Removed Options handout
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% use [nosolutions] flag to hide solutions.
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% use [solutions] flag to show solutions.
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\documentclass[
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solutions,
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unfinished
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]{../../resources/ormc_handout}
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\usepackage{../../resources/macros}
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\usepackage{mdframed}
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\newmdenv[
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topline=false,
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bottomline=false,
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rightline=true,
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leftline=true,
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linewidth=0.3mm,
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frametitle={Contract:},
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frametitlefont={\textsc},
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%
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skipabove=1mm,
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skipbelow=1mm,
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%
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innerleftmargin=2mm,
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innerrightmargin=4mm,
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leftmargin=2mm,
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rightmargin=2mm,
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]{contract}
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\uptitlel{Advanced 2}
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\uptitler{Fall 2023}
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\title{Options in Finance}
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\subtitle{
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Prepared by \githref{Mark} on \today{}
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}
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\begin{document}
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\maketitle
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\input{parts/0 intro}
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\input{parts/1 call}
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\input{parts/2 put}
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\input{parts/3 compound}
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\end{document}
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@ -1,19 +0,0 @@
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\section{Introduction}
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\definition{}
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An \textit{asset} is any resource that has economic value.\par
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Examples: gold, oil, grain, cash, real estate, treasury bonds, etc
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\definition{}
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A \textit{stock} is a particular type of asset.
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A share of stock represents \say{partial ownership} of a corporation.
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Like many assets, stocks are \textit{intangible}---they only exist on paper.
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\problem{}
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Let $\mathbb{X}$ be a stock, currently priced at $19\Rub$. \par
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Bogdan buys 10 shares of $\mathbb{X}$, and sells them after a month for $23\Rub$ per share. \par
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What was his net profit?
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\vfill
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\pagebreak
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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 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
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\section{Put Options}
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\definition{}
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A \textit{put option} is an agreement between a buyer (B) and a seller (S): \par
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\begin{contract}[frametitle={Contract: Put Option}]
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B pays S a premium $p$. \par
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In return, S agrees to buy a certain stock $\mathbb{X}$ from S for a fixed \say{strike price} $k$ at a future time $t$,
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if B decides to exercise this contract.
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\end{contract}
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As before, the \textbf{buyer} decides whether or not this contract is put into action. \par
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Also, note that B does not need to own any shares of stock to buy a put option. \par
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He may buy them whenever he wishes.
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\problem{}
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How is a put different from a call? \par
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What is S betting on? What is B betting on?
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\vfill
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\problem{}
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Suppose B paid $100\Rub$ for 300 put contracts on $\mathbb{X}$ at $17\Rub$.\par
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At time the contracts expired, the price of $\mathbb{X}$ was $20\Rub$.\par
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What is B's profit?
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\vfill
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\problem{}
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Plot profit curves for selling a put option, buying a put option,
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and buying a stock directly on the axis below.
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\begin{center}
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\begin{tikzpicture}
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\draw (0,0) -- (10, 0);
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\draw (0,-3) -- (0, 3);
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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 (6, 0) {$k$};
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\filldraw (6, 0) circle (0.5mm);
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\end{tikzpicture}
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\end{center}
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\vfill
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\pagebreak
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\section{Compound Strategies}
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\definition{}
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A \textit{covered call} is a trading strategy where one simultaneously
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buys a share of stock and sells a call option. When the contract
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expires, the stock is sold to the call buyer (if they choose
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to exercise their contract) or to the market (if they don't).
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\problem{}
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Say we set up a covered call by buying a share of $\mathbb{X}$ for $x_0$
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and selling a call option for $\mathbb{X}$ at $k$ for $p$. \par
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When our contract expires, $\mathbb{X}$
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is worth $x_1$.
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\vspace{2mm}
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What is the gross profit of a covered call?\par
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What is its net profit?\par
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\hint{Gross profit does not take setup cost into account. Net profit does.}
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\vfill
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\definition{}
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We say that trading strategy $A$ \textit{simulates} trading strategy
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$B$ if their net profits are equal.
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\problem{}
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Find a trading strategy that buys stock and call options
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to simulate a single put option with strike price $k$.
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\vfill
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\problem{}
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A \textit{straddle} is a trading strategy where one buys a call and a put
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with the same strike price and expiration. Plot the profit curve. \par
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What do you bet on when you buy a straddle?
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\begin{center}
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\begin{tikzpicture}
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\draw (0,0) -- (10, 0);
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\draw (0,-3) -- (0, 3);
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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 (5, 0) {$k$};
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\filldraw (5, 0) circle (0.5mm);
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\end{tikzpicture}
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\end{center}
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\vfill
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\pagebreak
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\definition{}
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A \textit{butterfly spread} is a trading strategy where one buys two
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calls with strike prices $k_1$ and $k_2$ and sells two calls with strike
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prices $\frac{k_1+k_2}{2}$.
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\problem{}
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When should you set up a butterfly spread? \par
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Find the payoff function.
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\vfill
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\vfill
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\pagebreak
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