Merge branch 'master' of ssh://git.betalupi.com:33/Mark/ormc-handouts

Advanced/Lambda Calculus/main.tex

@@ -58,6 +58,48 @@
 
 	\vspace{4mm}
 
+	\makeatletter
+	\if@solutions
+		\begin{instructornote}
+			\textbf{Context \& Computability:} (or, why do we need lambda calculus?)\par
+			\note{From Peter Selinger's \textit{Lecture Notes on Lambda Calculus}}
+
+			\vspace{2mm}
+
+			In the 1930s, several people were interested in the question: what does it mean for
+			a function $f : \mathbb{N} \mapsto \mathbb{N}$ to be computable? An informal definition of computability
+			is that there should be a pencil-and-paper method allowing a trained person to
+			calculate $f(n)$, for any given $n$. The concept of a pencil-and-paper method is not
+			so easy to formalize. Three different researchers attempted to do so, resulting in
+			the following definitions of computability:
+
+			\begin{itemize}
+				\item Turing defined an idealized computer we now call a Turing machine, and
+				postulated that a function is \say{computable} if and only
+				if it can be computed by such a machine.
+
+				\item G\"odel defined the class of general recursive functions as the smallest set of
+				functions containing all the constant functions, the successor function, and
+				closed under certain operations (such as compositions and recursion). He
+				postulated that a function is \say{computable} if and only
+				if it is general recursive.
+
+				\item Church defined an idealized programming language called the lambda calculus,
+				and postulated that a function is \say{computable} if and only if it can be written as a lambda term.
+			\end{itemize}
+
+			It was proved by Church, Kleene, Rosser, and Turing that all three computational
+			models were equivalent to each other --- each model defines the same class
+			of computable functions. Whether or not they are equivalent to the \say{intuitive}
+			notion of computability is a question that cannot be answered, because there is no
+			formal definition of \say{intuitive computability.} The assertion that they are in fact
+			equivalent to intuitive computility is known as the Church-Turing thesis.
+		\end{instructornote}
+		\vfill
+		\pagebreak
+	\fi
+	\makeatother
+
 	\input{parts/00 intro}
 	\input{parts/01 combinators}
 	\input{parts/02 boolean}

Advanced/Lambda Calculus/parts/00 intro.tex

@@ -178,7 +178,6 @@ Reduce the following expressions. \par
 	\textbf{Solution for $(I~I)$:}\par
 	Recall that $I = \lm x.x$. First, we rewrite the left $I$ to get $(\lm x . x )~I$. \par
 	Applying this function by replacing $x$ with $I$, we get $I$:
-
 	$$
 		I ~ I = 
 		(\lm x . \tzm{b}x )~\tzm{a}I =
@@ -193,11 +192,7 @@ Reduce the following expressions. \par
 			\draw[->,gray,shorten >=5pt,shorten <=3pt]
 				(a.center) to (b.east);
 		\end{tikzpicture}
-	$$
+	$$\null
-	\vspace{0.5mm}
-
-	So, $I~I$ reduces to itself. This makes sense, since the identity
-	function doesn't change its input!
 \end{examplesolution}
 
 
@@ -214,11 +209,20 @@ Rewrite the following expressions with as few parentheses as possible, without c
 Remember that lambda calculus is left-associative.
 \vspace{2mm}
 \begin{itemize}[itemsep=2mm]
-	\item $(\lm x. (\lm y. \lm (z. ((xz)(yz)))))$
+	\item $(\lm x. (\lm y. \lm z. ((xz)(yz))))$
 	\item $((ab)(cd))((ef)(gh))$
 	\item $(\lm x. ((\lm y.(yx))(\lm v.v)z)u) (\lm w.w)$
 \end{itemize}
 
+\begin{solution}
+	$(\lm x. ((\lm y.(yx))(\lm v.v)z)u) (\lm w.w) \implies (\lm x. (\lm y.yx) (\lm v.v)~z~u) \lm w.w$
+
+	\vspace{2mm}
+
+	It's important that a function's output (everything after the dot) will continue until we hit a close-paren.
+	This is why we need the parentheses in the above example.
+\end{solution}
+
 
 \vfill
 \pagebreak
@@ -317,17 +321,19 @@ We've already seen this on the previous page: $K$ takes an input $x$ and uses it
 You can think of $K$ as a \say{factory} that constructs functions using the input we provide.
 
 \problem{}<firstcardinal>
-Let $C = \lm f. \Bigl[\lm g. \Bigl( \lm x. [~ g(f(x)) ~] \Bigr)\Bigr]$. For now, we'll call it the \say{composer.}
+\vspace{1mm} % Slight gap for big paren
+Let $C = \lm f. \Bigl[\lm g. \Bigl( \lm x. [~ f(g(x)) ~] \Bigr)\Bigr]$. For now, we'll call it the \say{composer.} \par
+\note[Note]{We could also call $C$ the \say{right-associator.} Why?}
 
-\vspace{1mm}
+\vspace{3mm}
 
-Note that $C$ has three \say{layers} of curry: it makes a function ($\lm g$) that makes another function ($\lm x$). \par
+$C$ has three \say{layers} of curry: it makes a function ($\lm g$) that makes another function ($\lm x$). \par
 If we look closely, we'll find that $C$ pretends to take three arguments.
 
 \vspace{1mm}
 
 What does $C$ do? Evaluate $(C~a~b~x)$ for arbitary expressions $a, b,$ and $x$. \par
-\hint{Place parentheses first. Remember, function application is left-associative.}
+\hint{Evaluate $(C~a)$ first. Remember, function application is left-associative.}
 
 \vfill
 

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}

Advanced/Options in Finance/parts/0 intro.tex

@@ -0,0 +1,5 @@
+\section{Introduction}
+
+
+\vfill
+\pagebreak

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{}<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

Advanced/Origami/main.tex

@@ -370,7 +370,7 @@
 	\note{Proposed by Sunny}
 	Using a compass and ruler, find two circles tangent to a point D and lines AB and AC. (Problem of Appolonius, PLL case) \par
 	\hint{
-		All circles tangent to $AB$ and $AC$ are homothetic with centre at $A$. What does this mean? \par
+		All circles tangent to $AB$ and $AC$ are homothetic with centre at $A$. What does this mean? \\
 		Also, the angle bisector may help.
 	}
 

build.sh

@@ -36,12 +36,17 @@ function build() {
 	echo "|> Building ${job_name}..."
 	cd "${doc_dir}"
 
+	tectonic_args=(
+		--chatter minimal
+		--web-bundle "https://static.betalupi.com/tectonic/texlive2023-nopatch.tar"
+	)
+
 
 	# Build handout
 	echo "\\def\\argNoSolutions{1}\\input{${main_file}}" | \
 		tectonic \
+			"${tectonic_args[@]}" \
 			--outfmt pdf \
-			--chatter minimal \
 			-
 
 	stat=$?
@@ -60,8 +65,8 @@ function build() {
 	# Build solutions
 	echo "\\def\\argYesSolutions{1}\\input{${main_file}}" | \
 		tectonic \
+			"${tectonic_args[@]}" \
 			--outfmt pdf \
-			--chatter minimal \
 			-
 
 	stat=$?

resources/ormc_handout.cls

@@ -427,19 +427,15 @@
 		\centering
 
 		% Left and right headers (uptitle)
-		\def\uptitle{%
+		\if\@uptitlel\relax\else\textsc{\@uptitlel}\fi \hfill\null%
-			\if\@uptitlel\relax\else\textsc{\@uptitlel}\fi \hfill~%
+		\if\@uptitler\relax\else\textsc{\@uptitler}\fi \par
-			\if\@uptitler\relax\else\textsc{\@uptitler}\fi \par
-		}
-		\if\@uptitlel\relax % If neither header is defined,
-		\if\@uptitlel\relax % don't insert an upper title.
-			\def\@uptitle{\relax}
-		\fi\fi\uptitle{}
 
 		% Main title
 		\rule{\linewidth}{0.2mm} \par
-		\huge \@title \par \normalsize
+		\vspace{4mm}
-		\vspace{1ex} \@subtitle \par
+		{\huge \@title} \par
+		\vspace{2mm} \@subtitle \par
+		\vspace{1mm}
 		\rule{\linewidth}{0.2mm} \par
 
 		% Solution warning