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Show how an option's Greek letter risk measures are affected by the volatility smile.

FRM® Part II · Market Risk Measurement and Management · Volatility Smiles and Volatility Surfaces · FRM2.MR.17.h

The idea

Black-Scholes-Merton Greeks assume constant volatility. When implied volatility varies with strike and maturity, delta has an extra term. The effective delta is the model delta plus vega times the change in implied volatility per unit change in the asset price. If each option's implied volatility is assumed fixed whatever the price, which is called sticky strike, the extra term is zero. If the smile is instead a function of K/S, the extra term is not zero. For a downward-sloping skew in K/S, a rise in S lowers K/S and raises volatility, which pushes the call delta up. When observed volatility falls as the price rises, the effective delta is lower than the model delta. Vega also needs care. A parallel shift of the surface hides exposure to moves at particular strikes and maturities, so risk is measured bucket by bucket. Always state which assumption about the smile's movement you are using.

Formulas

Delta allowing for the volatility smile

Delta = DeltaBSM + Vega × (∂σ ÷ ∂S)

DeltaBSM
the Black-Scholes-Merton delta at the option's implied volatility
Vega
the option's vega
∂σ ÷ ∂S
the change in implied volatility per unit change in the asset price S

Under sticky strike the second term is zero. Always state the assumption about how the smile moves.

A practice question

From the question bank, with the reason each wrong option is wrong.

When implied volatility depends on the asset price, which expression gives an option's delta?

  1. Black-Scholes-Merton delta plus vega times the change in implied volatility per unit change in price. - correct
  2. Black-Scholes-Merton delta minus vega, whatever the shape of the smile.Why not: Subtracts vega as a constant; the adjustment is vega times the volatility's sensitivity to the price, which can have either sign.
  3. Black-Scholes-Merton delta multiplied by vega, which scales the hedge ratio for each unit of volatility.Why not: Multiplies two different sensitivities; the vega effect enters as an additive term.
  4. Black-Scholes-Merton delta unchanged, since the smile only affects vega and has no effect on delta.Why not: The smile makes delta change when volatility moves with the price; delta is unchanged only if volatility is independent of price.

The total derivative of the option price with respect to S has two parts: the direct effect (the model delta) and the effect through implied volatility, which is vega x d(sigma)/dS.

A harder one

A call has a Black-Scholes-Merton delta of 0.60 and a vega of 0.40 (price change per one volatility point). Observation suggests its implied volatility falls by 0.25 volatility points when the underlying rises by 1.00. What is the delta that allows for this smile effect?

  1. 0.70Why not: Adds 0.10 (0.40 x 0.25) instead of subtracting it; the implied volatility falls as the price rises.
  2. 0.60Why not: Ignores the change in implied volatility, which is the whole point of the adjustment.
  3. 0.35Why not: Subtracts the volatility change (0.25) directly without multiplying by vega.
  4. 0.50 - correct

Adjusted delta = 0.60 + vega x change in volatility per unit of S = 0.60 + 0.40 x (-0.25) = 0.50. When volatility falls as the underlying rises, the effective delta is lower than the model delta.

The hardest tier

An index is at 100. A 1-year call with strike 110 has an implied volatility of 20%, and the smile is stable in moneyness: sigma = 20% - 0.60 x (K/S - 1.10). The risk-free rate is zero and there are no dividends. The model delta is 0.35 and the model vega is 37.2 (price change for a volatility change of 1.00). What is the call's delta allowing for the smile, closest to?

  1. 0.35Why not: Reports the plain model delta and ignores that implied volatility changes with the index level.
  2. 0.60 - correct
  3. 0.11Why not: Subtracts the vega term; under this smile a higher S lowers K/S and so raises sigma, increasing delta.
  4. 0.25Why not: Reports only the smile adjustment (vega x d(sigma)/dS) and leaves out the model delta.

d(sigma)/dS = 0.60 x K/S² = 0.60 x 110/10000 = 0.0066. Adjustment = 37.2 x 0.0066 = 0.245. Delta = 0.353 + 0.245 = 0.60.

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