Black swans and white eagles: on mathematics and finance

Abstract

It is often objected that we cannot use mathematical methods in finance because (1) finance is dominated by unpredictable unique events (the black swans), (2) there are qualitative effects that cannot be quantified, and (3) the laws themselves of finance keep on changing. In this paper we discuss these three objections, offering arguments to reject them. We begin by reviewing the development of the physical sciences, pointing out parallels that are relevant for our discussion. Modern science has abandoned the objective of describing reality and has adopted an operational point of view that regards physical laws as tools to connect observations. Modern science is no longer deterministic, but has accepted a fundamental uncertainty in physical laws both at micro and macroscopic levels. Unpredictable pivotal events exist in the physical sciences as well in finance but this does not lead us to question the use of mathematics in the physical sciences. On the contrary, using principles of safe design, we try to understand how to avoid and contain unpredictability. Financial markets are manmade artifacts with, as actors, a large number of interacting agents. If we so wish, we can reduce the level of uncertainty present in markets: But if we try to do so describing financial markets with simple mathematical laws, we find that these laws are not stable but change over time, eventually with sudden structural breaks. This makes the use of mathematical finance difficult but not impossible. We can forecast human decision-making processes, crucial in forecasting financial markets, at the statistical level in aggregate. From an operational point of view, we have the tools to understand and describe the behavior of large number of interacting agents. At the present stage of development of our science, we need to use the mathematics of adaptive systems, changing mathematical models in function of different market states. However, reductionism to a small number of basic laws remains a fundamental objective of financial economics as it is in the physical sciences.