= Solution
{c}
At maturity, one stock share is worth $(X_T-m)^+/n$. For a nonnegative strike $K$, the <digital call option> therefore pays one exactly on
$$
\frac{(X_T-m)^+}n>K\quad\Longleftrightarrow\quad X_T>m+nK.
$$
Under the <risk-neutral measure>, $\log(X_T/x_0)$ is normal with mean $(r-\sigma^2/2)T$ and variance $\sigma^2T$. The <risk-neutral pricing> consequently gives
$$
\boxed{V_0=e^{-rT}\Phi\left(\frac{\log(x_0/(m+nK))+(r-\sigma^2/2)T}{\sigma\sqrt T}\right).}
$$
The effective company-value strike is $m+nK$, because both the outstanding debt and the number of shares matter. At $K=0$, the strict positive-share event remains $X_T>m$.
Back to article page
