Discrete time financial models

Financial economics and quantitative finance uses models of securities prices that must assume that either:

  1. Time is continuous: i.e. there is no smallest moment of time
  2. Time changes in discrete jumps

Discrete time models assume trades take place instantaneously, there is then a very short period of no trading, then trades take place again etc. This is a good approximation as the steps can be made arbitrary short without altering the models.

Discrete time models also, unlike most continuous time models, do not generally treat prices as continuous: the prices are “states of the world” of which there are a (very large!) finte number of possibilities. This is realistic as most securities prices move in ticks

Discrete time models generally require less advanced maths (relatively!) to understand, and many of the most important results of financial economics, such as CAPM have a discrete time derivation.

There are often both discrete time and continuous time versions of the same model, and the discrete time versions ofte converge (i.e. become) on the continuous time equivalent as the time is shortened to zero.

For practical modelling, rather than extending theory, discrete time models lend themselves to simulation techniques (e.g. Monte-Carlo). The actual implementation of continuous time models will also often require the use of computational methods that approximate continuous time with discrete steps: this is distinct from the use of discrete time in theory, and is merely a mathematical method, but show that discrete time is a useful and usable concept.