Universal structures in Mathematical Finance

The title of this proposal Universal structures in mathematical finance pertains literally to both, mathematics and finance. On the financial side we mean robust empirical features that hold universally across different financial markets, asset classes and in particular over time. On the mathematical side it concerns universally appearing model classes and probabilistic properties, inherent in many at first sight unrelated phenomena. This universality might sound surprising as financial markets certainly do not obey a law of nature as it is the case for instance in physics. However, even if finance rather appears as social phenomenon, universal market features do exist. Let us illustrate this by means of two important examples which arouse some of the most relevant questions in modern mathematical finance: first, the stability of capital distribution curves over time. These are curves that show the relative market capitalization of listed companies in ranked order. The market capitalization is a publicly known number: it is the number of outstanding shares times the current value of one share. The relative market capitalization is defined as the percentage of the market capitalization of a fixed company with respect to the capitalization of the whole market. The striking feature of these curves is their remarkably stable shape over the past 90 years, unperturbed by times of crisis or flourishing economy. This fundamental observation was the starting point for the mathematician R. Fernholz to develop stochastic portfolio theory about 20 years ago. Since then this fact has been detected in many circumstances, most recently also on the new market of crypto-currencies. The second universal phenomenon that we intend to investigate is called rough volatility. This paradigm asserts that volatility, i.e., the degree of variation of stock prices over time, is of a highly oscillatory nature, meaning that it fluctuates a lot, more than for instance Brownian particles. This behavior has been tested and confirmed recently for more than 2000 equities. Somewhat surprisingly we find a common mathematical framework for both phenomena. It allows to the treat universal phenomena from finance with universal mathematical methods, i.e., structures that appear over and over again in many different circumstances. It is the first time that two major fields of mathematical finance, i.e., stochastic portfolio theory and stochastic volatility modeling, are considered from one common perspective, namely from the perspective of the above described universal structures. We believe that other areas, for instance questions from systemic risk, can be considered in this new light as well.