We study risk-sharing equilibria with trading subject to small proportional transaction costs. We show that the frictionless equilibrium prices also form an "asymptotic equilibrium" in the small-cost limit. To wit, there exist asymptotically optimal policies for all agents and a split of the trading cost according to their risk aversions for which the frictionless equilibrium prices still clear the market. Starting from a frictionless equilibrium, this allows to study the interplay of volatility, liquidity, and trading volume.
(This is joint work with Johannes Muhle-Karbe, University of Michigan.)
 

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the “rough” regime of Hurst pa- rameter H < 1/2. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation es- timates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approxi- mation formulae from CLT type log-moneyness deviations of order t1/2 (recent works of Alo`s, Le ́on & Vives and Fukasawa) to the wider moderate deviations regime.

This is work in collaboration with C. Bayer, P. Friz, A. Gulsashvili and B. Stemper

In this lecture I intend to review a few selected recent results on numerical approximations for high-dimensional nonlinear parabolic partial differential equations (PDEs), nonlinear stochastic ordinary differential equations (SDEs), and high-dimensional nonlinear forward-backward stochastic ordinary differential equations (FBSDEs). Such equations are key ingredients in a number of pricing models that are day after day used in the financial engineering industry to estimate prices of financial derivatives. The lecture includes content on lower and upper error bounds, on strong and weak convergence rates, on Cox-Ingersoll-Ross (CIR) processes, on the Heston model, as well as on nonlinear pricing models for financial derivatives. We illustrate our results by several numerical simulations and we also calibrate some of the considered derivative pricing models to real exchange market prices of financial derivatives on the stocks in the American Standard & Poor's 500 (S&P 500) stock market index.

Generalising previous definability results in global fields using
quaternion algebras, I will present a technique for first-order
definitions in finite extensions of Q(t). Applications include partial
answers to Pop's question on Isomorphism versus Elementary Equivalence,
and some results on Anscombe's and Fehm's notion of embedded residue.

A variety of groups is an equationally defined class of groups, namely the class of groups in which each of a set of "laws" (or "identical relations") holds. Examples include the abelian groups (defined by the law $xy = yx$), the groups of exponent dividing $d$ (defined by the law $x^d$), the nilpotent groups of class at most some fixed integer, and the solvable groups of derived length at most some fixed integer. This talk will give an introduction to varieties of groups, and then conclude with recent work on determining for certain varieties whether, for fixed coprime $m$ and $n$, a group $G$ is in the variety if and only if the power subgroups $G^m$ and $G^n$ (generated by the $m$-th and $n$-th powers) are in the variety.

In this talk I will describe another strong link between the behaviour of a 3-manifold and the behaviour of its fundamental group- specifically the theorem that the group splits as a free product if and only if the 3-manifold may be divided into two parts using a 2-sphere inducing this splitting. This theorem is for some reason known as Kneser's conjecture despite having been proved half a century ago by Stallings.