Questions on Signal Processing / Non-Differentiable Manifolds / Cryptography / RenTech Strats

Hi, I am relatively new to Quantopian, so perhaps this question has been answered in one way or another in another thread. I come from a math, science background as well as fundamental investing background and am trying to wrap my mind around signal processing in the stock market. If you understand non-differentiable manifolds, cryptography and high energy physics please comment below with specific topics/research in those 3 fields that I may be scratching the surface of so I can do further research.

There are 2 questions I am trying to understand better:

1. If we take the stock market as a whole and think of all of the available alpha in it that can be derived from any number of strategies (Long / Short, convertible / M&A arbitrage, long only, pair trading, news articles as signals, price/volume), how do we think about the "exhaustion" of alpha that is achieved from many more people using any specific strategy. Specifically, how can this "exhaustion" be represented in mathematical or statistical terms? For example, we know price / volume is overused and we can sort of backtrack into the "exhaustion" of that strat by seeing lower returns of the strategy over time but it is not a precise science.

2. Along the same lines of the previous question, let's say we can represent the previously mentioned strategies (6 of them) as 6 different functions that are perhaps part of a larger universe of functions (strategies) that we have either not thought of yet or never will (perhaps they are hard to visualize and make sense of in our 3D + 1 world but would in more dimensions). Now my question is, how can this small subset of strategies (using 6 as an example but there many more) point us to either a larger universe of functions (or one wholistic function) that can represent signal processing in the stock market. I am curious to hear others' thoughts, but my intuition tells me we could take the strategies we know to be signals, represent them in mathematical form and use them to point us in the direction of the universe of this larger set of functions (it's kind of like solving a cryptography problem?) . If I understood my math and cryptography correctly, then the probability of finding and using the true wholistic market signal function (to make your trades) increases as you are essentially discovering more "secrets" which would be these strategies - perhaps it could also be said that as you find more and more of these orthogonal (an assumption) functions , you have a better idea of where the next orthogonal function would be? Wikipidia's secret sharing image shows this best - https://en.wikipedia.org/wiki/Secret_sharing. I'm basically theorizing that if you know enough of the functions to represent the planes in a "cube" then you can know what that cube - or stock market looks like and can make more accurate trades. I could also be completely wrong on the math or stats behind this.

Any thoughts/suggestions on the above would be appreciated. My interest in this comes from trying to backtrack into what Renaissance Technologies and the like do to generate alpha and I think places like that only do signal processing instead of trying to pursue any one specific strategy because why have one function of the entire universe when you can uncover the entire set by collecting enough data and be more accurate in your predictions in the stock market.

Andy

1 response

The type of exhaustion you are thinking about is sometimes called "alpha decay". I think a potential way of mathematically analyzing this could be:

1. Use some market model as a benchmark for expected returns (such as CAPM, Fama-French factor models, etc)
2. Compute the realized alphas of the strategy over some time frame (i.e. monthly)
3. Empirically estimate the distribution of realized alphas using non-parametric statistics such as Kernel Density Estimation. This would result in some function f(x).
4. Partition the x-axis into n linearly spaced points, and store the density at each point so we have a sequence ( f(x_1), ..., f(x_n) ). In other words, we have a n-dimensional vector that approximates f(x) if n is large enough.
5. Build a functional time series model of this n-dimensional alpha vector

Visually, we'd have a 3-dimensional surface with time axis, alpha axis, and density. Let's call this alpha axis x, the time axis y, and density z. If alpha decay exists, this surface may collapse towards x=0 z=1 as time increases.

In other words, take a bunch of distribution plots and stack them so we form a hill-like shape. If this hill starts to look less "hill-like"/concentrated around 0 returns, we may conclude that alpha decay has occurred.

As for the second question, I personally don't think the entire market could be collapsed into a single "holistic market signal function". At its core, markets are unsystematic and constantly evolving (i.e. look at market returns before and after the tech boom, 2008 recession, or growth of electronic trading). Thus as quants we can only hope to identify systematic patterns through data while keeping in mind that they are finite-lived due to fundamental changes to market regimes. If we took a cross-section of a market by holding time fixed, certain factor combinations (especially non-linear or higher dimensional) may "uncover" more signals as part of this theoretical holistic market signal function.