175 lines
4.4 KiB
TeX
175 lines
4.4 KiB
TeX
\section{Call Options}
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\definition{}
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A \textit{call option} is an agreement between a buyer (B) and a seller (S): \par
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\begin{contract}[frametitle={Contract: Call Option}]
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B pays S a premium $p$. \par
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In return, S agrees to sell B a certain stock $\mathbb{X}$ for a fixed \say{strike price} $k$ at a future time $t$.
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\end{contract}
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\problem{}<firstcall>
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B has ten call options for $\mathbb{X}$ at $23\Rub$. The current price of $\mathbb{X}$ is $20\Rub$. \par
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How much profit can B make if these contracts expire when $\mathbb{X}$ is worth $30\Rub$? \par
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\hint{When the contract expires, B can buy 10 shares of $\mathbb{X}$ at the price the contract set.}
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\begin{solution}
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B has the right to buy 10 shares of $\mathbb{X}$ at $23\Rub$. \par
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If B immediately sells them, his profit is $-230 + 300 = 70\Rub$
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\end{solution}
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\vfill
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\problem{}
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If B paid $10\Rub$ for the call options in \ref{firstcall}, how much money did he really make?
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\begin{solution}
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$-10 + (-230 + 300) = 60\Rub$
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\end{solution}
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\vfill
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\problem{}
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Now, suppose that B bought and sold $\mathbb{X}$ directly instead of using a call option. \par
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How much profit would B have made?
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\begin{solution}
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Buy for $200\Rub$, sell for $300\Rub$.\par
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$-200 + 300 = 100\Rub$
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\end{solution}
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\vfill
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\pagebreak
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\problem{}
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Suppose $\mathbb{X}$ is worth $x_0$ right now. \par
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Call options to buy $\mathbb{X}$ at strike price $k$ are sold for $p$.
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\begin{itemize}
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\item What is the set of B's possible profit if..
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\begin{itemize}
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\item B buys a call option?
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\item B buys $\mathbb{X}$ directly?
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\end{itemize}
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\hint{That is, what amounts of money can B make (or lose)?}
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\item Are call options priced above or below the price of their stock? Why?
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\item On the previous page, we saw that the profit
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made on a call option was much lower than the profit
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made by buying a stock directly.
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Why would anybody buy a call option?
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\end{itemize}
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\begin{solution}
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\textbf{Call Option:} $[p, \infty)$ \par
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If the price of $\mathbb{X}$ rises, there is no limit to how much money B can make. \par
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If the price falls, $B$ can choose to let his contract expire, losing only $p$.
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\vspace{2mm}
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\textbf{Direct:} $[x_0, \infty)$\par
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If the price of $\mathbb{X}$ rises, there is again no limit to how much money B can make. \par
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If the price falls, $B$ will lose everything he paid for his shares of $\mathbb{X}$.
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\vspace{2mm}
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Of course, call options are priced below their stock. There wouldn't be a reason to buy then
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if they were priced above!
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\end{solution}
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\vfill
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\problem{}
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Suppose $\mathbb{X}$ is worth $x_0$ right now. \par
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Call options to buy $\mathbb{X}$ for $k$ are sold for $p$. \par
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\vspace{2mm}
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Assume that S owns no stock---if B executes his contracts, she will buy stock and resell it to him. \par
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What are S's possible profits if she sells B a call option?
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\begin{solution}
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$(-\infty, ~p]$\par
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If the price of $\mathbb{X}$ rises, S will have to resell shares to B at a loss.
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If the price falls, B could choose to buy shares from S at a loss, but he won't.
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In this case, S only keeps the premium B paid for the contract.
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\end{solution}
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\vfill
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\pagebreak
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\problem{}
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How does the price of $\mathbb{X}$ at $t$ relate to the amount of
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profit B and S make? Complete the plots below.
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\null\hfill
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\begin{minipage}{0.48\textwidth}
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\begin{center}
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\begin{tikzpicture}
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\draw (0,0) -- (5, 0);
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\draw (0,-2) -- (0, 2);
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\node at (2.5, 2) {Profit plot for $B$};
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\node[
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anchor = south,
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rotate = 90
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] at (0,0) {\color{gray}Profit};
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\node[
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anchor = south west,
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] at (0, 0) {\color{gray}Price of $\mathbb{X}$ at $t$};
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\node[anchor = north] at (3, 0) {$k$};
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\filldraw (3, 0) circle (0.5mm);
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}{0.48\textwidth}
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\begin{center}
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\begin{tikzpicture}
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\draw (0,0) -- (5, 0);
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\draw (0,-2) -- (0, 2);
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\node at (2.5, 2) {Profit plot for $S$};
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\node[
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anchor = south,
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rotate = 90
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] at (0,0) {\color{gray}Profit};
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\node[
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anchor = south west,
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] at (0, 0) {\color{gray}Price of $\mathbb{X}$ at $t$};
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\node[anchor = north] at (3, 0) {$k$};
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\filldraw (3, 0) circle (0.5mm);
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\hfill\null
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When does B make a positive profit? When does S? \par
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Write an equation that calculates S and B's earnings given
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$p$, $k$, and the price of $\mathbb{X}$ at the time the contract expires.
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\vfill
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\pagebreak |