$1.$ 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 $=0).$