Financial Mathematics

The complexity, unpredictability and evolving nature of financial markets continues to provide an enormous challenge to mathematicians, engineers and economists to identify, analyze and quantify the issues and risks they pose. This has led to problems in stochastic analysis, simulation, differential equations, statistics and big data, and stochastic control and optimization (including dynamic game theory), all of which are reflected in the core of Financial Mathematics. Problems range from modeling a single risky stock and the risks of derivative contracts written on it, to understanding how intricate interactions between financial institutions may bring down the whole financial edifice and in turn the global economy, the problem of systemic risk. At the same time hitherto specialized markets, such as those in commodities (metals, agriculturals and energy), have become more financialized, which has led to our need to understand how financial reduced form models combine with supply and demand mechanisms. In its early days, Financial Mathematics used to rest on two pillars which could be characterized roughly as derivatives pricing and portfolio selection. In this article, we outline its development into broader and more modern topics including, among others, energy and commodities markets, systemic risk, dynamic game theory and equilibrium, and understanding the impact of algorithmic and high frequency trading. We also touch on the 2008 Financial Crisis, among others, and the extent to which such increasingly frequent tremors call for more mathematics, not less in understanding and regulating financial markets and products.