# Generalizing Binomial Method

- One step Binomial Method is simplistic
- Assumes just two values for the asset price is possible in the future

- More realism can be added by shortening the time intervals so that the calculations can allow for greater number of values for the asset price at expiration
- In our example if we allowed the stock to take values at the end of three months, we would have three values at the end of six months:

- To work out the equivalent upside and downside changes when we divide the period into two three-month intervals (h = 0.25), we use the same formula:
- 1 + upside change (3 months interval) = u = e
^{0.4069âˆš0.25 }= 1.226,=> upside change = 22.6% - 1 + downside change = d = 1/ u = 1/1.226 = 0.816, => downside change = 18.6%

- 1 + upside change (3 months interval) = u = e
- We get the following tree:

85 |
|||||

3 Months |
69.36 |
104.21 |
|||

-18.6% |
+22.6 |
||||

6 Months |
56.6 |
85 |
127.76 |
||

-18.6% |
+22.6 or -18.6% |
+22.6 |

- If the time intervals could be made extremely small, we would be able to account for a large number of changes in the share price
- With the help of computer programs available today the binomial method can be used with very small time intervals
- If one can think of infinitesimally small time intervals, one could have a continuum of stock prices as reflected in the plot below:

- The plot above reflects the log-normal distribution of stock prices
- It can take any value between 0 and infinity
- The fact that the stock price can never fall by more than 100 percent, but that there is a small chance that it could rise by much more than 100 percent is captured in this distribution.