Chronological snobbery has no place in the markets. Theorists of futurity mock fate when they decry that the ancients have little to say on matters of present importance. Experience and intuition converge to suggest that the oldest ideas have great value in the marketplace of ideas. Pending implementations of combinatorics in the development of Quantum AI is one such example of the enduring value of an ancient thing: in this case, the art of counting smartly, which is the basis of the combinatorial method, and is to a great degree the foundation of the theory of Ensembles that makes canonical mathematical physics possible. Valuation and exchange of derivatives is extensively based in the theory of Ordinary and Partial Differential Equations. Existence, Uniqueness, and Strange-Attractor-Free stability theorems guarantee the properties of solutions under given initial or boundary conditions.
Under ideal conditions, such initial data and the associated regularity conditions can be assumed. The utility of continuous dynamical systems is more limited in the messy world of real-life interactions.
In no sphere is this premise truer than in the modeling and exchange of options and forex instruments whose values at a given time are often determined by hypothetical states of the market. Stochastic Differential Equations form the basis of risk-modeling for such instruments and algorithmic trading systems employ numerical approximation to resolve such systems under varying degrees of uncertainty. As mathematical knowledge of physical systems advances, theoreticians may soon pioneer more stable and sophisticated paradigms to model market forces. These models may well be based upon models of Field Theory and Quantum Mechanics in which interactions are modeled upon the exchange of virtual particles rather than the classical phase-space orbits given by. Quantum computing will almost certainly advance to use in algorithmic trading. When this technology is implemented, it will likely provide sufficient utility that its adaptation across market sectors will be essentially ubiquitous.
The most ancient of counting techniques – pure algebraic combinatorics – is precisely what is needed to understand Quantum AI. Firms that lack the computational power afforded by Quantum computing would be in the stone-age compared to others who had adopted it and would have to rapidly adapt or perish. Its development on global exchanges will ring the death knell of semiclassical PDE as the main instrument of study in the exchange of derivatives. When microseconds matter, algorithmic efficiency becomes important. In the 1970s’ it was opined that massively parallel processing would render the gentle art of counting obsolete. Fifty years later, from the vista of advances in theoretical physics applied to data analysis, one can see that this thinking is essentially incorrect. The orbital mechanics that drove Cold-War space-races and the concomitant development of ballistic missiles were amenable to classical Differential Equations and Lagrangian Dynamics.
But how much more subtle are the workings of humans than the movements of the stars! Quantum computation requires combinatorial physics, and combinatorial physics requires classical combinatorics. Rather than heralding the irrelevance of classical pure mathematics in market analysis, postmodernity will very likely vindicate the utility of the most abstract combinatorics. Quasi-physical notions such as information pressure, volume, topology, and geometry will require adaptation of methods employed presently only in abstract physical contexts. Such contexts include the combinatorics of Field interactions and abstracta of geometry presently employed in the entropic and thermodynamic properties of Black Holes. It will be necessary for mathematicians, physicists, economists, and philosophers to work side-by-side to ensure the ethical operation of markets in a paradigm of decreased latency in trading and self-aware AI. Academicians and traders will have to collaborate to avert instabilities arising from problematic implementation of new technologies. The extent to which this collaboration will be forthcoming will soon be a matter of entrepreneurial and regulatory interest.
About the Author: Dr. Jonathan Kenigson, FRSA is a mathematician and the Academic Don of Athanasian Hall, Cambridge, Limited. Athanasian Hall is a university-independent think-tank comprised of leading scholars of the classical Quadrivium. By training, Dr. Kenigson is a pure mathematician.
