FRM Part II · FRM Exam Part II · Volatility Smiles and Volatility Surfaces
A desk holds a long position in a call option and uses a sticky-strike assumption, where each strike's implied volatility is fixed regardless of spot. Another desk uses a sticky-delta (floating smile) assumption, where implied volatility depends on moneyness. The market has a downward-sloping skew (implied volatility higher for lower strikes) and spot rises. Compared with sticky-strike, the sticky-delta delta of the call will be:
Sticky-delta gives a higher call delta. With a downward skew, a rising spot lowers strike over spot, which raises that strike's implied volatility. The positive volatility response multiplies vega and adds to Black-Scholes delta, whereas sticky-strike has no such term.
- ALower, because the option's strike becomes relatively more out-of-the-money in moneyness terms and volatility is lower... so the vega term subtracts
- BHigher, because as spot rises the strike becomes relatively lower in moneyness, so its implied volatility rises, adding a positive vega termCorrect
- CIdentical, because both models use the same Black-Scholes formula
- DZero, because sticky-delta removes the dependence on spot
Explanation
Under sticky-delta, volatility is a function of K/S. With a downward skew, a spot rise lowers K/S, so the option's implied volatility rises. Delta then equals Black-Scholes delta plus vega times a positive volatility sensitivity to spot, giving a higher delta than sticky-strike, where the vega term is zero. The first option reverses the direction.
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