In a binomial tree with \(n\) steps, let \(f_{j}=f_{u \ldots u d \ldots d}(j\) times \(u\) and \(n-j\) times \(d\) ). For a European Call option expiring at \(T=n \Delta t\) with strike \(K\), show that

\[f_{j}=\max \left\{S u^{j} d^{n-j}, 0\right\}\]

In particular, if \(u=\frac{1}{d}\), we have that

\[f_{j} \geq 0 \Leftrightarrow j \geq \frac{1}{2}\left(n+\frac{\log (K / S)}{\log (u)}\right)\]

(a) Is it true that the probability of positive payoff is the probability of \(S_{T} \geq\) \(K\) ? Justify your answer.

(b) Find a general formula for the present value of the option.