Markov Chain Properties

Markov chain properties describe systems where the probability of moving to the next state depends only on the current state, not on the sequence of events that preceded it. This memoryless property is a fundamental assumption in many financial models, including trinomial trees.

It greatly simplifies the math, as it means we do not need to track the entire history of an asset's price to calculate its future potential; we only need to know its current price. This allows for the efficient use of dynamic programming, as the value of an option at any node in a tree is independent of how the asset arrived at that node.

While some complex derivatives are path-dependent and require more sophisticated models, the Markovian assumption is sufficient for a vast range of financial instruments. Understanding these properties helps analysts identify when a model is appropriate and when the assumption of memorylessness might lead to inaccuracies in pricing or risk assessment.