An alternative to using \(d=1 / u\) in a binomial model is to use the available degree of freedom by setting \(p=1 / 2\).

(a) Let \(p=1 / 2\), and find the values of \(u\) and \(d\) that satisfy the matching conditions \(p U+(1-p) D=u \Delta t\) and \(p(1-p)(U-D)^{2}=\sigma^{2} \Delta t\), where \(U=\ln u\) and \(D=\ln d\). [For purposes of comparison, use \(\hat{u}\) and \(\hat{d}\) for the resulting alternate values of \(u\) and \(d\).]

(b) Which lattice approximation is preferable in applications, and why?