Instructions: This project is designed to help you apply what you have learned about optimal portfolio construction. You may use Excel, Stata, Matlab, R, etc. to complete the project. Regardless of which software you use, you must document with images each step. Brie y explain your work throughout the project: what is the purpose of the calculations you are doing? The main goal of the project is for you to understand the theory behind the Binomial Model of call option pricing, as well as nd good data sources and become familiar with using statistical software. Important: The project should look and read like a short paper. In particular, it should have an Introduction section, a Data section, a Results section etc. Please do not treat this project as a Problem Set.
1:Before we use the binomial model we need a way to choose reasonable values for the parameters u and d. The rst step in doing this is estimating , the standard deviation of the stock`s continuously compounded annual rate of return. Choose ONE stock that has been traded for more than 15 years and is di erent from one you used in Project 1. Same as in Project 1, given the wide variety of publicly traded stocks, it would be a tremendous coincidence if two students had the same stock. Use historical price data from the rst trading day of December in each year to get annual returns for this stock. Please use all available data (i.e. for stocks that have traded for more years you should have more observations). Provide a graph of the yearly stock return over time. In addition to the graph, provide a table of summary statistics on returns, including the mean, variance, skewness, kurtosis, median, interquartile range, and maximum and minimum values.
2:We now want to convert the annual returns that we have from above into continuously compounded annual returns. You may use a formula from the textbook for this, but please give some intuition as to where that formula is coming from. Find ^ the sample standard deviation of the continuously compounded annual returns. This is the unbiased estimate of the standard deviation of the continuosly compounded annual returns.
3:The binomial model we will use for this project is a 12-period model. Use the number of periods together with ^ to calibrate u and d, these are the factors that we will use to forecast the future possible prices of the stock. Let S0 be the Price of the Stock on Dec-1-16. Build a 12-period tree for the price of the stock where u and d are used to forecast the price on Jan-1-17, Feb-1-17, etc. up to Dec-1-17.
4:You need to price a call option on the stock with exercise price S0 and expiration date Dec-1-16. Build a tree that has C, the option price at the origin vertex and Cu12; Cu11d;; Cu10d2; : : : at the end vertices. Replace the notation for the end vertices with the option payo given the price forcast on Dec-1-17.
5:In order to use the bionomial model you need a risk-free interest rate. Instead of giving this exogenously, we would like to use an estimated rate from the data. Find the returns to one-month T-Bills over the past 10 years. Estimate the average monthly return of T-Bills given your data. Use this estimate as your monthly risk-free rate andnd the price of the call option.
6:Find the actual price of the call option on your stock with exercise price roughly S0. Compare the actual price to the price you computed. In the light of this comparison, what can you say about the e ectiveness of the Binomial Model in pricing call options? Summarize brie y what you learned from the experiment.