Cash-or-Nothing Digital Option Valuation and Delta

Scenario

A European cash-or-nothing digital call option pays a fixed monetary amount $Q$ at expiration date $T$ if and only if the underlying stock price exceeds the strike threshold $K$ ($S_T \ge K$), and pays $0$ otherwise. Assume the underlying asset price follows geometric Brownian motion under the risk-neutral measure $\mathbb{Q}$: $dS_t = r S_t dt + \sigma S_t dW_t^{\mathbb{Q}}$

Your Task

1. Formulate the risk-neutral expectation pricing integral for the digital call $V(S, t)$.
2. Derive the analytical closed-form pricing formula in terms of standard normal cumulative distribution $\Phi(d_2)$.
3. Compute the option's Delta $\Delta = \frac{\partial V}{\partial S}$ and interpret its behavior near maturity $T$ when $S \approx K$.

Constraints

• Constant risk-free interest rate $r > 0$ and volatility $\sigma > 0$.
• Fixed payout $Q > 0$.

Examples For $S = 100, K = 100, r = 0.05, \sigma = 0.20, T = 1.0, Q = 1.0$, $d_2 = \frac{0 + (0.05 - 0.02) \times 1}{0.20} = 0.15$. $V = e^{-0.05} \Phi(0.15) \approx 0.9512 \times 0.5596 = 0.5323$.