A digital call pays 1 if S_T exceeds 100 in one year. At that strike N(d2) = 0.45, the vanilla call has vega 39.8 per unit of volatility, implied volatility falls 0.20 volatility points per point of strike, and rates are zero. What is the digital worth?

A digital call pays 1 if S_T exceeds 100 in one year. At that strike N(d2) = 0.45, the vanilla call has vega 39.8 per unit of volatility, implied volatility falls 0.20 volatility points per point of strike, and rates are zero. What is the digital worth?

Approach: A digital is the negative derivative of the call price in the strike. Differentiate through the smile, so the strike enters both directly and through the implied volatility.

0.5296. The digital is the negative total derivative of the call price in the strike, and with a smile that derivative is C_K plus vega times dsigma/dK. The first term gives the flat volatility value N(d2) = 0.45. The skew term is 39.8*(-0.002) = -0.0796 per point of strike, so the digital is worth 0.45 + 0.0796 = 0.5296, which is 18% above the Black-Scholes number. The hedge is a call spread, (C(K - h/2) - C(K + h/2))/h, which prices the skew automatically because it is built from traded vanillas, so a desk quoting N(d2) sells the payoff eight cents cheap on every dollar of notional. What is left over is exposure to the skew at that strike moving, which the flat volatility model does not show at all.

Follow-up: How wide should the call spread be if the desk is charged for both the spread cost and the residual gap risk?

Key concepts: digital option, call spread, volatility skew, vega.