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I am studying a mathematics textbook on the modelling of stochastic systems. The textbook uses the price of a stock as an example of a continuous-time stochastic process: If $X(t)$ is the value of a stock at time $t$, then $\{ X(t), t \ge 0 \}$ is a continuous-time stochastic process with state space $[0, \infty)$. But in reality, stock price values are integer multiples of $0.01$. So does this mean that stock price values are examples of continuous-time stochastic processes, but have a discrete state space? Am I interpreting this correctly?

I would appreciate it if people could please take the time to clarify this.

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  • $\begingroup$ Or you could say that the stock value does exist in a continuous price space, but we then only observe it in a discrete space. $\endgroup$
    – will
    Dec 29 '19 at 14:12
  • $\begingroup$ @will Hmm, it seems a bit convoluted. So it is indeed correct to say that stock price values are examples of continuous-time stochastic processes, but have a discrete state space? $\endgroup$ Dec 29 '19 at 14:13
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    $\begingroup$ At the end of the day, the model is an approximation. Do we really care about this discrete vs. continuous space? does it make a difference to the properties of the underlying or derivatives priced based on it? $\endgroup$
    – will
    Dec 29 '19 at 14:16
  • $\begingroup$ @will Hmm, I'm more-so asking as a matter of mathematical accuracy, rather than practicality. $\endgroup$ Dec 29 '19 at 14:20
  • $\begingroup$ @ThePointer The model is a mathematical approximation of reality. If the underlying mathematics is correct (which it most likely is), then it's "mathematically accurate" regardless of experimental results. But the practical results generally demonstrate how accurately the model represents reality. A trading strategy is similar to an experiment in Physics where the predicted outcome is profits. A model that can't practically predict outcomes is probably not very accurate. $\endgroup$ Dec 30 '19 at 19:33
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Yes and no. Clearly, stock prices (or prices of any asset) are not observed continuously. This applies to both, the value (price) dimension and the time dimension.

This however does not mean that we can't model stock prices as a time and space continuous process. Frequently, time and space continuous approaches are more elegant and yield nicer results. Furthermore, continuous models are often limits of the discrete models. Anyway, the implementation of continuous models requires you to discretise the model (because you only have discrete data sets and computers can only work with discrete sets).

Note that we can, by the way, record prices with higher precision than cents or pence. Look at currencies which can be traded with a finer grid and we could (technically) use arbitrary fine partitions of the positive real axis. But yes, you can never observe a stock trading at \$$\pi$. But models are simply easier and nicer if you allow for a continuous range.

Whether you see stock prices as a continuous process which we merely record discretely or whether you believe stock prices are discrete objects which we simply model continuously is almost a philosophical question.

Here some examples:

  • Discrete time, discrete state space

Simple Random Walk

  • Discrete time, continuous state space

Gaussian Random Walk

  • Continuous time, discrete state space

Poisson Conting Process

  • Continuous time, continuous state space

Brownian motion

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