The price of a stock price is currently $$50.$ Over each of the next two three$-$month periods it

is expected to go up by $6\%$ or down by $5\%.$ The risk$-$free interest rate is $5\%$ per annum with

continuous compounding.

$($a$)$ Please use the two$-$period risk$-$neutral valuation to price a six$-$month Euro$-$

pean call option with a strike price of $$51.$ You are not required to draw the tree.

$($b$)$ Please use the two$-$period risk$-$neutral valuation to price a six$-$month Euro$-$

pean put option with a strike price of $$51.$ You are not required to draw the tree.

$($c$)$ verify that the prices of the European call and European put in $($a$)$ and $($b$)$

satisfy put$$call parity.