Consider a problem with competitive erosion due to substitute product arrivals $(\backslash$delta V $=0\mathrm{.}04)$

but with a stochastic development cost I. Noble is facing an investment opportunity to pay a

stochastic cost $($with time$-0$ value I$0=80$ M$)$ within $1$ year to acquire the project by T $=1($with time$-$

$0$ present value of cash flows V$0=100$ M$).$ The asset $($project$)$ value, V$,$ currently at $100$ M$,$ is

expected $($with equal actual probability, q $=0\mathrm{.}5)$ at the end of the period to either rise to V$+$

$=200$ M

$($double$)$ or fall to V$-$

$=50$ M $($half$).$ The standard deviation of project returns is $\backslash$sigma V $=69\mathrm{.}3\%.$ The

annual risk$-$free interest rate $($r$)$ over the period is $5\%.$ The project V has a dividend$-$like payout $($due

to competitive erosion$)$ of $\backslash$delta V $=0\mathrm{.}04.$ The cost I has a dividend$-$like payout of $\backslash$delta I $=0\mathrm{.}03$ and $\backslash$sigma I $=15\%.$

The correlation coefficient $(\backslash$rho $)$ between the rates of change in V and I is $0\mathrm{.}5.$

$($i$)$ Calculate the value of a European call option to invest in project V with a stochastic cost I

$($and $\backslash$delta V $=0\mathrm{.}04)$ within T $=1$ year using a relevant analytic formula. To calculate N$($d$)$ you can

use function NORMSDIST$($d$)$ in Exce

$($ii$)$ In the previous problem, determine the value of an American call option on project V with

maturity T $=1$ year using a $1-$D binomial tree with N $=2$ steps $($dt $=0\mathrm{.}5).$ You would need to

re$-$estimate the adjusted $($with $)$ u$,$ d and p $($using N $=2$ steps instead of $1$ step per year$).$

Again, V$0=100,$ I$0=80,\backslash$sigma V $=69\mathrm{.}3\%,\backslash$sigma I $=15\%,\backslash$delta V $=0\mathrm{.}04,\backslash$delta I $=0\mathrm{.}03,\backslash$rho $=0\mathrm{.}5,$ r $=5\%.$ Please

show all relevant calculations of intermediate parameters $($e$.$g$.,$ relative $\backslash$sigma $,$ adjusted u$,$ d$$ and

p$)$ and show the two$-$step binomial trees for project value V and for the investment option

$($first European and then American option$).$ Show the option value calculations along the tree

on the side.