Random Walk Subject to a Randomly Changing Environment

摘要

A common model for the changes over time of the price (or sometimes the logarithm of the price) of a commodity is the random walk model. This is a Markov model which supposes that the change in price in any time period is a random variable, independent of the past, and having a given distribution F. In this note, we propose a generalized model in which the distribution of price change at any time depends upon the (environmental state' at that time. That is, we suppose that if sub Sn and sub Yn represent the price and the environmental state at time in n then, given sub Yn = i, sub Sn+1 - Sn is a random variable with distribution Fi. We also suppose that the environmental state changes in a Markovian fashion. An application of this model to a stock option example is presented. (Author)