(Barone-Adesi and Whaley (1987)) We approximate the finite expiration American put option price with strike price \(K\) as

\[f(x, T) \simeq \begin{cases}\operatorname{BS}_{\mathrm{p}}(x, T)+\alpha\left(x / S^{*}\right)^{-2 r / \sigma^{2}}, & x>S^{*}, \tag{15.31}\\ K-x, & x \leqslant S^{*},\end{cases}\]

where \(\alpha>0\) is a parameter, \(S^{*}>0\) is called the critical price, and \(\operatorname{BS}_{p}(x, T)=\) \(\mathrm{e}^{-r T} K \Phi\left(-d_{-}(x, T)\right)-x \Phi\left(-d_{+}(x, T)\right)\) is the Black-Scholes put pricing function.

a) Find the value \(\alpha^{*}\) of \(\alpha\) which achieves a smooth fit (equality of derivatives in \(x\) ) between (15.31) and (15.32) at \(x=S^{*}\).

b) Derive the equation satisfied by the critical price \(S^{*}\).