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) oder in Präsenz (im Raum 1.115, Rudower  Chaussee 25) durchgeführt. Die entsprechende Ankündigung (ggf. mit ZOOM-Link) wird per e-mail versendet.
Zeit: Donnerstag, 17 Uhr s.t.









16 Uhr c.t.

Wei Xu  (Humboldt-Universität zu Berlin)

The Microstructure of Stochastic Volatility Models with Self-Exciting
Jump Dynamics


Abstract: We provide a general probabilistic framework within which we establish scaling limits for a class of continuous-time stochastic volatility models with self-exciting jump dynamics. In the scaling limit, the joint dynamics of asset returns and volatility is driven by independent Gaussian white noises and two independent Poisson random measures that capture the arrival of exogenous shocks and the arrival of self-excited shocks, respectively. Various well-studied stochastic volatility models with and without self-exciting price/volatility co-jumps are obtained as special cases under different scaling regimes. We analyze the impact of external shocks on the market dynamics, especially their impact on jump cascades and show in a mathematically rigorous manner that many small external shocks may trigger endogenous jump cascades in asset returns and stock price volatility.




17 Uhr s.t.

Ibrahim Ekren  (Florida State University)

Optimal transport and risk aversion in Kyle's model of informed trading


Abstract: We establish connections between optimal transport theory and the dynamic version of the Kyle model, including new characterizations of informed trading profits via conjugate duality and Monge-Kantorovich duality. We use these connections to extend the model to multiple assets, general distributions, and risk-averse market makers. With risk-averse mar- ket makers, liquidity is lower, assets exhibit short-term reversals, and risk premia depend on market maker inventories, which are mean re-verting. We illustrate the model by showing that implied volatilities predict stock returns when there is informed trading in stocks and options and market makers are risk averse.


17 Uhr s.t.

Eyal Neumann   (Imperial College London)

Trading with the Crowd


Abstract: We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact, along with taking into account a common general price predicting signal. The unique Nash-equilibrium strategies reveal how each agent’s liquidation policy adjusts the predictive trading signal to the aggregated transient price impact induced by all other agents. This unfolds a quantitative relation between trading signals and the order flow in crowded markets. We also formulate and solve the corresponding mean field game in the limit of infinitely many agents. We prove that the equilibrium trading speed and the value function of an agent in the finite N-player game converges to the corresponding trading speed and value function in the mean field game at rate O(N−2). In addition, we prove that the mean field optimal strategy provides an approximate Nash-equilibrium for the finite-player game.

This is a joint work with Moritz Voss.


17 Uhr s.t.

Evgueni Kivman (Humboldt-Universität zu Berlin) 

Optimal trade execution under small market impact and portfolio liquidation with semimartingale strategies


Abstract: We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with general semimartingale controls when the instantaneous impact factor converges to zero. Our results provide a unified framework within which to embed the two most commonly used modelling frameworks in the liquidation literature and show how liquidation problems with portfolio processes of unbounded variation can be obtained as limiting cases in models with small instantaneous impact as well as a microscopic foundation for the use of semimartingale liquidation strategies. Our convergence results are based on novel convergence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes.




Jörn Sass (TU Kaiserslautern) Vortrag fällt aus!

Utility Maximization in a Multivariate Black Scholes Type Market with Model Uncertainty on the Drift



17 Uhr s.t.


Blanka Horvath  (TU München)

Data - Driven Market Simulators  
& some applications of signature kernel methods in mathematical finance



Techniques that address sequential data have been a central theme in machine learning research in the past years. More recently, such considerations have entered the field of finance-related ML applications in several areas where we face inherently path dependent problems: from (deep) pricing and hedging (of path-dependent options) to generative modelling of synthetic market data, which we refer to as market generation.

We revisit Deep Hedging from the perspective of the role of the data streams used for training and highlight how this perspective motivates the use of highly accurate generative models for synthetic data generation. From this, we draw conclusions regarding the implications for risk management and model governance of these applications, in contrast to risk-management in classical quantitative finance approaches.

Indeed, financial ML applications and their risk-management heavily rely on a solid means of measuring (and efficiently computing) similarity-metrics between datasets consisting of sample paths of stochastic processes. Stochastic processes are at their core random variables on path space. However a consistent notion of and efficiently computable similarity-metrics for stochastic processes remained a challenge until recently. We propose such appropriate similarity metrics and contrast them with returns-based similarity metrics. Finally, we discuss the effect of incorporating the information structure (the filtration) of the market into these similarity metrics and the implications of such metrics on options prices.



17 Uhr s.t.

Alexander Schied (University of Waterloo) 

The Hurst roughness exponent and its model-free estimation



We say that a continuous real-valued function x admits the Hurst roughness exponent H if the pth variation of x converges to zero if p > 1/H and to infinity if p < 1/H. For the sample paths of many stochastic processes, such as fractional Brownian motion, the Hurst roughness exponent exists and equals the standard Hurst parameter. In our main result, we provide a mild condition on the Faber– Schauder coefficients of x under which the Hurst roughness exponent exists and is given as the limit of the classical Gladyshev estimates H_n(x). This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because no assumption whatsoever is made on the possible dynamics of the function x. Nonetheless, our proof is probabilistic and relies on a martingale that is hidden in the Faber–Schauder expansion of x. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators that are derived from the sequence (H_n)n∈N. We also discuss how a dynamic change in the Hurst roughness parameter of a time series can be detected. Our results are illustrated by means of high-frequency financial time series. This is joint work with Xiyue Han.


17 Uhr s.t.

Samuel Drapeau (Shanghai Jiao Tong University)

Robust Uncertainty Analysis



In this talk, we will showcase how methods from optimal transport and
distributionally robust optimisation allow to capture and quantify
sensitivity to model uncertainty for a large class of problems.
We consider a generic stochastic optimisation problem. This could be a mean-variance or a utility maximisation portfolio allocation problem, a risk measure computation, a standard regression or a deep learning problem. At the heart of the optimisation is a probability measure, or a model, which describes the system. It could come from data, simulations or a modelling effort for which
there is always exists a degree of uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated measure. Our main results provide explicit formulae for the first order correction to both the value function and the optimiser.
We further extend our results to optimisation under linear constraints.
Our sensitivity analysis of the distributionally robust optimisation
problems finds applications in statistics, machine learning,
mathematical finance and uncertainty quantification. In the talk, we will discuss several financial examples anchored in a one-step financial model and compute their sensitivity to model uncertainty. These include: option pricing, mean-variance portfolio selection, optimised certainty equivalent and similar risk assessments. We will also address briefly some other applications, such as explicit formulae for first-order approximations of square-root LASSO and
square-root Ridge optimisers and measures of NN architecture robustness
wrt to adversarial data.
This talk is based on joint works with Daniel Bartl, Jan Obloj and
Johannes Wiesel.


17 Uhr s.t.

Aurélien Alfonsi (CERMICS Marne-la-vallée)

A synthetic model for ALM in life insurance and numerical methods for SCR computation



We introduce a synthetic ALM model that catches the main specificity of life insurance contracts. First, it keeps track of both market and book values to apply the regulatory profit sharing rule. Second, it introduces a determination of the crediting rate to policyholders that is close to the practice and is a trade-off between the regulatory rate, a competitor rate and the available profits. Third, it considers an investment in bonds that enables to match a part of the cash outflow due to surrenders, while avoiding to store the trading history. We use this model to evaluate the Solvency Capital Requirement (SCR) with the standard formula, and illustrate the importance of matching cash-flows.

Then, we focus on the problem of evaluating the SCR at future dates. For this purpose, we study the multilevel Monte-Carlo estimator for the expectation of a maximum of conditional expectations. We obtain theoretical convergence results that complements the recent work of Giles and Goda. We then apply the MLMC estimator to the calculation of the SCR at future dates and compare it with estimators obtained with Least Squares Monte-Carlo or Neural Networks. Last, we discuss the effect of the portfolio allocation on the SCR at future dates.



Max Nendel (Universität Bielefeld) Vortrag wird verlegt!












Frank Riedel (Universität Bielefeld) Vortrag wird verlegt!




17 Uhr s.t.


Jörn Sass (TU Kaiserslautern) Vortrag entfällt!

Utility Maximization in a Multivariate Black Scholes Type Market with Model Uncertainty on the Drift



It is a by now classical observation that in a (realistic) financial market (model) simple portfolio strategies can outperform more sophisticated optimized portfolio strategies. For example, in a one period setting, the equal weight or 1/N-strategy often provides more stable results than mean-variance-optimal strategies. This is due to the fact that a good estimation of the mean returns is not possible for volatile financial assets. Pflug, Pichler and Wozabel (2012) gave a rigorous explanation of this observation by showing that for increasing uncertainty on the means the equal weight strategy becomes optimal in a mean-variance setting which is due to its robustness.

We aim at extending this result to continuous-time strategies in a multivariate Black-Scholes type market. To this end we investigate how optimal trading strategies for maximizing expected utility of terminal wealth under CRRA utility behave when we have Knightian uncertainty on the drift, meaning that the only information is that the drift parameter lies in a so-called uncertainty set. The investor takes into account that the true drift may be the worst possible drift within this set.
In this setting we can show that a minimax theorem holds which enables us to find the worst-case drift and the optimal robust strategy quite explicitly. This again allows us to derive the limits when uncertainty increases and hence to show that a uniform strategy is asymptotically optimal.

We also discuss the extension to a financial market with a stochastic drift process, combining the worst-case approach with filtering techniques. This leads to local optimization problems, and the resulting optimal strategy needs to be updated continuously in time. We carry over the minimax theorem for the local optimization problems and derive the optimal strategy. In this setting we show how an ellipsoidal uncertainty set can be defined based on filtering techniques and we demonstrate that investors need to choose a robust strategy to be able to profit from additional information.


17 Uhr s.t.


Zhenjie Ren (CEREMADE - Université Paris-Dauphine)











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