Products of trees for investment analysis

✍ Scribed by David G. Luenberger

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 294 KB
• Volume: 22
• Category: Article
• ISSN: 0165-1889
• DOI: 10.1016/s0165-1889(98)00018-9

✦ Synopsis

Binomial lattices are commonly used to represent the random return process of a single risky asset. 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.

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 and risk-neutral probabilities are not uniquely defined. It is shown, however, that under a condition that marginal utility is optimally independent, the risk-neutral probabilities are uniquely defined in the product tree. This condition is satisfied in particular if either (1) the single period utility is exponential, (2) the time between periods is very small, or (3) the optimal portfolio contains a zero level of some securities.

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.