Chapter 6
Continuous-Time Models and Their Applications
The price of a financial asset evolves over time and forms a stochastic process, which is a statistical term used to describe the evolution of a random variable over time. The observed prices are a realization of the underlying stochastic process. The theory of stochastic process is the basis on which the observed prices are analyzed and statistical inference is made.
There are two types of stochastic process for modeling the price of an asset. The first type is called the discrete-time stochastic process, in which the price changes at discrete time points. All the processes discussed in the previous chapters belong to this category. For example, the daily closing price of IBM stock on the New York Stock Exchange forms a discrete-time stochastic process. Here the price changes only at the closing of a trading day. Price movements within a trading day are not necessarily relevant to the observed daily price. The second type of stochastic process is the continuous-time process, in which the price changes continuously, even though the price is only observed at discrete time points. One can think of the price as the “true value” of the stock that always exists and is time varying.
For both types of process, the price can be continuous or discrete. A continuous price can assume any positive real number, whereas a discrete price can only assume a countable number of possible values. Assume that the price of an asset is a continuous-time ...
