A risky business: how to price derivatives
by Angus Brown
A general formula for the multi-period case
The price for the option in an 
period model is given by
![\[ C=\frac{1}{(1+r)^ n}\sum _{k=0}^ n {n \choose k} q^ k (1-q)^{n-k}y_{u^ k d^{n-k}}. \]](/MI/plus/issue49/features/brown/notationhtml1/images/img-0018.png)
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denotes the number of ways in which one can choose 
objects from a selection of objects
(called the binomial coefficient — you can read more in the Plus article Making the grade: Part II). Explicitly it is given by
![\[ {n \choose k} = \frac{n!}{k!(n-k)!}, \]](/MI/plus/issue49/features/brown/notationhtml2/images/img-0001.png)
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where
![\[ n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1. \]](/MI/plus/issue49/features/brown/notationhtml2/images/img-0002.png)
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The symbol 
stands for 
with a subscript consisting of 
and 
- these stand for the payoffs corresponding to the various combinations of good and bad periods.
The expression
![\[ \sum _{k=0}^ n {n \choose k} q^ k (1-q)^{n-k}y_{u^ k d^{n-k}} \]](/MI/plus/issue49/features/brown/notationhtml2/images/img-0009.png)
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means that you should sum the terms of the form
![\[ {n \choose k} q^ k (1-q)^{n-k}y_{u^ k d^{n-k}} \]](/MI/plus/issue49/features/brown/notationhtml2/images/img-0010.png)
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in turn with substituted by 0, 1, 2, etc, up to 
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