# What the most general but precise description one can make about mean-reversion and momentum strategies?

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.

• 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 – LazyCat Mar 1 '20 at 14:28
• I guess I mean people in finance who do not actually implement this stuff. So they "know" but don't actually know. – mathtick Mar 1 '20 at 15:41
• You should agree, it's hardly a quantitative question.. – LazyCat Mar 1 '20 at 19:17
• 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. – Chris Taylor Apr 3 '20 at 10:20