b$.$ A stock index has a current value of $7500$ points and is expected to increase or decrease by $10\%$ over each of the next two three$-$month periods. The dividend yield on the index is $2\%$ per annum and the risk$-$free rate of interest is $5\%$ with continuous compounding. The option contract size for options written on this index is $$100$ per index point.

$($i$)$ What is the price of a $6-$month European put option written on the index with a strike price of $8000?$

$($ii$)$ What is the price of a $6-$month American put option written on the index with a strike price of $8000?$

$($iii$)$ Use put$-$call parity to compute the price of $6-$month European call option written on the index with a strike price of $8000?$

$($iv$)$ If the dividend yield falls to zero, what is the price of the $6-$month European put option written on the index with a strike price of $8000?$