Binary Trees for pricing financial options
The binary Tree model for pricing options or other equity derivatives
assumes that the probability over time of each possible price follows a
binomial distribution. The basic assumption is that prices can move to only two
values, one up and one down, over any short time period.
The
price of the underlying is currently S_{0}. The
price can only take in the next period two values,
S_{u} and
S_{d}.

Figure 1: Binary Tree 
If
one can find all possible future price states, one can value a security by constructing
state price probabilities, which one can use to find
an artificial expected value of the underlying security, which is then discounted
at the risk free interest rate. The binomial framework is particularly simple,
since there are only two possible states. If we find the probability
q of one
state, we also find the probability of the other as (1q). Equation 1
demonstrates this calculation for the underlying security.
S_{0}= e^{r }(q S_{u}+(1q) S_{d}) 
(1) 
Now, any derivative security based on this underlying security can be priced using the same probability q. The contribution of binomial option pricing is in actually calculating the number q. To do valuation, start by introducing constants u and d implicitly defined by S_{u}= uS_{0} and S_{d}= dS_{0}, and you get a process as illustrated in figure 1, and calculate the probability q as
q = (e^{r}  d)/(u  d)
www.javaquant.net © 2006 H. Aliaga. Design by Cesar.
XHTML 1.0 Strict