Why Stock Prices Can Be Modeled with Continuous State Spaces
Summary
The document explains the distinction between how stock prices are observed and how they may be modeled. A stock price is recorded at discrete times and usually on a permitted price grid, yet a model can treat time and price as continuous variables. The response presents continuous models as mathematically convenient and notes that they can arise as limits of discrete models. In practice, a continuous model must still be implemented using discrete data and computation.
It illustrates the separate choices of time scale and state space with four process types: a simple random walk, a Gaussian random walk, a Poisson counting process, and Brownian motion. These examples show that a process may be continuous in one dimension and discrete in the other. The explanation is conceptual rather than an empirical study or pricing method. It does not specify a particular stock model or discuss how discretization affects estimates; the choice between treating prices as inherently discrete or as continuously modeled is left partly to modeling convention.
Key ideas
- Observed prices and observation times are discrete, but a model can represent both as continuous.
- Continuous formulations can be mathematically convenient and may serve as limits of discrete models.
- Computers and finite datasets require continuous models to be discretized in practice.
- Time continuity and state-space continuity are independent modeling choices.
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Full text
# Stock price value as a continuous-time stochastic process
# Stock price value as a continuous-time stochastic process
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.
## Answer by Kevin (score 3, accepted)
https://quant.stackexchange.com/a/50463
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 motionShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.