b$)$ You$$re planning to expand the operations of your

manufacturing plant by $50\%$ sometime in the next six months.

Suppose the value of the plant today is $100$m$.$ The cost from

expansion is $40$m$.$ Risk uncertainty in the operating environment

is estimated by an annualized variance of $25\%$ in cash flows. The

risk$-$free rate of return is $3\%$ per half annum. Use the binomial

option pricing approach with a time step of three months to

value the option to expand operations. Is there an optimal time

for taking such an action? Explain.