ABC$$s stock price is $$90.$ The stock price will either increase $10\%$ or decrease $8\%$ in the first month. If the price increases in the first month, it will go up by $$12$ or down by $$15$ in the second month. If the price decreases in the first year, it will go up by $$16$ or down by $$22$ in the second month. The monthly risk$-$ free interest rate is $3\%$ in the first month and $5\%$ in the second month.

a$)$ Using the replicating portfolio approach, calculate the value of a $2-$month ATM put option.

b$)$ Using the risk neutral probability approach, calculate the value of a $2-$month call option with a strike price of $$92.$

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