Hans Foellmer (Humboldt, Berlin)

Local vs Systemic Risk, Perceived Bubbles and Optimal Transport

ABSTRACT

The first two lectures discuss the quantification of financial risk in terms of monetary risk measures, both at the local and the systemic level.  The third lecture relates the perception of a bubble to a shift in the space of martingale measures, and this motivates a closer look at the optimal transport problem for measures on path space.

 
Lecture 1.  CONSISTENCY AND CONVERGENCE OF CONDITIONAL RISK MEASURES
(Monday May 15, 15:00-17:00, Lecture Theatre 140)
 
In the first lecture we discuss different notions of consistency for conditional convex risk measures. The resulting convergence results can be viewed as a non-linear extension of backward and forward martingale convergence.
 
 
Lecture 2.  SYSTEMIC RISK MEASURES AND THE APPEARANCE OF PHASE TRANSITIONS
(Monday May 22, 15:00-17:00, Lecture Theatre 140)

The second lecture focuses on systemic risk measures as introduced by Chen, Iyengar, and Moallemi (2013). We show how they are related to the spatial risk measures studied in [2], and we investigate the class of systemic risk measures that are consistent with a given local specification, that is, a given conditional risk assessment for local subsystems. This may be viewed as a non-linear version of a classical problem in Statistical Mechanics, namely the analysis of Gibbs measures consistent with a given family of local conditional probability distributions. In analogy to the classical case, we show that we may have a phase transition, that is, global systemic risk measures may not be uniquely determined by their local specification.

 
Lecture 3.  SHIFTING MARTINGALE MEASURES, PERCEIVED BUBBLES, AND OPTIMAL TRANSPORT
(Tuesday May 30, 10:00-12:00, Lecture Theatre 140)

In the third lecture we show how, in the context of an incomplete financial market model, a shift in the space of martingale measures may induce the birth of a perceived bubble. This is illustrated by a simple shift between two martingale measures such that the price process is uniformly integrable under one of them, but not under the other. To prepare for a more systematic study of dynamics on the space of martingale measures, we consider different versions of the optimal transport problem for measures on path space. 

 
REFERENCES:
 
[1] H. Föllmer, S. Weber: The Axiomatic Approach to Risk Measures for Capital Determination (with Stefan Weber). 
Annual Review of Financial Economics, Vol. 7, 301 - 337 (2015) 
 
[2] H. Föllmer, C. Klüppelberg: Spatial Risk Measures: Local Specification and Boundary Risk (with Clauda Klüppelberg). 
In: Stochastic Analysis and Applications 2014 - In Honour of Terry Lyons, Springer Proceedings in Mathematics \& Statistics 100, 307-326 (2014) 
 
[3] F. Biagini, H. Föllmer, S. Nedelcu: Shifting Martingale Measures and the Birth of a Bubble as a Submartingale (with Francesca Biagini and Sorin Nedelcu). Finance and Stochastics, Vol. 18, No. 2, 297 - 326 (2014)