We consider Geometric Mean Market Makers — a special type of Decentralized Exchange — with two types of users: liquidity takers and arbitrageurs. Liquidity takers trade at prices that can create arbitrage opportunities, while arbitrageurs align the exchange's price with the external market price. We show that in Geometric Mean Market Makers charging proportional transaction fees, Impermanent Loss can be super-hedged by a model-free rebalancing strategy. A joint work with Basile Maire, Marcus Wunsch.

Any model in finance is based on random variables and stochastic processes of any kind: continuous or discrete, one-dimensional or multi-dimensional, etc. The answer to any question arising is given via variety of properties of the involved probability distributions.

The following topics will be discussed: role of the moments of positive integer orders, infinite divisibility, nonlinear (Box-Cox) transformations, limit theorems. One of the goals is to explain why some distributions are M-determinate and others M-indeterminate. It is more than useful to see a complete picture of existing uncheckable and checkable conditions for either M-determinacy or M-indeterminacy. Here is an exciting question: How to “fight” with the M-indeterminacy in the most popular models such as Black-Scholes and others? The main ideas, techniques and results can and will be demonstrated by analysing in detail two of the most important distributions, the normal and the lognormal, and related classes of SDEs. Some facts are not so-well known and even look a little surprising. Along with classical results for M-(in)determinacy (Cramer, Hardy, Carleman, Krein, Heyde, Berg, Pakes), a few recent results will be reported. Challenging open questions will be outlined.

We consider the mean field game modeling of optimal cross-holding within a population of investors endowed with some idiosyncratic risk process. In this talk, we consider the extension to the situation where the individual risks are correlated through some common noise. Unlike the uncorrelated case, we recover here the trade off between average returns and risks, which is the main novel difficulty to solve the problem. Our main finding is that the mean field no-arbitrage condition imposes some structure restrictions on the model, which turn out to play a crucial role. Under these conditions, we provide a quasi-explicit solution for the mean field game of crossholding.

Automated Market Makers have emerged quite recently, and Uniswap is one of the most widely used platforms. This protocol is challenging from a quantitative point of view, as it allows participants to choose where they wish to concentrate liquidity. In this talk, we revisit Uniswap v3’s principles in detail to build an unambiguous knowledge base; we analyze the Impermanent Loss of a liquidity provider without assumptions about swap trades or other liquidity providers; we introduce the notion of a liquidity curve and show how to statistically replicate options with Uniswap v3; last, we provide a closed-form approximation formula for the collected fees.

My talk is about recent results we have obtained to evaluate super-hedging prices in general models without no-arbitrage condition. This is motivated by the desire to efficiently compute the prices as the traditional approach gives a dual characterization which is difficult to handle. Actually, our method is interesting because it is also relevant for non linear models.

I will review some recent results on claims reserving for life insurance policies with reserve-dependent payments driven by multi-state jump processes. As application I will discuss reserving under contract modification with and without maintaining actuarial equivalence.
If time permits, I will also discuss aggregation of prospective reserves for large and homogeneous insurance portfolios through mean-field approximations.