A stock price is currently $$40.$ It is known that at the end of $3$ months it will be either $$45$ or $$35.$ The risk$-$free rate of interest with quarterly compounding is $8\%$ per annum. Calculate the value of a $3-$month European put option on the stock with an exercise price of $$40.$ Verify that no$-$arbitrage arguments and risk$-$neutral valuation arguments give the same answers.