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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 0.4069√0.25 = 1.226,=> upside change = 22.6%
    • 1 + downside change = d = 1/ u = 1/1.226 = 0.816, => downside change = 18.6%
  • 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.





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