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\author{Jeremy Morris}
\title{Homework 1}

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\begin{definition}
\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 ratgher 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, $v_{3}(s,y) = (\frac{1}{4}y-4)^{+}$.}
\end{definition}

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