In the binomial model obtain the values of \(u, d\), and \(p\) given the volatility \(\sigma\) and the risk-free interest rate \(r\) for the following cases:

(a) \(p=\frac{1}{2}\).

(b) \(u=\frac{1}{d}\). Hint: \(E\left(S_{t+\Delta t}^{2}\right)=S_{t}^{2} e^{\left(2 r+\sigma^{2}\right) \Delta t}\)