I was reading a little bit today about the signal to noise ratio as it applies to sound, which uses a general equation:
, where
P = average power
A = amplitude measured as a quadratic mean

Signal To Noise is often referred to in finance, specifically when it comes to Black’s paper “Noise.” (in the Papers section)

While glancing through that and JSTOR, I came across Truman’s Theory of Noise in Trading paper (also in the Papers section), which used a comparable line of thought to that found in O’Hara’s book Market Microstructure Theory.

I can refer to the equations when I get near the book, but it has a number of them built upon two period models, much like Truman, that determine how a market maker may adjust the bid/ask spread based upon their interpretation of informed trading. Magnitude would play a role, meaning when a market maker felt trading was informed to a scale that would impact their inventory, the bid/ask spread would adjust by a larger amount to handle that. That reasoning is very intuitive, assuming the market maker is risk neutral.

On a tick frequency, could we get a signal to noise ratio based upon larger-than-normal moves, using this logic? I’ll look to find out. If we have:
,
then we could derive:
,
which results in:

The question then becomes if a ratio of 1 is the correct breakpoint, and whether or not the signal and noise measures should be aggregated over a set time bin. There are still a number of questions that relate to this, but it is a framework for starting.

Thanks to aiQUANT for his post on the Hurst Exponent. A copy of the paper he references is available here.

I tried to reproduce the results in excel but ran into a roadblock, mostly in understanding the log transformation. I wasn’t able to access the link referenced in the paper, but aiQUANT mentioned that there is a file on the Mathworks website, so I will check that out.

[Update1]
Bear Cave, I think a site[article] maintained[penned] by the author of the referenced Hurst paper, has a page about the Hurst exponent and its calculation. That information is available here.

Qian’s site also has the MATLAB code for the calculation. You can access the MATLAB file here.

I redid some of the calculations in the excel file, and now my results have narrowed the gap when compared to those presented in the original paper. I think I am messing up on the log transformation of R/S. I will look at the MATLAB code and see… right now I think I have the exponent estimated at around .59.

AUTOREGRESSIVE CONDITIONAL DURATION: A NEW MODEL FOR IRREGULARLY SPACED TRANSACTION DATA - Engle and Russell

This paper proposes a new statistical model for the analysis of data which arrive at irregular intervals. The model treats the time between events as a stochastic process and proposes a new class of point processes with dependent arrival rates. The conditional intensity is developed and compared with other self-exciting processes. Because the model focuses on the expected duration between events, it is called the autoregressive conditional duration (ACD) model. Asymptotic properties of the quasi maximum likelihood estimator are developed as a corollary to ARCH model results. Strong evidence is provided for duration clustering for the financial transaction data analyzed; both deterministic time-of-day effects and stochastic effects are important. The model is applied to the arrival times of trades and therefore is a model of transaction volume, and also to the arrival of other events such as price changes. Models for the volatility of prices are estimated with price-based durations, and examined from a market microstructure point of view.

KEYWORDS:Irregularly spaced time series data, dependent point process, high frequency data.