$($Keep four digits after the decimal point in calculation and round final answer to two places$)$

Call options expiring in one year is priced using a $3-$period binomial tree. You are given:

$($i$)u=1\mathrm{.}2,d=0\mathrm{.}8$

$($ii$)$ The current stock price is $1000.$

$($iii$)$ The continuously compounded dividend yield of the underlying stock is $0\mathrm{.}07.$

$($iv$)$ The continuously compounded risk $-$free interest rate is $0\mathrm{.}09$

Find

$(1)$ The premium of at$-$the$-$money European call option. $(2$pts$)$

$(2)$ The premium of at$-$the$-$money American call option. $(2$pts$)$

$(3)$ Is there early exercise in the option? If yes, find the early exercise node. What is the early exercise premium $($the difference between American option and the European option$).(2$pts$)$

$(4)$ If you are selling ten American call option at node u$,$ you use replicated portfolio to hedge your shorting, how many stocks and bonds you need to buy$/$sell in your hedging at time h $($for node $u?(2$pts$)$