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Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

FS Stochastische Analysis und Stochastik der Finanzmärkte

Bereich für Stochastik


Das Seminar wird ONLINE über ZOOM durchgeführt. (Der entsprechende Link kommt per e-mail)
Zeit: Donnerstag, 17 Uhr / 18 Uhr s.t.


Abstract and Zoom access details will be announced by email. To register for our e-mail list of research seminars, send an e-mail request with your university e-mail address  and your full name to bergmann@math.hu-berlin.de.








Renyuan Xu (University of Oxford)

Excursions in Math Finance


Abstract: The risk and return profiles of a broad class of dynamic trading strategies, including pairs trading and other statistical arbitrage strategies, may be characterized in terms of excursions of the market price of a portfolio away from a reference level. We propose a mathematical framework for the risk analysis of such strategies, based on a description in terms of price excursions, first in a pathwise setting, without probabilistic assumptions, then in a Markovian setting. We introduce the notion of $\delta$-excursion, defined as a path which deviates by $\delta$ from a reference level before returning to this level. We show that every continuous path has a unique decomposition into such $\delta$-excursions, which turns out to be useful for the scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss, and drawdown. As $\delta$ is decreased to zero, properties of this decomposition relate to the local time of the path. When the underlying asset follows a Markov process, we combine these results with Ito's excursion theory to obtain a tractable decomposition of the process as a concatenation of independent $\delta$-excursions, whose distribution is described in terms of Ito's excursion measure. We provide analytical results for linear diffusions and give new examples of stochastic processes for flexible and tractable modeling of excursions. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursions match those observed in a data set. This is based on joint work with Anna Ananova and Rama Cont (Oxford).


Christoph Frei (University of Alberta)

Principal Trading Arrangements: When Are Common Contracts Optimal?



Abstract: Many financial arrangements reference market prices that are yet to be realized at the time of contracting and consequently susceptible to manipulation. Two of the most common such arrangements are: (i) market-on-close contracts, which reference the price prevailing at the end of an execution window, and (ii) guaranteed VWAP contracts, which reference the volume-weighted average price (VWAP) prevailing over the execution window. To study such situations, we introduce a stylized model of financial contracting between a client, who wishes to trade a large position, and her dealer. Market-on-close contracts are generally not optimal in this principal-agent problem. In contrast, we provide conditions under which guaranteed VWAP contracts are optimal. These results question the usage of market-on-close contracts in practice, explain the usage of guaranteed VWAP contracts, and also suggest considerations for the design of financial benchmarks. The presentation is based on joint work with Markus Baldauf (University of British Columbia) and Joshua Mollner (Northwestern University).









Sergey Nadtochiy (Illinois Institute of Technology)

A simple microstructural explanation of the concavity of price impact


Abstract: I will present a simple model of market microstructure which explains the concavity of price impact. In the proposed model, the local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear, with a constant slope, which makes the model dynamically consistent. Nevertheless, the expected impact on midprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The main practical conclusion of the model is that, throughout a meta-order, the volumes at the best bid and ask prices change (on average) in favor of the executor. This conclusion, in turn, relies on two more concrete predictions of the model, one of which can be tested using publicly available market data and does not require the (difficult to obtain) information about meta-orders. I will present the theoretical results and will support them with the empirical analysis. 


18 Uhr s.t.


Jin Ma (University of Southern California)

On Set-valued Backward SDEs and Related Issues in Set-valued Stochastic Analysis


Abstract: In this talk we try to establish an analytic framework for studying Set-Valued Backward Stochastic Differential Equations (SVBSDE for short), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will be based on the notion of Hukuhara difference between sets, in order to com- pensate the lack of “inverse” operation of the traditional Minkowski addition, whence the vector space structure, in traditional set-valued analysis. We shall examine and establish a useful foundation of set-valued stochastic analysis under this algebraic framework, and identify the challenges that may arise in the study of SVBSDEs. This talk is based on the joint works with C. Ararat and W. Wu. 


18 Uhr s.t.


Jianfeng Zhang (University of Southern California)

Set Values of Mean Field Games


Abstract: When a mean field game satisfies certain monotonicity conditions, the mean field equilibrium is unique and the corresponding value function satisfies the so called master equation. In general, however, there can be multiple equilibriums, and in the literature one typically studies the asymptotic behaviors of individual equilibriums of the corresponding $N$-player game. We instead study the set of values over all (mean field) equilibriums, which we call the set value of the game. We shall establish two crucial properties of the set value: (i) the dynamic programming principle; (ii) the convergence of the set values from the $N$-player game to the mean field game. We emphasize that the set value is very sensitive to the choice of the admissible controls. For the dynamic programming principle, one needs to use closed loop controls (not open loop controls) and it involves some very subtle path dependence issue. For the convergence, one has to restrict to the same type of equilibriums for the $N$-player game and for the mean field game. The talk is based on a joint work with Zach Feinstein and Birgit Rudloff and another ongoing joint work with Melih Iseri.  



Xiaonyu Xia (Humboldt-Universität zu Berlin)

Portfolio Liquidation Games with Self-Exciting Order Flow


Abstract: We analyze novel portfolio liquidation games with self-exciting order flow. Both the $N$-player game and the mean-field game are considered. We assume that players' trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors each player solves a mean-field control problem. We characterize open-loop Nash equilibria in both games in terms of a novel mean-field FBSDE system with unknown terminal condition. Under a weak interaction condition we prove that the FBSDE systems have unique solutions. Using a novel sufficient maximum principle that does not require convexity of the cost function we finally prove that the solution of the FBSDE systems do indeed provide existence and uniqueness of open-loop Nash equilibria. The talk is based on joint work with Guanxing Fu and Ulrich Horst. 



Fred E.Benth (University of Oslo)

Pathwise Gaussian Volterra processes in Hilbert space


Abstract: We discuss a rough volatility model with fractional drift and noise allowing for more flexibility in modelling roughness. Motivated by an extension to infinite stochastic volatility models for commodity futures markets, we are led to a study of Gaussian Volterra processes. We suggest a definition of a pathwise stochastic integral based on combining the regularity of the kernel and the covariance of the noise. Likewise, we define pathwise integration with respect to multi-parameter covariance-like functions, and apply this to derive an explicit representation of the covariance of the Gaussian Volterra process. This is joint work with Fabian Harang (Oslo). 



Martin Larsson (Carnegie Mellon University)

Finance and Statistics: Trading Analogies for Sequential Learning


Abstract: The goal of sequential learning is to draw inference from data that is gathered gradually through time. This is a typical situation in many applications, including finance. A sequential inference procedure is `anytime-valid’ if the decision to stop or continue an experiment can depend on anything that has been observed so far, without compromising statistical error guarantees. A recent approach to anytime-valid inference views a test statistic as a bet against the null hypothesis. These bets are constrained to be supermartingales - hence unprofitable - under the null, but designed to be profitable under the relevant alternative hypotheses. This perspective opens the door to tools from financial mathematics. In this talk I will discuss how notions such as supermartingale measures, log-optimality, and the optional decomposition theorem shed new light on anytime-valid sequential learning. (This talk is based on joint work with Wouter Koolen (CWI), Aaditya Ramdas (CMU) and Johannes Ruf (LSE).) 



Cassandra Milbradt (Humboldt-Universität zu Berlin)

A cross-border market model


Abstract:On the XBID-market 13 European countries can trade electricity between each other. Like other intraday electricity markets, this is handled using a limit order book. However, cross-border trading is limited via the total amount of available transmission capacities during a trading session. We present a cross-border market model between two countries and want to give insight into the interactions on this market. We introduce a so-called reduced-form representation of the market and a capacity process which may restrict cross-border trades in each direction. Assuming that the capacity process is non-restricted, we are able to derive heavy traffic approximations of the standing volumes and the capacity process. We will further motivate a candidate for the heavy traffic approximation of the restricted market model. 



Mikhail Urusov(Universität Duisburg-Essen)

Optimal trade execution in an order book model with stochastic liquidity parameters


Abstract:We analyze an optimal trade execution problem in a financial market with stochastic liquidity. To this end we set up a limit order book model in which both order book depth and resilience evolve randomly in time. Trading is allowed in both directions. In discrete time, we discuss an explicit recursion that, under certain structural assumptions, characterizes minimal execution costs and observe some qualitative differences with related models. In continuous time, due to the stochastic dynamics of the order book depth and resilience, optimal execution strategies are typically of infinite variation, and the first thing to be discussed it how to extend the state dynamics and the cost functional to allow for general semimartingale strategies. We then derive a quadratic BSDE that under appropriate assumptions characterizes minimal execution costs, identify conditions under which an optimal execution strategy exists and, finally, illustrate our findings in several examples. This is a joint work with Julia Ackermann and Thomas Kruse. 


18 Uhr s.t.


Ruimeng Hu (University of California)

Convergence Of Deep Fictitious Play For Stochastic Differential Games


Abstract:In this talk, I will introduce the deep fictitious play (DFP), which is a novel machine-learning algorithm for finding Markovian Nash equilibrium of large N-player asymmetric stochastic differential games. By incorporating the idea of fictitious play, the algorithm decouples the game into N sub-optimization problems, and identifies each player’s optimal strategy with the deep backward stochastic differential equation method parallelly and repeatedly. I will show the proof of convergence of the fictitious play to the true Nash equilibrium, and show that the strategy based on DFP forms an \epsilon-Nash equilibrium. I will also discuss some generalizations by proposing a new approach to decouple the games and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems. 









Sebastian Jaimungal (University of Toronto)

Portfolio Optimisation within a Wasserstein Ball


Abstract:We consider the problem of active portfolio management where a loss-averse and/or gain-seeking investor aims to outperform a benchmark strategy's risk profile while not deviating too much from it. Specifically, an investor considers alternative strategies that have a specified copula with the benchmark and whose terminal wealth lies within a Wasserstein ball surrounding it. The investor then chooses the alternative strategy that minimises a distortion risk measure. We prove that an optimal dynamic strategy exists and is unique, and provide its characterisation through the notion of isotonic projections. Finally, we illustrate how investors with different risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped distortion risk measures, and lower- and upper-tail risk measures as examples. We find that investors' optimal terminal wealth distribution has larger probability masses in regions that reduce their risk measure relative to the benchmark while preserving some aspects of the benchmark. [ this is joint work with Silvana Pesenti, U. Toronto ] 

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