Noble oil & gas company has an opportunity to invest in a gas development project off the coast

of Cyprus within a year $($the underlying asset being a current claim to project cash inflows from

future sale of natural gas$)$ by paying as an exercise price the development cost of I $=80$ Million

$($M$)($suppose the cost remains constant during the year$).$ The current asset $($project$)$ value, V$0,$ is

expected at the end of the period $($at t $=1)$ to either rise to V$+$

$=200$ M with actual probability q $=$

$0\mathrm{.}5$ if future conditions are positive $(+)$ or fall to V$-$

$=50$ M with probability $1-$q $=0\mathrm{.}5$ if negative $(-).$

The annual risk$-$free interest rate $($r$)$ over the period is $5\%.$ Assume no competitive erosion to

project value $(\backslash$delta V $=0).$a$)$Suppose the annualized standard deviation on project value changes $(\backslash$sigma $)$ is estimated to be

$69\mathrm{.}3\%.$ Using one time step $($dt $=1),$ determine the multiplicative up $($u$)$ and down $($d$)$

parameters and the risk$-$adjusted $($or risk$-$neutral$)$ probability of an up move $($p$).$ Estimate the

present value $($at t $=0)$ of the project $$s expected cash inflows using the risk$-$adjusted

probabilities $($p$)$ and discounting at the risk$-$free rate $($r$).$ How do your answers for V$0$ in $($i$)$ and

$($iii$)$ above compare and why?

$($b$)$ The one$-$year lease allows Noble the right to wait to invest until next year, with no obligation

to do so$.$ Draw the binomial trees for the asset V and for the option C $($over the one$-$year

period$)$ and estimate the option values in the up $($C$+$

$)$ and down $($C$--)$ states at t $=1.$ What is

the current $($t $=0)$ value of the option to invest to develop the gas field provided by the lease

$($C$0)$ using the risk$-$adjusted $($risk$-$neutral$)$ valuation approach? What is the Expanded NPV$,$

net of the upfront $($time$-0)30$ M cost of the lease? Is it worth to pay $30$ M to buy the lease to

acquire the option to develop the gas field within a year?