$3.(10$ points$)$ Consider a European put option with expiration date $\backslash ($ T $\backslash )$ and exercise price $\backslash ($ X $\backslash ).$ Suppose that, at the current date $\backslash ($ t be out of the money in every possible state at date $\backslash ($ T $\backslash ),$ that is $\backslash ($ S$\_\{$T$\}>$X $\backslash )$ for all date$-$T states.

$($i$)$ Briefly explain why the current price of the put option should be zero. Derive the current date$-\backslash ($ t $\backslash )$ price of a European call option with the same expiration date and exercise price using the put$-$call parity relationship. Assume that the risk$-$free rate of return is $\backslash ($ r$\_\{$c$\}\backslash ),$ continuously compounded.

$($ii$)$ Verify that the price of the call option from $($i$)$ satisfies the general bounds on prices of European call options.