\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