Fitting power models

Market impact is often defined as relating the price difference to the volume (or POV) traded, i.e., \(S_T - S_0 = k v_T + \varepsilon\) That model assumes a normal price dynamics, which may or may not make sense depending on your time scale, but this could easily modify to assume a log-normal price dynamics by using the difference of the log of the prices instead.

However, another very common approach is to assume a power law for the market impact, i.e., something like \(S_T - S_0 = k v_T^\alpha + \varepsilon\) Now comes the questions of fitting that model. And this is what this post is about.

There are 2 approaches to go about that:

1. you could fit directly using nonlinear regression techniqes.
2. you could fit instead the log of that expression \(\log(S_T - S_0) = \log(k) + \alpha \log(v_T) + \varepsilon\) However, those two expressions are not equivalent, primarily because of their assumptions on the noise distributions. This is something that is explained in the introduction of that paper (note that the conclusions of that paper are heavily criticized by that other paper).

You can estimate the change in the noise variance when applying method 2 by using the Delta method. This is also discussed into that stackexchange question. In the most relevant answer, the suggestion is to first fit using the log-transform, then use the coefficients obtained with that method to start a nonlinear regression solved using Newton’s method.