Final Exam Aug. 29th Mathematical Foundations in Finance (FIN 500J) Summer, Sample Final Exam

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1 Final Exam Aug. 29th Olin Business School Yajun Wang Mathematical Foundations in Finance (FIN 500J) Summer, 2009 Sample Final Exam NAME (Print Clearly): Instructions 1. You have 90 minutes to complete this exam. 2. Try to answer each question in the space provided. If not enough space, you can use the back side of each page. Show all your work. 3. This is a closed book and closed-notes exam. But you are allowed to bring one page of A4 size cheat sheet. Please read and sign below the following statement. I agree not to discuss any aspect of this exam with anyone who has not taken this exam. I understand this exam must be entirely my own work and that the University s Code of Academic Integrity applies. Signature:

2 Final Exam Aug. 29th Part I. This part contains four problems. 1. For compute A 1 B T.(6 points) A = ( ) ( 2 5, B = 4 10 ) 2. Briefly describe the idea of Newton-Raphson method and use Newton-Raphson method to find a root of f(x) = x 3 x 1, use the initial point x 0 = 1 and do one iteration manually. (7 points)

3 Final Exam Aug. 29th For heat equation u t = β 2 u x 2 subject to the conditions u(x, 0) = f(x), u(0, t) = 0, u(1, t) = 0. Briefly describe the idea of explicit finite difference method for solving this heat equation. (7 points)

4 Final Exam Aug. 29th Part II. This part contains two problems. 1. A monopolistic producer of two goods, G 1 and G 2, has a joint total cost function, TC = 10Q 1 + Q 1 Q Q 2, where Q 1 and Q 2 denote the quantities of G 1 and G 2 respectively. If P 1 and P 2 denote the corresponding prices then the demand equations are P 1 = 50 Q 1 + Q 2, P 2 = Q 1 Q 2. (1)It can be shown that the profit of the producer is given by f(q 1, Q 2 ) = 40Q 1 Q Q 1 Q Q 2 Q 2 2. Use Lagrangian multipliers to find the maximum profit if the firm is contracted to produce a total of 15 goods of either type, i.e., Q 1 + Q 2 = 15. (10 points) (2)Check the second order conditions.(6 points) (3)Estimate the new optimal profit if the production quota rises by one unit. (2 points)

5 Final Exam Aug. 29th Suppose that X and Y have a continuous joint distribution for which the joint p.d.f. is defined as follows: { cy 2 0 x 2 and 0 y 1, f(x, y) = 0 otherwise. (1)Determine the value of c. (5 points) (ii)compute Pr(Y < 1 ). (4 points) 2 (iii) Compute Cov(X, Y ). (5 points)

6 Final Exam Aug. 29th Part III. This part contains two problems. 1. Solving the following Ordinary Differential Equations: (1) dy + 3 y = 4 3x, y(0) = 1. (12 points) dx 2 (2) d2 y dx dy dx + 5y = 2e 2x, y(0) = 1, dy (0) = 2. (13 points) dx

7 Final Exam Aug. 29th When we use separation of variables to solve the heat equation subject to the boundary conditions u t = 1 2 u 4 x2, for 0 < x < 10, t > 0 u (0, t) = 0, u(10, t) = 0, x and initial condition u(x, 0) = 5, we assume that u(x, t) = X(x)T(t), then we get T (t) 1 T(t) = X (x) X(x) = k. 4 We only consider the case of k < 0 in this problem. (1)Show that the boundary conditions become X (0) = 0 and X(10) = 0 if we want to find a non-trivial solution to this heat equation. Solve the ODE, X (x) = k with X(x) X (0) = 0 and X(10) = 0. Find k to get non-trivial solutions for X(x). (10 points)

8 Final Exam Aug. 29th T (t) 1 4 (2)Solve the ODE, = k, where k is what you got in (1). Then, write out the T(t) general form solution to this heat equation, u(x, t). (8 points) (3)Applying the initial condition to the general form solution in (2) to derive the coefficients in the general form solution in (2). (5 points) You may use the fact that if f(x) is a function on [0, L] and f(x) = n=0 a n cos (n+1 2 )πx then a n = 2 L f(x) cos (n + 1)πx 2 dx L L 0 L,