Exercise: Linearity and put-call parity from risk-neutral valuation

The risk-neutral valuation formula

is linear in $X$. Use this to derive put-call parity directly.

Tasks

1. Show that $V_0$ is linear: for any payoffs $X, Y$ and constants $a, b$, $V_0(aX + bY) = aV_0(X) + bV_0(Y)$.

2. Express the payoff $S_T - K$ as a linear combination of a call and a put on the same underlying with strike $K$.

3. Use linearity to compute $V_0(S_T - K)$ in two ways: (a) directly using risk-neutral valuation on the payoff; (b) via the call-put decomposition. Equate them to recover put-call parity.

4. Generalisation. Express the digital call payoff $\mathbb{1}_{\{S_T > K\}}$ as a limit of vertical spreads $((S_T - (K-h))^+ - (S_T - K)^+)/h$ as $h \to 0$. Use this and risk-neutral valuation to show the digital price is $e^{-rT}\mathbb{Q}(S_T > K)$.