Multi-Asset Digital Option

Consider two financial assets $S_1$ and $S_2$ evolving under the risk-neutral measure $\mathbb{Q}$ with Black-Scholes dynamics:

\frac{dS_i(t)}{S_i(t)} = (r - q_i)\,dt + \sigma_i\,dW_i(t), \quad i = 1,2

where $r$ is the constant risk-free rate, $q_i$ is the dividend yield of asset $S_i$ , $\sigma_i > 0$ is the volatility of , and is a two-dimensional Brownian motion with:

\langle W_1, W_2 \rangle_t = \rho \, t, \quad \rho \in (-1, 1)

Consider a digital option with maturity $T$ and payoff:

\mathbf{1}_{\{S_1(T) \geq S_2(T)\}}

Question: Determine the closed-form price of this option at $t = 0$ .

Multi-Asset Digital Option

Answer