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Added finance sections

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2
Advanced/Options in Finance/main.tex
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@ -86,5 +86,7 @@
\input{parts/0 intro}
\input{parts/1 call}
\input{parts/2 put}
\input{parts/3 compound}
\end{document}

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Advanced/Options in Finance/parts/0 intro.tex
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@ -1,5 +1,19 @@
\section{Introduction}
\definition{}
An \textit{asset} is any resource that has economic value.\par
Examples: gold, oil, grain, cash, real estate, treasury bonds, etc
\definition{}
A \textit{stock} is a particular type of asset.
A share of stock represents \say{partial ownership} of a corporation.
Like many assets, stocks are \textit{intangible}---they only exist on paper.
\problem{}
Let $\mathbb{X}$ be a stock, currently priced at $19\Rub$. \par
Bogdan buys 10 shares of $\mathbb{X}$, and sells them after a month for $23\Rub$ per share. \par
What was his net profit?
\vfill
\pagebreak

87
Advanced/Options in Finance/parts/1 call.tex
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@ -5,7 +5,7 @@ A \textit{call option} is an agreement between a buyer (B) and a seller (S): \pa
\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 $t$.
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}
@ -13,7 +13,7 @@ A \textit{call option} is an agreement between a buyer (B) and a seller (S): \pa
\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
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}
@ -51,14 +51,12 @@ How much profit would B have made?
\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 $p$.
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..
@ -66,10 +64,13 @@ Call options to buy $\mathbb{X}$ at $k$ are sold for $p$.
\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)?}
\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 Why would anybody buy a call option?
\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}
@ -96,21 +97,79 @@ Call options to buy $\mathbb{X}$ at $k$ are sold for $p$.
\problem{}
Suppose $\mathbb{X}$ is worth $x_0$ right now. \par
Call options to buy $\mathbb{X}$ at $k$ are sold for $p$. \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 re-sell it to him. \par
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}
$(-\infty, ~p]$
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
$(-\infty, ~p]$\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

55
Advanced/Options in Finance/parts/2 put.tex Normal file
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@ -0,0 +1,55 @@
\section{Put Options}
\definition{}
A \textit{put option} is an agreement between a buyer (B) and a seller (S): \par
\begin{contract}[frametitle={Contract: Put Option}]
B pays S a premium $p$. \par
In return, S agrees to buy a certain stock $\mathbb{X}$ from S for a fixed \say{strike price} $k$ at a future time $t$,
if B decides to exercise this contract.
\end{contract}
As before, the \textbf{buyer} decides whether or not this contract is put into action. \par
Also, note that B does not need to own any shares of stock to buy a put option. \par
He may buy them whenever he wishes.
\problem{}
How is a put different from a call? \par
What is S betting on? What is B betting on?
\vfill
\problem{}
Suppose B paid $100\Rub$ for 300 put contracts on $\mathbb{X}$ at $17\Rub$.\par
At time the contracts expired, the price of $\mathbb{X}$ was $20\Rub$.\par
What is B's profit?
\vfill
\problem{}
Plot profit curves for selling a put option, buying a put option,
and buying a stock directly on the axis below.
\begin{center}
\begin{tikzpicture}
\draw (0,0) -- (10, 0);
\draw (0,-3) -- (0, 3);
\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 (6, 0) {$k$};
\filldraw (6, 0) circle (0.5mm);
\end{tikzpicture}
\end{center}
\vfill
\pagebreak

76
Advanced/Options in Finance/parts/3 compound.tex Normal file
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@ -0,0 +1,76 @@
\section{Compound Strategies}
\definition{}
A \textit{covered call} is a trading strategy where one simultaneously
buys a share of stock and sells a call option. When the contract
expires, the stock is sold to the call buyer (if they choose
to exercise their contract) or to the market (if they don't).
\problem{}
Say we set up a covered call by buying a share of $\mathbb{X}$ for $x_0$
and selling a call option for $\mathbb{X}$ at $k$ for $p$. \par
When our contract expires, $\mathbb{X}$
is worth $x_1$.
\vspace{2mm}
What is the gross profit of a covered call?\par
What is its net profit?\par
\hint{Gross profit does not take setup cost into account. Net profit does.}
\vfill
\definition{}
We say that trading strategy $A$ \textit{simulates} trading strategy
$B$ if their net profits are equal.
\problem{}
Find a trading strategy that buys stock and call options
to simulate a single put option with strike price $k$.
\vfill
\problem{}
A \textit{straddle} is a trading strategy where one buys a call and a put
with the same strike price and expiration. Plot the profit curve. \par
What do you bet on when you buy a straddle?
\begin{center}
\begin{tikzpicture}
\draw (0,0) -- (10, 0);
\draw (0,-3) -- (0, 3);
\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 (5, 0) {$k$};
\filldraw (5, 0) circle (0.5mm);
\end{tikzpicture}
\end{center}
\vfill
\pagebreak
\definition{}
A \textit{butterfly spread} is a trading strategy where one buys two
calls with strike prices $k_1$ and $k_2$ and sells two calls with strike
prices $\frac{k_1+k_2}{2}$.
\problem{}
When should you set up a butterfly spread? \par
Find the payoff function.
\vfill
\vfill
\pagebreak