$2.$ Suppose someone longs a $$60$ stock when the risk$-$free rate is $6$ percent. The stock is held for $3$ months and can go up by $15\%$ of its value every month.

a$.$ If the investor purchased a call option with a $$70$ strike price, solve for the call option price using the binomial pricing model. Be sure to sketch the binomial lattice. SHOW YOUR WORK.

b$.$ If the investor purchased a put option also with a $$70$ strike price, solve for the put option price using the binomial pricing model. Be sure to sketch the binomial lattice.

c$.$ Compute the call gamma and the put gamma, using the numbers you have obtained from period two options. SHOW YOUR WORK.

d$.$ If the stock price changes in period $3,$ is the option$$s moneyness likely change from period $2?$ Explain. e$.$ Use the put$-$call parity relationship to test the binomial pricing model. SHOW YOUR WORK.