Question #$3$: Option Valuation $($Multi$-$State Approach$)[30$ Points$]$

Below is a two$-$period price tree for a share of stock in Qualys, Inc. $($QLYS$)$

$$171\mathrm{.}48$

Using the binomial model, calculate the value today of a call option on CYBR stock with a strike price of $$180.$ You may assume that the $6-$month risk$-$free rate is $3\mathrm{.}5\%.$

To solve this problem you will need to break$-$up the problem into a number of $2$ state models and work backwards. Let's go through each of the steps.

$($a$)$ Calculate the price of the option $6$ months from today $(C_U)$ if the stock price were to increase to $$209$ in $6$ months. $[10$ Points$]$

$($b$)$ Calculate the price of the option $6$ months from today $(C_d)$ if the stock price were to decrease to $$180\mathrm{.}50$ in $6$ months. Note: When calculating the hedge ratio round to $4$ decimal places. $[10$ Points$]$

$($c$)$ Calculate the price of the option today $($C$).($Hint: The value of the call $6$ months from now will be either be $C_u($the answer you found in Part $($a$))$ or $C_d($the answer you found in Part $($b$)).$ Note: When calculating the hedge ratio round to $4$ decimal places. $[10$ Points$]$