Problem $1.$

Consider a European call option on a non$-$dividend$-$paying stock where the stock price is $$40,$ the

strike price is $$40,$ the $($annualized$)$ risk$-$free rate is $4\%$ per year, the $($annualized$)$ volatility is $30\%,$

and the time to maturity is six months.

Calculate $u,d,$ and $q$ for a two timestep tree $($note: assume that the tree is "recombining"; thus,

$u=e^{\sqrt[\phantom{2}]{t}}$ and $d=e^{-\sqrt[\phantom{2}]{t}}.$

Using risk neutral valuation, price this call option using a two timestep tree.

Verify that the "Cox$-$Ross$-$Rubinstein compared with Black$-$Scholes$-$Merton spreadsheet" pro$-$

vides the same answer $($Important note: This spreadsheet includes Visual Basic macros. Upon

opening, you will receive a prompt to disable or enable macros; select "Enable Macros" to ensure

the full functionality of the spreadsheet for this assignment$).$

Use the "Cox$-$Ross$-$Rubinstein compared with Black$-$Scholes$-$Merton spreadsheet" to calculate

the price of this call option with $5,50,100,$ and $500$ timesteps.