From 7301f7c8c369b6143b8a458ca5b0433181c6dde0 Mon Sep 17 00:00:00 2001 From: mark Date: Tue, 24 Oct 2023 17:15:47 -0700 Subject: [PATCH] Added finance sections --- Advanced/Options in Finance/main.tex | 2 + Advanced/Options in Finance/parts/0 intro.tex | 14 +++ Advanced/Options in Finance/parts/1 call.tex | 87 ++++++++++++++++--- Advanced/Options in Finance/parts/2 put.tex | 55 ++++++++++++ .../Options in Finance/parts/3 compound.tex | 76 ++++++++++++++++ 5 files changed, 220 insertions(+), 14 deletions(-) create mode 100644 Advanced/Options in Finance/parts/2 put.tex create mode 100644 Advanced/Options in Finance/parts/3 compound.tex diff --git a/Advanced/Options in Finance/main.tex b/Advanced/Options in Finance/main.tex index e460da8..ac19364 100755 --- a/Advanced/Options in Finance/main.tex +++ b/Advanced/Options in Finance/main.tex @@ -86,5 +86,7 @@ \input{parts/0 intro} \input{parts/1 call} + \input{parts/2 put} + \input{parts/3 compound} \end{document} \ No newline at end of file diff --git a/Advanced/Options in Finance/parts/0 intro.tex b/Advanced/Options in Finance/parts/0 intro.tex index 62ff2e5..248e08d 100644 --- a/Advanced/Options in Finance/parts/0 intro.tex +++ b/Advanced/Options in Finance/parts/0 intro.tex @@ -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 \ No newline at end of file diff --git a/Advanced/Options in Finance/parts/1 call.tex b/Advanced/Options in Finance/parts/1 call.tex index 406b3a3..b33af94 100644 --- a/Advanced/Options in Finance/parts/1 call.tex +++ b/Advanced/Options in Finance/parts/1 call.tex @@ -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{} 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 \ No newline at end of file diff --git a/Advanced/Options in Finance/parts/2 put.tex b/Advanced/Options in Finance/parts/2 put.tex new file mode 100644 index 0000000..dfe5d91 --- /dev/null +++ b/Advanced/Options in Finance/parts/2 put.tex @@ -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 + diff --git a/Advanced/Options in Finance/parts/3 compound.tex b/Advanced/Options in Finance/parts/3 compound.tex new file mode 100644 index 0000000..09a68f2 --- /dev/null +++ b/Advanced/Options in Finance/parts/3 compound.tex @@ -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 +