# Newsletter of the Bachelier Finance Society

## Volume 12, Number 1, January 2020

### Results of the Elections 2019

The Bachelier Finance Society has elected a new incoming Vice–President, Jakša Cvitanić, and furthermore, five new Council members: Beatrice Acciaio, Pauline Barrieu, Masaaki Fukasawa, Kay Giesecke and Jan Obloj.
Our congratulations to all newly elected members and also our thanks to all those who indicated their willingness to serve as an officer of our Society.
We also want to thank the outgoing members of the Council for their commitment and contributions: Rama Cont, Paul Glasserman, Vicky Henderson, Jan Kallsen and Wim Schoutens.
https://www.bachelierfinance.org/forum/archives/6845

### 11th BFS World Congress, Hong Kong, 1-5 June 2020

The deadline for paper submissions is drawing closer – please remember to submit your paper until 31 January 2020 through the Congress website:
http://www1.se.cuhk.edu.hk/~bfs2020/

### JOB POSTINGS

The aim of these postings is to create a forum for the dissemination of information on academic and industrial positions related to mathematical finance, across different disciplines and different geographical regions. Please submit any job advertisements you are aware of to jobads@bachelierfinance.org, preferably in plain text and sending the link to the website containing all the information. Updates and new items appear continuously at: bachelierfinance.org/forum/jobs

PhD positions
Berlin and Oxford

Lecturer/Assistant Professor
University College Dublin

Postdoctoral Associate position
Boston University

### Announcements

Call for Submissions – Nicola Bruti Liberati Prize
https://www.bachelierfinance.org/nicola-bruti-liberati-prize
Theses defended in 2018 and 2019 must be submitted no later than January 31, 2020.

### BOOKS & JOURNALS

The Society maintains a list of books, book reviews and journals at: https://www.bachelierfinance.org/publications. Members who would like to have their books added to the website, should please let us know.

### Interview with Nicole El Karoui

Would you like to tell us about your youth? How did you develop your interest in mathematics?

I grew up in Nancy in eastern France. At school, like every good student at that time, I studied German, Latin and Greek. I started studying mathematics at high school. After having hesitated between philosophy and mathematics, I finally decided to enter “classes préparatoires scientifiques”. This is where I found the pleasure of doing mathematics, like a pianist who, practicing scales for years, plays his first piece. I realized that mathematical abstraction made it possible to free oneself from the real and to bring together domains that have nothing in common. When I was a student at “Ecole Normale Supérieure”, it was a “love at first sight” for probability. The hazard can be measured! This allows a new way of looking at and of understanding the unfathomable world of the uncertainty. Reconciling hazard and geometry, what a daring vision! Only mathematical abstraction enables such a leap. Looking at the world differently, breaking the codes to capture the essence, abstracting from the reality, and then going back to the original world: these are the essential ideas that inspired my research in mathematics. A very uncertain and exciting quest.

Where did you get your PhD from and on what topic? What were the next steps in your academic career?

I had the chance to start my career in research by doing a thesis on Markov stochastic processes, more precisely on “Processus de diffusion associés à un opérateur elliptique dégénéré et à une condition frontière” [1], defended in 1971. Although it was unusual at that time, the thesis was jointly written. With Bernard Roynette and Hervé Reinhard, we were working together every day and all of our papers were signed by the three of us. Our efficiency came from our different ways of doing mathematics (which has nothing to do with gender). It was an extraordinary time, which gave me the necessary confidence to move ahead in mathematical research, despite the rather poor encouragement from the environment. After the thesis, I participated in the great adventure of the “general theory of processes”, under the main impulse of P.A. Meyer in Strasbourg and J. Neveu in Paris.

Then I got a Professor position at Le Mans University in 1973, a small institution created a few years earlier. It was very interesting because we had to create everything from scratch, especially in probability. I was conducting research on stochastic optimization at the Jussieu Probability Laboratory, to which I was also affiliated. I created a working group on “control theory and filtering” in Paris, which brought together a lot of people from small neighboring structures.

In 1979 I came back to Paris as the responsible person for studies at “Ecole Normale Fontenay aux Roses”, where I introduced in the curriculum probability theory and applied mathematics. Then I moved to a Professor position at University Pierre et Marie Curie (Paris 6), where I co-founded with Helyette Geman the Master “Probability and Finance” in 1989. I became Professor at “Ecole Polytechnique” in 1997, where I created a mathematical finance team. Since 2007 I am Emeritus Professor at Paris 6.

In the course of your long and intense career, you have made many fundamental contributions to our field. Looking back in perspective, is there a common thread among your diverse research interests? What are the research topics that you found most fascinating?

I believe that there are some common elements in my research, even if they are not always manifestly evident. In one of my very first papers [1] I studied (jump)-diffusion processes reflected on a boundary and, in this context, the concept of local time appears naturally. More generally, the link between local times and optimal stopping problems, showing the contribution of the «general theory of processes», led me to deepen the abstract theory of stochastic control [2]. The same tools also played a crucial role when I worked on reflected BSDEs [3], or more singular problems as for example in [4]. In the same vein – even though it did not encounter much success – is the paper on the max-plus decomposition of supermartingales [5], a technique that in my view leads to interesting results. In general, I should say that optimization problems have been a constant presence in my research, and most of the problems studied in finance essentially amount to optimization problems.

How did you develop an interest in mathematical finance?

In 1988, somewhat by chance, I decided to do a sabbatical semester in a bank to experience the professional world. My internship project was in statistics, but I soon discovered that there was a team within the bank using Brownian motion for their work. What a surprise! The first paper I read in finance was a preprint of the pioneering paper by Heath, Jarrow and Morton on term structure modeling, that circulated in the banks already in the years 1988-1989 (it was published in 1992). In 1986-1987 the financial markets had just been created in France. We were in the midst of a technological revolution, the first PCs started to appear, and new ideas were emerging in finance. The feeling was like being in a start-up. Banks needed mathematicians to describe the erratic movements of the markets. I did not know a word in finance (neither in French, nor in English!), but my mathematical culture was very close to what markets needed and the probabilistic tools were ready – I had been doing stochastic control for more than 10 years at that time point. I firstly started working on interest rate models. We were developing models for bonds and using the data on the French government bonds. I did some consulting at Caisse des dépôts, where I continued to discuss on a regular basis with practitioners specialized in interest rates and debt.

You mentioned earlier the Master program “Probability and Finance”. This Master program, also known as “Master El Karoui”, enjoys a worldwide reputation in our field, as documented for instance by the article that appeared in the Wall Street Journal in 2006. How did this Master program develop? In your view, what are its distinguishing features?

It all started in 1989, after my personal experience in the financial industry. Mathematics was playing an increasingly important role in finance and we realized that a specific master program was missing and would have attracted the attention of scientific students. At the beginning, this idea did not encounter positive reactions from my colleagues at Paris 6, but I did not give up and the Master program in “Probability and Finance” eventually started, with the first students graduating in 1990. It was one of the very first programs in the field of mathematical finance and it started attracting brilliant students. In 2006, the Wall Street Journal published its article, trying to understand why so many trading floors were populated by French graduates. In 2008, just after the burst of the global financial crisis, we reached the highest number of students. The program had a somehow rigid structure, strongly rooted on mandatory foundational courses such as stochastic calculus, with the aim of providing a sound probabilistic education, and completed by practical courses. I believe that this marked mathematical focus has been one of the distinguishing characteristics of this program. At the same time, we always kept a close connection with the financial industry and the evolution of the markets.

In your view, did probability mark a significant and lasting impact on the financial industry?

I believe that one of the lasting contributions of mathematics to the financial industry is the creation of precise vocabulary and tools for discussing and analyzing critical aspects of finance. Mathematics helped to create a common background for the quantitative desks of financial institutions and a more rigorous way to analyze concepts. Obviously, it also provides tools for simulation and risk management, always taking into account the permanent evolution. At first sight, this may look as a modest contribution, but it should not be underestimated, especially now in the Big Data era. At every level, we are surrounded by risk and uncertainty. Mathematics can make us aware of this fundamental fact and help us when taking decisions in a risky and uncertain environment.

What do you think are the most important challenges to be faced by mathematical finance in the near future?

I still have the impression that mathematical finance did not take fully into account the fundamental changes that happened in the markets after the global financial crisis in 2008 and that sometimes the research that is done is a bit far from the real market challenges. In particular, we did not make much progress on the very complex issue of financial regulation. This is of course partially due to the fact that there are very few mathematicians involved in creating regulatory policies, which is mainly the realm of economists and lawyers.

The derivative markets changed substantially and will continue to change in the coming years. In particular, there is also the question of currencies, with the emergence of online trading based on cryptocurrencies. Another issue is of course the big data and the value of having access and being in possession of data. There is also the (political) question of dropping the policy of zero interest rates and addressing instead the problem of the public debt. The mathematical finance community certainly has the knowledge and the skills to treat such issues.

One thing I would like to say is that in the social and economic world, such as the finance world, there is no universal truth. We always have to be ready to change, but in an intelligent way and identifying precisely the problem we have to solve (which is not always easy!).

You have recently been working on longevity risk, and the intersection/interaction of financial and actuarial mathematics. What are the challenges there? How do you see this relation evolving in the future?

It has been now more than 10 years, during which period I supervised 3 PhD theses. Already a while ago, we started discussing with insurance companies because they were selling longevity products, such as variable annuities for example, that needed to be correctly priced and methods from financial mathematics lent themselves very well to that purpose. This led to our first survey article [5] on longevity modeling, published in 2012. Studying the longevity of a population is not just a question of mathematical model and one should take into account all the factors and the information on what happened in the past (changes in education, public policies, health standards etc.). To apply methods from mathematical finance to insurance, there is a need to understand precisely which tools can be used and how (understanding the risk-neutral pricing in particular) given that the financial and the insurance markets and their specific problems are nevertheless very different. The main challenge in insurance mathematics lies in the fact that insurance is a long-established and well regulated industry, and it is difficult to introduce much innovation in practice. These are rather small steps aside from common industry practice.

Do you think it is difficult to be a woman in mathematics?

The world of mathematics is still very masculine, in its rules and selection criteria, and not ready to be challenged. In a scientific career, there are crucial moments where women have to be in a positive dynamics towards mathematics, otherwise they may give up. They have a retraction force towards the reality of everyday life which brings out the question “is it reasonable to spend so much time doing math?” It is therefore very important to encourage girls to do mathematics and to provide them successful role models with which they can identify. This is how we can deconstruct stereotypes and preconceptions. Attitudes are slowly improving, but there is still a lot of work to be done and we should remain vigilant.

There is a major need for women in science and in particular in mathematics. The revolution of Big Data makes this matter of great urgency. Humanity cannot deprive itself of half of its intelligence. To make men and women work together in science seems essential, because complementarity is a strength, as evidenced by my own experience.

[1] N. El Karoui (1975), Processus de reflexion dans R^n, Sém. Probab. IX, pp. 534-554, Springer;
M. Chaleyat-Maurel, N. El Karoui, B. Marchal (1980), Reflexion discontinue et systèmes stochastiques, Ann. Probab., 8: 1049-1067.

[2] N. El Karoui (1981), Les aspects probabilistes du contrôle stochastique, École d’Été de Probabilités de Saint-Flour IX-1979, pp. 73-208, Springer.

[3] N. El Karoui, S. Peng, M.C. Quenez (1997), Backward stochastic differential equations in finance, Math. Finance, 7: 1-71.
N. El Karoui, C. Kapoudjian, E. Pardoux (1997), Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, Ann. Probab., 25: 702-737.

[4]. P. Bank, N. El Karoui (2004), A stochastic representation theorem with applications to optimization and obstacle problems, Ann. Probab., 32: 1030-1067;
N. El Karoui, I. Karatzas (1994), Dynamic allocation problems in continuous time, Ann. Appl. Probab., 4: 255-286.

[5]. N. El Karoui, A. Meziou (2008), Max-plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance, Ann. Probab., 36: 647-697.

[6]. P. Barrieu, C. Hillairet, H. Bensusan, N. El Karoui, S. Loisel, C. Ravanelli, Y. Salhi (2012), Understanding, modeling and managing longevity risk: key issues and main challenges. Scand. Actuar. J., 3: 203-231.

The interview was conducted by Claudio Fontana (University of Padova), Zorana Grbac (Université de Paris) and Caroline Hillairet (ENSAE Paris).

## High-Dimensional Probability

### reviewed by Omiros Papaspiliopoulos (ICREA, Universitat Pompeu Fabra and Barcelona Graduate School of Economics​)

“Let us summarize our findings. A random projection of a set $$T$$ in $$R^n$$ onto an $$m$$-dimensional subspace approximately preserves the geometry of $$T$$ if $$m$$ ≳ $$d(T)$$. For smaller $$m$$, the projected set $$PT$$ becomes approximately a round ball of diameter $$\sim w_s(T)$$ and its size does not shrink with $$m$$.”

These are the last three sentences in Vershynin’s excellent textbook (Chapter 11, Section 11.3).

The fact that I picked the last lines of a 300-page monograph on modern probability and most of you can learn something important from such an extract says a lot about this book and its priorities. The only quantity needing explanation is $$w_s(T)$$, the so-called spherical width of a set (which I return to below). The extract also says quite a bit about the contents of this book, where concentration inequalities, random projections, notions of sizes of sets and the probabilistic method are big recurring themes.

A good textbook is as much about learning as about learning something specific. Vershynin’s High-Dimensional Probability is a good textbook. When developing a topic, it starts from the simplest idea, it examines its weaknesses and builds up to a better idea; this is superbly done when bounding the tail probabilities of binomial distributions in Chapter 2. It always prioritises high-level narrative to technical details; the reader never loses sight of the main theme, arguments are kept to their essence, side results are given as exercises and important special cases are given priority over the most general statements. Intuition is at least as important as the techniques; this is usually the hardest to communicate in a book, compared for example to a classroom presentation, but it comes across beautifully in this book, as for example in the proof of the “decoupling” theorem in Chapter 6. Finally, it shows sympathy to the reader, when sympathy is due (and much appreciated!): “The definition of the VC dimension may take some time to fully comprehend. We work out a few examples to illustrate this notion.” (Chapter 8).

One fundamental theme in the book is that of concentration inequalities, that is non-asymptotic bounds on the deviation of a random quantity from its mean. This is superbly developed from elementary principles in Chapter 2 building up to results on small and large deviations. Such inequalities are used in Chapter 3 to obtain results on concentration of norm for random vectors, which in turn lead to the first “high dimensional probability” results in the book concerning the concentration of measure in different high-dimensional distributions. There are further developments in Chapter 5 where, with the help of some isoperimetric inequalities that are recalled in the chapter, concentration inequalities for Lipschitz functions of random variables are derived.

Another broad theme is that of geometric and combinatorial notions of “size” of a set. Chapter 4 introduces and studies the notions of $$\epsilon$$-nets, covering and packing numbers. A multiscale version of $$\epsilon$$-nets, known as chaining, is developed in Chapter 8, where also the VC dimension is introduced, explained and related to packing. Chapter 7 introduces the notions of Gaussian and spherical width of a set, the concept of the stable dimension of a set (and its surprising and sharp contrast from that of the familiar linear-algebraic notion of dimension) as well as that of stable rank of a matrix; interestingly, the notion of stable dimension is introduced at this level of generality for the first time in this book.

Random matrices and random projections is another recurring topic. Not in the sense of the Tracy-Widom distribution and related ideas, which are not at all covered in this textbook. Neither in terms of randomized linear algebra, which is also not discussed at all but I think it would fit nicely given the existing structure. Rather, about norms of random matrices (Chapter 4) -bounds on eigenvalues and eigenvectors are obtained by means of basic perturbation theory- concentration inequalities for sums of random matrices (Chapter 5) and quadratic forms (Chapter 6), dimension reduction by random projections and the well-known Johnson-Lindenstrauss lemma (Chapters 5-7-9-11, a development that concludes with the sentences that open this review).

The full title of the book is “High-Dimensional Probability, An Introduction with Applications in Data Science”. I think the book could do without the subtitle (the “An Introduction … Science”), which is what we would call “aggressive marketing”. Still, there are some nice passages on applications of high-dimensional probability to machine learning and high-dimensional statistics. My two preferred are the matrix completion in Chapter 6 and those on compressed sensing and the lasso in Chapter 10.

Vershynin’s textbook is one that everyone who works in modern stochastics should be familiar with. Parts of the book could and probably should be incorporated in any modern intermediate probability course – especially those targeted to statistical machine learning students. Interestingly, measure-theoretic probability is never really required in this textbook. This potentially opens the possibility of designing and redesigning graduate probability courses along the narrative in this textbook for graduate programs in Data Science.

Maybe there is more in that subtitle than “aggressive marketing”!

### UPCOMING CONFERENCES

This list contains conferences related to mathematical finance that take place in the next three months. A full list is available at https://www.bachelierfinance.org/conferences. Please let us know of conferences we are not aware of and include a URL for the event.

The 2020 AMS Short Course on “Mean Field Games: Agent Based Models to Nash Equilibria”
January 13–14, 2020
Denver CO, USA

Leeds Winter School on Theory and Practice of Optimal Stopping and Free Boundary Problems
January 13–17, 2020
Leeds, UK

January 14–17, 2020
Paris, France

19th Winter School on Mathematical Finance
January 20–22, 2020
Lunteren, The Netherlands

21st Workshop on Quantitative Finance QFW2020
January 29–31, 2020
Napoli, Italy

13th Actuarial and Financial Mathematics Conference
February 6–7, 2020
Brussels, Belgium

Energy Finance Italia Workshop (EFI5)
February 10–11, 2020
Rome, Italy