Value a one-year digital call paying $1 if a $100 stock finishes above 100, with 20% volatility and zero rates. Then say how a skew of -0.10 volatility points per point of strike changes the price implied by the replicating call spread.

Value a one-year digital call paying $1 if a $100 stock finishes above 100, with 20% volatility and zero rates. Then say how a skew of -0.10 volatility points per point of strike changes the price implied by the replicating call spread.

Approach: The digital is the negative derivative of the call price in strike. Price it flat first, then add the term that comes from the volatility itself changing with strike.

0.4602. With zero rates d_2 = (ln(S/K) - 0.5*sigma^2*T)/(sigma*sqrt(T)) = -0.02/0.20 = -0.1, and the flat-volatility digital is the risk-neutral probability of finishing above the strike, N(d_2) = N(-0.1) = 0.4602. The digital is the limit of a tightening call spread, -dC/dK, and once volatility depends on strike the chain rule adds a term: digital = N(d_2) - Vega*(d sigma/dK). Here vega is about 40 per unit of volatility, or 0.40 per volatility point, and d sigma/dK = -0.0010 per point of strike, so the correction is +0.04, and a 1-point wide call spread prices the digital about 0.04 higher than the flat number. With a steeper skew the effect is much larger, so quoting a digital off a single at-the-money volatility understates it in any market with downside skew and leaves the seller short the wing they never priced.

Follow-up: You are short a digital and hedge it with a 1-point call spread. What happens to your risk as expiry approaches with spot pinned at the strike?

Key concepts: digital option, call spread, skew, risk-neutral probability, vega.