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Disclaimer: the question is similar to

Can momentum strategies be quantitative in nature?

and (to an extent)

What is the expected return I should use for the momentum strategy in MV optimization framework?

However (on the surface of it), I did not quite find a desirable answer.

Can a momentum strategy be expressed as something to the effect of $$r = \beta_0 + \beta_1 \frac{\mathrm{ten\_day\_moving\_average}}{\mathrm{hundred\_day\_moving\_average}} + \epsilon$$ or any other combination of predictors ($x_{2}$, $x_{3}$, ...); or maybe as some nonlinear model?

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Yes, they can be quantitative in nature. A way to think about this is to set a period of time to analyze the return and then to set it up as a differential equation model and estimate the unknown parameters. If you think the rate of increase in the return is linear, i.e. the momentum is linear then try running a linear regression model and run diagnostics.

For a first attempt try running some simulation models where you use historical data from another asset with similar prior to purchase return behavior and expected momentum behavior. Use your data to calculate your return (r), which is based on when you purchase and sell the asset and how you are calculating your return and whether you return is absolute or relative (you can simulate different scenarios and different calculations for r).

I also suggest adding in some relative performance measures to explore the notion of 'poor prior performance of a given asset' when building your simulation model.

The multivariate optimization occurs when you solve for the unknown parameters of the quantitative model. The validity of the optimization strategy depends on what method you use to solve and what assumptions you make for the model, such as whether variance is fixed and finite and furthermore whether the return is stochastic.

To be clear, I have no comment on the validity of the theory of momentum or its use for trading strategies.

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  • $\begingroup$ A dissenting view: because a momentum strategy has a go/no go character, switching abruptly between long and not long, it is not particularly helpful to analyze the ratio $\frac{\mathrm{ten\_day\_moving\_average}}{\mathrm{hundred\_day\_moving\_average}}$ which varies continuously. To put it in stochastic control theory terms, the control function is discontinuous in this problem. $\endgroup$
    – nbbo2
    May 26, 2017 at 15:10

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