\section{Call Options}

\definition{}
A \textit{call option} is an agreement between a buyer (B) and a seller (S): \par

\begin{contract}[frametitle={Contract: Call Option}]
	B pays S a premium $p$. \par
	In return, S agrees to sell B a certain stock $\mathbb{X}$ for a fixed \say{strike price} $k$ at a future time $t$.
\end{contract}




\problem{}<firstcall>
B has ten call options for $\mathbb{X}$ at $23\Rub$. The current price of $\mathbb{X}$ is $20\Rub$. \par
How much profit can B make if these contracts expire when $\mathbb{X}$ is worth $30\Rub$? \par
\hint{When the contract expires, B can buy 10 shares of $\mathbb{X}$ at the price the contract set.}

\begin{solution}
	B has the right to buy 10 shares of $\mathbb{X}$ at $23\Rub$. \par
	If B immediately sells them, his profit is $-230 + 300 = 70\Rub$
\end{solution}


\vfill



\problem{}
If B paid $10\Rub$ for the call options in \ref{firstcall}, how much money did he really make?

\begin{solution}
	$-10 + (-230 + 300) = 60\Rub$
\end{solution}


\vfill




\problem{}
Now, suppose that B bought and sold $\mathbb{X}$ directly instead of using a call option. \par
How much profit would B have made?

\begin{solution}
	Buy for $200\Rub$, sell for $300\Rub$.\par
	$-200 + 300 = 100\Rub$
\end{solution}



\vfill
\pagebreak


\problem{}
Suppose $\mathbb{X}$ is worth $x_0$ right now. \par
Call options to buy $\mathbb{X}$ at strike price $k$ are sold for $p$.

\begin{itemize}
	\item What is the set of B's possible profit if..
	\begin{itemize}
		\item B buys a call option?
		\item B buys $\mathbb{X}$ directly?
	\end{itemize}
	\hint{That is, what amounts of money can B make (or lose)?}

	\item Are call options priced above or below the price of their stock? Why?
	\item On the previous page, we saw that the profit
	made on a call option was much lower than the profit
	made by buying a stock directly.
	Why would anybody buy a call option?

\end{itemize}


\begin{solution}
	\textbf{Call Option:} $[p, \infty)$ \par
	If the price of $\mathbb{X}$ rises, there is no limit to how much money B can make. \par
	If the price falls, $B$ can choose to let his contract expire, losing only $p$.

	\vspace{2mm}

	\textbf{Direct:} $[x_0, \infty)$\par
	If the price of $\mathbb{X}$ rises, there is again no limit to how much money B can make. \par
	If the price falls, $B$ will lose everything he paid for his shares of $\mathbb{X}$.

	\vspace{2mm}

	Of course, call options are priced below their stock. There wouldn't be a reason to buy then
	if they were priced above!
\end{solution}


\vfill

\problem{}
Suppose $\mathbb{X}$ is worth $x_0$ right now. \par
Call options to buy $\mathbb{X}$ for $k$ are sold for $p$. \par

\vspace{2mm}
Assume that S owns no stock---if B executes his contracts, she will buy stock and resell it to him. \par
What are S's possible profits if she sells B a call option?

\begin{solution}
	$(-\infty, ~p]$\par
	If the price of $\mathbb{X}$ rises, S will have to resell shares to B at a loss.
	If the price falls, B could choose to buy shares from S at a loss, but he won't.
	In this case, S only keeps the premium B paid for the contract.
\end{solution}

\vfill
\pagebreak

\problem{}
How does the price of $\mathbb{X}$ at $t$ relate to the amount of
profit B and S make? Complete the plots below.

\null\hfill
\begin{minipage}{0.48\textwidth}
	\begin{center}
		\begin{tikzpicture}
			\draw (0,0) -- (5, 0);
			\draw (0,-2) -- (0, 2);

			\node at (2.5, 2) {Profit plot for $B$};


			\node[
				anchor = south,
				rotate = 90
			] at (0,0) {\color{gray}Profit};

			\node[
				anchor = south west,
			] at (0, 0) {\color{gray}Price of $\mathbb{X}$ at $t$};

			\node[anchor = north] at (3, 0) {$k$};
			\filldraw (3, 0) circle (0.5mm);
		\end{tikzpicture}
	\end{center}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
	\begin{center}
		\begin{tikzpicture}
			\draw (0,0) -- (5, 0);
			\draw (0,-2) -- (0, 2);

			\node at (2.5, 2) {Profit plot for $S$};

			\node[
				anchor = south,
				rotate = 90
			] at (0,0) {\color{gray}Profit};

			\node[
				anchor = south west,
			] at (0, 0) {\color{gray}Price of $\mathbb{X}$ at $t$};

			\node[anchor = north] at (3, 0) {$k$};
			\filldraw (3, 0) circle (0.5mm);
		\end{tikzpicture}
	\end{center}
\end{minipage}
\hfill\null

When does B make a positive profit? When does S? \par
Write an equation that calculates S and B's earnings given
$p$, $k$, and the price of $\mathbb{X}$ at the time the contract expires.


\vfill
\pagebreak