ABC$$s stock is trading at $$63$ per share. The stock price will either increase or decrease by $$10$ in

the first month. If the price increases in the first month, it will go up by $20\%$ or down by $10\%$ in

the second month. If the price decreases in the first month, it will go up by $40\%$ or down by

$20\%$ in the second month. The monthly risk$-$free interest rate is $2\%$ for the first month and $3\%$

for the second month.

a$)$ Using the replicating portfolio approach, calculate the value of a $1-$month ATM put option.

b$)$ Using the risk neutral probability approach, calculate the value of a $2-$month ATM call

option.