You must agree with me that seeing the prices of stocks tied into knots and links in the market, like shoelaces, is particularly odd. What knots and links have to do with stocks in the first place you might ask?! The answer can be deciphered in the recent eBook–Geometry and Topology of the Stock Market fot the Quantum Computer generation of quants- and it will change your perception of stock market forever.
Although knots are known since the dawn of humankind, and extensively used in everyday life, the mathematical knot theory is only over a century old and was born out of Lord Kelvin vision to model the atom. Lord Kelvin suggested that different atoms were different knots tied in the ether. In the effort of creating a table of elements based on knotted atoms researchers had to distinguish between types of knots and create a table of distinct knots. These efforts lead to discovery of polynomial invariants in the work of Alexander, Conway, Jones and recently many others.
Along the time knot theory found fruitful applications in many fields of research such as biochemistry in reveling the DNA structure, chemistry, quantum field theory and lately an impressive insight in quantum computation. Finance is the latest domain knot theory touched.
A simple and elegant arrangement of stock components of a portfolio (market index-DJIA), has led in the above mentioned paper to the construction of crossing of stocks diagram. Such diagram for a fraction of 4 DJIA components, considering their prices in the interval from 15/05/2013 to 6/07/2013 is shown in figure 1.
Figure 1. Crossings of stocks prices
The crossing stocks method revealed hidden remarkable geometrical and topological aspects of stock market. As is depicted in figure 2, by weighting the stock crossings with the impact every stock has in the crossing, the braid of stocks can be drawn.
Figure 2. From crossing stocks to braided stocks
Closure of a braid is associated with a knot or link of stocks. To distinguish the type of knot stock market form due to quotations of prices, the Jones polynomial is employed. In figure 3 the braded stocks closure results in link, named in topology as Hopf link.
Figure 3. Linked stocks in the market
Apart from the beauty of the idea of a topological stock market the practical applications of this concept must be questioned. I should emphasize here the connection of Jones polynomial with statistical mechanics of phase transition models can be exploited to anticipate and may be even prevent the flash crashes high frequency trading methods can create.
The resemblance of the braiding stocks with the logic gates of the topological quantum computer is a second hint about the ways braided and knotted stocks topology could be applied to financial realities. Under this scenario the Jones polynomial of the knotted stock market acts, making a parallel with the common financial literature, in a topological quantum computation as a counterpart of a classical technical indicator in trading the stock market, but this will be the subject of a future post.