$1)$ S $=52\mathrm{.}50,$ X $=52\mathrm{.}50,$ r $=4\%,$ T $=3$ months, and $=\mathrm{.}15.($a$)$ Find the values of u and d for the one$-$step CRR model. $($b$)$ Draw the one step lattice identifying the security prices and the values for Cu and Cd$.($C$)$ Determine the probabilities and the price of the call option. $($D$)$ Using Put$-$Call Parity, find the value of P$.$

#$2$ S $=52\mathrm{.}50,$ X $=52\mathrm{.}50,$ r $=4\%,$ T $=3$ months, and $=\mathrm{.}15.($a$)$ Find the values of u and d for the two$-$step CCR model. $($B$)$ Draw the two$-$step lattice identifying the security prices and the values for Cuu, Cud, and Cdd$.($C$)$ Determine the probabilities and the price of the call option. $($D$)$ Using Put$-$Call Parity, find the value of P$.$

#$3$ S $=47\mathrm{.}50,$ X $=52\mathrm{.}50,$ r $=4\%,$ T $=3$ months, and $=\mathrm{.}15.($a$)$ Do the values of u and d change for either the one$-$step or two$-$step models? $($b$)$ Do the probabilities change? $($C$)$ For the one$-$step model, find C$.($D$)$ For the two$-$step model, Find C $($E$)$ Why does C have zero value for the one$-$step model but positive value for the two$-$step model?

#$4$ S $=47\mathrm{.}50,$ X $=50,$ r $=4\%,$ T $=3$ months, and $=\mathrm{.}15.($a$)$ Using the Black Scholes Model, find the values of C and P$.($B$)$ If the market price of the call of option is C $=0\mathrm{.}48,$ what volatility is being used? $($C$)$ What are the hedge ratios $($deltas$)$ for the call and put options? What do they mean? $$Note:$$This is called a split$-$strike strategy.