# Financial Mathematics and Simulation MATH Spring 2011 Homework 2

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1 Financial Mathematics and Simulation MATH Spring 2011 Homework 2 Due Date: Friday, March 11 at 5:00 PM 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. 1 Theoretical Calculations 1.1 Gaussianity of Random Sums of Random Variables (40 points) Consider a sum of random variables: Z = N j=1 X j 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: P (N = n) = e m nm, n = 0, 1, 2,... m! 1

2 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 } N j=1 is symmetric (so that P (X j > a) = P (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 kurtosis K = µ Z,4 (µ Z,2 ) 2, where µ Z,n is the nth order cumulant of Z. This is defined as: µ Z,n ( i) n dn dk log φ Z(k) n (1) k=0 where is the characteristic function of Z. φ Z (k) Ee ikz (2) a. (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. b. (15 points) Choose some simple, symmetric, non-gaussian probability distribution for the {X j } N j=1, 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. c. (10 points) Discuss your above results. 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 jth 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). 2

3 1.2 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 (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 (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... ). 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 jth student in the ith class of the instructor of that class will be given by some function h i (t, W ij (t)), where h i (t, m) are deterministic functions and the {W ij (t)} i=1,...,c,j=1,...,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: W ij (t) describes some sort of state of mind of the jth student in the ith class, evolving as a function of time t. For example, 3

6 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. e. (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). f. (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. g. (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. h. (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? i. (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? 2 Numerical Computations 2.1 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. a. (15 points) Specifically, divide time into small intervals of width 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 t} for integer values of j. Plot a continuous-time approximation of the Wiener process by linearly interpolating 6

7 between the values computed at the points {j t}. Provide a few example plots with different values of t. b. (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 t 0. Support your statements with careful mathematical calculations. c. (10 points) Calculate the total variation and quadratic variation of your random walk approximations over a unit interval, and describe how it behaves as t is reduced. How do your observations relate to the corresponding behavior of the Wiener process? 3 Mathematical Problems 3.1 Conditional Covariance Formula (15 points) Define Cov (X, Y G) = E[(X µ X G )(Y µ Y G ) G], where µ X G = E(X G) and µ Y G = E(Y G) and G is a sub σ-algebra of the full σ algebra of measurable sets on the probability space. Derive a formula for Cov (X, Y ) in terms of the conditional expectations E(X G) and E(Y G) as well as the conditional covariance Cov (X, Y G). 7

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