Financial Mathematics is the field of applied mathematics that involves defining problems in finance and providing solutions using methods that draw from probability, statistics, differential equations, optimization, numerical methods, and data science.

The primary emphasis in financial mathematics is the derivation of the mathematical models that confirm the intuition from financial economics. For example, the seminal case of the Black-Scholes-Merton model, and its many extensions such as stochastic volatility, pure jump processes, and collateral funding, is built around the no-arbitrage assumption and assumes as given the evolution of the stock price in order to find the prices of derivative securities.

The unifying premise for financial mathematics is more than just a collection of techniques applied to a common problem area. Rather, it quantifies and enables much of the modern interplay in global markets among companies, investors, and financial agents, often constrained or constructed by the actions of central banks, regulators and governments. Global financial institutions develop and provide products and services that are vital to the course of capital allocation, investment, and risk transfer. None of this could occur without the sophisticated approaches enabled by financial mathematics which have evolved over the past 25 years.

## Primary Areas of Research

Hopkins Engineering faculty research in financial mathematics focuses primarily in the following areas:

Studies of the markets for oil, metals, agriculture, cryptocurrencies, electricity and alternative energy, from extraction or production through delivery and usage – the so-called ‘supply chain’ and its financing issues.

Extending and proposing new models with realistic and desirable financial properties and then employing various tools from stochastic calculus to PDEs and Monte-Carlo methods to find ‘no-arbitrage’ prices of derivatives. Many problems are still open in the case of incomplete markets.

To the extent that financial markets are not perfectly efficient, statistics combined with financial theory offers the possibility of producing positive excess returns on a risk-adjusted basis.  Classical methods such as value investing and factor models are now supplemented by modern approaches using machine learning, natural language processing and ESG investing.

Risk management focuses on quantifying various risks that financial players are subject to and defines ways to mitigate and reduce them.  Examples include credit risk and systemic risk.

This area focuses on understanding and solving investors’ fundamental problem of wealth maximization in various settings, and then deriving the resulting price models, especially in the relevant and unsolved context of market incompleteness.

## Related Courses

Complete descriptions appear in the course catalog.

View the semester course schedule.

• EN.553.4/613 Applied Statistics and Data Analysis I
• EN.553.4/614 Applied Statistics and Data Analysis II
• EN.553.4/627   Stochastic Processes and Applications to Finance
• EN.553.4/628   Stochastic Processes and Applications to Finance II
• EN.553.4/629   Introduction to Research in Discrete Probability
• EN.553.4/633   Monte Carlo Methods
• EN.553.4/636   Introduction to Data Science
• EN.553.4/639   Time Series Analysis
• EN.553.4/641   Equity Markets and Quantitative Trading
• EN.553.4/642   Investment Science
• EN.553.4/644   Introduction to Financial Derivatives
• EN.553.4/645   Interest Rate and Credit Derivatives
• EN.553.4/646  Risk Measurement/Management in Financial Markets
• EN.553.4/647 Quantitative Portfolio Theory and Performance Analysis
• EN.553.4/648   Financial Engineering and Structured Products