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Unformatted text preview: IEOR 4404 Practice Final Solutions Simulation December 8, 2008 Prof. Mariana OlveraCravioto 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 lognormal with parameters ( , 2 ). To estimate and 2 we can compute the joint MLEss for and 2 using iid lognormal 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 lognormal 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 lognormal distribution is given below. f ( x ) = 1 x 2 e (ln x ) 2 2 2 , x &gt; 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...
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 Spring '08
 MarianaOlveraCravioto
 Normal Distribution, Probability theory, PRACTICE FINAL SOLUTIONS

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