L4

In this talk, I consider the problem of pricing and (statically)
hedging short-term contingent claims written on illiquid or
non-tradable assets.
In a first part, I show how to find the best European payoff written
on a given set of underlying assets for hedging (under several
metrics) a given European payoff written on another set of underlying
assets -- some of them being illiquid or non-tradable. In particular,
I present new results in the case of the Expected Shortfall risk
measure. I also address the associated pricing problem by using
indifference pricing and its link with entropy.
In a second part, I consider the more classic case of hedging with a
finite set of simple payoffs/instruments and I address the associated
pricing problem. In particular, I show how entropic methods (Davis
pricing and indifference pricing à la Rouge-El Karoui) can be used in
conjunction with recent results of extreme value theory (in dimension
higher than 1) for pricing and hedging short-term out-of-the-money
options such as those involved in the definition of Daily Cliquet
Crash Puts.

In this talk, I consider  the problem of
hedging European and Bermudan option with a given probability. This 
question is
more generally linked to portfolio optimisation problems under weak
stochastic target constraints.
I will recall, in a Markovian framework, the characterisation of the 
solution by
non-linear PDEs. I will then discuss various numerical algorithms
to compute in practice the quantile hedging price.

This presentation is based on joint works with B. Bouchard (Université 
Paris Dauphine), G. Bouveret (University of Oxford) and ongoing work 
with C. Benezet (Université Paris Diderot).

We formulate and solve a class of Backward Stochastic Differential Equations (BSDEs) driven by the compensated random measure associated to a given marked point process on a general state space. We present basic well-posedness results in L 2 and in L 1 . We show that in the setting of point processes it is possible to solve the equation recursively, by replacing the BSDE by an ordinary differential equation in between jumps. Finally we address applications to optimal control of marked point processes, where the solution of a suitable BSDE allows to identify the value function and the optimal control. The talk is based on joint works with Marco Fuhrman and Jean Jacod. 

This paper formulates an utility indifference pricing model for investors trading in a discrete time financial market under non-dominated model uncertainty.
The investors preferences are described by strictly increasing concave random functions defined on the positive axis. We prove that under suitable
conditions the multiple-priors utility indifference prices of a contingent claim converge to its multiple-priors superreplication price. We also
revisit the notion of certainty equivalent for random utility functions and establish its relation with the absolute risk aversion.

 In the first part of the talk, we present some recent and new developments in the theory of control and optimal stopping with nonlinear expectations. We first introduce an optimal stopping game with nonlinear expectations (Generalized Dynkin Game) in a non-Markovian framework and study its links with nonlinear doubly reflected BSDEs. We then present some new results (which are part of an ongoing work) on mixed stochastic stochastic control/optimal stopping problems (as well as stochastic control/optimal stopping game problems) in a non-Markovian framework and their relation with constrained reflected BSDEs with lower obstacle (resp. upper obstacle). These results are obtained using some technical tools of stochastic analysis. In the second part of the talk, we discuss applications to the $\cal{E}^g$ pricing of American options and Game options in complete and incomplete markets (based on joint works with M.C.Quenez and Agnès Sulem).
 

We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a tractable family of functionally generated portfolios which often improves the classical spread while still being tractable, more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

This talk explains how to formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. Here the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG ", we hence provide generic results to address these " MFG of controls ", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.

In this presentation, we investigate the spectrum of the Neumann-Poincaré operator associated to a periodic distribution of small inclusions with size ε, and its asymptotic behavior as the parameter ε vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the `trivial' eigenvalues 0 and 1, and of a subset which stays bounded away from 0 and 1 uniformly with respect to ε. This non trivial part is the reunion of the Bloch spectrum, accounting for the collective resonances between collections of inclusions, and of the boundary layer spectrum, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light about the homogenization of the voltage potential uε caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possibly negative) conductivity a surrounded by a dielectric medium, with unit conductivity.

Broken cryptographic algorithms and hardness assumptions are a constant
threat to real-world protocols. Prominent examples are
hash functions for which collisions become known, or number-theoretic
assumptions which are threatened by advances in quantum computing.
Especially when it comes to key exchange protocols, the switch to
quantum-resistant primitives has begun and aims to protect today’s
secrets against future developments, moving from common Diffie–Hellman
based solutions to Learning-With-Errors-based approaches. Remarkably,
the authentication step in such protocols is usually still carried out
with quantum-vulnerable signature schemes. The intuition here is that
the adversary would need to break this protocol primitive today, without
having quantum power yet. The question we address here is if this
intuition is justified, and if so, if we can show this rigorously. We
particularly consider the authenticated variant of the recently
introduced post-quantum secure key exchange protocol NewHope (Alkim et
al., USENIX Security 2016), as well as by TLS 1.3, which is currently
being developed by the Internet Engineering Task Force.