Products of Trees for Investment Calculations

David G. Luenberger
EES/OR, Stanford University
dgl@lis.stanford.edu

Abstract

The use of binomial or multinomial trees to represent uncertainty is a common and important tool in investment analysis. It is simple to represent the random evolution of the price of a single security in this way. When several different securities are considered simultaneously, it is more difficult to construct a multinomial tree that appropriately represents the joint probabilistic evolution of these securities. One way to construct such a representation is to form a product of the trees, constructed from simple trees that individually represent the separate securities. Such a product tree can faithfully capture the joint probability distribution of the security prices.

An apparent disadvantage of the product representation is that the number of nodes at each step of the process is then greater than the number of available securities, and hence replication arguments do not apply. This means that risk-neutral probabilities are not uniquely defined on the overall product tree. It can be shown, however, that under a condition that marginal utility is optimally independent, the risk-neutral probabilities are uniquely defined in the product tree. This special condition is satisfied under a few special but common circumstances, including the following three cases: (1) the single period utility is exponential, (2) the optimal portfolio contains a zero level of some securities, or (3) the time between periods is very small.

This result provides a simple means for representing the prices of several securities in a single tree and for small numbers of securities the method forms a simple and practical method of analysis. Furthermore, the construction is useful for theoretical developments or for pedagogical purposes because continuous-time results can be derived easily without use of multidimensional Ito calculus.