Session 11

Exercise Session 11, November 29th ,2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua Chen (Cissy), Natalia Guseva Problem 1 A stock price is currently at S0 = $20: It is known that at the end of every one year the stock price will either move up by 10% or move down by 10%: Consider a three year horizon. Assume the probability of an up movement is P (U ) = 50%; and a down movement is P (D) = 50%: 1. Compute the expected value and variance for stock price at time three. 2. What is P (S3 > 20)? 3. What is P (S3 > 20 j down movement in the …rst year)? 4. What is the sample space? 5. Is the stock price at time two a random variable? 6. What is the

algebra generated by S2 , denoted by (S2 )?

7. If we only know (S2 ), do we have enough information about how the stock price has evolved? 8. If we know how the stock price has evolved up to time two, do we know exactly the stock price at time two? p Problem 2 Find P (X > Y ) if X and Y are jointly distributed with pdf: f (x; y) = x + y; 0 x 1; 0 y 1: Problem 3 Let X1 ; X2 and X3 be uncorrelated random variables with mean and variance 2 : Find in terms of and 2 , Cov(X1 + X2 ; X2 + X3 ) and Cov(X1 + X2 ; X1 X2 ): Problem 4 Suppose that conditional density for Y given X = x is fY jx (y=x) = 2y+4x 1+4x 1+4x and marginal density of x is fX (x) = 3 ; for 0 < x < 1 and 0 < y < 1: Find the joint density fXY (x; y), marginal density of Y : fY (y) and conditional density for X given Y = y : fXjy (x=y): Problem 5 Find the moment generating function for the following distributions: (a) bernoulli(p), (b) exp( ), (c) N ormal( ; 2 ) and N ormal(0; 2 ) and compute the …rst and second moments.

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