# Pricing digital options in discrete time

I am stuck in this exercise from my textbook:

Consider a one-period market model with $N+1$ assets: a bond, a stock and $N-1$ call options. The prices of the bond are $B_0=1$ and $B_1 = 1+r$, where $r$ is a constant. The prices of the stock are given by a constant $S_0$ and a random variable $S_1$ taking values in $\{0, 1, \ldots, N-1, N \}$ for a given integer $N \geq 4$. Finally, let the time-0 price of the call option with strike $K \in \{ 1, \ldots, N-1 \}$ be denoted by $C(K)$. Now we introduce a contingent claim with time-1 payout $\xi_1 = g(S_1)$, where $g$ is the function $$g(M) = \mathbf{1}_{ \{M = K_0 \} }, \quad 0 \leq K_0 \leq N.$$ Assuming that the market has no arbitrage, we want to find the time-0 price $\xi_0$ in the following cases: $$2 \leq K_0 \leq N-2 \, ; \quad K_0 = N-1 \, ; \quad K_0 = 0 .$$

Let $Y$ be the state price density of the market such that $Y_0 =1$. We know that $$\mathbb{E} [ YS_1 ] = S_0, \quad \mathbb{E} [ Y ( S_1 - K)^{+} ] = C(K), \text{ for } K \in \{1, \ldots, N-1 \}.$$

But how can we compute $$\xi_0 = \mathbb{E}[ Y \mathbf{1}_{ \{ S_1 = K_0 \} }] \quad ?$$

The claim payoff you describe, $g(M)$, looks to me like a tight butterfly spread that pays off only in one state of the world. Can't you just replicate that by short two calls with strike $K_0$ and long two calls, with strikes one either side at $K_0\pm 1$? Then the price of your option would be $C(K_0+1)+C(K_0-1)-2\cdot C(K_0)$.