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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 4.
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Given November 5, due November 19.
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{\bf Objective:} Basics of stochastic differential equations.
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\begin{description}
\item[(1)]
The solution of the model stochastic differential equation
\begin{equation}
dX = XdZ \;\; , \;\;\;\; X(0) = 1
\end{equation}
is given by
\begin{equation}
X(t) = \exp\left( Z(t) - t/2 \right) \;\; .
\end{equation}
Here $Z(t)$ is a standard Brownian motion with $Z(0) = 0$. From (2) we get
the formula for the density function for $X(t)$:
\begin{equation}
u(x,t) = \frac{1}{\sqrt{ 2 \pi t}} \frac{1}{x}
\exp \left( \frac{-\left( \log(x) + t/2 \right)^2}{2t} \right) \;\; .
\end{equation}
(Check this yourself.) The forward Euler approximation gives
$X_k \approx X(t_k)$ whose density function is $u(x,t_k,\Delta t)$.
Use the histogram method to compute $u(x,t,\Delta t)$ and then
compare it to the exact
answer, $u(x,t)$ for $t=1$ and $\Delta t$ small but not too small;
compare the curves. If the bins are too small, this will be impossible,
it will require too large a sample size. Use values $\Delta t$,
$\Delta t/2$, $\Delta t/4$, and so on to verify that $u(x,t,\Delta t)$
is a first order accurate approximation to $u(x,t)$, as curves.
\item[(2)]
Redo the above with the complication that $X$ is absorbed if $X(t) \geq 2$
for any $t \leq 1$. This time, do not use a formula for the exact
density, $u(x,t)$, when you determine the accuracy of $u(x,t,\Delta t)$.
In the forward Euler approximation you implement the absorption by
absorbing a walker if $X_k \geq 2$ for any $k$ with $t_k \leq 1$.
\item[(3)]
Now turn (1) into a (simplified) stochastic volatility model by writing
\begin{equation}
dX = \sigma(t) dZ_1(t) \;\; , \;\;\;\; X(0) = 1 \;,
\end{equation}
\begin{equation}
d\sigma(t) = \gamma \sqrt{\sigma} dZ_2(t) \;\; , \;\;\;\; \sigma(0) = 1 \;.
\end{equation}
Here $Z_1(t)$ and $Z_2(t)$ are independent standard Brownian motions.
For $\gamma = 0$, the $X$ process is as before. Give Monte Carlo estimates
of
\begin{displaymath}
F(\gamma, T, K) = \mbox{\bf E}\left[ \left( X(T) - K \right)_+\right] \;\; ,
\end{displaymath}
for parameter values in the ranges: $.1 \leq \gamma \leq 5$,
$.1 \leq T \leq 5$, and $.2 \leq K \leq 1$.\footnote{Please do not hand in
hundreds or even tens of plots representing every possible combination of
parameters.}
Use two variance reduction strategies, antithetic variates and control
variates. For the control variate case, the exactly solvable problem is
the case $\gamma = 0$. The solution for this case is given by the
Black--Scholes formula with $r=0$ and $\sigma = 1$. Control variates
work well when the approximation is good. Antithetic variates
work well when the function being "expected" is nearly linear as a function
of $Z$ over the range of variation of $Z$. For which combinations of
parameters do you expect these conditions to hold? Comment on how well
the numerical experiments agree with your expectations.
\end{description}
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