116 lines
3.0 KiB
TeX
116 lines
3.0 KiB
TeX
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\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 commodity $\mathbb{X}$ for a fixed 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 $30\Rub$? \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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Given the results of the previous problems, why would anybody buy a call option?
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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 $k$ 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 he 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 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 $\mathbb{X}$ at $k$ 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 re-sell 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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$(-\infty, ~p]$
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If the price of $\mathbb{X}$ rises, S will have to re-sell shares to B at a loss. \par
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If the price falls, B could choose to buy shares from S at a loss, but he won't. \par
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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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