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Is there anything about this metaphor of momentum and mean-reversion in markets that is more subtle, more general. What factors are amenable to the interpretation?

Are people almost always referring to this kind of setup with log prices $x_t$, latent "momentum" $v_t$, latent mean $m_t(X_t)$, $X_t \equiv \{X_s;s\leq t\}$?

$$ \begin{align*} \text{d} x_t &= v_t \text{d} t + B_x \text{d} W^x_t\\ \text{d} v_t &= \left(a(X_t) \left(m_t(X_t) - x_t\right) + b(X_t) v_t \right) \text{d} t + B_v \text{d} W^v_t \end{align*} $$

If this is what they mean, then the only restriction from a linear dynamics model (in $(x, v)$) is that the dynamics of $x$ does not depend on $v$. This kind of idea makes sense in physics but probably not so much in financial markets.

Or perhaps some people actually are talking some bigger statement about stationarity and decomposition of the processes.

This is a question about the semantics AND the interpretability of variables.

UPDATE: Best approximation of the semantics so far. I've also tried to clean up the question. Remember this is a poorly-phrased question about semantics more than a conceptual question about modelling. Maybe should be in quant-meta but that doesn't exist.

So I think the best general approach is to understand that one must define a measurable notion of "stationarity" in order to define the class of viable factors/features. For example see https://onlinelibrary.wiley.com/doi/full/10.1002/sta4.125 for an example of scoring methods.

If you learn "stationary" features, you can then learn the coefficients of the prediction problem given those features, potentially trading off degree of stationarity for predictive power though that might be a bad idea in practice.

If you set the sign of the features by constraining the derivative w.r.t last price to be positive, you can probably start to say something about mean-reversion vs momentum (per factor) based on the sign of the coefficients you learn in your filtering/prediction problem.

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    $\begingroup$ What people? If you're asking about market news, they don't have any underlying model and simply mean that if the market was going down it will continue to do so (momentum). If it's an academic paper, they usually specify the model they are talking about $\endgroup$
    – LazyCat
    Commented Mar 1, 2020 at 14:28
  • $\begingroup$ I guess I mean people in finance who do not actually implement this stuff. So they "know" but don't actually know. $\endgroup$
    – safetyduck
    Commented Mar 1, 2020 at 15:41
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    $\begingroup$ You should agree, it's hardly a quantitative question.. $\endgroup$
    – LazyCat
    Commented Mar 1, 2020 at 19:17
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    $\begingroup$ They are deliberately general, to avoid being tied down to a specific model. By "momentum" most people mean ${\rm E}(R_{t-t,t}|R_{t-n,t-1}>0)>0$ and by "reversion" people mean ${\rm E}(R_{t-t,t}|R_{t-n,t-1}>0)<0$ where $R_{a,b}$ is some return between $a$ and $b$ (e.g. price return, or change in interest rate, or return relative to an index) and $n$ is a timescale. It's perfectly possible for a market to have both momentum-like and reversion-like dynamics if they are operating on different timescales, or are time-varying in their strength, or interact with exogenous variables. $\endgroup$
    – Chris Taylor
    Commented Apr 3, 2020 at 10:20

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By mean reversion, people usually mean to say that some price process is second order stationnary -- even though they do not always know the technical term for it. It's only vague in the sense that this defines a class of processes, but doesn't pin down any specific process.

By momentum, people usually mean a process which follows trends: upward movements tend to lead to more upward movements and downward movements to more downward movements. This one is more vague.

In both cases, people tend to use this in reference to prices and in both cases it hints to arbitrage opportunities.

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  • $\begingroup$ Cool, but this implies the structure above I think. Make no assumption about noise and just structure the signal. $\endgroup$
    – safetyduck
    Commented Mar 1, 2020 at 9:48
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    $\begingroup$ It does not imply the structure above. You're assuming continuous time affine dynamics. Clearly, it's not the only class of model that fits these descriptions. $\endgroup$
    – Stéphane
    Commented Mar 1, 2020 at 17:26
  • $\begingroup$ I get the feeling that most basic folks just do these kind of constructive descriptions but but they don't actually define anything. There must be something in the desription of the problem that is momentum like and something that is fundamental "price" like. These are probably latent params for interpretability purposes. But these form the dynamics of the system which is effectively symplectic plus noise. Something like this is probably the general statement. $\endgroup$
    – safetyduck
    Commented Apr 3, 2020 at 9:25
  • $\begingroup$ The impression I'm getting @mathtick is that you are very smart, you probably have a lot of qualifications, are good at math and know many programming languages, but that you have spent very little time either working in finance or trading with your own money. $\endgroup$
    – Chris Taylor
    Commented Apr 3, 2020 at 10:25
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    $\begingroup$ @ChrisTaylor How is that relevant to the question being asked? $\endgroup$
    – Stéphane
    Commented Apr 3, 2020 at 17:20

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