We investigate the statistical evidence for the use of ‘rough’ fractional pro- cesses with Hurst exponent H < 0.5 for modeling the volatility of financial assets, using a model-free approach. We introduce a non-parametric method for estimating the roughness of a function based on discrete sample, using the concept of normalized p-th variation along a sequence of partitions. Detailed numerical experiments based on sample paths of fractional Brownian motion and other fractional processes reveal good finite sample performance of our estimator for measuring the roughness of sample paths of stochastic pro- cesses. We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corre- sponding to a Hurst exponent significantly smaller than 0.5. Comparison of roughness estimates for realized and instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irre- spective of the roughness of the spot volatility process, realized volatility always exhibits ‘rough’ behaviour with an apparent Hurst index H < 0.5. These results suggest that the origin of the roughness observed in realized volatility time series lies in the estimation error rather than the volatility process itself.

We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting the temporal dynamics of the time series distribution. We estimate the drift function from data samples by nonparametric, e.g. kernel regression methods, and the simulation of the SB diffusion yields new synthetic data samples of the time series.The performance of our generative model is evaluated through a series of numerical experiments. First, we test with autoregressive models, a GARCH Model, and the example of fractional Brownian motion, and measure the accuracy of our algorithm with marginal, temporal dependencies metrics, and predictive scores. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets.

Overnight rates, such as the Secured Overnight Financing Rate (SOFR), are central to the current reform of interest rate benchmarks. A feature of overnight rates is the presence of jumps and spikes occurring at predetermined dates due to monetary policy interventions and liquidity constraints. This corresponds to stochastic discontinuities (i.e., discontinuities occurring at ex-ante known points in time) in their dynamics. We propose a generalized Heath-Jarrow-Morton (HJM) setup allowing for stochastic discontinuities and characterize absence of arbitrage. We extend the classical short-rate approach to accommodate stochastic discontinuities, developing a tractable setup driven by affine semimartingales. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Furthermore, we investigate hedging in the sense of local risk-minimization when the underlying term structures feature stochastic discontinuities. Based on joint work with Z. Grbac and T. Schmidt.

We discuss dynamic risk assessment and critically consider its commonly assumed features. Then we discuss the different time horizons appearing in some context such as pensions where both short and long scales appear. In this respect we identify horizon risk as the possibility to make a mistake in using a risk measure that is not adequate to the actual time horizon. We shall introduce horizon longevity as an evaluation of such horizon risk. We give examples of dynamic risk measures able to capture horizon risk and constructed via backward stochastic differential equations. In addition we plunge the problem of long horizon into the framework of interest rate uncertainty and see how we can design risk measure that are able to take care of this uncertainty as well.

„Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascod, so pages of formulae will not get a definite result out of loose data“ (Huxley).

So, we must try to put a good grain in mathematical mills. In other words, the adequacy of the mathematical model is no less important than the correctness of the formal mathematical analysis.

The paper discusses the definitions of Pure and Applied Mathematics and mathematical model, as well as the points of view on the subject of the classics - Poincaré, Lyapunov, Lord Rayleigh. Examples include problem of truncation, continualization and splashes, Navier-Stokes equations, Kirhhoff’s and Bolotin’s approximations in nonlinear dynamics of continuous systems, etc.

The main conclusion can be formulated as follows. Any correct mathematical model is asymptotic. It includes as an inseparable part the asymptotic estimates and constraints on which it is based. Only in this case it is formulated quite definitely as a mathematical one.

The talk is devoted to a new type of derivatives - decentralized options. We’ll discuss the existing models of decentralized exchanges, dive into details of concentrated liquidity mechanism with Uniswap V3 examine and draw the parallels between a liquidity provider’s position in a CLMM with a fixed price range and a perpetual option. Ideas about option premium calculation will be provided.

Static potential games, pioneered by Monderer and Shapley (1996) are non-cooperative games in which there exists an auxiliary function called static potential function, so that any player's change in utility function upon unilaterally deviating from her policy can be evaluated through the change in the value of this potential function. The introduction of the potential function is powerful as it simplifies the otherwise challenging task of searching for Nash equilibria in multi-agent non-cooperative games: maximizers of potential functions lead to the game's Nash equilibria.

In this talk, we propose an analogous and new framework called α-potential game for dynamic N-player games, with the potential function in the static setting replaced by an α-potential function. We will present the analytical criteria for any game to be an α-potential game, and identify several important classes of Markov α-potential games.

We will provide detailed analysis for games with mean-field interactions, distributed games, and the crowd aversion games, in which α is shown to depend on the number of players, the admissible policies, and the cost structure. We will also show the changes of α from open-loop to closed-loop settings.

Nash equilibria and Pareto optimization are two distinct concepts in multi-criteria decision making. It is well known that the two concepts do not coincide. However, in this work we show that it is possible to characterize the set of all Nash equilibria for any non-cooperative game as the set of all Pareto optimal solutions of a certain vector optimization problem. This characterization opens a new way of computing Nash equilibria. It allows to use algorithms from vector optimization to compute resp. to approximate the set of all Nash equilibria, which is in contrast to the classical fixed point iterations that find just a single Nash equilibrium. Examples are given, first in the linear case. Then, the convex case is considered and an algorithm is proposed that computes a subset of the set of epsilon-Nash equilibria such that it contains the set of all (true) Nash equilibria for convex games with either independent convex constraint sets for each player, or polyhedral joint constraints. Furthermore, the computation of the set of Nash equilibria of bi-matrix games is discussed.

This is joint work with Zachary Feinstein, Niklas Hey, and Andreas Löhne.

Decentralized exchanges are a key concept in DeFi. They often employ automated market makers, which are algorithms that pool liquidity and make it available to users of the DEX at automatically determined prices. This study categorizes DEX players into Platformers, Liquidity Providers, Arbitrageurs, and Traders, gives the model which describes the behavior and optimization of those players and analyzes how the structure of the DEX is affected by exogenous circumstances. Numerical examples intuitively illustrate some new findings, especially on exchange fee rates.

This is based on joint works with Takanori Adachi, Chiaki Hara and Tomooki Yuasa.