Following a quick introduction to derivatives markets and the classic theory of valuation, we describe the changes triggered by post 2007 events. We re-discuss the valuation theory assumptions and introduce valuation under counterparty credit risk, collateral posting, initial and variation margins, and funding costs. A number of these aspects had been investigated well before 2007. We explain model dependence induced by credit effects, hybrid features, contagion, payout uncertainty, and nonlinear effects due to replacement closeout at default and possibly asymmetric borrowing and lending rates in the margin interest and in the funding strategy for the hedge of the relevant portfolio. Nonlinearity manifests itself in the valuation equations taking the form of semi-linear PDEs or Backward SDEs. We discuss existence and uniqueness of solutions for these equations. We present an invariance theorem showing that the final valuation equations do not depend on unobservable risk free rates, that become purely instrumental variables. Valuation is thus based only on real market rates and processes. We also present a high level analysis of the consequences of nonlinearities, both from the point of view of methodology and from an operational angle, including deal/entity/aggregation dependent valuation probability measures and the role of banks treasuries. Finally, we hint at how one may connect these developments to interest rate theory under multiple discount curves, thus building a consistent valuation framework encompassing most post-2007 effects.

Damiano Brigo, Joint work with Andrea Pallavicini, Daniele Perini, Marco Francischello. 

We argue that Credit Default Swap (CDS) premia for safe-haven sovereigns, like Germany and the United States, are driven to a large extent by regulatory requirements under which  derivatives dealing banks have an incentive to buy CDS to hedge counterparty credit risk of their counterparties.
We explain the mechanics of the regulatory requirements and develop a model in which derivatives dealers, who have a derivatives exposure with sovereigns, need CDS for capital relief. End users without exposure to the sovereigns sell the CDS and require a positive premium equivalent to the capital requirement. The model's predictions are confirmed using data on several sovereigns.

Joint with OMI

A well-known result of Landsberger and Meilijson says that efficient risk-sharing rules for univariate risks are characterized by a so-called comonotonicity condition. In this talk, I'll first discuss a multivariate extension of this result (joint work with R.-A. Dana and A. Galichon). Then I will discuss the restrictions (in the form of systems of nonlinear PDEs) efficient risk sharing imposes on individual consumption as a function of aggregate consumption. I'll finally give an identification result on how to recover preferences from the knowledge of the risk sharing (joint work with M. Aloqeili and I. Ekeland).

The talk is concerned with adapted solution of a multi-dimensional BSDE with a "diagonally" quadratic generator, the quadratic part of whose iith component only depends on the iith row of the second unknown variable. Local and global solutions are given. In our proofs, it is natural and crucial to apply both John-Nirenberg and reverse Holder inequalities for BMO martingales. 

We provide a result on prospect theory decision makers who are naïve about the time inconsistency induced by probability weighting. If a market offers a sufficiently rich set of investment strategies, investors postpone their trading decisions indefinitely due to a strong preference for skewness. We conclude that probability weighting in combination with naïveté leads to unrealistic predictions for a wide range of dynamic setups. Finally, I discuss recent work on the topic that invokes different assumptions on the dynamic modeling of prospect theory.

Equations of quantum mechanics in the semiclassical regime present an enduring challenge for numerical analysts, because their solution is highly oscillatory and evolves on two scales. Standard computational approaches to the semiclassical Schrödinger equation do not allow for long time integration as required, for example, in quantum control of atoms by short laser bursts. This has motivated our approach of asymptotic splittings. Combining techniques from Lie-algebra theory and numerical algebra, we present a new computational paradigm of symmetric Zassenhaus splittings, which lends itself to a very precise discretisation in long time intervals, at very little cost. We will illustrate our talk by examples of quantum phenomena – quantum tunnelling and quantum scattering – and their computation and, time allowing, discuss an extension of this methodology to time-dependent semiclassical systems using Magnus expansions

This talk focuses on the direct search method, arguably one of the simplest optimization algorithms. The algorithm minimizes an objective function by iteratively evaluating it along a number of (polling) directions, which are typically taken from so-called positive spanning sets. It does not use derivatives.

We first introduce the worst case complexity theory of direct search, and discuss how to choose the positive spanning set to minimize the complexity bound. The discussion leads us to a long-standing open
problem in Discrete Geometry. A recent result on this problem enables us to establish the optimal order for the worst case complexity of direct search.

We then show how to achieve even lower complexity bound by using random polling directions. It turns out that polling along two random directions at each iteration is sufficient to guarantee the convergence
of direct search for any dimension, and the resultant algorithm enjoys lower complexity both in theory and in practice.

The last part of the talk is devoted to direct search based on inaccurate function values. We address three questions:
i) what kind of solution can we obtain by direct search if the function values are inaccurate? 
ii) what is the worst case complexity to attain such a solution? iii) given
the inaccuracy in the function values, when to stop the algorithm in order
to guarantee the quality of the solution and also avoid “over-optimization”?

This talk is based on joint works with F. Delbos, M. Dodangeh, S. Gratton, B. Pauwels, C. W. Royer, and L. N. Vicente.

I will review some recent work on the problem of reliable automatic detection of blow-up behaviour for nonlinear parabolic PDEs. The adaptive algorithms developed are based on rigorous conditional a posteriori error bounds. The use of space-time adaptivity is crucial in making the problem computationally tractable. The results presented are applicable to quite general spatial operators, rendering the approach potentially useful in informing respective PDE theory. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including ones exhibiting regional blow-up. 

The talk will introduce the concepts of multilevel optimization and motivate them in the context of problems arising from the discretization of infinite dimensional applications. It will be shown how optimization methods can accomodate a number of useful (and classical) ideas from the multigrid
community, and thereby produce substantial efficiency improvements compared to existing large-scale minimization techniques.  Two different classes of multilevel methods will be discussed: trust-region and linesearch algorithms.
The first class will be described in the context of a multilevel generalization of the well-known trust-region-Newton method.  The second will focus on limited-memory quasi-Newton algorithms.  Preliminary numerical results will be presented which indicate that both types of multilevel algorithms may be practically very advantageous.