Question $4$

The US$-$based XYZ Bank has a foreign asset which is worth $20$ million. The current spot exchange rate is $$\frac{1\mathrm{.}10}{}.$ XYZ bank wants to hedge the Foreign Exchange risk of this asset over the next month using option. It buys a $1-$month vanilla European put option whose underlying is the spot exchange rate $\frac{\text{\$}}{},$ with the strike price $$\frac{1\mathrm{.}10}{}$ and the size of the option $20$ million. The premium of this put option is $$10,000.$

$($a$)(10$ marks$)$ What is the gain or loss of XYZ bank from this hedged position $($foreign asset and option$)$ if the exchange rate becomes $$\frac{1\mathrm{.}11}{}$ after one month? What is the gain or loss if the exchange rate becomes $$\frac{1\mathrm{.}09}{}$ after one month?

$($b$)(10$ marks$)$ If the exchange rate after one month is a random variable $$\frac{e^x}{},$ where $x$ follows a normal distribution with mean $0\mathrm{.}08$ and variance ${0\mathrm{.}02}^2,$ what is the $95-$th percentile of the profit from this hedged position?