pronounced downward skew (higher implied volatility for low strikes). For a deep out-of-the-money put, which statement best describes the effect of the skew on the hedge?

The Black-Scholes delta holds volatility fixed, but under a skew implied volatility changes when the index moves. The true hedge ratio therefore includes a volatility-driven component, so the simple delta can misstate the correct hedge for out-of-the-money puts.

  1. AThe skew has no effect on delta because delta depends only on the stock price
  2. BThe Black-Scholes delta using the strike-specific implied volatility ignores how volatility moves when the index moves, so the hedge ratio can differ from the true sensitivityCorrect
  3. CThe skew guarantees that the Black-Scholes delta overstates the hedge ratio for every maturity
  4. DThe skew makes gamma negative for all long option positions

Explanation

With a skew, implied volatility changes as the underlying price moves (sticky-strike versus sticky-delta behavior). Plain Black-Scholes delta holds volatility fixed, so it misses the volatility-induced price change. The true delta therefore differs from the Black-Scholes delta. Claiming no effect is wrong because the hedge depends on how volatility co-moves with the price.

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