How Mean Reversion and Momentum Describe Price Processes

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


# 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.

## Answer by Stéphane (score 3, accepted)

https://quant.stackexchange.com/a/51422

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.