From ea3ef136fa52ae83e6aa2f211369b365757dfecf Mon Sep 17 00:00:00 2001 From: Mark Date: Tue, 24 Oct 2023 10:21:56 -0700 Subject: [PATCH] Added first section of option handout --- Advanced/Options in Finance/main.tex | 90 ++++++++++++++ Advanced/Options in Finance/parts/0 intro.tex | 5 + Advanced/Options in Finance/parts/1 call.tex | 116 ++++++++++++++++++ 3 files changed, 211 insertions(+) create mode 100755 Advanced/Options in Finance/main.tex create mode 100644 Advanced/Options in Finance/parts/0 intro.tex create mode 100644 Advanced/Options in Finance/parts/1 call.tex diff --git a/Advanced/Options in Finance/main.tex b/Advanced/Options in Finance/main.tex new file mode 100755 index 0000000..e460da8 --- /dev/null +++ b/Advanced/Options in Finance/main.tex @@ -0,0 +1,90 @@ +% use [nosolutions] flag to hide solutions. +% use [solutions] flag to show solutions. +\documentclass[ + solutions +]{../../resources/ormc_handout} +\usepackage{../../resources/macros} + +\usepackage{mdframed} +\usepackage{pgf} + +% Ruble symbol with tweaks +\DeclareRobustCommand*{\Rub}{% + \begingroup + \dimendef\H=0 % + \settoheight\H{P}% + \begin{pgfpicture}% + \pgfsetlinewidth{.1\H}% + \pgfsetrectcap + \pgfsetmiterjoin + \pgfmoveto{\pgfpoint{0pt}{0.07\H}}% Move to bottom of main line + \pgflineto{\pgfpoint{0pt}{.90\H}}% P main line + \pgflineto{\pgfpoint{.3\H}{.90\H}}% P top line + \pgfpatharc{90}{-90}{.21\H}% P circle + \pgflineto{\pgfpoint{-.08\H}{.48\H}}% P bottom line + \pgfmoveto{\pgfpoint{-.08\H}{.31\H}}% Bonus line move + \pgflineto{\pgfpoint{.34\H}{.31\H}}% Bonus line draw + \pgfusepath{stroke}% + \pgfmoveto{\pgfpoint{-.23\H}{0pt}}% Before space + \pgfmoveto{\pgfpoint{0.55\H}{0pt}}% After space + \end{pgfpicture}% + \endgroup +} + +% Ruble symbol, per official Kremlin specification +% +%\DeclareRobustCommand*{\Rub}{% +% \begingroup +% \dimendef\H=0 % +% \settoheight\H{P}% +% \begin{pgfpicture}% +% \pgfsetlinewidth{.1\H}% +% \pgfsetrectcap +% \pgfsetmiterjoin +% \pgfmoveto{\pgfpoint{0pt}{0.05\H}}% +% \pgflineto{\pgfpoint{0pt}{.95\H}}% +% \pgflineto{\pgfpoint{.35\H}{.95\H}}% +% \pgfpatharc{90}{-90}{.225\H}% +% \pgflineto{\pgfpoint{-.05\H}{.5\H}}% +% \pgfmoveto{\pgfpoint{-.05\H}{.34\H}}% +% \pgflineto{\pgfpoint{.38\H}{.34\H}}% +% \pgfusepath{stroke}% +% \pgfmoveto{\pgfpoint{-.175\H}{0pt}}% +% \pgfmoveto{\pgfpoint{.7\H}{0pt}}% +% \end{pgfpicture}% +% \endgroup +%} + +\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} \ 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 new file mode 100644 index 0000000..62ff2e5 --- /dev/null +++ b/Advanced/Options in Finance/parts/0 intro.tex @@ -0,0 +1,5 @@ +\section{Introduction} + + +\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 new file mode 100644 index 0000000..406b3a3 --- /dev/null +++ b/Advanced/Options in Finance/parts/1 call.tex @@ -0,0 +1,116 @@ +\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{} +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 \ No newline at end of file