16. A project generates annual cash ows Ct, received at the end of the year. The cash- ow is based on market conditions and changes from year to year as follows: Ct+1 = 2 + 0:8Ct + 20t+1 where N(0; 1) is drawn randomly from a normal distribution with zero mean and unit variance. This cash- ow-generating process continues year after year even if the plant is closed. You have to decide when to keep the plant open. Shut-down cost is 5 and start-up cost is 6. The rst year's cash- ow is 10. The project is such that one can start and stop it in any year. Find a rule of the following form: nd triggers a; b within which the project will operate. The limit a is such that if the project is not in progress it will start when the cash- ow level crosses above a. The level b is the stop limit, i.e., if the project is in progress it is worthwhile to stop it when the cash- ow level drops below b. The goal is to nd a rule that maximizes the average cash ow over time. [Hint: think of a way to solve this problem using Monte Carlo simulation]. 17. In this problem, you will download market data and generate a state space for valuation purposes. Please carry out the following set of steps: (a) Download 5 years of monthly stock price data from the web. You may use a convenient source such as Yahoo! Finance. Do this for 10 stocks. (b) Convert the stock price data into returns. (c) Compute the mean stock returns for each stock, and the covariance matrix of returns for all the stocks. Sundaram & Das: Derivatives - Problems and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 (d) Use the covariance matrix and mean returns to generate the state space of stock returns using the Gram-Schmidt decomposition technique. (e) Price an option that pays o $1 million when the stock return exceeds 10% on more than 5 stocks. (f) What business decision might this option pricing problem provide you an insight into? 18. You have current wealth of $100. You are o ered a venture in which you may with equal probability, double your money or halve it. If your utility is the square root of your wealth, would you take this venture? 19. You have developed a new material called gossamer, which has demand characteristics closely related to the markets for gold and silver. The prices of these commodities at the end of the year are forecast to be as follows: Material Low Demand Prices High Demand Prices Gold 300 400 Silver 4 8 Gossamer 50 80 The input raw materials to make Gossamer cost $65. Do you think this is a project you would be interested in pursuing?
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