I have added a fully functional Models page to the foquant.com website. This page will be updated frequently with new financial models as I standardize and generalize existing ones, and create others. To kick things off, I have posted the first model, Black Scholes Model No Div Yld, which is a standard BSM Model used on an underlying that does not pay a dividend. Of course, variations of this model will be uploaded, including those that accept a dividend-paying underlying. Expansions will be uploaded as well, such as one that will calculate the greeks.

Whenever a new model is uploaded, I will create a post with its name and a brief description, and will add that post to the Models category in addition to the Models page.

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.

THE ECONOMETRICS OF ULTRA-HIGH-FREQUENCY DATA - Engle

Abstract:

Ultra-high-frequency data is defined to be a full record of transactions and their associated characteristics. The transaction arrival times and accompanying measures can be analyzed as marked point processes. The ACD point process developed by Engle and Russell (1998) is applied to IBM transactions arrival times to develop semiparametric hazard estimates and conditional intensities. Combining these intensities with a GARCH model of prices produces ultra-high-frequency measures of volatility. Both returns and variances are found to be negatively influenced by long durations as suggested by asymmetric information models of market micro-structure.

KEYWORDS: transactions data, point processes, hazard functions, survival models, ACD, volatility, ARCH, GARCH, market micro-structure.

Engle draws heavily on his earlier work Engle and Russell (1998). I will try to find, review, and upload this paper. He also draws upon the theories presented in O’Hara’s text, Market Microstructure Theory.

The findings are what would be expected, adding validity to the model.

The durations show very substantial differences over the day and have the typical pattern of high activity at the beginning and end of the day. At the open the average time between trades is about 10 seconds while at lunch it rises to almost 40 seconds. The volatility effect is much smaller but shows the same pattern. At lunch volatility is about 40% of the morning peak. The volatility per trade, which is not shown, is lowest at the open and highest at the end of the day.