Q-30.

$u = 48/40 = 1.2$

$d = 32/40 = 0.8$

$P_{\text{up}} = \frac{e^{r\Delta t} - d}{u - d} = \frac{e^{6.2\% \times 0.5} - 0.8}{1.2 - 0.8} = 57.87\%$

$f = [(50 - 48) \times 0.5787 + (50 - 32) \times (1 - 0.5787)] \times e^{-6.2\% \times 0.5} = 8.47$

Early Exercise, therefore, the value of this American put is 10.