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

Under the minimum variance delta approach, a risk manager adjusts the Black-Scholes delta for the smile. For a European call, the adjusted delta is Delta_MV = Delta_BS + Vega × (∂E[σ_imp]/∂S). Delta_BS = 0.55, vega = 12 per 1.00 of volatility, and the model expects implied volatility to fall by 0.0004 for each 1.00 increase in the asset price (so ∂σ/∂S = -0.0004). What is the adjusted delta?

The adjusted delta is 0.5452. The vega adjustment equals 12 times minus 0.0004, which is minus 0.0048, and this is added to the Black-Scholes delta of 0.55. A falling implied volatility as the price rises lowers the effective delta slightly.

  1. A0.5452Correct
  2. B0.5548
  3. C0.5020
  4. D0.5980

Explanation

The adjustment is Vega × ∂σ/∂S = 12 × (-0.0004) = -0.0048. Adjusted delta = 0.55 - 0.0048 = 0.5452. The option 0.5548 adds instead of subtracting, ignoring the negative sign of the volatility response.

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