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# fm457y12h2 - 3 The value of the MNO stock is given by a...

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RECOMMENDED HOMEWORK PROBLEMS 2 INTRODUCTION TO FINANCIAL MATHEMATICS MTH 457-001 SPRING 2012 PROF. G ´ ABOR FRANCSICS D310 WELLS HALL, 353-7962, [email protected] 1. PROBLEMS from PROBABILITY . 1. Compute the mean and the variance of a log-normal random variable. 2. Let X be a standard normal random variable. Compute (a) P ( X < - 0 . 5), (b) P ( | X | < 2 . 3). 3. Let X be a standard normal random variable. Show that P ( | X | > x ) = 2 P ( X > x ) by using a substitution or a graph, for example. 4. Let X be a normal random variable with mean μ and variance σ 2 . Let Y be the random variable Y = a + bX for some numbers a 6 = 0, b . Find the values of a , b that give Y the same distribution as X . 5. Give an example for two normal random variables, X and Y with parameters μ , σ such that the sum X + Y does not have normal distribution. 6. A portfolio contains a long position of 100 shares of the ABC company, a short position of 20 shares of the MNO company, and a \$10 , 000 cash position. The value of the ABC stock is given by a lognormal variable X with parameters μ = 0 . 1, σ 2 = 0 .
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Unformatted text preview: 3. The value of the MNO stock is given by a lognormal variable Y with parameters μ = 0 . 2, σ 2 = 0 . 25. The stocks ABC and MNO behave independently. Let the random variable Z be the value of the portfolio. (a) Give a formula for Z . (b) Calculate the expected value and the variance of the portfolio. 2. A MODEL for the MOVEMENT of STOCK PRICES . 7. Let S (0) be the initial price of a stock. Let S ( n ) be the price of the same stock at the end of the n th week, n ≥ 1. Assume that the evolution of these prices follows the rule that S ( n +1) S ( n ) are independent log-normal variables with lognormal parameters μ = 0 . 0165, σ = 0 . 0730 for n ≥ 0. (a) What is the probability that the price of the stock at the end of the ﬁrst week is higher than the initial price? (b) What is the probability that the price of the stock at the end of the second week is higher than the initial price? 1...
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