Серия встреч с ведущими российскими и зарубежными учеными в области финансовой и актуарной математики

ГЛОБАЛЬНЫЙ СЕМИНАР

За три года работы семинар стал ключевым событием финансовой математики в России

Более **50 спикеров**

Докладчики —**ведущие мировые ученые и специалисты из индустрии**

География экспертов и слушателей —**от Лос-Анджелеса до Сиднея**

**Семинар является открытым** для всех, кому интересна финансовая математика

Докладчики —

География экспертов и слушателей —

РУКОВОДИТЕЛЬ СЕМИНАРА

Кабанов Юрий Михайлович

д.ф.-м.н., профессор

Председатель Совета директоров

Научный директор Фонда

Член Academia Europaea

Председатель Совета директоров

Научный директор Фонда

Член Academia Europaea

24 | 02 | 2024

Миряна Григорова

Optimal stopping and non-zero-sum games of stopping:

Bermudan strategies meet non-linear evaluations

Bermudan strategies meet non-linear evaluations

Тема:

15:00 (МСК)

Уорикский университет, Великобритания

We address an optimal stopping problem over the set of Bermudan-type stopping strategies (which we understand in a more general sense than the stopping strategies for Bermudan options in finance) and with non-linear operators (non-linear evaluations) assessing the rewards, under general assumptions on the non-linear operators. We provide a characterization of the value family V in terms of a suitably defined non-linear Snell envelope of the pay-off family. We establish a Dynamic Programming Principle. We provide an optimality criterion in terms of a non-linear martingale property of V on a stochastic interval. We investigate the non-linear martingale structure, and we show that, under suitable conditions, the first time when the value family coincides with the pay-off family is optimal. The reasoning simplifies in the case where there is a finite number, say n, of pre-described stopping times, where n does not depend on the state of nature. We will also discuss a non-zero-sum non-linear game with Bermudan stopping strategies, for which we show the existence of a Nash equilibrium point, via a recursive procedure. We provide examples of non-linear operators from the stochastic control and mathematical finance literature, which enter our framework.

The talk is based on joint works with Marie-Claire Quenez (Paris) and Peng Yuan (Warwick).

The talk is based on joint works with Marie-Claire Quenez (Paris) and Peng Yuan (Warwick).

02 | 03 | 2024

РАма конт

Rough Volatility: Fact or Artefact?

Тема:

15:00 (МСК)

Оксфордский университет, Великобритания

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.

09 | 03 | 2024

Хуен фам

Nonparametric generative modeling for time series via Schrödinger bridge

Тема:

15:00 (МСК)

Университет Париж Сите, Франция

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.

16 | 03 | 2024

Клаудио фонтана

Interest rate modeling beyond stochastic continuity

Тема:

15:00 (МСК)

Падуанский университет, Италия

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.

23 | 03 | 2024

Джулия ди Нунно

Fully-dynamic risk measures: horizon risk and interest rate uncertainty

Тема:

Университет Осло, Норвегия

15:00 (МСК)

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.

30 | 03 | 2024

Игорь Васильевич Андрианов

Mathematical Models in Pure and Applied Mathematics

Тема:

15:00 (МСК)

д.ф.-м.н., профессор

„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.

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.

Ростислав Березовский

06 | 04 | 2024

Decentralized exchange liquidity based options

Тема:

15:00 (МСК)

Head of Research, Hash CIB

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.

13 | 04 | 2024

Синь гуо

Alpha Potential Games: A New Paradigm for N-play Games

Тема:

15:00 (МСК)

Калифорнийский университет, США

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.

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.

20 | 04 | 2024

Биргит Рудлофф

Epic Math Battles: Nash vs Pareto

Тема:

15:00 (МСК)

Венский экономический университет, Австрия

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.

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

27 | 04 | 2024

Кохта Такехара

Analysis on Structure of Decentralized Exchanges: Decision Making for Liquidity Providers and Platformers

15:00 (МСК)

Тема:

Токийский столичный университет, Япония

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.

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

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