A risky business: how to price derivatives

December 2008

A general formula for the multi-period case

The price for the option in an $n$ period model is given by

  \[ C=\frac{1}{(1+r)^ n}\sum _{k=0}^ n {n \choose k} q^ k (1-q)^{n-k}y_{u^ k d^{n-k}}. \]    
Here ${n \choose k}$ denotes the number of ways in which one can choose $k$ objects from a selection of $n$ objects (called the binomial coefficient — you can read more in the Plus article Making the grade: Part II). Explicitly it is given by
  \[ {n \choose k} = \frac{n!}{k!(n-k)!}, \]    
where
  \[ n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1. \]    
The symbol $y_{u^ k d^{n-k}}$ stands for $y$ with a subscript consisting of $k$ $u's$ and $n-k$ $d's$ - these stand for the payoffs corresponding to the various combinations of good and bad periods.
The expression
  \[ \sum _{k=0}^ n {n \choose k} q^ k (1-q)^{n-k}y_{u^ k d^{n-k}} \]    
means that you should sum the terms of the form
  \[ {n \choose k} q^ k (1-q)^{n-k}y_{u^ k d^{n-k}} \]    
in turn with $k$ substituted by 0, 1, 2, etc, up to $n$.

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