Recover full tick data from missing tick data

Question

Due to some economics/regime problem, I can only have access to non full-tick data from an exchange.

To make the problem precise, a full tick data $X$ is a series of $(t_i,p_i,v_i)$ for $0 \leq i \leq N$ where $t_i$ is the timestamp, $p_i$ is the price, $v_i$ is the deal volume.

The data that I could only see is a lower resolution $\hat{X}$ of $X$, in the sense that, I can only observe the market in a sequence $j_1 < j_2 < \ldots < j_m$ and get the data like: (the sequence is not necessary deterministic or in fixed interval)

$(\hat{t_{j_k}},\hat{p_{j_k}},\hat{v_{j_k}})$ where $\hat{t_{j_k}} = t_{j_k}, \hat{p_{j_k}} = p_{j_k}$, but $$\hat{v_{j_k}} = \sum_{i=j_{k-1}+1}^{j_k} v_i$$

For instance, if the true data $X$ is:

$(0,100,1) \\ (1,102,2) \\ (2,101,1)$

I may only see the lower resolution one $\hat{X}$ as

$(0,100,1) \\ (2,101,3)$

or

$(1,102,3) \\ (2,101,1)$

The question is..

1. Suppose I only have one source of $\hat{X}$, what is the best way to recover most missing tick? I know this may be a bad question, as information has already been lost. I think I need to add some model assumption for this problem from Bayesian point of view, any reference for this?

2. Suppose I have two different source of $\hat{X}$, and because of random nature of the missing ticks, two source would be different. Any method to recover it?

P.S. I think I can think the tick data as a one-dimensional image, and lower resolution data is a pixelized version of real image data, and apply some image processing technique on it, any idea?

Answer 1

If you're missing ticks, then no technique will get those ticks back.

If you have two sources, then designate one source as the primary feed and then fill-in gaps from the secondary feed. Of course, you'll have to mind the timestamps when determining whether the secondary feed can be used properly.

Answer 2

Obviously merging two streams is harmless and it should be done. But it's hard to advise you regarding the "interpolation" methods you can use to generate the ticks without knowing why you need this. The reason is that any method will introduce a certain bias to the data. Therefore, it very much depends on what are you going to do with your altered data on the next step.

Some links regarding the interpolation methods that you can find useful:

• Take a look at the book An Introduction to High-Frequency Finance (preview in Google Books available)
• Olsen Data is using linear interpolation