Java Quant

            Financial Quantitative Algorithms

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 S0. The price can only take in the next period two values, Su and Sd.

\begin{figure}\begin{center}
\begin{picture}(100,80)(0,-40)
\put(-20,-3){$S_0$}
...
...put(53,23){$uS_0$}
\put(53,-27){$dS_0$}
\end{picture} \end{center}
\end{figure}

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 (1-q). Equation 1 demonstrates this calculation for the underlying security.

S0= e-r (q Su+(1-q) Sd)

(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 Su= uS0 and Sd= dS0, and you get a process as illustrated in figure 1, and calculate the probability q as

                                     q = (er - d)/(u - d) 

 

 


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