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# IE 5553 Homework 4, due April 7 at 6:10pm 1. (40 pts) Consider the UNH stock trading data that are posted on the class web site. Suppose that on...

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IE 5553 Homework 4, due April 7 at 6:10pm 1. (40 pts) Consider the UNH stock trading data that are posted on the class web site. Suppose that on March 19, 2010, your portfolio consists of 1 share of stock and one call option with a strike price of \$32 and a duration of six months (125 trading days). For this problem, you are to obtain a 95% prediction interval for the value of the portfolio in six months. To estimate the price of the stock six months into the future, use the multiplicative model that was discussed in class, and that also appears in section 6.8 of [2]. That is, S n = S 0 e X 1 + X 2 + ... + X n , n 0 where S i is the price of the stock on day i and X i is a random disturbance on day i . The X i are distributed normal with mean μ and variance σ 2 . The policy for exercising the call option is to wait until the last possible day and then exercise the option if the stock price is greater than the strike price. Here are the steps you need to perform: (a) (10 pts) Using the historical data, estimate the mean μ and standard deviation σ of the random disturbance. (b) (10 pts) Use Monte Carlo simulation to estimate the price of the stock in six months (125 days). That is, six months from the last trading day in the data, which is March 19, 2010. (c) (10 pts) Compute the cash ±ow from the call option. (d) (5 pts) Obtain a 95% prediction interval for the value of the portfolio (one share of stock plus one call option.) (e) (5 pts) Use the bootstrap technique to assess the accuracy of your estimate for the standard deviation σ that was computed in step 1a. That is, use the bootstrap to compute the standard error of σ . 2. (30 pts) Trumpco stock closed today at a price of \$56. From recent history, suppose that the yearly total return is μ = . 12 and the standard deviation of the yearly total return is σ = . 30. Suppose now that you engage in the following strategy: buy one call option with strike price K 1 = \$52 and premium p 1 = \$3 . 05. sell two call options with strike price K 2 = \$62 and premium p 2 = \$1 . 20 (each). buy one call option with strike price K 3 = \$72 and premium p 3 = \$ . 85. The duration for all of the above call options is six months. Use Monte Carlo simulation to estimate the value of this portfolio in six months time (i.e. 125 trading days). You should obtain a range of possible values from the simulation. (Perform at least 1000 replications.) (a) (10 pts) Make a histogram of the simulation results, i.e. a histogram of the possible portfolio values. (b) (10 pts) Qualitatively describe the value of this portfolio. (c) (10 pts) What is the probability that this portfolio will lose money? 3. (30 pts) A bank is planning to install an automated teller station and must choose between buying and installing one Zippytel machine or two Klunkytel machines [1]. Although one Zippy machine costs twice as much to purchase and maintain as one Klunky, the Zippy works twice as fast. Since the total cost of the two systems is the same to the bank, the manager would like to determine which system will provide the best performance in terms of customer waiting times. Customers arrive according to a Poisson process with rate 1 per minute (that is, the times between arrivals are distributed exponential with mean 1 minute.) One Zippy machine (system 1) can serve customers such that the service times are distributed exponential with mean .9 minutes. Alternatively in System 2, each Klunky machine can serve customers such that the service times are distributed exponential with mean 1.8 minutes. In system 2 there will be a single FIFO queue (see the ²gure below).
For both systems, the frst customer arrives to fnd an empty and idle system. The per±ormance measure o± interest is the expected average delay in queue o± the frst 100 customers. Denote this quantity by d 1 (100) and d 2 (100) ±or system 1 and system 2, respectively. Note that the per±ormance measure o± interest does not include the time in service, only the time in queue. Zippy System 1 System 2 Klunky Klunky Model each scenario using Arena. Make n independent replications o± each system ±or n = 10, 50, 100, and 1,000. Note that the output o± each replication will contain the average o± 100 wait times. Compare the di²erence in per±ormance between the two systems by computing a confdence interval ±or the di²erence in the expected average delay in queue. What does it mean i± the confdence interval contains zero? Report your fndings and recommend which o± the two systems gives a better per±ormance measure. What do you notice about the confdence interval, and the quality o± your recommendation, as n increases? References [1] Averill M. Law and W. David Kelton. Simulation Modeling and Analysis . McGraw Hill, second edition, 1991. [2] Sheldon M. Ross. Simulation . Elsevier Academic Press, Inc., ±ourth edition, 2006.

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