The current stock price stands at $$20.$ Over the next three$-$month period, it's anticipated to either increase by $10\%$ or decrease by $10\%.$ The risk$-$free interest rate is $10\%$ per year with quarterly compounding.

Part I.

Utilizing a one$-$step binomial tree model, determine the fair price of a derivative contract with a payoff given by S$\_$T$^2$ after three months, where S$\_$T represents the stock price at maturity.