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\title{Derivatives Lab}
\author{Session 14}
\date{November 7, 2011}
\maketitle

\def\e{\mathrm e}
\def\d{\mathrm d}

\begin{enumerate}

\item Approximate the It\=o integral
\[
  I = \int_0^T X(t-) \, \d W(t)
    = \lim_{N\to\infty} \sum_{i=0}^{N-1} X(t_i) \, (W(t_{i+1}) -
    W(t_i))
\]
and the Stratonovich integral
\[
  W = \int_0^T X(t) \, \d W(t)
    = \lim_{N\to\infty} 
      \sum_{i=0}^{N-1} X\Bigl(\frac{t_{i+1}+t_i}2 \Bigr) 
      \, (W(t_{i+1}) - W(t_i)) \,,
\]
where $W(t)$ denotes standard Brownian motion, $t_i = i \, \Delta t$
with $\Delta t = T/N$, and we choose, as an example, $X(t)=W(t)$.   Do
they converge to the same value as $N \to \infty$?

\item It is known that the stochastic differential equation
\begin{gather*}
  \d S(t) = \mu \, S(t) \, \d t + \sigma \, S(t) \, \d W(t) \,,
  \tag{*} \\
  S(0) = S_0
\end{gather*}
is solved by geometric Brownian motion
\[
  S(t) = S_0 \, \e^{(\mu - \sigma^2/2) \, t + \sigma \, W(t)} \,.
\]
\begin{enumerate}
\item Use the Euler--Maruyama method to solve (*) and compare pathwise
against the exact solution.
\item Find the \emph{strong order of convergence}, i.e., an exponent
$p$ such that
\[
  \mathbb{E} [\lvert S_N - S(T) \rvert]
  \leq c \, \Delta t^p \,
\]
where $S(T)$ denotes true geometric Brownian motion and $S_N$ its
Euler--Maruyama approximation at the final time $T$.

\item Find the \emph{weak order of convergence}, i.e., an exponent $q$
such that
\[
  \lvert \mathbb{E}[S_N] - \mathbb{E}[S(T)] \rvert
  \leq c \, \Delta t^q \,.
\]

\end{enumerate}
\end{enumerate}

\end{document}

