PracticeFinalSol

PracticeFinalSol - IEOR 4404 Practice Final Solutions...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: IEOR 4404 Practice Final Solutions Simulation December 8, 2008 Prof. Mariana Olvera-Cravioto Page 1 of 13 Practice Final Solutions This exam is open book/notes/handouts/homework. You are allowed to use a calculator, but not a computer. Write all answers clearly and in complete sentences. All answers should be supported by analysis or an argument. This exam has a total of 120 points and you have 3 hours to complete it. This examination consists of 13 printed pages. First Name: Last Name: I II III Total Honor Code I resolve i ) not to give or receive aid during this examination, and ii ) to take an active part in seeing that other students uphold this Honor Code. Signature: 2 IEOR 4404, Practice Final Solutions Problem I [20 points] Suppose that you are interested in modeling the price of a stock that is assumed to evolve as a geometric Brownian motion. According to this model, the price of the stock at the beginning of the k th day you observe is S k = S exp { X 1 + X 2 + + X k } where the X i s are iid normal random variables. In order to estimate the mean and variance of the X i s you record the price of the stock for 30 days, that is, you collect the values S ,S 1 ,...,S 30 . (a) [10 pts] Describe the steps you would follow to find the parameters and 2 of the normal distribution that best fits the X i s implied by your observations. Note that the data you have are not the X i s themselves. Solution: Option 1: Define X i = log( S i /S i- 1 ), for i = 1 , 2 ,...,k . Then, the X i s are iid normal with parameters ( , 2 ). To estimate and 2 we can compute the joint MLEs for and 2 using iid normal random variables X 1 ,...,X k . From homework 10 we know that the corresponding MLEs are = X ( k ) and 2 = 1 k k X i =1 ( X i- X ( k )) 2 An alternative to estimate 2 is to use the sample variance S 2 ( k ) = 1 k- 1 k X i =1 ( X i- X ( k )) 2 Option 2: Define Y i = S i /S i- 1 , for i = 1 , 2 ,...,k . Then, the Y i s are iid log-normal with parameters ( , 2 ). To estimate and 2 we can compute the joint MLEss for and 2 using iid log-normal random variables Y 1 ,...,Y k . The corresponding MLEs are given in part (b) of this problem. IEOR 4404, Practice Final Solutions 3 (b) [10 pts] Note that if X Normal( , 2 ), then Y = exp( X ), is a log-normal random variable with parameters ( , 2 ). Find the joint maximum likelihood estimators for and 2 . Assume that you have iid observations Y 1 ,...,Y n . The density function of the log-normal distribution is given below. f ( x ) = 1 x 2 e- (ln x- ) 2 2 2 , x > Solution: The likelihood function is L ( , 2 ) = k Y i =1 1 Y i 2 e- (ln Y i- ) 2 2 2 Instead of maximizing L ( , 2 ), we maximize ln L ( , 2 ), which is given by ln L ( , 2 ) = k X i =1 ln 1 Y i 2 e- (ln Y i- ) 2 2 2 = k X i =1- ln Y i- 1 2 ln 2- ln 2 - (ln Y i- ) 2 2 2 =- k X i...
View Full Document

Page1 / 13

PracticeFinalSol - IEOR 4404 Practice Final Solutions...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online