Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / B / 4 / d / Solution

At maturity, one stock share is worth $(X_T-m{)}^{+}/n$. For a nonnegative strike $K$, the digital call option therefore pays one exactly onUnder the risk-neutral measure, $\log (X_T/x_0)$ is normal with mean $(r-{\sigma }^2/2)T$ and variance ${\sigma }^{2}T$. The risk-neutral pricing consequently givesThe 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 $X_T>m$.