1. Using Probability Distributions for Project 2

2. Class Project How can probability distributions help us price a stock option? Last time we computed the weekly ratios of closing prices based on our historical data. A lot of ratios + chaotic ordering to values = Reason to create histogram!! Because our weekly ratios can take on an infinite number of values, we must treat it as a continuous random variable.

3. Class Project—Random Variables We will be using the following variables in our project. Each one will be explained as we go along in this presentation: C=closing price per share of our stock on expiration date R=ratio of future to present values R m =average value of all weekly ratios of adjusted closing prices r rf =risk-free rate R rf =the ratio of future to present values of an investment growing at the risk-free rate R norm =the normalized ratios The random variables, C, R, and R norm are all continuous random variables

4. Class Project According to assumption 1 of our project, we have no way of knowing E(C) from the use of our historical data. Assumption 2 says that we can use historical data to predict a stock’s volatility. In this project, we’ve been looking at the weekly ratios of adjusted closing prices to measure a stock’s volatility  Remember: A lot of ratios + chaotic ordering to values Unfortunately, these ratios are too reliant upon the underlying growth of our historical data.

5. Class Project Let R be a continuous random variable for the weekly ratios of closing prices of your particular stock. Our calculations showed us that the average value of all weekly ratios of closing prices, R m, for Walt Disney was 1.0019. This number told us that on average, Walt Disney stock went up by 0.19 % per week. Recall also that our risk-free rate, which we will denote as r rf, is 4%. The weekly ratio corresponding to this weekly rate is e 0.04/52. We call R rf = e 0.04/52  1.0007695 the risk-free weekly ratio for the Walt Disney option

6. Class Project Notice that R m is greater than R rf. So what!?  R m is based on the underlying historical data of our particular stock  Assumption 3 says that all stocks are assumed to have the same rate of return.  Assumption 4 says that rate of return is the risk-free rate

7. Class Project Depending on your stock’s historical data R m may be higher or lower than R rf We need to find a way to bring all stocks to some common means of comparison so that the average weekly ratios, R m, will be the same as the risk-free weekly ratio, R rf. For example, consider two possible savings accounts. Account A compounds interest quarterly at a rate of 4%. Account B compounds interest monthly at a rate of 3.9%. Which account is more likely to accrue more interest? To bring these two accounts to some common means of comparison, we would need to look at the effective annual yield. The same idea is analogous for our options project

8. Class Project--Normalizing In our Project, we need to eliminate the inherent growth rate for our particular stock. This growth rate is embedded in our R m. This process is called normalizing. Normalization is done by adjusting the observed ratios, so that their average is the same as the risk-free weekly ratio, R rf. We must reduce our observed ratios by the difference R m  R rf We let R norm be the continuous random variable of normalized ratios, and write R norm = R  (R m  R rf ).

9. Class Project--Normalizing Note that the average value of our normalized ratios will be the average of the original ratios minus (R m  R rf ), which will be equal to R m  (R m  R rf ) = R rf. This was our goal in normalization. Each team should now normalize the ratios of their company’s stock. Create a histogram plot of the relative frequencies of R norm

10. Class Project Recall that R was a continuous random variable for the ratios of weekly adjusted closing prices of your stock. Now that we’ve adjusted these ratios through normalizing, R norm must also be a continuous random variable Because of this R norm, we would like to know the probability density function for this random variable. If we know this, we can then get a sense of what the closing price might be for Walt Disney stock for each week and thus on the expiration date of our option.

11. Class Project Unfortunately, we don’t have a “nice” formula for the p.d.f. All we have is the histogram that we made for our normalized ratios. Is there a way we can use this? YES! But we need to learn about something called random sampling and simulation