Tasmania is a state within Australia, currently producing electricity using a single source of energy

based on hydropower $($water$)$ at a fixed cost I $=100.$ The government of Tasmania considers adding

two additional modes of energy production that will allow it to switch production to the cheapest of

two new energy sources $($V$1,$ V$2)$ in year $2.$ The current cost of producing energy based on the first

additional source, geothermal energy $($V$1),$ is $95,$ with payout $\backslash$delta $1=0\mathrm{.}03$ and volatility $\backslash$sigma $1=0\mathrm{.}25$; the

current cost of producing with the second mode, adding a link connecting to the electricity network

of neighboring Australian states that will allow buying electricity based on the electricity prices in

Australia $($V$2),$ is $105,$ with $\backslash$delta $2=0\mathrm{.}05$ and $\backslash$sigma $2=0\mathrm{.}10.$ The correlation coefficient $\backslash$rho between the two

new energy sources is $0\mathrm{.}5,$ and the risk$-$free interest rate in Tasmania, r$,$ is $5\%.$ Evaluate this $2-$year

European option to switch energy modes using a two$-$dimensional $(2-$D$)$ binomial lattice tree with N

$=2$ steps$($i$)$ V$1$ and V$2$ after two steps $($moving to the right and down as in the missing black$-$out box a$.$ on

the next page$)$;

$($ii$)$ The option payoff formula $($equation in symbols$)$ and the relevant value $($number$)$ for the

European option value to switch from the current hydropower to the cheapest of the two new

energy modes at maturity $($t $=2)$; $($see the missing info. in black box b$.$ to the right$)$;

$($iii$)$ The risk$-$adjusted probability if the first asset moves down and the second moves up $($pdu$)$

$($see missing black box c$.).($You are given two other probabilities: puu $=0\mathrm{.}35125,$ pud $=$

$0\mathrm{.}12625)$;

$($iv$)$ The current $($time $0)$ value of this European option to switch $($see black box d$.$ at the bottom$)$