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% Homework for the course "Computational Methods in Finance",
% Fall semester, 1997, Jonathan Goodman.
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{\scriptsize Computational Methods in Finance, Fall 1997} \hfill
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Assignment 2.
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Given September 17, due October 1.
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{\bf Objective:} Basic familiarity with finite differences.
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A certain illegal organization establishes an illegal interest rate,
$R(t)$. This rate satisfies the stochastic differential equation
\begin{displaymath}
dR = \alpha \sqrt{R} dt + \sigma \sqrt{R} dZ \;\; .
\end{displaymath}
Here, $t$ and is measured in years and $\alpha$ and $\sigma$ are
measured accordingly. Presently, $R = 10$. The organization
also indulges in gambling. It will sell a wager that pays one
dollar after a year if $\left| R(1) - 10 \right| \leq A$. It will
not pay if $\left| R(t) - 10 \right| \geq 5$ at any time $0 \leq t \leq 1$
(a ``knockout'' feature in its language and on Wall street).
\begin{description}
\item[(1)]
Write the backward equation to compute the present expected value of the
wager. Neglect discounting factors. Take $A=2$, $\alpha = 1$, and
$\sigma = 2$. Solve the backward equation using the forward Euler
finite difference method with various values of $\Delta r$ and $\Delta t$
but with the ratio $\Delta t / \Delta r^2$ fixed.
\begin{description}
\item[a.]
Do a numerical convergence study using your code to determine the order
of accuracy of your method. Does it agree with the theoretical order of
accuracy?
\item[b.]
What is a reasonable $\Delta r$ if you want $.1\%$ accuracy in the answer.
\end{description}
\item[(2)]
Write the forward equation for the probability density of $R(t)$. Solve
it by forwards Euler. Use the methodology of part 1 to choose a reasonable
step size. Compute the expected value of the wager by integrating the
probability density at time $t=1$ against the payout.
\item[(3)]
We want to evaluate the sensitivity $\Lambda = \partial_{\sigma} f$.
\begin{description}
\item[a.]
Use the code for part 1a and finite differences to evaluate $\Lambda$
for $\sigma = 2$.
\item[b.]
Write the backward equation satisfied by $\partial_{\sigma}f$ and solve it
directly. Does the result agree with part a? What is its order of accuracy?
\end{description}
\item[(4)]
For the same values of $\sigma$ and $\alpha$, compute and plot the expected
value as a function of $A$ in the range $1 \leq A \leq 3$.
Try to adapt the two methods in part 3 to compute $\Omega$ (the last possible
Greek), the derivative with respect to $A$.
\end{description}
\noindent
{\bf Note on programming.} This is partly an exercise in software engineering.
If you code from scratch for each part, it will be tedious. Plan the
work so that routines (such as that for convergence analysis) can be used
over.
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