3-DIMENSIONAL LATTICE MODELS

By Espen Gaarder Haug

3-dimensional binomial and trinomial models are extremely flexible models that can be used for valuation of a large number of different options on 2 correalted assets.

THE IDEA BEHIND THE BINOMIAL PYRAMID

The idea is basically to combine 2 binomial trees into a 3 diminesional binomial model, as illustrated below:

This is also known as the binomial pyramid model. By combining two binomial trees in a special way we can easily taking into account the correlation between the 2 assets. The figure below illustrates the 3-dimensional binomial pyramid with 3 time steps.

The binomial pyramid model is simply a discretisation of 2 correlated geometric Brownian motions. To get accurate option values one typically have to use a minimum of 20 time steps. In practice this is naturally done in a computer based model.

VALUATION OF COMPLEX PAYOFF PROFILES

Let's say you want to value a Mad-Max option paying a pre-specified cash amount K=8 at maturity if asset one is at X1=100 and asset two simultaniousy is at X2=150. From this Max point (payoff K) the payoff decrease gradually in each direction. Well in other words you want a payoff profile as illustrated below:

A contour surface of the same payoff profile.

More precisely you want to value a "Mad-Max" option with the payoff profile below:

Payoff = K* (Max(0,((X1+X2)/alpha)^2-Max(S1-X1,X1-S1)^betaA

- Max(S2-X2,X2-S2)^betaB))/((X1+X2)/alpha)^2

where S1 is the price of asset one and S2 is the price of asset 2. K is the pre-specified cash amount. X1 and X2 defines the cross point for maximum payoff K. The alpah, bataA and betaB parameters defines the shape of the payoff. In the figure above I used S1=100, S2=150, X1=100, X2=150, K=8, alpha=9, beatA=1.8, betaB=2. Assuming time to maturity T=0.5, s1= 20%, s2= 30%, correlation coefficient of 0.54, and 50 time steps I get a value of 2.79665 . Using a Tempus C++ Excel add-inn function such a calculation only takes a fraction of a second.

For more information behind the 3-dimensional binomial model see:

Rubinstein, M, 1994: "Return to OZ," Risk Magazine, 7(11).

Haug, E. 1997: The Complete God To Option Pricing Formulas, McGraw-Hill New York.