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

FS Stochastische Analysis und Stochastik der Finanzmärkte

Bereich für Stochastik


P. BANK, Ch. BAYER, Ch. BELAK, D. BECHERER, P. FRIZ, U. HORST, D. KREHER


 
Das Seminar findet im Wintersemester 2022/23 in Präsenz an der HU Berlin, Institut für Mathematik, Hörsaal 1.115 (RUD 25, 12489 Berlin) statt. 
 
Zeit: Donnerstag, 16.15 Uhr und 17.15 Uhr

         Kaffee und Tee ab 15.45 Uhr im Raum 1.214 Johann von Neumann - Haus, Rudower Chaussee 25

 

 

 

 

 

 

 

27.10.2022

16.15 Uhr

Chiheb Ben Hammouda (RWTH Aachen)

Smoothing Techniques Combined with Hierarchical Approximations for Efficient Option Pricing

        

Abstract: 

When approximating the expectation of a functional of a stochastic process, in particular for option pricing purposes, the performance of numerical integration methods based on deterministic quadrature, quasi-Monte Carlo (QMC), or multilevel Monte Carlo (MLMC) techniques may critically depend on the regularity of the integrated. To overcome this issue,  we introduce in [1,2,3] different smoothing techniques.

In the first part of the talk, we will discuss our novel numerical smoothing approach [1,2]  in which we combine root-finding methods with one-dimensional integration with respect to a single well-selected variable, focusing on cases where the discretization of the asset price dynamics is necessary. We prove that, under appropriate conditions, the resulting function of the remaining variables is highly smooth, affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e., Brownian bridge and Richardson extrapolation on the weak error). Our analysis in [1] demonstrates the advantages of combining numerical smoothing with ASGQ and QMC methods over ASGQ and QMC methods without smoothing, and the Monte Carlo approach.  Moreover, our analysis in [2] shows that our numerical smoothing improves the robustness (the kurtosis at deep levels becomes bounded) and complexity of the MLMC method. In particular,  we recover the optimal MLMC complexities obtained for Lipschitz functionals.

In the second part of the talk, we will discuss our efficient Fourier-based method in [3] for pricing European multi-asset options under Lévy models. Given that the integrand in the frequency space often has higher regularity than in the physical space, we extend the one-dimensional Fourier valuation formula to the multivariate case and employ two complementary ideas. First, we smooth the Fourier integrand via an optimized choice of the damping parameters based on a proposed heuristic optimization rule.  These parameters ensure integrability and control the regularity class of the integrand. Second, we use sparsification and dimension-adaptivity techniques to accelerate the convergence of the numerical quadrature in high dimensions.  We demonstrate the advantages of adaptivity and our damping parameter rule on the numerical complexity of the quadrature methods. Moreover, we reveal that our approach achieves substantial computational gains compared to the Monte Carlo method for different dimensions and parameter constellations. 

References:[1] Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone. ”Numerical Smoothing with Hierarchical Adaptive Sparse Grids and Quasi-Monte Carlo Methods for Efficient Option Pricing.” arXiv preprint arXiv:2111.01874 (2021). Accepted in Quantitative Finance Journal (2022).[2] Christian Bayer, Chiheb Ben Hammouda, and Raúl Tempone. "Multilevel Monte Carlo Combined with Numerical Smoothing for Robust and Efficient Option Pricing and Density Estimation." arXiv preprint arXiv:2003.05708 (2022).[3] Christian Bayer, Chiheb Ben Hammouda, Antonis Papapantoleon, Michael Samet, and Raúl Tempone. "Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models." arXiv preprint arXiv:2203.08196 (2022).

 

27.10.2022

17.15 Uhr

 

Marc Sedjro (AIMS South Africa)

Conservation laws arising in the study of forward-forward Mean Field Games

 

Abstract: 

In this talk, we introduce several models of the so-called forward-forward Mean-Field Games (MFGs). These models arise, for example, in the study of numerical schemes to approximate stationary states of MFGs. We establish a link between the forward-forward Mean-Field Games and a class of hyperbolic conservation laws. Furthermore,  we show how these models are connected to certain nonlinear wave equations. Finally, we investigate the existence of solutions and examine their long-time limit properties.

Joint work with Diogo Gomes and Levon Nurbekyan.

 

10.11.2022

16.15 Uhr

Tanut (Nash) Treethantiploet (University of Edingburgh)

Exploration vs Exploitation: From stochastic control theory to Reinforcement Learning

                             

Abstract: 

In stochastic control theory, the model is assumed to be known by the decision-makers; whereas, in Reinforcement Learning (RL), the decision-makers should infer the models and makes decisions through statistical estimates. In short, we may interpret the RL problem as a stochastic control problem, where the model uncertainty must be taken into consideration on top of the controller risk. This additional model uncertainty leads to the exploration-exploitation trade-off where the controller faces a dilemma between following the optimal control strategy obtained from the estimated model (exploitation), and making the decision to understand the model more (exploration). 

In this talk, we will discuss such a phenomenon in the context of the linear-convex episodic reinforcement learning problem. We propose a simple approach to balance such a trade-off obtained through the identifiability and sensitivity of the control problem. We will also discuss how exploration can be achieved through the execution of a solution to a relaxed control problem.

 

 

10.11.2022

17.15 Uhr

Hao Xing (Boston University)

The Dark Side of Circuit Breakers

 

Abstract: 

Market-wide trading halts, also called circuit breakers, have been widely adopted as part of the stock market architecture, in the hope of stabilizing the market during dramatic price declines.  We develop an intertemporal equilibrium model to examine how circuit breakers impact market behavior and welfare.  We show that a circuit breaker tends to lower the level of price and significantly alters its dynamics.  In particular, as the price approaches the circuit breaker, its volatility rises drastically, accelerating the chance of triggering the circuit breaker -- the so-called ``magnet effect''.  In addition, returns exhibit increasing negative skewness and positive drift, while trading activity spikes up.  Our empirical analysis finds supportive evidence for the model's predictions.  Moreover, we show that a circuit breaker can affect the overall welfare either negatively or positively, depending on the relative significance of investors' trading motives for risk sharing vs.\ irrational speculation. This is a joint work with Hui Chen, Anton Petukhov, and Jiang Wang.

 

 

24.11.2022

16.15 Uhr

 

Martin Wahl (HU Berlin)

Optimal estimation for linear SPDEs from multiple measurements

 

Abstract:

We consider the problem of parameter estimation for a second order linear stochastic partial differential equation (SPDE). Observing the solution to the SPDE continuously in time and averaged in space over a small window at multiple locations, we construct estimators for the diffusivity, transport and reaction coefficients, and show that their rates of convergence depend on the respective differential order. Moreover, we prove that these rates are minimax-optimal and establish sufficient and necessary conditions for consistent estimation. The proof of the minimax lower bounds relies on an explicit analysis of the reproducing kernel Hilbert space (RKHS) of a general stochastic evolution equation, and may be of independent interest. This is joint work with Randolf Altmeyer and Anton Tiepner.

 

24.11.2022

17.15 Uhr

 

ACHTUNG !! DIESER VORTRAG ENTFÄLLT !!!

Dylan Possamai (ETH Zürich)

Non-asymptotic convergence rates for mean-field games: weak formulation and McKean–Vlasov BSDEs

 

Abstract:

This work is mainly concerned with the so-called limit theory for mean-field games. Adopting the weak formulation paradigm put forward by Carmona and Lacker, we consider a fully non-Markovian setting allowing for drift control, and interactions through the joint distribution of players’ states and controls. We provide first a new characterisation of mean-field equilibria as arising from solutions to a novel kind of McKean–Vlasov back- ward stochastic differential equations, for which we provide a well-posedness theory. We incidentally obtain there unusual existence and uniqueness results for mean-field equilibria, which do not require short-time horizon, separability assumptions on the coefficients, nor Lasry and Lions’s monotonicity conditions, but rather smallness conditions on the terminal reward. We then take advantage of this characterisation to provide non-asymptotic rates of convergence for the value functions and the Nash-equilibria of the N-player version to their mean-field counterparts, for general open-loop equilibria. This relies on new back- ward propagation of chaos results, which are of independent interest. This is a joint work with Ludovic Tangpi.

 

08.12.2022

16.15 Uhr

Yan Dolinsky (Hebrew University Jerusalem)

Utility indifference pricing with high risk aversion and small linear price impact

 

Abstract: We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options and we compute their non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. Moreover, we find explicitly a family of portfolios which are asymptotically optimal.

 

08.12.2022

17.15 Uhr

Matteo Burzoni (University of Milan)

A Tikhonov Theorem for McKean Vlasov SDEs and an application to mean-field control problems

 

Abstract: We present a stochastic Tikhonov theorem for two-scales systems of SDEs, which cover the case of McKean-Vlasov SDEs. Our approach extends and generalizes previous results on two-scales systems of SDEs without mean-field interaction. As an application we provide a novel method for approximating the solution of certain systems of FBSDEs, related to the Pontryagin maximum principle, which is new even for the case without mean-field interaction. This is a joint work with A. Cosso.

 

 

 

 

12.01.2023

16.15 Uhr

 

Wei  Xu (Humboldt-Universität zu Berlin)

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12.01.2023

17.15 Uhr

Emma Hubert (Princeton University)

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26.01.2023

16.15 Uhr

 

 

Ralf Wunderlich (BTU Cottbus)

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26.01.2023

17.15 Uhr

David Itkin (Imperial College London)

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09.02.2023

16.15 Uhr

 

 

Rouyi Zhang (Humboldt-Universität zu Berlin)

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09.02.2023

17.15 Uhr

 

 

Alain Rossier (University of Oxford)

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Interessenten sind herzlich eingeladen.

 

 

 


Für Rückfragen wenden Sie sich bitte an:

Frau Sabine Bergmann

bergmann@math.hu-berlin.de
Telefon: 2093 45450
Telefax: 2093 45451