3DIMENSIONAL LATTICE MODELS By Espen Gaarder Haug
3dimensional 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 3dimensional 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 MadMax option paying a prespecified 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 "MadMax" option with the payoff profile below: Payoff = K* (Max(0,((X1+X2)/alpha)^2Max(S1X1,X1S1)^betaA  Max(S2X2,X2S2)^betaB))/((X1+X2)/alpha)^2 where S1 is the price of asset one and S2 is the price of asset 2. K is the prespecified 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 addinn function such a calculation only takes a fraction of a second. For more information behind the 3dimensional binomial model see: Rubinstein, M, 1994: "Return to OZ," Risk Magazine, 7(11). Haug, E. 1997: The Complete God To Option Pricing Formulas, McGrawHill New York.

copyright Espen Haug 1998 2004 all rights reserved
