Prof. Herdegenm, what is your working group researching?
My working group deals with very different topics from financial mathematics with four main focuses:
- Portfolio optimization in financial markets (with transaction costs)
- Equilibrium models for financial markets and stochastic forward-backward equations
- Risk measures
- (Automatic) Market Making
Roughly speaking, the first topic is about how a layman or institutional investor can optimally invest their money in different market situations given different risk preferences. Mathematically, various financial instruments (e.g. shares) are described by stochastic differential equations and the portfolio optimization problem as a stochastic control problem. A particular challenge here is to take transaction costs (such as broker fees) into account appropriately. The solution combines methods from different areas of mathematics (stochastic analysis, linear algebra, ordinary and partial differential equations, numerics), which I find particularly appealing in this problem.
In the second topic, we try to find out how prices on financial markets (e.g. share prices) are formed through the interplay of supply and demand. Mathematically, this is a fixed point problem on an infinite dimensional space and therefore relatively complex. Many fundamental questions remain unanswered. A relatively new approach is to describe the problem as a system of (coupled) stochastic forward-backward equations, which in itself is an exciting area in stochastic analysis.
In the third topic, we deal very fundamentally with the question of how to accurately measure financial risk in the financial sector and thus contribute to the long-term stability of the financial system. It turns out that “risk” has different dimensions and that you can reduce ‘risk’ (of one kind) by increasing “risk” (of another kind) - which is probably often done in practice, but does not necessarily contribute to the stability of the financial system. Mathematically, we examine various (new) axioms that a “risk measure” fulfills or does not fulfill, and what effects this has on the capital requirements of financial companies. We mainly use methods from functional analysis.
In the last topic, we examine both traditional electronic trading platforms such as limit
order books and new decentralized trading platforms such as automatic market makers based on
blockchain technology. The latter platforms in particular are virtually unregulated and offer both
opportunities and major risks for naive investors. We try to understand the strengths and
weaknesses of these new trading platforms and how they can be improved if necessary.
Mathematically, we use methods from stochastic control theory to solve these questions.
What do you find particularly exciting about your work?
For me, my profession is a vocation and I consider it a great privilege to be able to work in research and teaching. This includes the following three points in particular.
Firstly, I am very grateful for the great intellectual freedom I have in my research. As a financial mathematician in a company, I might earn a lot more money, but I wouldn't be able to choose the problems I work on and, above all, I wouldn't be able to think about them in the depth and detail that is possible in a university context.
Secondly, I really enjoy interacting with very different groups of people (students, PhD students, local and international researchers) and working in a very international environment (my 5 PhD students come from 5 different countries). I perhaps enjoy interacting with students in lectures, seminars and final theses the most.
Thirdly, I consider it a huge privilege to be paid essentially for thinking - and I am always amazed that this thinking is then also of practical use. The famous astronomer and mathematician Johannes Kepler once described his fascination with mathematical/scientific research in a letter to the Bavarian Chancellor Herwart von Hohenburg as follows: "[The laws of nature] are within the power of human comprehension. God wanted us to recognize them by creating us in His image so that we would come into the fellowship of His thoughts." I can only agree with this.
What would you like to focus on in your teaching?
First of all, I want to give my lectures in such a way that the students are enthusiastic about the beauty of mathematics, specifically stochastics and financial mathematics. To do this, I try to constantly rethink how I can present the topics in an exciting, elegant and clear way - especially in “standard lectures” (I am allowed to read measure and probability theory this summer semester).
Furthermore, I would like to contribute my expertise in financial mathematics and stochastic
analysis and regularly give lectures such as Financial Mathematics 1 (end of Bachelor), Financial
Mathematics II (beginning of Master) or Stochastic Analysis (beginning of Master), and offer
seminars, advanced lectures and theses based on these.
Finally, internationality is very important to me in an academic context.
What prior knowledge should students have in order to enter your field of research as part of a Bachelor's, Master's or doctoral thesis?
The most important prerequisite for me is an enthusiasm for stochastics and financial
mathematics topics as well as an openness to learning new methods and techniques from different
areas of mathematics (and beyond).
For a Bachelor's thesis, measure and probability theory is essential, and functional analysis
and financial mathematics I are also desirable. Programming skills (especially in Python) are not
absolutely necessary, but are an advantage for many topics.
For a Master's thesis, you should also have attended advanced courses in the field of stochastic analysis and financial mathematics.
A great enthusiasm for the topic is of central importance for a doctorate. You should also have stamina and an interest in mathematical teaching.
Prof. Martin Herdegen
Research Group for Stochastics and Application
Institute for Stochastics and Applications