$($Ch $12)(25$ points$)$ The volatility of a non$-$dividend$-$paying stock, whose price is $$78,$ is $30\%.$

The risk$-$free rate is $3\%$ per annum $($continuously compounded$)$ for all maturities.

$($a$)$ Calculate values for $u,d,$ and $p$ when a two$-$month time step is used. $(6$ points$)$

$($b$)$ What is the meaning of $p?(2$ points$)$

$($c$)$ What is the value of a four$-$month European call option with a strike price of $$80$ given by a two$-$step binomial tree? $(7$ points$)$

$($d$)$ Continue from part $($c$),$ and suppose a trader sells $10,000$ call options. What position in the stock is necessary to hedge the trader's position at the time of the trade initiated? $(4$ points$)$

$($e$)$ Continuing from part $($d$),$ how many shares of the stock are needed to hedge the position for the second two$-$month period? Consider both the case where the stock price moves up during the first period and the case where it moves down during the first period. $(6$ points$)$