Added first section of option handout

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Mark 2023-10-24 10:21:56 -07:00
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% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[
solutions
]{../../resources/ormc_handout}
\usepackage{../../resources/macros}
\usepackage{mdframed}
\usepackage{pgf}
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\newmdenv[
topline=false,
bottomline=false,
rightline=true,
leftline=true,
linewidth=0.3mm,
frametitle={Contract:},
frametitlefont={\textsc},
%
skipabove=1mm,
skipbelow=1mm,
%
innerleftmargin=2mm,
innerrightmargin=4mm,
leftmargin=2mm,
rightmargin=2mm,
]{contract}
\uptitlel{Advanced 2}
\uptitler{Fall 2023}
\title{Options in Finance}
\subtitle{
Prepared by \githref{Mark} on \today{}
}
\begin{document}
\maketitle
\input{parts/0 intro}
\input{parts/1 call}
\end{document}

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\section{Introduction}
\vfill
\pagebreak

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\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 commodity $\mathbb{X}$ for a fixed 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 $30\Rub$? \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
Given the results of the previous problems, why would anybody buy a call option?
\pagebreak
\problem{}
Suppose $\mathbb{X}$ is worth $x_0$ right now. \par
Call options to buy $\mathbb{X}$ at $k$ 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 he make (or lose)?}
\item Are call options priced above or below the price of their stock? Why?
\item 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 $\mathbb{X}$ at $k$ are sold for $p$. \par
\vspace{2mm}
Assume that S owns no stock---if B executes his contracts, she will buy stock and re-sell it to him. \par
What are S's possible profits if she sells B a call option?
\begin{solution}
$(-\infty, ~p]$
If the price of $\mathbb{X}$ rises, S will have to re-sell shares to B at a loss. \par
If the price falls, B could choose to buy shares from S at a loss, but he won't. \par
In this case, S only keeps the premium B paid for the contract.
\end{solution}
\vfill
\pagebreak