Signal To Noise in High Frequency Trading

by theorangedog on Dec.20, 2007, under Skills

I was reading a little bit today about the signal to noise ratio as it applies to sound, which uses a general equation:
$SNR(dB) = 10log_{10}(frac{P_{s}}{P_{n}}) = 20log_{10}(frac{A_{s}}{A_{n}})$ , 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:
$r_{i} = ln(frac{x_{i}}{x_{i-1}})$ ,
then we could derive:
$SNR = frac{r_{i}}{sigma_{r}}$ ,
which results in:
$SNR = left{begin{array}{2}<br /> signal & mbox{ if $frac{r_{i}}{sigma_{r}} g 1$};\<br /> noise & mbox{if $frac{r_{i}}{sigma_{r}} leq 1$}.end{array} right.$

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.

:High Frequency, Signal to Noise, Time Series, Trading, Volatility

14 comments for this entry:

rod
December 20th, 2007 on 2:40 pm
The SNR is a valuable figure of merit in Electrical Engineering, where one deals with stationary signals. When I say stationary signals, I say signals whose statistical properties (such as mean, variance, autocorrelation) are time-invariant. In the physical world one can (fortunately) deal with such signals because Physical laws allow it.

In Finance, the SNR can be used as a figure of merit, but it’s quite a voodoo trick. Why should signals be stationary in finance? After all, doesn’t volatility change over time? One could have a volatility estimator and then estimate the SNR based on it, but there will be a time delay, and time delays + hi-freq finance don’t go well together.

In Engineering one can come up with models that work. In Finance, it seems that whatever model we can think of, it’ll always be a crude, ugly approximation. It sucks.
foq
December 20th, 2007 on 3:08 pm
Thanks for the note rod, a couple of points:

1. I absolutely forgot to mention the part about stationary signals. I read that and just didn’t mention it. It makes me wonder, then, if a test for stationarity would aid in the reliability of a SNR value. Or, would mean-adjusted returns make it more linear, as in the rescaled-range portion of the Hurst equation.

2. The timing issue is one I didn’t think of how to address off the top of my head. For example, even if a series is stationary, volatility inherently changes if the tick jumps. However, is the change large enough to be considered material if the expected holding period is short enough? That would probably also depend upon expected profit. Is there a workable relationship between time/tick intervals and the, well, volatility of volatility?

3. I agree on your last point… all models are simply estimates. A professor recommended I read O’Hara’s book, and I noticed that almost all equations, which were well developed, relied upon a set of unrealistic assumptions, much like Black Scholes. Still, I think there is systematic profit to be made, even if it is timeframe or regime dependent.

As always, I appreciate your feedback.
aiQUANT
December 21st, 2007 on 3:08 pm
Rod,
I can give you one example in electrical engineering where SNR estimation is applied to non-stationary signal. Consider an OFDM signal transmitting from base station to mobile receiver. Given that the channel is wireless you get fading and doppler shifts at the point of reception i.e. non-stationary OFDM reception. Would you say applying SNR to such a scenario is flawed? And yet this method works elegantly for equaliser tracking etc and all other things a demodulator needs to do!

And then the issue of lag - at the high frequency level, a good SNR estimator will give you an accurate measurement within 4 ticks. At the low frequency level the same estimator will give you a measurement within 4 price bars. Is 4 ticks really too long to wait?
DavidF
January 28th, 2008 on 8:48 am
HI guy,

can you more clear up derivatino part of ri? And how large data arrayvof tick i can have for good recognize signal. thanks
foq
January 28th, 2008 on 2:49 pm
$\small r_i$ is just a return figure, used as an alternative to $\small p_t-p_{t-1}$ or $\small\frac{p_t}{p_{t-1}}-1$ .

As far as the size of the data array is concerned… that widely varies. Some measures I use go back 100 bars that measure xx numbers of contracts, others go back 8 5-minute bars. As for backtesting and strategy testing, that also varies. Some argue using 5 to 7 years is the best option, others argue it is important to use an extended period in different market regimes. I don’t have a hard and fast method as of yet, and while I have studied it to some degree, I plan on exploring it further in about two months.
DavidF
January 29th, 2008 on 12:56 am
Thx fro your reply. Can I distrub again? The part wich i dont understand is $SNR = \frac{r_{i}}{\sigma_{r}}$ term of Ro r.

thanks DavidF
foq
January 29th, 2008 on 5:25 am
Hello DavidF,
This may be the weakness of mimeTex compared to LaTeX, as mimeTex isn’t as neat and clean… the bottom figure is actually sigma, not rho. I’m sure that clarifies it for you, but just in case, $\small\sigma_r$ is simply the standard deviation of the return series.

Also, I haven’t tested this idea - it was just something that came into my head one day, as I had been reading some basic signal processing material, that I wanted to note. I’ve been playing with the terms slightly and that has to do with my relative volatility post that was put up recently. I plan on spending more time on this between other studies.

If you have any other comments or questions, feel free to comment again.
DavidF
January 29th, 2008 on 7:39 am
Thx again,

i was supposed that i will be stdev…
Im worknig on simple trading metod with random entry. the concept you can find on:

***link removed by request***
data are from Gain Capital web page

if i do not mistake the concept has great results.

… i am interested in you web page.
David
foq
January 29th, 2008 on 8:24 am
Thanks for sharing David - I’ll be sure to check the link out
DavidF
February 5th, 2008 on 7:06 am
Hi, can you delete my previous post wiht web link?

I have tryed determine nois o FX series and it works fine. Signals can by used for buy or sell I have add chart and it is nice visible on it.

DavidF
foq
February 6th, 2008 on 5:00 pm
sure, I’ll take it off David… sorry for the delay, I was out of town for just shy of a week.
DavidF
February 7th, 2008 on 1:50 am
HI, i am interested in your page, how way and with conditions i can read paper section? THX DavidF
foq
February 7th, 2008 on 8:23 pm
Hello DavidF,
The Papers page is password protected as it is my library of academic papers that I have used in projects. Its not a collection of my working papers, and as of yet I don’t have any published papers.

The page is really for my personal use as the materials are often times protected through distributors such as Blackwell and JSTOR.
Marian
February 18th, 2008 on 12:28 pm
http://www.ederman.com/new/docs/qf-market_bubbles.pdf