Simulation 1
A forest consists of 1,000 trees forming a perfect 50 x 20 rectangle. The northwestern (top-left) corner tree catches fire. Wind blows from the west, therefore, the probability that any tree catches fire from its burning left neighbor is 0.8. The probabilities to catch fire from trees immediately to the right, above, or below are all equal 0.3.
1. Conduct a Monte Carlo study to estimate the probability that more than 30% of the forest will eventually be burning. With probability 0.95, your answer should differ from the true value by no more than 0.005.
2. Based on the same study, predict the total number of affected trees X.
3. Estimate Std(X) and comment on the accuracy of your estimator of X.
4. What is the probability that the actual number of affected trees differs from your estimator by more than 25 trees?
5. A wooden house is located in the northeastern corner of the forest. Would you advise the owner that her house is in real danger?
Simulation 2
You can choose to invest your money in one particular stock or put it in a savings account. Your initial capital is 1000 $. The initial stock price is 100 $ . The interest rate r is 0.5% per month and does not change.
Your stochastic model for the stock price is as follows:
- next month the price is the same as this month with probability 1/2,
- with probability 1/4 it is 5% lower,
- and with probability 1/4 it is 5% higher.
This principle applies for every new month. There are no transaction costs when you buy or sell stock. Your investment strategy for the next 5 years is:
- convert all your money to stock when the price drops below 95 $,
- and sell all stock and put the money in the bank when the stock price exceeds 110 $. Describe how to simulate the results of this strategy for the model given.