Here are some “helpful hints” pertaining to problem set 8, which is due at the beginning of class on Tuesday, October 31.
1. In problem 1, by assuming that markets are open 365 days per year, this implies that the “delta t” for 1 day is simply 1/365 = .00274.
2. Problem 2 asks for probability distributions. Given that the company’s cash position follows a generalized Wiener process with a drift rate of .2 per month and a variance rate of .5 per month (with an initial cash position of $3 million), then the probability distributions for 1 month, 6 months, and 1 year are all normal, and it’s your job to determine what the parameter values are for the mean and variance of the company’s cash position. Note that the generalized Wiener process implies that the expected value of the cash position will equal the initial value of the cash position ($3 million) plus multiplied by the number of months (1, 6, and 12), and the variance of the cash position will be the monthly variance rate multiplied by the number of months. The probability of a negative cash position in x months time is calculated by finding the z statistic, i.e., z(x) = (0 – (3 + .2(x))/(), and then plugging the z statistic into Excel or relying upon the Standard Normal Distribution Function (“z”) Table.
Perhaps the most challenging aspect of problem 2 is part 3: “At what time is the probability of a negative cash position the greatest?” The trade-off here is that as time passes, the expected value of the cash position increases linearly, whereas the volatility increases in a non-linear fashion. To solve this problem, note that the probability of the cash position being negative is maximized when (3 + .2(x))/()) is minimized (note that this will yield the most negative possible value for z(x)).
3. The key to solving problem 3 is to recognize that since the stock price is lognormally distributed, the natural logarithm of the stock price is normally distributed. Equation 8 from my “Applying Ito’s Lemma to determine the parameters of the probability distribution for the continuously compounded rate of return” reading shows the distribution in this case:
All you have do here is calculate the z statistic, which corresponds to the ratio of the difference between the natural logarithm of 100 minus the expected value of the natural logarithm of the stock price, divided by the standard deviation of the natural logarithm of the stock price. Since you are interested in knowing the probability that the stock price will exceed $100 in 2 years, this corresponds to 1 minus the probability of this z statistic.