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\title{Financial Mathematics and Simulation\\
MATH 6740--1 -- Spring 2011 \\
Homework 2}
\date{Due Date: Friday, March 11 at 5:00 PM}
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\maketitle
This homework has 170 points plus 20 bonus points available but, as always, homeworks are graded out of 100 points. Full credit will generally be awarded for a solution only if it is both correctly and efficiently presented using the techniques covered in the lecture and readings, and if the reasoning is properly explained. If you used software or simulations in solving a problem, be sure to include your code, simulation results, and/or worksheets documenting your work. If you score more
than 100 points, the extra points do count toward your homework total. For full credit on this homework, make use of the measure-theoretic versions of conditional probability and conditional expectation when appropriate.
\section{Theoretical Calculations}
\subsection{Gaussianity of Random Sums of Random Variables (40 points)}
Consider a sum of random variables:
\begin{displaymath}
Z = \sum_{j=1}^{N} X_j
\end{displaymath}
where the $ X_j $ are independent, identically distributed random
variables, and $ N$ is either a fixed deterministic constant $ N = m $
or a random variable with Poisson distribution with mean $ m$:
\begin{equation*}
P (N=n) = \expe^{-m} \frac{n^m}{m!}, n=0,1,2,\ldots
\end{equation*}
The
aim of this question is to investigate how well $ Z $ can be modeled
by a Gaussian random variable when $ N $ is deterministic or random.
We will restrict attention to the case in which the (common)
probability distribution of the $ \{X_j\}_{j=1}^N $ is symmetric (so
that $ \Prob(X_j > a) = \Prob(X_j < - a) $ for any real $ a $). Then
the random variable $ Z $ readily can be shown to be symmetric. A
standard measure for the deviation of a symmetric probability
distribution from a
Gaussian shape is through the \emph{kurtosis}
\begin{displaymath}
K = \frac{\mu_{Z,4}}{(\mu_{Z,2})^2},
\end{displaymath}
where $ \mu_{Z,n} $ is the $n$th order cumulant of $ Z $. This is defined as:
\begin{equation}
\mu_{Z,n} \equiv (-\mathi)^n \left. \frac{\difd^n}{\difd k^n} \log \phi_Z (k)\right|_{k=0}
\end{equation}
where
\begin{equation}
\phi_Z (k) \equiv \bExp \expe^{\mathi k Z}
\end{equation}
is the \emph{characteristic function} of $ Z $.
\begin{enumerate}
\item (\textbf{15 points})
Calculate the kurtoses of $ Z $ for both the deterministic and
Poisson random models of $ N $ when the $ \{X_j\} $ are drawn from a common
Gaussian distribution with mean zero. Plot the kurtosis in each case
as a function of $ m $.
\item (\textbf{15 points}) Choose some simple, symmetric,
non-Gaussian probability distribution for the
$ \{X_j\}_{j=1}^N $, and then
calculate the kurtoses of $ Z $ for both the deterministic and
Poisson random models of $ N $. Plot the kurtosis in each case
as a function of $ m $.
\item (\textbf{10 points}) Discuss your above results.
\end{enumerate}
Such statistical studies have relevance in finance,
where one wishes to know whether the statistical models can be
reasonably modeled by Gaussian random variables. Asset prices have
both Gaussian and non-Gaussian features, depending on how one collects
the statistics. Two alternative approaches are to consider how an asset price
fluctuates over a
fixed number of transactions or
over a fixed time period. The above results can be connected to such
data analysis if we think of $ N$ as the number of transactions and $
X_j $ as the price change due to the $j$th transaction. As you would learn in
a class on stochastic processes (such as the one to be offered in the fall), the Poisson distribution
would arise naturally if transactions were occurring at random times, but with
some well-defined average frequency and no memory (a fully Markovian renewal process).
\subsection{Information Flow in Decision Markets (75 points plus 20 bonus points)}
A recent innovation based on the ``wisdom of the crowds" concept is the use of decision markets to gauge true public opinion. The idea is to use market mechanisms and some sort of (often financial) incentive to motivate participants to register their true beliefs (in a possibly relatively anomalous way) and efficiently aggregate and weight the opinions of various participants, based on their self-assessed confidence. Perhaps the most famous example are the decision markets managed at the University of Iowa (\url{http://en.wikipedia.org/wiki/Iowa_Electronic_Markets}) where markets are estabiished for various political questions, like whether a given candidate will win a given election. Closer to home, Profs. Sanmay Das and Malik Magdon-Ismail in the computer science department have been analyzing decision markets and developing algorithms for trying to tease out the true beliefs (in an aggregate sense) of participants in contexts beyond politics.
As an experiment, they have been running decision markets for the purpose of dynamically assessing instructors in several computer science classes. The basic idea is to motivate students to provide honest course evaluations in real time, throughout the semester. Since setting up a financial market for this purpose would be tantamount to a gambling racket, which is probably illegal in New York State, the incentive scheme used is a bit indirect. For the purposes of this problem, I won't worry about these legal issues, etc., and modify what they are actually doing. You can consult the link to RPI course/instructor market on
(\url{http://www.cs.rpi.edu/~magdon/}) if you really want to know what they do for real.
For each participating instructor, a stock is created. At the end of the semester, when final instructor evaluations are submitted, each instructor's stock will be worth \$100 times their evaluation score, which is obtained by taking the average of the evaluation score submitted by each student in the class via the IDEA survey or whatever they're about to replace it with. These evaluation scores are numbers between 1.0 and 5.0, intended to be an increasing function of instructor quality (or easiness or comedic ability or\ldots).
Suppose the decision market involves $ c $ classes, and for simplicity we
assume each class has the same number of students $ s $. Assume students do not add or drop these classes
once the semester has begun. Now, the honest evaluation of the $j$th student in the $i$th class of the instructor of that class will be given by some function $ \evali (t,\Wij (t)) $, where $ \evali (t,m) $ are deterministic functions and the $ \{\Wij (t)\}_{i=1,\ldots,c,j=1,\ldots,s} $ are
independent Wiener processes. For the moment assume no person is in more than one of the classes under consideration; we will relax this unrealistic assumption later. The way to interpret this model is as follows: $ \Wij (t) $ describes some sort of state of mind of the $j$th student in the $i$th class, evolving as a function of time $ t $. For example, it could incorporate issues such as the amount of effort they are spending on the class, illness, state of relationship with a significant other or a pet, level of sleep deprivation, effectiveness of medication, etc. The deterministic functions $ \evali (t,m) $ describes how a student's evaluation of the instructor of the $i$th class depends on time $ t $ and his state of mind $ m $. The explicit dependence on time in $ \evali (t,m) $ corresponds to systematic tendencies, such as the evaluation of the instructor falling after a difficult test, which are apart from the student's state of mind. The dependence on $i $ (via the index) corresponds to systematic differences in instructor quality.
Apart from each of these students' honest evaluations of their instructors, the $j$th student in the $i$th class adopts a strategy $ \stratij $ with regard to the instructor market, which we will treat as time-independent random variables. We consider these strategies as random since they may not be explicitly revealed, and we allow the $ \{\stratij\}_{i=1,\ldots,c; j=1,\ldots,s} $ to depend on each other (as friends may influence each others' strategies.) For now we won't specify explicitly how the $ \stratij $ map onto a specific strategy, but will simply assert that such a mapping exists. (For example, one value of $ \stratij $ may correspond to ``Continually adjust one's holdings of one's own instructor's stock so that the number of shares held is the closest integer to 93 times the difference between one's current evaluation of the instructor and the current trading value of the stock (and hold no shares when this difference is negative)." Another value might correspond to ``Each day at 1 AM that the stock of an instructor from a different class is higher than 4.0, generate a binomially distributed random variable from 20 trials with success probability equal to the difference of the stock value from 4.0, and buy that many shares of that other instructor's stock." If you think such a coding of strategies is not possible, see the proof of G\"{o}del's incompleteness theorem.) The value of the stock of each instructor $ i $ at a given moment of time will be determined by some market-clearing mechanism that depends on the number of shares of that stock each student wishes to buy or sell at that moment of time. We won't be explicit about this mechanism other than assuming it operates effectively instantaneously (so that all buyers and sellers have their requests quickly satisfied by some ``market maker.") In particular, we will neglect bid-ask spreads and simply assume each stock buys or sells at its current value. We will also neglect participation by people who are not students in the classes under consideration.
Discuss each of the following questions, with explanation for your reasoning, when the strategies of the students are of the following type:
\begin{itemize}
\item A general reasonable strategy (no constraints or special assumptions beyond those stated above)
\item Each strategy is encoded by a pair of real-valued random variables ($ \stratij = (\ownij, \othij) $), which are
to be interpreted: ``The $j$th student in the $i$th class continually adjusts his holdings so that the number of shares he holds in his instructor's stock is the closest integer to $ \ownij $ times the difference between his evaluation of his instructor and the current value of that stock (and hold no shares when this difference is negative). That same student continually adjusts his holdings in the number of shares in other instructors' stocks so that the number of shares he holds of the stock of instructor $ \ip \neq i $ is $ \othipj $ times the difference between the average of the current values of all the instructors' stocks and the current value of the stock of instructor $ \ip $
(and hold no shares when this difference is negative)."
\item A modification of the previous strategy wherein ``the average of the current values of all the instructors' stocks" is replaced by ``the best estimate of the evaluation score of instructor $ \ip $ at the end of the course." To make this latter quantity meaningful (rather than a wild guess), suppose the students have excellent information about the material and tests of all classes, as well as about all instructors. So they would have some idea of how a typical student would evaluate the instructor of any given class,
but they can't expected, however, to know how the particular students in the class this particular semester will react.
\end{itemize}
But first (\textbf{10 points}), provide a specific mathematical formulation for the third strategy. Stay within the modeling framework laid out above.
\begin{enumerate}
\item (\textbf{10 points}) Describe all natural filtrations in this model with respect to which the price of a given instructor's stock is adapted. Which one would you suggest is most natural for encoding its dependency on information flow? Explain your reasoning.
\item (\textbf{10 points}) Describe all natural filtrations in this model with respect to which the number of shares of stock held by the $j$th student in the $i$th class of his own instructor's stock is adapted. Which one would you suggest is most natural for encoding its dependency on information flow? Explain your reasoning.
\item (\textbf{5 points}) Describe all natural filtrations in this model with respect to which the number of shares of stock held by the $j$th student in the $i$th class of the stock for instructor $\ip \neq i $ is adapted. Which one would you suggest is most natural for encoding its dependency on information flow? Explain your reasoning.
\item (\textbf{10 points}) Suppose we are at a time $ t $ midway through the semester; express as precisely as you can an estimate for the value of a given student's portfolio at the end of the semester (when the evaluation scores are finalized and the decision market is liquidated.), assuming the student stops making any more trades after time $ t $, and simply holds the stocks he has. Include a measure of the uncertainty of your estimate. Think about this from two perspectives: the estimate the student would make, and the estimate some expert observer (who is thoroughly familiar with the class and instructors, etc.) would make. Maybe one or the other won't be meaningful in the current model; if so explain why. Your answers may not in all cases be fully explicit, since we haven't completely specified all aspects of the model above, but be as specific as you can for full credit.
\item (\textbf{20 bonus points}) Revisit the previous question, but now assuming the student will continue trading after time $ t $, and the goal is now to estimate the value of the portfolio he will have at the end of the semester (which will generally not be the same portfolio he is holding at time $ t $).
\item (\textbf{10 points}) Describe under what conditions the price of an instructor's stock will be a Markov process with respect to one of the natural filtrations in the model (and specify the filtration(s)). Answer the same question for the number of shares of stock of a given instructor a given student holds.
\item (\textbf{10 points}) Describe under what conditions the price of an instructor's stock will be a martingale with respect to one of the natural filtrations in the model (and specify the filtration(s)). Answer the same question for the number of shares of stock of a given instructor a given student holds.
\item (\textbf{5 points}) How would your above answers change if some students were in classes of more than one instructor participating in the decision market during a given semester?
\item (\textbf{5 points}) How would your above answers change if a group of students from the various classes talked to each other about their current evaluation of their instructors?
\end{enumerate}
\section{Numerical Computations}
\subsection{Discrete Time Approximation of Wiener Process (40 points)}
Write a computer program which generates a discrete-time approximation
to the Wiener process
by approximating it with a random walk.
\begin{enumerate}
\item (\textbf{15 points})
Specifically, divide time
into small intervals of width $ \Delta t $, and simulate (in a precise
way) the increments the Wiener process undergoes over each interval.
That is, you should be accurately simulating the statistics of the
Wiener process on the discrete points $ \{j \Delta t\} $ for integer
values of $ j $. Plot a continuous-time approximation of the Wiener
process by linearly interpolating between the values computed at the
points $ \{j \Delta t\} $. Provide a few example plots
with different values of $ \Delta t $.
\item (\textbf{15 points})
Discuss to what extent your random walk approximation has the properties defining the Wiener process. Which of these properties are exactly true for your random walk approximation, and which are only approximately true? Show that the approximate properties become exact in the limit $ \Delta t \downarrow 0 $. Support your statements with careful mathematical calculations.
\item (\textbf{10 points})
Calculate the total variation and quadratic variation of your random walk approximations over a unit interval, and describe how it behaves as $ \Delta t $ is reduced. How do your observations relate to the corresponding behavior of the Wiener process?
\end{enumerate}
\section{Mathematical Problems}
\subsection{Conditional Covariance Formula (15 points)}
Define
\begin{displaymath}
\Cov (X,Y|\cG) = \bExp [(X - \mu_{X|\cG}) (Y-\mu_{Y|\cG})|\cG],
\end{displaymath}
where $ \mu_{X|\cG} = \bExp (X|\cG) $ and $ \mu_{Y|\cG} = \bExp (Y|\cG) $ and $ \cG $ is a sub $\sigma$-algebra of the full $\sigma $ algebra of measurable sets on the probability space.
Derive a formula for $ \Cov (X,Y) $ in terms of the conditional expectations $ \bExp (X|\cG) $ and $ \bExp (Y|\cG) $ as well as the conditional covariance $ \Cov (X,Y|\cG) $.
\end{document}