A stock price is currently e $40.$ It is known that in $6$ months it will be either e $50$ or e $30.$ The

risk free$-$rate of interest with semi$-$annual compounding is $4\%$ per annum.

$($a$)$ Calculate the value of a $6$ month European put option on the stock with an exercise price of e $40.$

$($b$)$ Verify that no$-$arbitrage arguments and risk$-$neutral valuation arguments give the same answe