From Bachelier to Dupire via optimal transport

Famously, mathematical finance was started by Bachelier in his 1900 PhD thesis where – among many other achievements – he also provided a formal derivation of the Kolmogorov forward equation. This also forms the basis for Dupire’s (again formal) solution to the problem of finding an arbitrage-free model calibrated to a given volatility surface. The latter result has rigorous counterparts in the theorems of Kellerer and Lowther. In this survey article, we revisit these hallmarks of stochastic finance, highlighting the role played by some optimal transport results in this context.

An Italian perspective on the development of financial mathematics from 1992 to 2008

This paper is intended to be a survey of the development of financial mathematics as seen through the events that I organised, and partly co-organised, between 1992 and 2008. These events all took place in Italy between 1992 and 2003, while in 2008 I was involved in the organisation of an entire special semester in Linz (Austria); this semester is included here because it marks quite well the state-of-the-art of the period just before the so-called big financial crisis that lasted from, roughly, 2008 to 2012. Even if the survey may be affected by my personal views, it can still be seen as reflecting the actual global development since what I am going to describe here concerns major occurrences. For completeness, I also mention, although only briefly, some events that took place in Italy during the given period, but where I was not personally involved.

My journey through finance and stochastics

This year, Finance and Stochastics celebrates its 25th anniversary. The journal provides a platform for the community of researchers on which they can publish their ideas and results.Publication is an outcome of research which may be conducted for a number of years before it reaches the required maturity. I find this research process to be very important. Unfortunately, it is almost impossible to decode it from reading the research publications. This special issue of Finance and Stochastics gives me an opportunity to focus on it. I am grateful I can present my personal memory of this process. Understanding why questions are asked and how the answers are found is critical.

Complete and competitive financial markets in a complex world

We investigate the possibility of completing financial markets in a model with no exogenous probability measure, with market imperfections and with an arbitrary sample space. We also consider whether such an extension may be possible in a competitive environment. Our conclusions highlight the economic role of complexity.