Digital Option Pricing in Black-Scholes

Consider a risky asset $(S_t)_{t \geq 0}$ evolving under the risk-neutral measure $\mathbb{Q}$ according to the Black-Scholes model with constant risk-free rate $r$ and constant volatility $\sigma > 0$ :

dS_t = r S_t \, dt + \sigma S_t \, dW_t^{\mathbb{Q}}

A digital (binary) call option with maturity $T$ and strike $K > 0$ has payoff:

\Psi(S_T) = \mathbf{1}_{\{S_T \geq K\}}

Question: Give a closed-form formula for the price $D_t$ of this option at time $t \leq T$ .

Digital Option Pricing in Black-Scholes

Answer