For a stock, you are given:

$($i$)$ The current stock price is $50.$

$($ii$)$ At the end of three months the stock price will be either $45$ or $55.$

$($iii$)$ The stock pays dividends at a rate proportional to its price. The dividend yield is $4\%.$

$($iv$)$ The continuously compounded risk$-$free interest rate is $3\%.$

Consider a $3-$month $53-$strike European put on the stock.

$(1)$ Find $$ for the put option.

$(2)$ Apply replicating portfolio method and risk$-$neutral method to calculate the time$-0$ price of

the put option.