Regression in liquidity risk model of Jarrow/Protter

Question

In the paper "Liquidity Risk and Risk Measure Computation" authors describe a linear supply curve model for liquidity risks in presence of market impact, i.e. impact-affected asset price $S(t,x)$ is proportional to unaffceted price $S(t,0)$ and to traded volume $x$ with some coefficient $\alpha$.

Under the diffusion (with constant drift $\mu$ and volatility $\sigma$) assumption for unaffected price process, the model parameter $\alpha$ is estimated through the regression on returns (see (9)):

$$\log\left(\frac{S(t_2,x_{t_2})}{S(t_1,x_{t_1})}\right) \simeq \int_{t_1}^{t_2}(\mu-\frac{1}{2}\sigma^2)dt + \int_{t_1}^{t_2}\sigma dW_t + \alpha(x_{t_2}-x_{t_1})$$

There is a couple of basic questions that comes up regarding this regression:

1. Should one use the signed values for the buy/sell volumes $x_{t_1}$ and $x_{t_2}$? In the section 2 of the article this is mentioned, however the regression on $x_{t_2}-x_{t_1}$ seems to be complicated in some cases. For example, if there are only buy trades of the same size or when one has only external trades data with no indication of buy/sell available.

2. How should we treat $\int_{t_1}^{t_2}(\mu-\frac{1}{2}\sigma^2)dt$ term? Should we estimate it in the same regression or estimate it separately using unaffected price time series $S(t,0)$?

3. Is it fine to use intraday data throughout a certain period for this regression (e.g. shouldn't one try to account for price jumps between trading days)?

Answer 1

I really like Philip P's work, but frankly I do not believe this paper is his best one. It is understandable you do not catch how to use it: there is no dataset in the paper, and the orders of magnitude of $\sigma dW$ and $\delta_t x$ are so different.

My suggestions:

• some components are missing, $x$ should be a point process, for instance an Hawkes process. Have a look at Market impacts and the life cycle of investors orders (Bacry, Iuga, Lasnier, L) if you want to see how to capture market impact on an informed database (i.e. you have 100% of the information on trades: their side, the metaorder they belong to, etc).

• There is a recent paper by Réné Carmona (and Webster): The Self-Financing Equation in High Frequency Markets. It is shown how to use raw market data to write properly the self financing equation needed in math finance. It contains Jarrow and Protter viewpoint.