L1

Binary Cash-or-Nothing Option

Junior Quant · Derivatives Pricing

K$1pays $1if S_T > KStock price S_TPayoff

Binary cash-or-nothing: pays $1 if S_T > K, zero otherwise. Price it and derive its delta.

Question

Price a binary cash-or-nothing option under Black-Scholes. Derive its delta. How does delta behave as expiry approaches for an at-the-money binary, and why does this matter for hedging?