Constructing Binomial Trees

1. Pick two stocks, one more volatile and one less so.
2. Construct a two-period binomial tree for each stock.
3. Based on the binomial trees in step 2, price (a) an European at-the-money put option and (b) an American at-the-money put option for each stock. You can choose any time to maturity (T), e.g., 1 month, 3 months, or 1 year. The strike price can be the stock close price on the day you start your assignment. There are in total 4 options you need to price.
4. Discuss your results. For example, which stock has a more expensive European put option? For the same stock, which option (European or American) is more expensive?

When constructing the binomial trees, you can make any assumptions you want, as long as you can justify them (e.g., show the website from which you download the data or simply provide verbal explanation of why you make this assumption).
Some website you can use (I took Apple as an illustration):
1. For the dividend yield, you can look up the “Forward Annual Dividend Yield” in Statistics in Yahoo Finance: https://finance.yahoo.com/quote/AAPL/key-statistics?tilip=AAPL When constructing the binomial trees, you can make any assumptions you want, as long as you can back them up with evidence and reasoning (e.g., show the website from which you download the data or simply provide verbal explanation of why you make this assumption).
Some websites that you can utilize (I’ll use Apple as an example) are as follows:
If you want to know the dividend yield, you may look up the “Forward Annual Dividend Yield” in the Statistics section of Yahoo Finance at this link: https://financial.yahoo.com/quote/AAPL/key-statistics?tilip=AAPL
2. You can estimate the annual volatility using the procedure on pages 48-50 of the attached slides. For the historical stock prices, you can look up Historical Data in Yahoo Finance. You can use the “close price”. In order to make calculations easier, you can change the frequency to “monthly” so that you only need 12 data points: https://finance.yahoo.com/quote/AAPL/history?period1=1616892666&period2=1648428666&interval=1mo&filter=history&frequency=1mo&includeAdjustedClose=true
3. The risk free interest rate can be found at: https://ycharts.com/indicators/3_month_treasury_rate
This is an annualized interest rate, so don’t forget to adjust for the length of each step.

The final report doesn’t need to be long. You only need to show the binomial trees (including the stock price and the European&American option price at each node), the final option prices, and your discussions. You can use any software (excel, word, etc) you want.

Hints on constructing a binomial tree:
1. Estimate the historical annualized volatility of the stock.
2. Calculate u and d using the formulas on page 44.
3. A two-period binomial tree has 3 time points: t0, t1, and t2 (the maturity time)
4. It’s easier to use the risk neutral probability to price the option (page 34 of the slides).