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% Homework for the course "Stochastic Calculus",
% Fall semester, 2004, Jonathan Goodman.
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{\scriptsize Stochastic Calculus, Fall 2004
(http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc2004/)} \hfill
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\large
Assignment 4.
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\noindent
Given October 1, due SOctober 14. Last revised, October 1.\\
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{\bf Objective:} Gaussian random variables.
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% The questions
\begin{description}
\item[1.] Suppose $X \sim {\cal N}(\mu,\sigma^2)$. Find a formula for
$E[e^X]$. Hint: write the integral and complete the square in the
exponent. Use the answer without repeating the calculation to get
$E[e^{aX}]$ for any constant $a$.
\item[2.] In finance, people often use $N(x)$ for the CDF (cumulative
distribution function) for the standard normal. That is, if
$Z\sim {\cal N}(0,1)$ then $N(x) = P(Z\leq x)$. Suppose $S=e^X$ for
$X \sim {\cal N}(\mu,\sigma^2)$. Find a formula for $E[\max(S,K)]$
in terms of the $N$ function.
(Hint: as in problem 1.)
This calculation is part of the Black--Scholes theory of the value
of a vanilla European style call option. $K$ is the known strike price and
$S$ is the unknown stock price.
\item[3.] Suppose $X=(X_1,X_2,X_3)$ is a 3 dimensional Gaussian random variable
with mean zero and covariance
$$
E[XX^*] = \left(\begin{array}{ccc} 2 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2 \\ \end{array} \right) \; .
$$
Set $Y = X_1 + X_2 - X_3$ and $Z = 2 X_1 - X_2$.
\begin{description}
\item[a.] Write a formula for the probability density of $Y$.
\item[b.] Write a formula for the joint probability density for $(Y,Z)$.
\item[c.] Find a linear combination $W = aY+bZ$ that is independent of $X_1$.
\end{description}
\item[4.]
Take $X_0 = 0$ and define $X_{k+1} = X_k + Z_k$, for $k = 0, \ldots,n-1$.
Here the $Z_k$ are iid.\ standard normals,
so that
\begin{equation}
X_k = \sum_{j=0}^{k-1}Z_j \; .
\end{equation}
Let $X \in R_n$ be the vector $X = (X_1,\ldots,X_n)$.
\begin{description}
\item[a.]Write the joint probability density for $X$ and show that $X$
is a multivariate normal.
Identify the $n \times n$ tridiagonal matrix $H$ that arises.
\item[b.] Use the formula (1) to calculate the variance of $X_k$ and the
covariance $E[X_j,X_k]$.
\item[c.] Use the answers to part b to write a formula for the elements of
$H^{-1}$.
\item[d.] Verify by matrix multiplication that your answer to part c is correct.
\end{description}
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