1. Binomial Distribution Derivation of the Estimating Formula for u an d ESTIMATING u AND d

2. Estimating u and d The estimating equations for determining u and d are obtained by mathematically solving for the u and d values which make the statistical characteristics of a binomial distribution of the stock’s logarithmic returns equal to the characteristic's estimated value.

3. The resulting equations that satisfy this objective are: Equations

4. Terms

5. Logarithmic Return The logarithmic return is the natural log of the ratio of the end-of-the-period stock price to the current price:

6. Annualized Mean and Variance The annualized mean and variance are obtained by multiplying the estimated mean and variance of a given length (e.g, month) by the number of periods of that length in a year (e.g., 12). For an example, see JG, pp. 167-168. The annualized mean and variance are obtained by multiplying the estimated mean and variance of a given length (e.g, month) by the number of periods of that length in a year (e.g., 12). For an example, see JG, pp. 167-168.

7. Example: JG, pp. 168-169. Using historical quarterly stock price data, suppose you estimate the stock’s quarterly mean and variance to be 0 and.004209. The annualized mean and variance would be 0 and.016836. If the number of subperiods for an expiration of one quarter (t=.25) is n = 6, then u = 1.02684 and d =.9739.

8. Estimated Parameters: Estimates of u and d:

9. Call Price The BOPM computer program (provided to each student) was used to value a $100 call option expiring in one quarter on a non- dividend paying stock with the above annualized mean (0) and variance (.016863), current stock price of $100, and annualized RF rate of 9.27%.

10. BOPM Values

11. u and d for Large n In the u and d equations, as n becomes large, or equivalently, as the length of the period becomes smaller, the impact of the mean on u and d becomes smaller. For large n, u and d can be estimated as:

12. Binomial Process The binomial process that we have described for stock prices yields after n periods a distribution of n+1possible stock prices. This distribution is not normally distributed because the left-side of the distribution has a limit at zero (I.e. we cannot have negative stock prices) The distribution of stock prices can be converted into a distribution of logarithmic returns, gn:

13. Binomial Process The distribution of logarithmic returns can take on negative values and will be normally distributed if the probability of the stock increasing in one period (q) is.5. The next figure shows a distribution of stock prices and their corresponding logarithmic returns for the case in which u = 1.1, d =.95, and So = 100.

15. Binomial Process Note: When n = 1, there are two possible prices and logarithmic returns:

16. Binomial Process When n = 2, there are three possible prices and logarithmic returns:

17. Binomial Process Note: When n = 1, there are two possible prices and logarithmic returns; n = 2, there are three prices and rates; n = 3, there are four possibilities. The probability of attaining any of these rates is equal to the probability of the stock increasing j times in n period: pnj. In a binomial process, this probability is

18. Binomial Distribution Using the binomial probabilities, the expected value and variance of the logarithmic return after one period are.022 and.0054:

19. Binomial Distribution The expected value and variance of the logarithmic return after two periods are.044 and.0108:

20. Binomial Distribution Note: The parameter values (expected value and variance) after n periods are equal to the parameter values for one period time the number of periods:

21. Binomial Distribution Note: The expected value and variance of the logarithmic return are also equal to

22. Deriving the formulas for u and d The estimating equations for determining u and d are obtained by mathematically solving for the u and d values which make the expected value and variance of a binomial distribution of the stock’s logarithmic returns equal to the characteristic's estimated value.

23. Deriving the formulas for u and d

24. Derivation of u and d formulas Solution:

25. Annualized Mean and Variance Equations