Question $4$: Option valuation

Assume that the anticipated future dividends from PayNoDividendsPal is zero $($that is$,$ all positive earnings are plowed back into the firm$).$

a$)$ Compute the value of a European call option written on that stock if

$$ current share price $($S$0)$ is $$50$

$$ exercise or strike price $($K$)$ is $$50$

$$ stock has a standard deviation $($volatility$)$ of $25\%$ per year

$$ call option matures in $4$ months $($T$)$

$$ risk$-$free interest rate per annum is $3\mathrm{.}5\%$

b$)$ How large is the time value of the option? Is the option $($deep$)$ in$-$the$-$money, at$-$themoney, or $($deep$)$ out$-$of$-$the money? Motivate your answers.

c$)$ Draw a figure showing the payoff and profit diagram of the option position for different stock prices $($of PayNoDividendsPal$).$ Clearly mark the break$-$even price. The figure can be a simple but conceptually correct illustration that you can draw by hand.

d$)$ On the same diagram as in $($c$),$ draw a figure that shows the value of the call option for different stock prices $($of PayNoDividendsPal$).$ The figure can be a simple but conceptually correct illustration that you can draw by hand.