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\author{Jeremy Morris}
\title{Homework 9}

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\item[Exercise 4.6 (Estimating American put prices.)]
	\textit{For each $n$, where $n = 0,1,\ldots,N$, let $G_{n}$ be a random variable depending on the first $n$ coin tosses.  The time-zero value of a derivative security that can be exercised at any time $n \leq N$ for payoff $G_{n}$ but must be exercised at time $N$ if it has not been exercised before that time is \[V_{0} = \max_{\tau \in S_{0},\tau \leq N}\tilde{E}\left[\frac{1}{(1+r)^{\tau}}G_{\tau}\right]\]}
	\begin{enumerate}
	\item \textit{Consider $G_{n} = K-S_{n}$, the derivative security that permits its owner to sell one share of stock for payment $K$ at any time up to and including $N$, but if the owner does not sell by time $N$, then she must do so at time $N$.  Show that the optimal exercise policy is to sell the stock at time zero and that the value of this derivative security is $K - S_{0}$. (assume $r \geq 0$.)} \label{q1}

	According to the algorithm described for determining the risk-neutral price of the American option, we should exercise at time $0$ if $G_{0} \geq V_{0}$.  We know that $G_{0} = K - S_{0}$.  And we calculate $V_{0}$ using the risk-neutral formula below.
	\begin{align*}
	V_{0} & = \frac{1}{1+r}\left[\tilde{p}(K-uS_{0}) + \tilde{q}(K-dS_{0})\right] \\
	 & = \frac{1}{1+r}\left[\tilde{p}K-\tilde{p}uS_{0} + \tilde{q}K-\tilde{q}dS_{0}\right] \\
	 & = \frac{1}{1+r}\left[K(\tilde{p}+\tilde{q}) - S_{0}(\tilde{p}u + \tilde{q}d)\right] \\
	 & = \frac{1}{1+r}\left[K - S_{0}(1+r)\right] \\
	 & = \frac{K}{1+r} - S_{0}
	\end{align*}
	And 
	\begin{align*}
	G_{0} = K - S_{0} & \geq \frac{K}{1+r} - S_{0} = V_{0}
	\end{align*}
	so that we should always exercise at time $0$.

	\item \textit{Explain why a portfolio that holds the derivative security in (\ref{q1}) and a European call with strike $K$ and expiration time $N$ is at least as valuable as an American put struck at $K$ with expiration time $N$.  Denote the time-zero value of the European call by $V_{0}^{EC}$ and the time-zero value of the American put by $V_{0}^{AP}$.  Conclude that the upper bound\[V_{0}^{AP} \leq K-S_{0}+ V_{0}^{EC}\]on $V_{0}^{AP}$ holds.}

	If we look at the payoff function of the security in (\ref{q1}) and a European call, we have
	\begin{align*}
	K - S + \max{\{S-K,0\}} & = \left\{ \begin{array}{ll}
		0 & S > K \\
		K-S & S < K
	\end{array} \right.
	\end{align*}
	which is exactly the payoff of the American put.  In this case, we have a little more flexibility with the derivative in (\ref{q1}) so that the value of the American put should be at most the value of (\ref{q1}).

	\item \textit{Use put-call parity to derive the lower bound on $V_{0}^{AP}$ : \[\frac{K}{(1+r)^{N}} - S_{0} + V_{0}^{EC} \leq V_{0}^{AP}.\]}

	The put-call parity says
	\begin{align*}
	P + S &= C + \frac{K}{(1+r)^{N}}
	\end{align*}
	where $P$ is a put option and $C$ is a call option.  So that the inequality here can be rewritten to fit into the put-call parity
	\begin{align*}
	\frac{K}{(1+r)^{N}} + V_{0}^{EC} \leq V_{0}^{AP} + S_{0}
	\end{align*}
	Then we see that the American put should have a larger value since it has more flexibility than the the discounted cash plus the European call.

	\end{enumerate}
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