Current Stock price $S(0)$ is $$60$; risk$-$free rate is $3\%($simple rate$).$ There are two possible outcome in one year, stock price might go up $40\%($to $$84)$ or go down $40\%($to $$36).$ Consider both a call and a put with the same strike price of $$60$ and one year to maturity.

a$.$ You estimate that the probability of stock price to go up is $90\%$ and probability to go down is about $10\%.$ Compute the price of call and put and also expected returns on the stock, call, put and a straddle $($combination of a call and put with same strike of $$60)$ using the binomial tree model. $(6)$

b$.$ However you are not sure about your estimates of the probabilities and decide to check out your results under different scenarios. You write down the following scenarios $(P$ is the probability of stock price to go up to $$84)$: $(18)$

$\backslash$table$[[$Prob$,1-$Prob,call price,put price,$\backslash$table$[[$Stock$],[$return$]],\backslash$table$[[$Call$],[$return$]],$Put return,$\backslash$table$[[$Straddle$],[$return$]]],[0\mathrm{.}1,0\mathrm{.}9,,,,,,],[0\mathrm{.}3,0\mathrm{.}7,,,,,,],[0\mathrm{.}5,0\mathrm{.}5,,,,,,],[0\mathrm{.}5375,0\mathrm{.}4625,,,,,,],[0\mathrm{.}7,0\mathrm{.}3,,,,,,],[0\mathrm{.}8,0\mathrm{.}2,,,,,,],[0\mathrm{.}9,0\mathrm{.}1,,,,,,]]$

Optional question: What is your intuition on the expected returns on the call, the put and the straddle. $(2)$