:: Collaborations ::

:: Using Binomial Model to Price Derivatives ::

The Binomial Model is one of a set of numerical procedure used for pricing derivative securities. Others include finite difference schemes and Monte- Carlo methods. Binomial model agrees asymptotically with the Black-Scholes model if constructed assuming the same conditions. However it is more versatile, in that it can be used to for pricing American options and complex derivative securities.

As an illustration we shall consider the pricing of a European option. By descretizing the time to maturity of the option into N interval each of duration , we can model the movement of a stock price S by a geometric random walk. At each interval S can increase  to uS with probability p, or fall to vS with probability 1-p.  The constants u, v and p are chosen such that the geometric random walk is consistent with the geometric Brownian motion described by the stochastic differential equation (SDE):

The expectation and the second moment of the stock price after one interval can be found from this construction.

We can equivalently find these moments from the SDE. The second moment is achieved from computing the term of an expansion of F(S)= using Ito's Lemma (see article).

Assuming one knows and then equating the corresponding moments we see that we have two equations and three unknowns, namely u,v and p.

However the parameter p is not independent from the other two if it to describe the same process as the SDE. Consider fixing the value of v. Then we have a range of choices for u each with an appropriate value for p. The larger the value u the smaller the probability p.

Our choice of u with its appropriate p is such that  uv=1 so that we can reduce the problem to a quadratic equation for u, and thus find u and v in terms of , and .

So given the above construction how do we price the option?

Assuming we know the option value at expiry say. We  construct a portfolio at time t consisting of one option and shorting Δ stocks of the underlying S.

Depending if S rises or falls during the next interval, the option value changes to respectively. These values are known since they are defined at the expiry time T. So our portfolio changes to either . If we choose our portfolio to be risk neutral then its value will remain the same regardless of whether the asset rise or fall. This gives that

Risk neutrality also means that our portfolio is guaranteed to increases in value as if it were money invested in a bank with interest rate r.  At expiry the value of the portfolio is its original value plus any risk interest earned in the interval .

Finally we find that

The Binomial method works as follows:

1. Knowing the initial value of the underlying S(t=0), construct a binomial tree for the future values of S up to the expiry time T.

2. For each of these possible final values of S we evaluate the option value at expiry V(S,T).  These make up a set of   at time T, from which we can work backwards to determine all the values of V(S,T- ).  These in turn make up our new set of  , and the process is repeated up to t=0.

This gives a tree or table of values of the option value at different times for the possible values of S.

Written by Raffaello Vardavas

Back for more