Consider a binomial world in which a stock can go up in value by $35\%$ or down by $15\%$ over the next year. The stock is currently trading at $$45.$

The annual risk$-$free rate is $5\%.$

A$.(*)$ What is the risk$-$neutral probability that the stock will go up over the next year?

$\%$

$($Use at least four decimals for intermediate calculations, and write your final answers in percentage with two decimals$)$

B$.{(}^{**})$ What is the value of a Call option that expires in one year and has a strike price of $$55?$ $

$($Use at least four decimals for intermediate calculations, and write your final answers with two decimals$)$

C$.(*)$ What is the value of a Put option that expires in one year and has a strike price of $$55?$ $

$($Use at least four decimals for intermediate calculations, and write your final answers with two decimals$)$

D$.$ Suppose we know that the true $($not risk$-$neutral$)$ probability $q$ that the stock will go up is $42\%,$ then the $($annual$)$ expected return of the

stock $r_S$ can be computed as

$45=\frac{0\mathrm{.}42(451\mathrm{.}35)+0\mathrm{.}58(450\mathrm{.}85)}{1+r_S}=>r_S=6\%$

What is the $($annual$)$ expected return of the Call option $r_C?$

$\%$

$($Use at least four decimals for intermediate calculations, and write your final answers in percentage with two decimals$)$

What is the $($annual$)$ expected return of the Put option $r_P?$

$\%$

$($Use at least four decimals for intermediate calculations, and write your final answers in percentage with two decimals$)$