Consider a European call option on a non$-$dividend$-$paying stock where the stock price is $$33,$ the strike price is $$36,$ the risk$-$free rate is $6\%$ per annum, the volatility is $25\%$ per annum and the time to maturity is $6$ months.

$($a$)$ Calculate u and d for a one$-$step binomial tree.

$($b$)$ Value the option using a non arbitrage argument.

$($c$)$ Assume that the option is a put instead of a call. Value the option using the risk neutral approach.

$($d$)$ Verify that the European call and European put prices found in $($b$)$ and $($c$)$ satisfy the put$-$call parity.