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\author{Jeremy Morris}
\title{Homework 2}

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\begin{description}
\item[Exercise 1.8 (Asian option)] \textit{Consider the three-period model of Example 1.2.1, with $S_{0}=4,u=2,d=1/2$, and take the interest rate $r=1/4$, so that $\tilde{p}=\tilde{q}=1/2$.  For $n=0,1,2,3$, define $Y_{n}=\sum_{k=0}^{n}S_{k}$.  Consider and Asian call option that expires at time three and has strike $K=4$.  This is like a European call, except the payoff of the option is based on the average stock price rather than the final stock price.  Let $v_{n}(s,y)$ denote the price of this option at time $n$ if $S_{n}=s$ and $Y_{n}=y$.  In particular, $v_{3}(s,y) = (\frac{1}{4}y-4)^{+}$.}
\begin{enumerate}
\item \textit{Develop an algorithm for computing $v_{n}$ recursively.  In particular, write a formula for $v_{n}$ in terms of $v_{n+1}$.} \label{first}
\begin{align*}
v_{n}(s,y) & = \frac{1}{1+r}\left[\tilde{p}v_{n+1}\left(2s,y + 2s\right) + \tilde{q}v_{n+1}\left(\frac{1}{2}s,y+\frac{1}{2}s\right)\right]
\end{align*}
\item \textit{Apply the algorithm developed in (\ref{first}) to compute $v_{0}(4,4)$, the price of the Asian option at time 0.}

I wrote a perl program to calculate this recursively, the answer it calculated was $1.216$.  (Code available on request.)

\item \textit{Provide a formula for $\delta_{n}(s,y)$, the number of shares of stock that should be held by the replicating portfolio at time $n$ if $S_{n}=s$ and $Y_{n}=y$.}
\begin{align*}
\delta_{n}(s,y) & = \frac{v_{n+1}(2s,y+2s) - v_{n+1}(\frac{1}{2}s,y+\frac{1}{2}s)}{2(n+1)s - \frac{1}{2}(n+1)s} \\
 & = \frac{v_{n+1}(2s,y+2s) - v_{n+1}(\frac{1}{2}s,y+\frac{1}{2}s)}{\frac{3}{2}(n+1)s} \\
 & = \frac{2}{3}\frac{v_{n}(s,y)}{(n+1)s}
\end{align*}
\end{enumerate}

\item[Exercise 1.9 (Stochastic volatility, random interest rate)]\textit{Consider a binomial pricing model, but at each time $n \geq 1$, the ``up factor'' $u_{n}$, the down factor $d_{n}$ and the interest rate $r_{n}$ are allowed to depend on $n$ and on the first $n$ coin tosses $\omega_{1}\omega_{2}\ldots\omega_{n}$.  The initial up factor $u_{0}$, the initial down factor $d_{0}$ and the initial interest rate $r_{0}$ are not random.  More specifically, the stock price at time one is given by
\begin{align*}
S_{1}(\omega_{1}) & = \left\{ 
\begin{array}{ll} 
	u_{0}S_{0} & \textrm{if }\omega_{1} = H \\
	d_{0}S_{0} & \textrm{if }\omega_{1} = T  
\end{array}
\right.
\end{align*}
and, for $n \geq 1$, the stock price at time $n+1$ is given by
\begin{align*}
S_{n+1}(\omega_{1}\omega_{2}\ldots\omega_{n+1}) & = \left\{ 
\begin{array}{ll} 
	u_{n}(\omega_{1}\omega_{2}\ldots\omega_{n+1})S_{n}(\omega_{1}\omega_{2}\ldots\omega_{n+1}) & \textrm{if }\omega_{n+1} = H \\
	d_{n}(\omega_{1}\omega_{2}\ldots\omega_{n+1})S_{n}(\omega_{1}\omega_{2}\ldots\omega_{n+1}) & \textrm{if }\omega_{n+1} = T  
\end{array}
\right.
\end{align*}
One dollar invested in or borrowed from the money market at time zero grows to an investment or debt of $1+r_{0}$ at time one, and , for $n \geq 1$ one dollar invested in or borrowed from the money market at time $n$ grows to an investment or debt of $1 + r_{n}(\omega_{1}\omega_{2}\ldots\omega_{n+1})$ at time $n+1$.  We assume that for each $n$ and for all $\omega_{1}\omega_{2}\ldots\omega_{n+1}$, the no-arbitrage condition holds.
}
\begin{enumerate}
\item \textit{Let $N$ be a positive integer.  In the model just described, provide an algorithm for determining the price at time zero for a derivative security that at time $N$ pays off a random amount $V_{N}$ depending on the result of the first $N$ coin tosses.}

Here we use the same form of the recursive equation used above.
\begin{align*}
V_{n} & = \frac{1}{1+r_{n+1}}\left[\tilde{p}V_{n+1}(H) + \tilde{q}V_{n+1}(T)\right] \\
 & = \frac{1}{1+r_{n+1}}\left[ \left( \frac{1+r_{n+1}-d_{n+1}}{u_{n+1} - d_{n+1}} \right) V_{n+1}(H) + \left( \frac{u_{n+1}-1-r_{n+1}}{u_{n+1} - d_{n+1}} \right)V_{n+1}(T)\right] 
\end{align*}
Where we have dropped the $\omega$ notation for readabilty.

\item \textit{Provide a formula for the number of shares of stock that should be held at each time $n$ by a portfolio that replicates the derivative security $V_{n}$.}

Again, this formula follows the same format as in the previous question.
\begin{align*}
\Delta_{n} & = \frac{V_{n+1}(H) - V_{n+1}(T)}{S_{n+1}(H) - S_{n+1}(T)} \\
 &= \frac{V_{n}(\omega_{1}\omega_{2}\ldots\omega_{n}H) - V_{n}(\omega_{1}\omega_{2}\ldots\omega_{n}T)}{S_{n}(\omega_{1}\omega_{2}\ldots\omega_{n}H) - S_{n}(\omega_{1}\omega_{2}\ldots\omega_{n}T)}
\end{align*}

\item \textit{Suppose the initial stock price is $S_{0} = 80$, with each head the stock price increases by $10$, and with each tail the stock price decreases by $10$.  Assume the interest rate is always zero.  Consider a European call with the strike price $K=80$, expiring at time five.  What is the price of this call at time zero?}

I wrote another perl program (altered the one from the first question) to come up with the answer $9.375$.  (Code available on request.)

\end{enumerate}

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