$($Ch $12)(16$ points$)$ A stock price is currently $$50.$ It is known that at the end of two months, it will be either $$53$ or $$48,$ each with a $50\%$ probability in the real world. The risk$-$free interest rate is constant at $10\%$ per annum with continuous compounding. $($i$)$ Use a one$-$step binomial tree and the no$-$arbitrage arguments $($to build a risk$-$free portfolio using a unit of option and some units of underlying stocks$)$ to analyze the option values in the following questions, and $($ii$)$ Verify that riskneutral valuation arguments give the same answers derived from the no$-$arbitrage arguments.

$($a$)$ What is the value of a two$-$month European call option with a strike price of $$49?(8$ points$)$

$($b$)$ What is the value of a two$-$month European put option with a strike price of $$49?(8$ points$)$