Data-Driven Models & Mathematical Finance: Apposition or Opposition?

Abstract

The aftermath of the financial crisis of 2009 as well as the multiple Flash Crashes of the early 2010s resulted in social uproars in the general population and ethical malaises in the scientific community [15, 9, 11, 10] which triggered noticeable changes in Quantitative Finance (QF).

More specifically, QF was instructed to change [16, 17, 18] and become more realistic as opposed to more convenient. The concurrent rise of Big Data (BD) [19] and Data Science (DS) [20] contributed to facilitating these changes. More specifically, in terms of defining new models, we saw a significant increase in the use of Machine Learning (ML) overtaking traditional Mathematical Finance (MF) models. In this thesis we consider the impact of such data-driven modelling transition in finance. In order to illustrate these changes the thesis is divided into two parts, each consisting of three and four chapters.

The first part of the thesis consists of examples in which BD has been exposing the limitations of traditional Financial Mathematics assumptions. Specifically, we develop in that context the Cointelation [11, 10, 5], the IVP [12, 13, 6], the modified Heston [8] and the Responsible VaR [7] models, all data driven modifications of distinguished Financial Mathematics models. We also illustrate how the sum of traditional Financial Mathematics and ML methods can be larger than their individual parts. For instance, we expose how Deep Learning by constraints and Stochastic Calculus can, with the help of feature engineering, allow us to formalize useful dynamical strategies [5].

In the second part, we take a bottom-up approach to algorithmic trading and introduce the High Frequency Financial Trading Ecosystem (HFTE) [4] and illustrate some intriguing connections to the world of evolutionary dynamics. We introduce the concept of path of interaction [4, 3] as a way to test concepts such as strategy invasion. We then explore the challenges associated with properly regulating the algorithmic trading markets, in the era of flash crashes, by formalizing a particle filter methodology [3].

Keywords: Machine Learning, Quantitative Finance, Mathematical Finance, Data Science, Cointelation, Implied Volatility, Agent Based Models