Consider a similar problem to invest in developing a natural resource field but over $2$ periods.

The investment involves buying a lease that costs I$0=30$ M upfront $($at t $=0)$ that provides a

compound option $($first option leading to a subsequent option$)$ giving the right at t $=1$ to invest I$1=$

$80$ M to receive $($at t $=1)$ the base$-$scale $($first$-$stage$)$ project cash flow value V plus an option to

subsequently $($at t $=2)$ make an additional investment I$2=40$ M $($decision$-$tree like representation$)$

to expand $($processing$)$ capacity by $50\%.$ Assume dt $=1$ step per year, multiplicative up and down

parameters, u $=2$ and d $=0\mathrm{.}5,$ and risk$-$adjusted $($risk$-$neutral$)$ probability p $=0\mathrm{.}367,$ r $=0\mathrm{.}05.$

Draw three binomial trees $($over t $=0,1,2)$ as follows:

$($three parts, $10$ marks all$)$

$($a$)$ One binomial tree for the underlying project value $($V$)($assume V$0=100$ M$)$

$($b$)$ a second binomial tree for the last $($expansion$)$ option, showing the expansion

option payoff at t $=2($write the formula$).$ What is the current $($t $=0)$ value of

the option to expand capacity by $50\%$ by paying cost I$2=40$ M at t $=2?$

$($c$)$ a third binomial tree for the first $($compound$)$ option to invest in the base scale

project in the first stage at t $=1($showing its payoff at t $=1$ to pay I$1=80$ M to

acquire the first$-$stage base project value V plus the follow$-$on expansion

option$).$ What is the current $($t $=0)$ value of the first $($compound$)$ option?In above, show the appropriate values $($numbers$)$ on all nodes $($states$)$ on all three

binomial trees and show your basic calculations e$.$g$.,$ for the option payoffs and

calculated option values along each binomial tree on the side $($margin$).$What is the time$-0$ net value $($Expanded$-$NPV$)$ of this investment opportunity $($after

accounting for the upfront lease cost of I$0=30$ M$)?$ Should you acquire the lease?

What is the maximum you would pay for it$?$

$($iv$)$ Now suppose you are to solve the above $2-$year problem using N $=4$ steps to divide

the $2-$year maturity $($of the option to expand$),$ using annualized $\backslash$sigma of $0\mathrm{.}693.$ What will

then be the multiplicative up and down parameters u and d and the risk$-$adjusted

probability p$,$ assuming no competitive damage or other form of $$dividend yield$(\backslash$delta $=$

$0)?$ What would p be instead if there is a $4\%$ competitive erosion in project value V $(\backslash$delta

$=0\mathrm{.}04)?$