FAM @ TU Wien: VS-MFP

Seminar around once a month on Thursdays, 15:00-18:00, Uni Wien (on-site)

2023-11-30
University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, SR13, 2nd floor

15:15

Alexander Kolesnikov (HSE University)
Transportation problem and auction theory
Abstract: A remarkable relation between auction theory (the case of one bidder) and optimal transportation was discovered in the seminal paper Daskalakis-Deckelbaum-Tzamos, Strong Duality for a Multiple-Good Monopolist, Econometrica (2017). We will talk about recent developments of this theory for the case of multiple bidders. In particular, we discuss duality, numerical results and relation to other problems from economy, such as Beckmann's continuous transportation and monopolist problems. The talk is partially based on our recent work with Aleh Tsyvinski, Fedor Sandomirskiy and Alexander Zimin.

2023-11-09
University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, SR13, 2nd floor

15:30

Andreas Søjmark (LSE London)
Endogneous distress contagion in a dynamic interbank model.
Abstract: In this talk, I will introduce a simple interbank model with stochastic dynamics and multiple maturities, allowing us to study the systemic risk-aware term structure for interbank claims. To account for informational contagion, we consider a mark-to-market valuation of interbank assets which turns out to involve a non-standard forward-backward mechanism, since conditional probabilities of future solvency are required to determine today's balance sheets. The outcome is a form of distress contagion that acts as a stochastic volatility term in the capital of each bank, leading, endogenously, to both volatility clustering and a marked downside ‘leverage effect’. Moreover, we will see the possibility of an inverted term structure arising for the entire system solely from excessive volatility of a core group of banks. This is based on joint work with Zach Feinstein.

16:15

Gregoire Loeper (BNP Paribas, previously Monash Univ.)
Black and Scholes, Legendre and Sinkhorn
Abstract: This talk will be a unified overview of some recent contributions in financial mathematics. The financial topics are option pricing with market impact and model calibration. The mathematical tools are fully non-linear partial differential equations and semi-martingale optimal transport. Some new and fun results will be a Black-Scholes-Legendre formula for option pricing with market impact, a Measure Preserving Martingale Sinkhorn algorithm for martingale optimal transport, and a lognormal version of the Bass Martingale.

17:00

Sascha Desmettre (JKU Linz)
Equilibrium Investment with Random and State-Dependent Risk Aversion
Abstract: We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example, that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by its expectation. If time permits, we also elaborate on what happens when the risk aversion is driven by a factor process.

2023-10-12
University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, SR13, 2nd floor

15:45

Jan Kallsen (Kiel University)
Should I invest in the market portfolio? - A parametric approach
Abstract: This study suggests a parsimonious stationary diffusion model for the dynamics of stock prices relative to the entire market. Its aim is to contribute to the questions how to choose the relative weights in a diversified portfolio and, in particular, whether the market portfolio behaves close to optimally in terms of the long-term growth rate.
Specifically, we introduce the elasticity bias as a measure of the market portfolio's suboptimality. We heavily rely on the observed long-term stability of the capital distribution curve, which also served as a starting point for the Stochastic Portfolio Theory in the sense of Fernholz.

16:30

Hanspeter Schmidli (University of Cologne)
Stabilizing the surplus process through the control of Drawdowns
Abstract: The drawdown is the loss of the surplus process compared its historical maximum. In order to stabilize the surplus one tries to keep the surplus close the its maximum; that is, keeping the drawdown small. We use proportional reinsurance to control the drawdown and measure the time the drawdown spends below a predefined barrier. If the time in drawdown is the only criterion, the maximum will never increase under the optimal strategy, see Brinker and Schmidli (2022). To avoid this, we maximise simultaneously the increase of the maximum. Dependent on how we weight the two contradicting criteria, we obtain different strategies, of which some are surprising. The talk is based on joint work with Leonie Brinker.
Brinker, L.V. & Schmidli, H. (2022). Optimal discounted drawdowns in a diffusion approximation under proportional reinsurance. J. Appl. Probab. 59, 527-540.

Past Talks / Summer Term 2023

Seminar around once a month on Thursdays, 15:30-18:30, TU Wien (on-site)

2023-06-15, 15:30-17:45 CEST¹⁾
TU Wien, Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, yellow section, 7th floor, seminar room DB gelb 07

15:30

Han Gan (Waikato University, NZ)
Stationary distribution approximations for two-island and seed bank models
Abstract: In this talk we will discuss two-island Wright-Fisher models which are used to model genetic frequencies and variation for subdivided populations. One of the key components of the model is the level of migration between the two islands. We show that as the population size increases, the appropriate approximation and limit for the stationary distribution of a two-island Wright-Fisher Markov chain depends on the level of migration. In a related seed bank model, individuals in one of the islands stay dormant rather than reproduce. We give analogous results for the seed bank model, compare and contrast the differences and examine the effect the seed bank has on genetic variability. Our results are derived from a new development of Stein's method for the two-island diffusion model and existing results for Stein's method for the Dirichlet distribution

16:15

Paul Hager (HU Berlin)
Unified Signature Cumulants and Generalized Magnus Expansions
Abstract: Signature cumulants, defined as the tensor-logarithm of expected signatures of semimartingales, are seen to satisfy a fundamental functional relation. This equation, in a deterministic setting, contains Hausdorff’s differential equation, which itself underlies Magnus’ expansion. The (commutative) case of multivariate cumulants arises as another special case and yields a new Riccati-type relation valid for general semimartingales. Here, the accompanying expansion provides a new view on recent “diamond" and "martingale cumulants" (Alos et al ’17, Lacoin et al ’19., Friz et al. ’20) expansions. We will further discuss possible applications in finance.

17:00

Joaquin Fontbona (University of Chile)
Quantitative mean-field limit for interacting branching diffusions
Abstract: We establish an explicit rate of convergence for some systems of mean-field interacting diffusions with logistic binary branching, towards solutions of nonlinear evolution equations with non-local self-diffusion and logistic mass growth describing their large population limit. The proof relies on a novel coupling argument for binary branching diffusions based on optimal transport, allowing us to sharply mimic the trajectory of the interacting branching population by means of a system of independent particles with suitably distributed random space-time births. We are thus able to derive an optimal convergence rate, in the dual bounded-Lipschitz distance on finite measures, for the empirical measure of the population, from the known convergence rate in Wasserstein distance of empirical distributions of i.i.d. samples. Our approach and results extend propagation of chaos techniques and ideas from kinetic models, to stochastic systems of interacting branching populations, and appear to be new in this setting, even in the simple case of pure binary branching diffusions. Joint work with Felipe Muñoz-Hernandez

2023-03-30, 15:30-17:30 CEST²⁾
TU Wien, Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, yellow section, 7th floor, seminar room DB gelb 07

15:30

Alfred Müller (University of Siegen)
Decisions under uncertainty: sufficient conditions for almost stochastic dominance
Abstract: Decision making under risk involves a ranking of distributions, which is typically based on a method for assigning a real number to a distribution using a risk measure, a premium principle or a context of expected utility. As it is typically difficult to assess a concrete risk measure or utility function it is a well established idea to use stochastic dominance rules in form of stochastic orders to compare distributions. However, it is often equally difficult to completely specify a distribution. Therefore it is an interesting question whether one can derive unambiguous decisions under partial knowledge of the distributions. In this talk we in particular address this question under the condition that we only know the mean and variance of the involved distributions or that we know the marginal distributions but not the copulas in a multivariate context. Under such assumptions we derive sufficient conditions for concepts of almost stochastic dominance that are based on restrictions on marginal utilities.
The talk is based on joint work with Marco Scarsini, Ilia Tsetlin and Robert L. Winkler.

16:30

Birgit Rudloff (WU Vienna)
Epic Math Battles: Nash vs Pareto -news for convex games-
Abstract: 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.
The characterization holds for all non-cooperative games (non-convex, convex, linear).
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.
This computation is straight forward in the linear case. In this talk we will discuss recent results in the convex case. 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.

Past Talks / Winter Term 2022

Seminar around once a month on Thursdays, 15:30-18:30, TU Wien (on-site)

2023-02-09, 16:00 CET³⁾
University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, 2nd floor, seminar room 7

Hans Föllmer (HU Berlin)
Optimal Transport, Entropy, and Risk Measures on Wiener space
Abstract: We discuss the interplay between entropy and optimal couplings on Wiener space. In particular we prove a new rescaled version of Talagrand’s transport inequality, and we take a look at the corresponding risk measures.

2023-01-26, time between 15:00 and 18:00 CET⁴⁾ t.b.a.
WU Vienna, Welthandelsplatz 1, 1020 Wien, ground floor, seminar room D4.0.039

15:30

Sascha Desmettre (JKU Linz)
Rough Volatility Modeling and Portfolio Optimization.
Abstract: This talk focuses on the modeling of and portfolio optimization in financial markets with different rough models for the volatility. In the first part we deal with the (one-dimensional) stochastic volatility model of Heston, which is extended to fractional and rough market dynamics by applying the Riemann-Liouville fractional integral operator and the Marchaud fractional derivative to the classical Cox-Ingersoll-Ross process. Using a Markovian representation, followed by a reasonable quantization of the underlying probability measures, we show that it is possible to cast the problem into the classical stochastic control framework. We deduce a Feynman-Kac representation for these one-dimensional fractional and rough market models and solve the corresponding continuous time Merton portfolio optimization problems for power utility. In the second part we are again concerned with portfolio selection for an investor with power utility, but now in a multi-dimensional rough stochastic environment. In particular we investigate Merton’s, portfolio problem for different multivariate Volterra models. Based on the classical Wishart model, we introduce a new matrix-valued stochastic volatility model, where the volatility is driven by a Volterra-Wishart process. In contrast to the first part, the solution methods are now based on the calculus of convolutions and resolvents. The resulting optimal strategy can be expressed in terms of the solution of a multivariate Riccati-Volterra equation, extending existing results to the multivariate case, avoiding restrictions on the correlation structure.

16:15

Florian Huber (University of Vienna)
Interacting particles and market capitalization curves
Abstract: Motivated by the robustness of the so-called market capitalization curve, our goal is to study the behaviour of equity market models on a macroscopic scale. This is done by extending the volatility stabilized market models studied by Fernholz and co-authors and allowing for simple correlation structure induced by a common noise term. Letting the number of companies approach infinity, we show that the limit of the empirical measure of the N-company system converges to the unique solution of a degenerate, non-linear SPDE. The obtained limit also possess a representation as a conditional probability of the solution to a certain McKean-Vlasov SDE.
This is joint work in progress with Christa Cuchiero.

17:00

Thomas Simon (University of Lille, FR)
On the log-concavity of the Mittag-Leffler distribution
Abstract: The Mittag-Leffler random variable is the positive random variable whose moment generating function is the classical Mittag-Leffler function with parameter α ∈ (0,1). In this talk, we consider basic distributional properties of this random variable which appears in various domains of probability theory. In particular, we show that its density is log-concave if and only if α ≤ α_★, where α_★ = 0.771667... is some threshold defined implicitly with the Gamma functions, and that this property is equivalent to the reciprocal convexity on the negative half-line of the associated Mittag-Leffler function. We will also discuss some extensions of these results to the more general Wright functions of the second kind and their associated two-parameter Mittag-Leffler functions.

2022-12-01, time between 15:00 and 18:00 CET⁵⁾
TU Wien, Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, yellow section, 7th floor, seminar room DB gelb 07

15:15

Nurtai Meimanjan (WU Vienna)
Computation of systemic Risk Measures: A Mixed-Integer Programming Approach
Abstract: Systemic risk is concerned with the instability of a financial system whose members are interdependent in the sense that the failure of a few institutions may trigger a chain of defaults throughout the system. Recently, several systemic risk measures have been proposed in the literature that are used to determine capital requirements for the members subject to joint risk considerations. We address the problem of computing systemic risk measures for systems with sophisticated clearing mechanisms. In particular, we consider an extension of Rogers-Veraart network model where the operating cash flows are unrestricted in sign. We propose a mixed-integer programming problem that can be used to compute clearing vectors in this model. Due to the binary variables in this problem, the corresponding (set-valued) systemic risk measure fails to have convex values in general. We associate nonconvex vector optimization problems to the systemic risk measure and provide theoretical results related to the weighted-sum and Pascoletti-Serafini scalarizations of this problem. Finally, we test the proposed formulations on computational examples and perform sensitivity analyses with respect to some model-specific and structural parameters.

16:00

Andreas Celary (WU Vienna)
Regime-switching affine term structures
Abstract: We consider an HJM model setting for Markov-chain modulated forward rates. The underlying Markov chain is assumed to induce regime switches on the forward curve dynamics. Our primary focus is on the interest rate and energy futures markets. After deriving HJM-drift conditions for the two markets, we prove under the assumption of affine structure for the term structure that the forward curves are solutions to specific systems of ODEs that can be solved explicitly in many cases. This allows for a tractable model setting, and we present an algorithm for obtaining consistent forward curve models within our framework. We conclude by presenting some simple numerical examples.

2022-11-03, 15:30 - (max.)18:30 CET⁶⁾, seminar room DB gelb 07
TU Wien, Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, yellow section, 7th floor

15:30

Camilla Damian (TU Wien)
Invisible Infections: A Partial Information Approach for Estimating the Transmission Dynamics of the Covid-19 Pandemic
Abstract: We provide a discrete-time model for the evolution of a pandemic, which represents an adaptation of the well-know SIR model for epidemics to a partial information setting. In particular, we assume that the current number of infected people is unknown and use nested particle filtering to estimate the effective reproduction rate and the parameters governing its dynamics. We present an application where our observations consist of the time series of Covid-19 positive tests in Austria.

16:30

Larry Goldstein (University of Southern California, US)
Zero Biased Enhanced Stein Couplings
Abstract: Stein's method for distributional approximation has become a valuable tool in probability and statistics by providing finite sample distributional bounds for a wide class of target distributions in a number of metrics. A key step in popular versions of the method involves making couplings constructions, and a family of couplings of Chen and Roellin vastly expanded the range of applications for which Stein's method for normal approximation could be applied. This family subsumes both Stein's classical exchangeable pair, and the size bias coupling. A further simple generalization includes zero bias couplings, and also allows for situations where the coupling is not exact. The zero bias versions result in bounds for which often tedious computations of a variance of a conditional expectation is not required. An example to the Lightbulb process shows that even though the method may be simple to apply, it may yield improvements over previous results that had achieved bounds with optimal rates and small, explicit constants.

Past Talks / Summer Term 2022

Seminar around once a month on Thursdays, 15:30-18:30, TU Wien (on-site)

Th, 23.06.2022
seminar room DB gelb 10
TU Wien, 1040, Wiedner Hauptstr. 8, Freihaus, yellow area, 10th floor

15:00
Çağın Ararat (Bilkent University, Ankara, Turkey)
Dynamic mean-variance problem: recovering time-consistency
Abstract: The dynamic mean-variance problem is a well-studied optimization problem that is known to be time-inconsistent. The main source of time-inconsistency is that the family of conditional variance functionals indexed by time fails to be recursive. We consider the mean-variance problem in a discrete-time setting and study an auxiliary dynamic vector optimization problem whose objective function consists of the conditional mean and conditional second moment. We show that the vector optimization problem satisfies a set-valued dynamic programming principle and is time-consistent in a generalized sense. Moreover, its weighted sum scalarizations are closely related to the mean-variance problem through simple nonlinear transformations. This is at the cost of using stochastic and time-varying weights in the mean-variance problem. We also discuss the relationship between our results and some recent results in the literature that discuss the use of time-varying weights under special dynamics. Finally, in a finite probability space, we propose a computational procedure that relies on convex vector optimization and convex projection problems, and we use this procedure to calculate time-consistent solutions in concrete market models. Joint work with Seyit Emre Düzoylum (UC Santa Barbara).

15:45
Zinoviy Landsman (University of Haifa, Israel)
The Minimum Variance Squared Distance Risk Functional
Abstract: In this paper, we introduce a novel multivariate functional that represents a position where the intrinsic uncertainty of a system of mutually dependent risks is maximally reduced. The proposed multivariate functional defines the location of the minimum variance of squared distance (LVS) for some n-variate vector of risks X. We compute the analytical representation of LVS(X), which consists of the location of the minimum expected squared distance, LES(X), covariance matrix A, and a matrix B of the multivariate central moments of the third order of X. From this representation it follows that LVS(X) coincides with LES(X) when X has a multivariate symmetric distribution, but differs from it in the non-symmetric case. As LES(X) is often considered a neutral multivariate risk measure, we show that LVS(X) also possesses the important properties of multivariate risk measures: translation invariance, positive homogeneity, and partial monotonicity. We also study the mean-variance approach based on the balanced sum of an expectation and a variance of the square of the aforementioned Euclidean distance and control for the closeness of LES(X) and LVS(X). The proposed theory and the results are distribution free, meaning that we do not assume any particular distribution for the random vector X. The results are demonstrated with real data of Danish fire losses.

Th, 09.06.2022, 15:10-17:45
seminar room DB gelb 10
TU Wien, 1040, Wiedner Hauptstr. 8, Freihaus, yellow area, 10th floor

15:10
Zachary Feinstein (Stevens Institute of Technology, USA)
Deep learning the efficient frontier of convex vector optimization problems with applications to finance
Abstract: In this talk, we propose a neural network architecture to approximate the weakly efficient frontier of convex vector optimization problems satisfying Slater's condition. The proposed machine learning methodology provides both an inner and outer approximation of the weakly efficient frontier, as well as an upper bound to the error at each approximated efficient point. In numerical case studies we demonstrate that the proposed algorithm is effectively able to approximate the true weakly efficient frontier of convex vector optimization problems. This remains true even for large problems (i.e., many objectives, variables, and constraints) and thus overcoming the curse of dimensionality. Special attention is paid to the mean-variance and mean-risk problems in finance.

16:00
Sonja Cox (University of Amsterdam, NL)
Simulation of a diffusion first-passage time via a Fokker-Planck equation
Abstract: First-passage times of diffusions are used in mathematical psychology to account for behavioral data from two-alternative forced choice tasks. I will explain how these first-passage times are used, and then discuss the well-known fact that first-passage times of a diffusion can be described via a Fokker-Planck equation. Finally, I will discuss numerical methods tailor-made to deal with the Fokker-Planck equations arising from first-passage time models used in mathematical psychology.
Joint work with Udo Böhm, Gregor Gantner, and Rob Stevenson.

16:30
Asma Kheder (University of Amsterdam, NL)
Utility maximisation and change of variable formulas for time-changed dynamics
Abstract: In this paper we derive novel change of variable formulas for stochastic integrals w.r.t. a time-changed Brownian motion where we assume that the time change is an increasing stochastic process with finitely many jumps in a bounded set of the positive half-line and is independent of the Brownian motion. As an application we consider the problem of maximising the expected utility of the terminal wealth in a semimartingale setting, where the semimartingale is written in terms of a time-changed Brownian motion and a finite variation process. To solve this problem, we use an initial enlargement of filtration and our change of variable formulas to shift the problem to a maximisation problem under the enlarged filtration for models driven by a Brownian motion and a finite variation process. The latter problem can be solved by using martingale properties. Then applying again a change of variable formula, we derive the optimal strategy for the original problem for a power utility and for a logarithmic utility.

17:00
Leonie Brinker (University of Cologne, DE)
Stochastic Optimisation of Drawdowns via Dynamic Reinsurance Controls
Abstract: The drawdown of a stochastic process is the absolute distance to its running maximum and can be interpreted as a path-dependent measure of risk. In this talk, we consider a stochastic control problem inspired by the real-world question of how to reinsure in an ‘optimal’ way. Here, the notion of optimality is based on the minimisation of the ‘expected time in (critical) drawdown’ under dynamic controls, i.e. the time during which the drawdown process exceeds a predefined, ‘critical’ threshold d > 0. By exploiting connections to Laplace transforms of passage times, Hamilton–Jacobi–Bellman equations, Gerber-Shiu functions and reflected stochastic differential equations, we find the value functions and the optimal strategies for the Cramér-Lundberg and the Brownian risk model.

Th, 05.05.2022
seminar room DB gelb 10
TU Wien, 1040, Wiedner Hauptstr. 8, Freihaus, yellow area, 10th floor

16:00
Kristof Wiedermann (FAM @ TU Wien)
An SMP-Based Algorithm for Solving the Constrained Utility Maximization Problem via Deep Learning
Abstract: We consider the utility maximization problem under convex constraints with regard to theoretical results which allow the formulation of algorithmic solvers which make use of deep learning techniques. In particular for the case of random coefficients, we prove a stochastic maximum principle (SMP) generalizing the SMP proved by Li and Zheng (2018). We use this SMP together with the strong duality property for defining a new algorithm, which we call deep primal SMP algorithm. Numerical examples illustrate the effectiveness of the proposed algorithm. Moreover, our numerical experiments for constrained problems show that the novel deep primal SMP algorithm overcomes the deep SMP algorithm's (see Davey and Zheng (2021)) weakness of erroneously producing the value of the corresponding unconstrained problem. Furthermore, in contrast to the deep controlled 2BSDE algorithm from Davey and Zheng (2021), this algorithm is also applicable to problems with path dependent coefficients. Finally, we propose a learning procedure based on epochs, which improved the results of our algorithm even further. Implementing a semi-recurrent network architecture for the control process turned out to be also a valuable advancement.

16:45
Anke Wiese (Heriot-Watt University Edinburgh, UK)
Series Expansion and Direct Inversion for the Heston Model
Abstract: The Heston model is one of the most well-known stochastic volatility models. It models jointly the evolution of the price process of an investment asset and the stochastic variance of the asset's log-returns. The variance process is given as a Cox-Ingersoll-Ross process, also known as mean-reverting square root process, a process used widely in financial and other applications. While the Heston model provides tractability to a certain extent, its numerical treatment is well-known to be very challenging. Key components in the model are the variance process and its time integral conditioned on the variance values at the end points of the integral. We derive a new series representation for the latter quantity. For this we explore the connection of the CIR process to squared Bessel processes and bridges. The new representation has the advantage that truncation errors decay exponentially, and that building blocks of this series are random variables that are largely independent of the model parameters. Based on this new representation, we derive high-accuracy direct inversion methods that enable the efficient sampling of the Heston model. This talk is based on joint work with SJA Malham and J Shen.

Past Talks / Winter Term 2021

Seminar around once a month on Thursdays, 15:30-18:30, University of Vienna (on-site if possible)

Th, 03.02.2022
16:30 (UTC+1 = CET), online via Zoom

Xin Zhang (University of Vienna)
Expert Prediction Problem
Abstract: This talk focuses on expert prediction problem with finite horizon, which is formulated as a zero sum game between a player and an adversary. By considering a scaled game, the value function of discrete games converges to the viscosity solution of a PDE. We explicitly solve this nonlinear PDE with N = 4 experts. By showing that the solution is C^2, we are able to show that the comb strategies, as conjectured in “Towards Optimal Algorithms for Prediction with Expert Advice” by Peres et al., form an asymptotic Nash equilibrium. We also prove the “Finite vs Geometric regret” conjecture proposed in Peres et al. for N = 4, and show that this conjecture in fact follows from the conjecture that the comb strategies are optimal. This talk is based on a joint work with Erhan Bayraktar and Ibrahim Ekren.

Th, 18.11.2021
lecture hall 14, Oskar-Morgenstern-Platz 1, 2nd floor and online via Zoom

16:45-17:30 (UTC+1 = CET)

Gabriela Kováĉova (WU Vienna)
Time consistency of mean-risk problem and a set-valued Bellman's principle
Abstract: Selecting a portfolio of risky assets which maximizes the expected terminal values at the same time as it minimizes portfolio risk is a classical problem in Mathematical Finance known as the mean-risk problem. The usual approach in the literature is to combine the mean and the risk to obtain a problem with a single objective. In a dynamic setting this scalarization, however, comes at the cost of time inconsistency.
We show that these difficulties disappear by considering the problem in its natural form, that is, as a bi-objective optimization problem. As such the mean-risk problem can be shown to satisfy an appropriate notion of time consistency, closely related to existence of a moving scalarization. Additionally, we show that the mean-risk problem satisfies a Bellman's principle appropriate for a bi-objective optimization problem: a set-valued Bellman's principle.
The talk is based on join work with Birgit Rudloff.

17:30-18:15 (UTC+1 = CET)

Stefan Rigger (University of Vienna)
Optimal bailout strategies and the drift-controlled probabilistic supercooled Stefan problem
Abstract: We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash to a subset of the entities in order to limit defaults to a given proportion. We prove that the value of the agent's control problem converges as the number of agents goes to infinity, and the limit satisfies a drift-controlled version of the probabilistic supercooled Stefan problem. Solving the associated HJB equations numerically suggests that the agent's optimal strategy is to subsidise banks whose asset values lie in a non-trivial time-dependent region. Finally, we study a linear-quadratic version of the model where instead of the terminal losses, the agent optimises a terminal cost function of the equity values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically.

Th., 14.10.2021

15:45-16:30 (UTC+2 = CEST), lecture hall 8, Oskar-Morgenstern-Platz 1, 1st floor

Julia Eisenberg (TU Wien)
Dividend maximisation with negative and positive preference rates: a behaviouristic interpretation
Abstract: In this talk, we look at a dividend maximisation problem under a Brownian surplus and a Markov-switching preference rate model. The preference rate can attain two values - a positive and a negative.
First, we discuss the optimal dividend payout strategy for the setting with a classical ruin concept - the ruin is declared when the surplus becomes negative. In the second part, the setting will be modified by a Parisian ruin with an exponential delay - the ruin is declared if the process stays negative during an exponentially distributed time interval.
In the first case, the optimal strategy turns out to be of a barrier type, being a finite barrier during the positive rate phases and infinite barrier (no dividends are paid) during the negative phases.
We show that the finite barrier is a monotone function of regime switching intensities.
In the case of the Parisian delay, the optimal strategy depends on the relation between the expected income rate and the parameter of the exponential delay. The cases of long, medium and short expected delays have to be considered separately in order to find explicit expressions for the value function and the optimal strategy (remaining of a barrier type for short and medium delays).
If the expected delay is too long, the optimal strategy in the negative state can change from not paying dividends to a band strategy.
Joint work with Leonie Brinker.

16:45-17:30 (UTC+2 = CEST), lecture hall 5, Oskar-Morgenstern-Platz 1, ground floor

Francesca Primavera (University of Vienna)
Lévy type signature models
Abstract: Signature models have recently entered the ﬁeld of Mathematical Finance. However, despite the presence of jumps in ﬁnancial data, the signature models for asset prices proposed so far have only dealt with the continuous-path setting. Based on recent results on the signature of càdlàg paths, we deﬁne signature-based models which include jumps. The approach that we follow consists of parameterizing the model itself or its characteristics as linear functions of the signature of an augmented Lévy process, interpreted as market’s primary underlying process. We show that, in this contest, ﬁrst principles, like absence of arbitrage, still apply. Finally, we prove that the signature of a generic ℝ^d-valued Lévy process is a polynomial process on the extended tensor algebra and derive its expected value via polynomial technology. This result, when applied to the market’s primary process, yields a compact pricing formula, used in the calibration of the model to market data.
This talk is based an ongoing joint work with Christa Cuchiero and Sara Svaluto-Ferro.

Past Talks / Summer Term 2021

online seminar around once a month on Thursdays, 15:30-18:30, live stream via Zoom.

2021-05-27, 15:30-18:30 CEST¹⁾, online seminar:

15:30
Benedict Bauer (TU Wien)
Self-similar Gaussian Markov processes
Abstract: We give a representation for centered self-similar Gaussian Markov processes which yields short proofs on non-Markovianity results concerning variants of fractional Brownian motion.

16:00
Aleksandar Arandjelovic (TU Wien)
Deep hedging in continuous time
Abstract: We present some recent results that extend the deep hedging methodology from discrete to continuous time. After defining an algorithmically generated class of processes, we establish the universal approximation property of this class in several topological spaces. As an application, we study deep hedging under convex risk measures in continuous time.

16:40
Verena Köck (WU Vienna)
Solving partial-integro differential equations in finance and insurance: a deep learning approach
Abstract: Partial integro-differential equations (PIDEs) appear in many applications that are related to insurance or finance, such as pricing models or stochastic optimization problems. Frameworks that take many different economical factors or assets into account lead to high-dimensional PIDEs, that are typically not explicitly solvable. In the literature there are already methods to tackle PDEs with deep-learning based methods, yet on the other hand, the literature on PIDEs is scarce. Besides of the problem of "the curse of dimensionality", the majority of approximation methods only provide a solution for a single space point. We propose a deep neural network (DNN) algorithm for solving parabolic partial integro-differential equations with boundary conditions in high dimension and apply the method on several examples. Specifically, the solution u(t,x), x \in \Xi to a PIDE is computed for a fixed time t \in [0,T] and a subset \Xi \subset \R^d. Hereby we concentrate on insurance and finance related problems.

17:30
Guido Gazzani (University of Vienna) and
Sara Svaluto-Ferro (University of Vienna)
Universal signature-based models: theory and calibration
Abstract: Universal classes of dynamic processes based on neural networks and signature methods have recently entered the area of stochastic modeling and Mathematical Finance. This has opened the door to robust and more data-driven model selection mechanisms, while first principles like no arbitrage still apply.
Here we focus on signature SDEs whose characteristics are linear functions of a primary underlying process, which can range from a (market-inferred) Brownian motion to a general multidimensional tractable stochastic process. The framework is universal in the sense that any classical model can be approximated arbitrarily well and that the model characteristics can be learned from all sources of available data by simple methods. Indeed, we derive formulas for the expected signature in terms of the expected signature of the primary underlying process. These formulas enter directly in the calibration procedure to option prices, while time series data calibration just reduces to a simple regression.
The talk is based on a joint work with Christa Cuchiero.

2021-04-29, 15:30-18:30 CEST²⁾, online seminar:

Friedrich Hubalek (TU Wien)
Comparing binomial and Gaussian tails with an application to utility maximization
Abstract: We give a rather fine comparison of binomial and Gaussian distribution functions on the whole real line and discuss some related quite classical results.
As an application we answer an open question from recent work of David Kreps and Walter Schachermayer on the convergence of the solution of a discrete-time utility maximization problem for a sequence of binomial models to the Black-Scholes-Merton model for general utility functions.
This is based on joint work with Walter Schachermayer.

Benjamin Robinson (University of Vienna)
Optimal control of martingales in a radially symmetric environment
Abstract: We consider a control problem for multidimensional martingales in a radially symmetric environment. We provide an explicit construction of the value function by reducing the problem to a one-dimensional switching problem with two behaviour regimes. Under mild conditions on the cost function, we can see that the optimal martingale is Markovian. For a particular class of cost functions, however, we conjecture that any Markov martingale is suboptimal. In support of this conjecture, we prove that an SDE describing the optimal behaviour does not admit a strong solution. In this case, we also require results on Brownian filtrations in order to find the value function. This is based on joint work with Alexander Cox.

Zach Feinstein (Stevens Institute of Technology)
Equilibrium Inverse Demand Functions
Abstract: In this talk we present an equilibrium formulation for price impacts. This is motivated by the Buhlmann equilibrium in which assets are sold into a system of market participants and can be viewed as a generalization of the Esscher premium. Existence and uniqueness of clearing prices for the liquidation of a portfolio are studied. Additional properties of the liquidation value of a portfolio are studied, e.g., monotonicity and concavity. Price per portfolio unit sold is also presented. In special cases we study price impacts generated by market participants who follow the exponential utility and power utility.
This is joint work with Maxim Bichuch.

2021-03-25, 15:30-18:30 CET³⁾, online seminar:

Birgit Rudloff (WU Vienna)
Multivariate dynamic programming- from dynamic Nash games to the Mean-Risk problem
Abstract: In several time-inconsistent problems, the time-inconsistency is due to the fact the underlying problem is multi-variate in some sense. For example, the mean-risk problem has two underlying objective functions; maximizing performance measures often involves a ratio of two functions; Nash equilibria in a dynamic game might not be unique and attain different values; a dynamic risk measure in a market with frictions is set-valued.
What unifies these examples is that one can formulate these problems with a set-valued value function that accounts for these different types of multi-variateness.
And it can be shown that this value function is recursive under mild assumptions. Thus, these problems are actually time-consistent in a set-valued sense. Practical implications and economic interpretations are discussed. Examples are given, where this set-valued Bellman's principle is implemented to numerically solve the problem.

Stefan Gerhold (TU Wien)
Asymptotic pricing of VIX options under rough volatility
Abstract: Small-maturity asymptotics for VIX options in rough volatility models are known to be a challenging problem. In this talk we report on recent results for two concrete models. For the rough Heston model, we show that similar methods as for the vanilla smile lead to a large deviations principle for the VIX smile. For the rough Bergomi model, we use some tools from the theory of Gaussian processes to establish estimates that are so far a bit weaker than an LDP. Joint work in progress with Benedict Bauer, Martin Forde and Benjamin Smith.

Walter Schachermayer (University of Vienna)
Faking Brownian Motion with continuous Markov martingales
Abstract: We present an example with the properties mentioned in the title.
Joint work with M. Beiglböck and G. Pammer.

Past Talks / Winter Term 2020

hybrid seminar around once a month on Thursdays, 15:30-18:30,
lecture hall HS 03, Univ. of Vienna, 1090 Wien, Oskar-Morgenstern-Platz 1, ground floor,
& live stream via Zoom.

2021-01-28, 15:30-18:30 CET¹⁾, online seminar:

Thorsten Schmidt (University of Freiburg)
No Arbitrage in Insurance and equity-linked life insurance
Abstract: This paper is an attempt to study fundamentally the valuation of insurance contracts. We start from the observation that insurance contracts are inherently linked to financial markets, be it via interest rates, or – as in hybrid products, equity-linked life insurance and variable annuities – directly to stocks or indices. By defining portfolio strategies on an insurance portfolio and combining them with financial trading strategies we arrive at the notion of insurance-finance arbitrage (IFA). A fundamental theorem provides two sufficient conditions for presence or absence of IFA, respectively. For the first one it utilizes the conditional law of large numbers and risk-neutral valuation. As a key result we obtain a simple valuation rule, called QP-rule, which is market consistent and excludes IFA.
Utilizing the theory of enlargements of filtrations we construct a tractable framework for general valuation results, working under weak assumptions. The generality of the approach allows to incorporate many important aspects, like mortality risk or dependence of mortality and stock markets which is of utmost importance in the recent corona crisis. For practical applications, we provide an affine formulation which leads to explicit valuation formulas for a large class of hybrid products.
This is joint work with Philippe Artzner, Karl-Theodor Eisele and Moritz Ritter.

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).)

Deborah Dormah Kanubala (Academic City University College, Accra, Ghana)
Machine Learning in Option Pricing
Abstract: Option pricing is an important study in contemporary finance and it has been extensively studied both in academic literature and industry. Artificial Neural Network is a type of machine learning algorithm that is capable of learning non-linearity in data. This talk, therefore, focuses on leveraging artificial neural networks to pricing European call options. The talk will first give a broad overview of what machine learning is and its types, provide a high-level overview of financial derivatives with the main focus on options. Finally, it would price European call option using artificial neural network and evaluate the performance of the model on both the train and test data sets.

2020-12-17, 15:30-18:30, online seminar:

Julio Backhoff (University of Twente)
On the small noise behaviour for convex BSDE
Abstract: We discuss the deterministic limit for a class of convex backward differential equations (BSDE) as the volatility of the driving Brownian motion goes to zero. To a certain degree, this provides a generalization of the classical Schilder Theorem in large deviations theory, where the role of the cumulant generating function is replaced by a more general risk measure. We also discuss how Gaussian fluctuations may also appear in this limit result, after identifying the correct scaling. Time permitting, we finalize with a discussion on the analogue of Sanov's theorem in the setting of BSDE. Based on joint work with D. Lacker and L. Tangpi, as well as communications with M. Shkolnikov.

Jana Hlavinova (WU Vienna)
Elicitability and Identifiability of Systemic Risk Measures
Abstract: Identification and scoring functions are statistical tools to assess the calibration and the relative performance of risk measure estimates, e.g. in backtesting. A risk measure is called identifiable (elicitable) if it admits a strict identification function (strictly consistent scoring function). We consider measures of systemic risk introduced by Feinstein, Rudloff and Weber. Since these are set-valued, we work within the theoretical framework of Fissler, Frongillo, Hlavinová and Rudloff for forecast evaluation of set-valued functionals. We construct oriented selective identification functions, which induce a mixture representation of (strictly) consistent scoring functions. Their applicability is demonstrated with a simulation study.

Junjian Yang (TU Wien)
On the planning problem in mean-field games
Abstract: In this talk, we consider a generalization planning problem introduced by P.L. Lions: given a family of marginal distributions μ=(μ_t)_t, find a specification of the game problem which induces μ_t at the mean-field game equilibrium. Since this is related to mean-field games (MFG), I would like to give a short introduction to MFG and the relation to mean-field planning problems.

2020-11-19, 15:30-18:30, online seminar:

Paul Eisenberg (WU Vienna)
Integer constraint trading
Abstract: We investigate discrete time trading under integer constraints, that is, we assume that the offered goods or shares are traded in entire quantities instead of the usual real quantity assumption. For rational asset prices this has little effect on the core of the theory of no-arbitrage pricing. For price processes not restricted to the rational numbers, a novel theory of integer arbitrage free pricing and hedging emerges. We establish an FTAP, involving a set of absolutely continuous martingale measures satisfying an additional property. Finally, we discuss superhedging with integral portfolios. Finally, we investigate quadratic hedging error under integer constraint traiding.

Christoph Gerstenecker (TU Wien)
Stochastic Volterra equations and rough volatility
Abstract: We want to construct a new rough volatility model inspired by the well-known 3/2 model. We will start with a very short introduction to rough volatility. Then, a standard way to make an Ito diffusion model rough is to multiply the integrands with a weakly singular kernel, resulting in a stochastic Volterra integral equation. This motivates us to learn more about and discuss SVIEs, especially under worse regularity and growth conditions for the integrands to be able to introduce more sophisticated models than models just conveniently satisfying Lipschitz and linear growth conditions.

Gudmund Pammer (University of Vienna)
The Wasserstein space of Filtered Processes
Abstract: Sequential decision making is often based on mathematical models, hich may or may not adequately portray reality. To have confidence in the proposed action it is crucial to understand the sensitivity of these decisions with respect to the underlying modeling assumptions. The leading theme of my PhD is to investigate topologies for stochastic processes, which are suitable for sequential decision making in discrete time. Several mathematicians with vastly different backgrounds came up with topologies to tackle this question and astonishingly, in discrete time on the space of stochastic processes equipped with their canonical filtration, they all coincide. We also discuss the extension of this topology to stochastic processes with arbitrary filtrations.

2020-10-15, 15:30-18:30, hybrid seminar:

Aleksandar Arandjelovic (TU Wien)
Deep portfolio optimization in financial markets with a large trader
Abstract: We consider a financial market model in which market participants - which we call large traders - face liquidity risk. Based on the theory of nonlinear stochastic integration, we first revisit the construction of corresponding price and wealth processes that are affected by trading strategies. We then discuss a method which allows us to identify optimal large trader strategies from a utility maximization point of view. After discussing several examples and highlighting phenomena that arise due to the market impact of the large trader in the form of differing levels of market instability, we move to an algorithmic approach inspired by the concept of deep hedging. Upon encoding market incompleteness into our models, we discuss two important points. While we do find reasonably generalized deep neural strategies in comparison to classical liquid settings, we also highlight some problems that arise with this approach. (Joint work with Thorsten Rheinländer.)

Stefan Rigger (University of Vienna)
Propagation of minimality in the supercooled Stefan problem
Abstract: The one-dimensional one-phase Supercooled Stefan Problem is a PDE problem with free boundary which serves as a model for the freezing of supercooled liquids. Under certain conditions, this model will exhibit blow-up in finite time. Following the methodology of Delarue, Nadtochiy and Shkolnikov, we construct solutions to the Supercooled Stefan Problem through the Fokker-Planck equation associated to a stochastic process that solves a certain McKean-Vlasov equation. This technique allows us to define solutions globally even in the presence of blow-ups. Solutions to the associated McKean-Vlasov equation can be constructed via an approximating particle system, and we prove Propagation of Chaos. The particle system in question appears in the literature on systemic risk, establishing the connection of the aforementioned results to Mathematical Finance. Finally, we prove a conjecture of Delarue, Nadtochiy and Shkolnikov, relating the solution concepts of so-called minimal and physical solutions, showing that minimal solutions of the McKean-Vlasov equation are physical whenever the initial condition is integrable.

Kevin Kurt (WU Vienna)
Markov-modulated Affine Processes
Abstract: We introduce Markov-modulated affine processes, a class of stochastic processes that emerge from affine processes by allowing some of their coefficients to be a function of an exogenous Markov process. Our proposed extension preserves the tractability of affine processes and allows for richer models in various financial applications. We prove existence of Markov-modulated affine processes via a martingale problem approach. Our framework unifies and generalizes a number of established models. On top of that, we introduce novel models, which are capable of capturing empirical features of financial data that are not explainable by means of standard affine processes, but are at the same time easy to calibrate.

Past Talks / Summer Term 2020

Regular Time & Location:
Thursdays, 16:15-(max.)17:45, seminar room DC rot 07 (TU Wien),
Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, red section, 7th floor.

Further talks have been cancelled due to the COVID-19 pandemic!

2020-02-11:
Elisa Alòs (Universitat Pompeu Fabra, Barcelona, ES)
Discretization errors in variance swaps

Abstract: We study an Edgeworth-type refinement of the central limit theorem for the discretization error of Ito integrals. Toward this end, we introduce a new approach, based on the anticipating Ito formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula. As an application, we study the difference between continuously and discretely monitored variance swap payoffs under stochastic volatility models.

Past Talks / Winter Term 2019

Regular Time & Location:
Thursdays, 16:30-(max.)18:00, lecture hall HS11, (Univ. Vienna),
Oskar-Morgenstern-Platz 1, 1090 Wien, 2nd floor.

2020-01-30:
Eduardo Abi Jaber (Université Paris 1 Panthéon-Sorbonne, FR)
Linear-Quadratic control of stochastic Volterra equations

Abstract: We treat Linear-Quadratic control problems for a class of stochastic Volterra equations of convolution type. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than 1=2 as a special case. We prove that the value function is of linear quadratic form with a linear optimal feedback control, depending on non-standard infinite dimensional Riccati equations, for which we provide generic existence and uniqueness results. Furthermore, we show that the stochastic Volterra optimization problem can be approximated by conventional finite dimensional Markovian Linear Quadratic problems, which is of crucial importance for numerical implementation.
Joint work with Enzo Miller and Huyên Pham.

2020-01-16:
Thorsten Rheinländer (TU Wien)
On pathwise stochastic integration

Abstract: We will review an idea due to K. Bichteler who uses a space-time grid to construct the Ito-Integral via a Riemann sum sampled at hitting times (and then stopping times) of the space grid. This concept has been greatly generalized, and has applications in optimal control, deep learning, as well as certain SPDE's.
In the second part, we will review a certain circle of ideas originating from the 'Indian School' based at that time in Strasbourg, where the Ito formula has been derived via the Tanaka formula (normally it is vice-verse), and taken to a countable Hilbertian space structure on distribution space (which is similar to Colombeau algebras).
This will be illustrated with countable, but numeruous applications, and some easy-to-formulate but yet surprisingly unresolved questions.
Everything is based on countless inspiring discussions with Friedrich Hubalek and Paul Eisenberg.

2020-01-09:
Wei Xu (HU Berlin, DE)
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 tigger endogenous jump cascades in asset returns and stock price volatility.
This talk is based on a recent joint work with Prof. Ulrich Horst (HU-Berlin).

2019-12-12:
Kaitong Hu (Ecole Polytechnique, FR)
Mean-field Langevin dynamics and its Applications in Deep Learning

Abstract: We present a probabilistic analysis of the long-time behaviour of the non-local, diffusive system, namely the Mean-field Langevin dynamics. Our goal is to provide a theoretical underpinning for the convergence of stochastic gradient type of algorithms widely used for non-convex learning tasks in deep learning. We first show that the corresponding optimization problem can be lifted to infinite-dimensional measures space and by doing so, the energy function has a unique minimiser which can be characterized by a first-order condition. We then show that the marginal laws induced by the Mean-field Langevin dynamics converges exponentially to the stationary distribution which is exactly the minimiser of the energy functional. In particular, the mean-field Langevin system has a gradient flow structure in the convex case, e.g. two-layer overparameterized neural network. More generally speaking, the mean-field Langevin system can also be viewed as a feasible continuous-time numerical algorithm for computing optimal control in high dimensional problems.

2019-12-05:
Stefan Gerhold (TU Wien)
Dynamic trading under integer constraints

Abstract: We first review results on arbitrage theory for some notions of "simple" strategies, which do not allow continuous portfolio rebalancing by arbitrary amounts. Then, the focus of the talk is on trading under integer constraints, that is, we assume that the offered goods or shares are traded in integer quantities instead of the usual real quantity assumption. For finite probability spaces and rational asset prices this has little effect on the core of the theory of no-arbitrage pricing. For price processes not restricted to the rational numbers, a novel theory of integer arbitrage free pricing and hedging emerges. We establish an FTAP, involving a set of absolutely continuous martingale measures satisfying an additional property. The set of prices of a contingent claim is not necessarily an interval, but is either empty or dense in an interval. We also discuss superhedging with integral portfolios.
Joint work with Paul Eisenberg.

2019-11-19
Paul Eisenberg (University of Liverpool, UK)
Modelling energy futures markets and finite dimensional approximations

Abstract: In this talk we discuss mathematical modelling of energy futures and briefly discuss some modelling decissions. Then we turn our attention to finite dimensional approximation of these models. From an economical perspective one should reject any model which allows for arbitrage. For this reason we deal with the question of finding an approximation of arbitrage-free finite dimensional models to a given one. The main problem is that finite-dimensionality enforces an invariance condition and the no-arbitrage condition enforces that a certain SPDE is solved under an equivalent measure. These together implies that the approximation models must be realised via so-called finite dimensional realisations of some SPDE. The problem is to ensure that the amount of these finite dimensional realisation models is rich enough to achieve an approximation to a given potentially infinite-dimensional model.

2019-11-19
Christian Schmeiser (Uni Wien)
Can entropy dissipation for Markov processes by time reversal be generalized to nonlinear evolution problems?

Abstract: The formal derivation of the dissipation equation for the relative entropy of a pair of Markov processes with the same Generator will be explained. Examples for an analogous structure for nonlinear evolution problems for probability densities will be discussed.

2019-10-17:
Mathias Beiglböck (University of Vienna)
An introduction to weak adapted topologies

Abstract: Several researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. We find that all of these seemingly independent approaches define the same topology in finite discrete time.
Moreover, we explain how optimal transport theory can be used obtain a compatible metric that is both tractable and highly relevant for a number of questions in mathematical finance.

2019-10-10:
Benjamin Jourdain (CERMICS, FR)
Differentiability of the squared quadratic Wasserstein distance

Abstract: Any optimal coupling for the quadratic Wasserstein distance between two probability measures with finite second order moments is the composition of a martingale coupling with an optimal transport map. We check the existence of optimal couplings in which this map gives the unique optimal coupling between the two probability measures for which it is optimal. Next, we prove that the squared quadratic Wassertein distance is differentiable with respect to one of its arguments iff there is a unique optimal coupling between this argument and the other one and this coupling is given by a map.

Past Talks / Summer Term 2019

Regular Time & Location:
Thursdays, 16:15-(max.)17:45, seminar room DC rot 07 (TU Wien),
Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, red section, 7th floor.

2019-04-18, 15:45, ESI:
Josef Teichmann (ETH Zurich, CH)
Machine Learning in Finance

Abstract: We consider certain learning tasks, which appear, e.g., in mathematical finance, from the point of view of controlled differential equations. By means of hypo-ellipticity results, certain universal expansions and corresponding transport equations, we shed some new light on generic learning algorithms and their amazing efficiency.
(Joint work with Christa Cuchiero and Martin Larsson.)

2019-04-04:
Hanna Wutte (ETH Zurich, CH)
Randomized shallow neural networks and their use in understanding gradient descent

Abstract: Today, various forms of neural networks are trained to perform approximation tasks in many fields (including Mathematical Finance). However, it has been questioned how much training really matters, in the sense that randomly choosing subsets of the networks weights and training only a few leads to an almost equally good performance. This motivates us to analyse properties of the optimizers found by the gradient descent algorithm frequently employed to perform the training task. In particular, we consider (shallow) neural networks in which weights are chosen randomly and only the last layer is trained. We believe, that the resulting optimizer converges to the smooth spline interpolation of the training data as the number of hidden nodes tends to infinity. This might give valuable insight on the properties of the solutions obtained using gradient descent methods in general settings.

Past Talks / Winter Term 2018/19

Regular Time & Location:
Thursdays, 16:30-(max.)18:00, lecture hall HS13 (Univ. Vienna),
Oskar-Morgenstern-Platz 1, 1090 Wien, 2nd floor.

2019-01-24:
Nathael Gozlan
Optimal transport for barycentric transport costs

Abstract: We will present a variant of the optimal transport problem where elementary mass transports are penalized through their barycenters. After recalling some general duality results obtained in collaboration with P-M Samson, C. Roberto, Y. Shu and P. Tetali, we will present a recent result with N. Juillet describing optimal transport plans for the quadratic barycentric cost. If time permits, we will also recall how these transport costs are naturally connected to concentration of measure phenomenon and functional inequalities for convex functions.

2019-01-17:
Maike Klein (FAM @ TU Wien)
Optimal Stopping Problems with Expectation Constraints

Abstract: We consider the problem of optimally stopping a continuous-time process with a stopping time satisfying an expectation constraint.
By introducing a new state variable, we show that one can transform the problem into an unconstrained control problem. Moreover, we characterize the value function in terms of the dynamic programming equation, which turns out to be an elliptic, fully non-linear partial differential equation of second order.
In a second approach we use results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures.
This talk is based on a joint work with Stefan Ankirchner, Nabil Kazi-Tani and Thomas Kruse.

2019-01-10:
Christian Bayer (WIAS)
A regularity structure for rough volatility

Abstract: A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extreme skews that were thought to be outside the scope of stochastic volatility. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In this paper we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provides a new and powerful tool to analyze rough volatility models.
(Joint work with Peter K. Friz, Paul Gassiat, Jörg Martin, and Benjamin Stemper.)

2018-12-13:
Oleg I. Klesov (National Technical University of Ukraine)
Law of the iterated logarithm for inverse subordinators and some applications in risk models

Abstract: The law of the iterated logarithm is proved for inverse subordinators. The limsup is nonrandom in contrast to what is claimed by some other authors. An application to some risk models is discussed.

2018-12-06:
Halil Mete Soner (ETH Zürich, CH)
Pricing-hedging Duality

Abstract: The convex duality between the super-hedging functional and the pricing operators is a central result in mathematical finance. It is also closely related to the fundamental theorem of asset pricing. Often both results are proved simultaneously as was the case in the seminal paper of Delbaen and Schachermayer. In this talk, we consider a rather general class of financial markets and investigate the validity of the duality. In particular, we present the recent results for model independent duality on the Skorokhod space. The talk will be based on several joint papers with Yan Dolinsky from Hebrew University, Patrick Cheridito and Matti Kiiski from ETH and David Proemel from Oxford.

2018-11-29:
Andrey Pilipenko (National Technical University of Ukraine)
Functional limit theorems for perturbed random walks

Abstract: We study a random walk over the integer points of the real line. Jumps of the random walk outside the membrane (a fixed "locally perturbing set") are i.i.d., have zero mean and finite variance, whereas jumps from the membrane have other distributions which may be different for different points of the membrane. The invariance principle is obtained under standard scaling of time and space. In particular, if jumps from the membrane have finite means, then the limit process turns out to be a skew Brownian motion.

2018-11-22:
Thilo Meyer-Brandis (LMU Munich, DE)
Contagion and Systemic Risk in Heterogeneous Financial Networks

Abstract: One of the most defining features of modern financial networks is their inherent complex and intertwined structure. In particular the often observed core-periphery structure plays a prominent role. Here we study and quantify the impact that the complexity of networks has on contagion effects and system stability, and our focus is on the channel of default contagion that describes the spread of initial distress via direct balance sheet exposures. We present a general approach describing the financial network by a random graph, where we distinguish vertices (institutions) of different types – for example core/periphery – and let edge proba- bilities and weights (exposures) depend on the types of both the receiving and the sending vertex. Our main result allows to compute explicitly the systemic damage caused by some initial local shock event, and we derive a complete characterization of resilient respectively non-resilient financial systems. Due to the random graphs approach these results bear a considerable robustness to local uncertainties and small changes of the network structure over time. In particular, it is possible to condense the precise micro-structure of the network to macroscopic statistics. Applications of our theory demonstrate that indeed the features captured by our model can have significant impact on system stability; we derive resilience conditions for the global network based on subnetwork conditions only.

2018-10-30, Tuesday (TU Wien, Freihaus, Sem.R. DA grün 06A):
Zivorad Tomovski (University in Skopje, MK)
Recent advanced of fractional calculus operators

Abstract: We present a generalization of Hilfer derivatives in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Furthermore, we show some applications of these generalized Hilfer–Prabhakar derivatives in classical equations of mathematical physics such as the heat and the free electron laser equations, and in difference–differential equations governing the dynamics of generalized renewal stochastic processes, like fractional Poisson Processes etc.

2018-10-25:
Dominykas Norgilas (University of Warwick, UK)
The left-curtain martingale coupling and the American put in the presence of atoms

Abstract: In a two-period setting we derive the model-independent upper bound of the American put option. The model associated with the highest price of the American put is the extended left-curtain martingale coupling. Moreover we derive the cheapest superhedge.

2018-10-18:
Johannes Wiesel (University of Oxford, UK)
Statistical estimation of superhedging prices

Abstract: In this talk I discuss statistical estimation of superhedging prices using historical stock returns in a frictionless market with d traded assets. I first introduce a simple plugin estimator based on empirical measures and show it is consistent but lacks suitable robustness. This issue is then addressed by improved estimators which use a larger set of martingale measures defined through a tradeoff between the radius of Wasserstein balls around the empirical measure and the allowed norm of martingale densities. I also give results regarding the convergence of superhedging strategies and different hedging criteria.

Past Talks / Summer Term 2018

Regular Time & Location:
Thursdays, 16:30-(max.)18:00, seminar room DC rot 07 (TU Wien),
Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, red section, 7th floor.

2018-07-05:
Khaled Bahlali (University of Toulon, FR)
Unbounded Quadratic BSDEs Existence by domination

Abstract: We introduce a domination argument which roughly expresses that : if we can dominate the parameters (terminal value and driving coefficient) of a Quadratic BSDE from above and from below by those of two BSDEs having an ordered solution, then also our original Quadratic BSDE has a solution. This result will be presented in a general setting, that is without integrability of the solutions. No integrability condition on none of the terminal data of the three involved BSDEs is needed. Neither continuity nor constraints on the growth are required to the dominating coefficients. Next, we consider a large class of quadratic BSDE which which englobe the classical ones. The domination argument allows us to show that the solvability of this class of BSDEs can be reduced to the solvability of a simple BSDE whose generator is zero. This allows to deduce the condition we should impose to the terminal value. The method we propose neither uses a priori estimates nor approximations.

2018-06-28:
Archil Gulisashvili (Ohio University, US)
Volterra type fractional stochastic volatility models. Small-noise and small-time asymptotic formulas for the implied volatility

Abstract: We study fractional stochastic volatility models for the asset price, in which the volatility process is a positive continuous function of a continuous fractional stochastic process. One of the main results discussed in the present talk is a generalization of small-noise and small-time large deviation principles for the log-price process due to M. Forde and S. Zhang. We assume that the function in the definition of the volatility process satisfies a relatively mild condition expressed in terms of its local modulus of continuity, while the fractional process is a Volterra type Gaussian process. The assumptions used by Forde and Zhang are more restrictive. As an application, we obtain asymptotic formulas for binary options, call and put pricing functions, and the implied volatility in certain mixed regimes.

2018-06-21:
Oleg Szehr (University of Vienna)
Supervised learning and optimal transport

Abstract: Recently methods of optimal transport have found wide applications in machine learning.
In this talk we review some basics of optimal transportation and show how they can be applied, particularly in view of supervised learning.
We discuss the duality method of optimization and show a representer theorem for a Wasserstein loss.
The talk concludes with a review of Otto's calculus on the Wasserstein-2 manifold of probability measures and by highlighting a link to stochastic processes.

2018-06-14:
Stephan Eckstein (University of Konstanz, DE)
Superhedging and distributionally robust optimization with Neural Networks

Abstract: The focus of the talk is to numerically solve superhedging problems related to optimal transport, distributionally robust optimization problems over Wasserstein balls, as well as certain combinations of the two. The main solution approach uses neural networks, which is described and analyzed in detail and showcased with several examples. Further, the approach is compared with linear programming methods, whereby strengths and weaknesses are identified and discussed.

2018-06-07:
Lorenzo Mercuri (University of Milan, IT)
On properties of the Mixed Tempered Stable distribution and its Multivariate Version

Abstract: In this talk, a review of the univariate Mixed Tempered Stable is given and some new results on the asymptotic tail behaviour are derived. The multivariate version of the Mixed Tempered Stable, which is a generalisation of the Normal Variance Mean Mixtures, is discussed. Characteristics of this distribution, its capacity in fitting tails and in capturing dependence structure between components are investigated.
Joint work with Asmerilda Hitaj and Edit Rroji.

2018-05-24:
Matteo Burzoni (ETH Zürich, CH)
Arbitrage, Hedging and all that: lessons learned from the absence of a probability

Abstract: In this talk we discuss how the no arbitrage theory is strongly connected to the so-called Martingale Selection Problem (MSP). Given a collection of random sets V=(Vt) the MSP consists in finding a stochastic process S taking values in V and such that S is a martingale under a measure Q. We obtain robust (pointwise) versions of the Fundamental Theorem of Asset Pricing in examples spanning frictionless, proportional transaction cost and illiquidity markets with possible trading constraints. In a frictionless framework, we also discuss pointwise hedging versus quasi-sure hedging and under which conditions they coincide.

2018-05-09 (Wednesday, 14:00, seminar room 4, Univ. Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, 1st floor):
Tahir Choulli (University of Alberta, CA)
Martingales classification and decomposition for progressively enlarged filtrations with applications to finance and insurance

Abstract: Our starting point is a filtration F, satisfying the usual conditions, which is defined on a probability space and models the flow of information of an initial market model. To this stochastic basis, we consider a general random time T that represents a default time of a firm, a death time of an insured/agent, and/or an occurrence time of an event that might affect the initial market somehow. Given that the death time can not be seen before its occurrence, the progressively enlargement of F with T, denoted hereafter by G, seems tailored-made for modelling the flow of information that incorporates both F and the information about T as it occurs. Thus, our principal goal lies in understanding how the class of martingales with respect to G and that live on [0,T] can be described. In other words, we would like to answer the following questions: Besides the G-martingales intrinsic to the F-martingales stopped at T (given for instance via the Doob-Meyer decomposition of F-martingales stopped at T), are there other new type(s) of G-martingales that can not be explained through the flow F? How many are there such additional G-martingales? How can one classify all G-martingales depending on the different uncertainties in T? Is there any G-martingales basis that allows us to represent any G-martingale with respect to it?
The applications of these are numerous and multifold. For the moment, we focus on the applications in credit risk (the case of default) and life insurance (mortality an longevity risk). The first applications of our classification and decomposition can be summarized as follows: 1) Decompose and classify risks under G (risk coming from both default/mortality and initial market model F) and hence to well and efficiently manage it, 2) Describe the dynamics and the stochastic structures for the default/death derivatives, which is highly important for the mortality/longevity securitization. 3) Completely and fully describe the set of all deflators (local martingale densities/measures) under G. 4) Describe explicitly and completely, using the flow F only, the log-optimal portfolio and the numeraire portfolio under G.
This talk is based on joints works with Catherine Daveloose, Michele Vanmaele (Ghent University), and Sina Yansori (UofA).

2018-04-26:
Claudio Fontana (Paris Diderot University (Paris VII), FR)
The Value of Informational Arbitrage

Abstract: In the context of a general semimartingale model of a complete market, we aim at answering the following question: How much is an investor willing to pay for learning some inside information that allows to achieve arbitrage? If such a value exists, we call it the value of informational arbitrage.In particular, we are interested in the case where the inside information yields arbitrage opportunities but not unbounded profits with bounded risk. In the spirit of Amendinger et al. (2003), we provide a general answer to the above question by relying on an indifference valuation approach. To this effect, we establish some new results on models with inside information and study optimal investment-consumption problems in the presence of initial information and arbitrage, also allowing for the possibility of leveraged positions. We characterize when the value of informational arbitrage is universal, in the sense that it does not depend on the preference structure. Our results are illustrated with several explicit examples. Based on joint work with H.N. Chau and A. Cosso.

2018-04-12:
Christoph Kühn (Johann Wolfgang Goethe-Universität, DE)
How local in time is the no-arbitrage property under capital gains taxes?

Abstract: In most countries, trading gains have to be taxed. Tax systems are usually realization based, i.e., gains on assets are taxed when assets are sold and not when gains actually accrue. Thus, for positive interest rates, there is an incentive to realize losses immediately and to defer the realization of profits.
In this talk, we discuss the arbitrage theory for models with taxes. In frictionless financial markets, no-arbitrage is a local property in time. This means, a discrete time model is arbitrage-free if and only if there does not exist a one-period-arbitrage. With capital gains taxes, this equivalence fails. For a model with a linear tax and one non-shortable risky stock, we introduce the concept of robust local no-arbitrage (RLNA) as the weakest local condition which guarantees dynamic no-arbitrage. Under a sharp dichotomy condition, we prove (RLNA). Since no-one-period-arbitrage is necessary for no-arbitrage, the latter is sandwiched between two local conditions, which allows us to estimate its non-locality.
Furthermore, we construct a stock price process such that two long positions in the same stock hedge each other. This puzzling phenomenon is used to show that no-arbitrage alone does not imply the existence of an equivalent separating measure if the probability space is infinite. Finally, we show how the tax payment stream can be constructed in continuous time for all adapted trading strategies with caglad paths, that means beyond strategies of finite variation.

2018-03-15:
Matthias Meiners (University of Innsbruck)
Convergence of Biggins' martingales at complex parameters

Abstract: In several models of applied probability such as Pólya urns, search trees, and fragmentation processes, the limiting behavior of quantities of interest is described by distributions on the complex plane that solve smoothing equations.
In my talk, I will consider such complex smoothing equations and explain the relation to convergence of Biggins' martingales in the branching random walk at complex parameters. For those martingales, Biggins (1992) proved local uniform convergence at complex parameters within a certain open domain. I will explain how critical smoothing equations are related to martingale convergence on the boundary of this domain. If time permits, I will also address rates of convergence.
The talk is based on joint work with Alexander Iksanov (Kyiv), Konrad Kolesko (Innsbruck) and Sebastian Mentemeier (Dortmund).

2018-03-08:
Philipp Grohs (University of Vienna)
Approximation theory, Numerical Analysis and Deep Learning

Abstract: The development of new classification and regression algorithms based on deep neural networks – coined “Deep Learning” – revolutionized the area of artificial intelligence, machine learning, and data analysis. More recently, these methods have been applied to the numerical solution of high dimensional partial differential equations with great success.
This talk will start with a brief introduction to machine learning and deep learning. Then we will show that the problem of numerically solving a large class of (high-dimensional) PDEs (such as linear Black-Scholes or diffusion equations) can be cast into a classical supervised learning problem which can then be solved by deep learning methods. Simulations suggest that the resulting algorithms are vastly superior to classical methods such as finite element methods, finite difference methods, spectral methods, or sparse tensor methods. In particular we empirically observe that these algorithms are capable of breaking the curse of dimensionality. In the last part of the talk we will present theoretical results which confirm this observation.
This is joint work with Dennis Elbrächter (University of Vienna), Arnulf Jentzen (ETH Zürich) and Christoph Schwab (ETH Zürich).

Past Talks / Winter Term 2017/18

Regular Time & Location:
Thursdays, 16:30-(max.)18:00, seminar room SR11 (Univ. Vienna),
Oskar-Morgenstern-Platz 1, 1090 Wien, 2nd floor.

2018-02-01:
Ludovic Tangpi (University of Vienna)
Representations of some non-exponential functions of Wiener process and applications

Abstract: On a filtered probability space driven by a Brownian motion, we present variational representation results for some non-exponential functions of Brownian motion. These representations have a variety of applications. In this talk we will mostly dwell on consequences in terms of large deviations. The results make ample use of the theory of backward stochastic differential equations.

2018-01-25:
Omar El-Euch (Université Pierre et Marie Curie - Paris 6, FR)
Multi-factor approximation of rough volatility models

Abstract: Rough volatility models are very appealing because of their remarkable fit of both historical and implied volatilities. However due to the non-Markovian and non-semimartingale nature of the volatility process, there is no obvious way to simulate efficiently such models, which makes risk management of derivatives an intricate task. In this paper, we design tractable multi-factor stochastic volatility models approximating rough volatility models and enjoying a Markovian structure. Furthermore, we apply our procedure to the specific case of the rough Heston model. This enables us to derive a numerical method for solving fractional Riccati equations appearing in the characteristic function of the log-price in this setting.

2017-12-20:
Alexander Cox (University of Bath, UK)
Measure-value martingales (or martingale measures), and financial applications

Abstract: We give a gentle introduction to measure-valued martingales, and outline some applications in mathematical finance, and particularly to robust pricing and optimal Skorokhod embedding problems. (Based loosely on joint work with Bayraktar, Beiglboek, Huesmann, Kallblad & Stoev).

2017-12-14:
Ilya Molchanov (University of Bern, CH)
The semigroup of metric measure spaces and its infinitely divisible measure

Abstract: The family of metric measure spaces can be endowed with the semigroup operation being the Cartesian product. The aim of this talk is to arrive at the generalisation of the fundamental theorem of arithmetic for metric measure spaces that provides a unique decomposition of a general space into prime factors. These results are complementary to several partial results available for metric spaces (like de Rham's theorem on decomposition of manifolds). Finally, the infinitely divisible and stable laws on the semigroup of metric measure spaces are characterised.
Joint work with S.N. Evans (Berkeley).

2017-12-07:
Tongseok Lim (University of Oxford, UK)
Dual attainment problem for Martingale Optimal Transport

Abstract: The Martingale Optimal Transport (MOT) problem is a variant of the Optimal Transport problem where the underlying process (X,Y) is assumed to be a martingale. MOT problem was inspired by mathematical finance community as it is closely connected to the Model-Independent option pricing. In particular, the dual problem of MOT can be interpreted as finding optimal super / subhedging strategies for an option, thus the existence of dual optimisers, i.e. the dual attainment problem, is important.

When the underlying asset (X,Y) is real-valued, the dual attainment problem has been thoroughly studied by Beiglboeck-Juillet and Beiglboeck-Nutz-Touzi. However, little is known when (X,Y) is vector valued, meaning that the option c(X,Y) depends on many assets, which is often the case in financial markets.

At the heart of the dual attainment problem is a simple-looking control problem of convex functions which satisfy a certain integral bound. We shall focus on this issue, and other issues for dual attainment problem as time permits.

2017-11-30:
Julio Backhoff (TU Wien)
Martingale Benamou-Brenier: a probabilistic perspective

Abstract: In classical optimal transport, the contributions by Benamou, Brenier and McCann (among others) regarding the time-dependent version of the problem, have had a lasting impact in the field and led to many applications. It is remarkable that this is achieved even if in continuoustime classical optimal transport mass/particles only travel in straight lines. This fails to happen when we consider (continuous-time) martingale optimal transport. In this talk we discuss the existence of a martingale analogue to McCann's interpolation and the Benamou-Brenier formula from a probabilistic point of view. This remarkable martingale is characterized by very natural optimality and geometric properties, leading us to say that it provides a canonical martingale way to connect two measures in convex order.

2017-11-09:
Walter Schachermayer (University of Vienna)
The amazing power of dimensional analysis: Quantifying market impact

Abstract: A basic problem when trading in financial markets is to analyze the prize movement caused by placing an order. Clearly we expect - ceteris paribus - that placing an order will move the price to the disadvantage of the agent. This price movement is called market impact.

Following Kyle and Obizhaeva we apply dimensional analysis - a line of arguments wellknown in classical physics - to analyze to which extent the square root law applies. This universal law claims that the market impact is proportional to the square root of the size of the order. The mathematical tools of this analysis reside on elementary linear algebra.

Joint work with Mathias Pohl, Alexander Ristig and Ludovic Tangpi.

2017-10-19:
Beatrice Acciaio (LSE, UK)
Dynamic Cournot-Nash equilibrium via non-anticipative optimal transport.

Abstract: I will consider Cournot-Nash equilibrium problems in a dynamic setting, where each agent faces a cost that is composed by an idiosyncratic part depending on its own type and action, and a mean-field term depending on the actions distribution over all agents. The tools used in order to get existence and uniqueness come from dynamic optimal transportation of non-anticipative nature.

The talk is based on an ongoing project with Julio Backhoff Veraguas.

2017-10-12:
Gaoyue Guo (University of Oxford, UK)
Numerical computation of martingale optimal transportation

Abstract: We provide a numerical method for solving the martingale optimal transport problem. The scheme considers the approximation of marginal distributions, through which the primal problem could be approximated by a LP problem with the relaxation of martingale constraint.

2017-10-05:
Oleg I. Klesov and Elena Timoshenko (National Technical University of Ukraine)
Generalized renewal processes with applications to stochastic differential equations

Abstract: The first part of the talk is a short introduction to the theory of dual objects by an example of a random walk and the corresponding renewal process. Some examples in probability theory, number theory etc. are also discussed. Asymptotic properties of dual objects are described and a link to the theory of pseudo-regularly varying functions is exhibited. Introducing the so-called asymptotically quasi-inverse functions, we show how asymptotic behavior of various functionals of stochastic processes (like first exit time, last exit time, sojourn time etc.) can be derived in a universal way from the corresponding properties of processes themselves.

The second part of talk contains an interesting application of the general results to studying the almost sure asymptotic behavior of solutions of stochastic differential equations. Some sufficient conditions are obtained, under which the exact order of growth of a solution of a stochastic differential equation is determined by that of a solution of the corresponding ordinary differential equation.

Past Talks / Summer Term 2017

Regular Time & Location:
Thursdays, 16:30-18:00, seminar room DC rot 07 (TU Wien),
Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, red section, 7th floor.

2017-07-19, Wed., 11:00-12:00, seminar room DB gelb 04 ("Freihaus" building, yellow section, 4th floor):
Peter Friz (TU Berlin, DE)
General semimartingales and rough paths

Abstract: We revisit some classical results of Kurtz, Protter, and Pardoux concerning stability of stochastic differential equations and put them in perspective with latest results on (cadlag) rough paths.
Joint work with A. Shekhar, I. Chevyrev and H. Zhang.
The talk is intended to be a rough paths teaser for non-specialists.

2017-06-29, 17:15:
Ramin Okhrati (University of Southampton, GB)
Hedging of defaultable securities under partial asset observation

Abstract: We investigate a hedging problem of certain defaultable securities through local risk minimization approach assuming partial accounting data. More precisely, in addition to the risk of default, we suppose that investors face lagged data, i.e. they receive information with some delay. In our analysis, different levels of information are distinguished including full market, company’s management, and investors information. We obtain semi-explicit solutions to locally risk minimizing strategies from investors perspective where the results are presented according to the solutions of partial differential equations. In obtaining the main results of this paper, minimal equivalent local martingale measures are not used; instead, we apply a filtration expansion theorem that determines the canonical decomposition of martingales in an investors enlarged filtration.

2017-06-29, 16:30:
Elisa Alos (Universitat Pompeu Fabra, Barcelona, ES)
On the relationship between implied volatilities and volatility swaps: a Malliavin calculus approach

Abstract: This work is devoted to studying the difference between the fair strike of a volatility swap and the at-the-money implied volatility (ATMI) of a European call option. It is well-known that the difference between these two quantities converges to zero as the time to maturity decreases. We make use of a Malliavin calculus approach to derive an exact expression for this difference. This representation allows us to establish that the order of the convergence is different in the correlated and in the uncorrelated case, and that it depends on the behavior of the Malliavin derivative of the volatility process. In particular, we will see that for volatilities driven by a fractional Brownian motion, this order depends on the corresponding Hurst parameter H. Moreover, in the case H ≥ 1/2, we develop a model-free approximation formula for the volatility swap, in terms of the ATMI and its skew.
Joint work with Kenichiro Shiraya, University of Tokyo.

2017-06-22:
Châu Ngọc Huy (Alfréd Rényi Institute of Mathematics, HU)
On fixed gain recursive estimators with discontinuity in the parameters.

Abstract: In this talk, we estimate the tracking error of a fixed gain stochastic approximation scheme. The underlying process is not assumed Markovian, a mixing condition is required instead. Furthermore, the updating function may be discontinuous in the parameter.

2017-06-20, 16:30 in seminar room DA grün 06A (Freihaus, green section, 6th floor)
Alexander Steinicke (University of Graz, AT)
Backward Stochastic Differential Equations and Applications

Abstract: Stochastic differential equations (SDEs) are useful for modeling a tremendous amount of phenomena, where random effects over time are involved. Following the usual procedure, we start with an initial condition at time zero and obtain at time T a random variable X(T), the solution of our SDE. The situation is different if one looks at the situation backward in time: If we start with a given random value at time T, are we able to find a deterministic value X(0) by following the dynamics of a stochastic differential equation, backward in time? This type of problem is called a backward stochastic differential equation (BSDE) and has been introduced in 1971 by Bismut in the context of stochastic control. From then on, BSDEs became more and more important for various applications and their systematic study began in the early 90's. In this talk I will introduce standard BSDEs and outline how they appear e.g. in pricing of contingent claims, stochastic control beyond Markovianity, Feynman-Kac representation for PDEs or utility maximization. Moreover, I will present treatment of BSDEs in simple cases and give an overview about my current field of interest within BSDE-theory.

2017-06-08:
Alexander Drewitz (Universität zu Köln, DE)
Sign clusters of the Gaussian free field percolate on ℤ^d, d ≥ 3

Abstract: We consider level set percolation of the Gaussian free field in the Euclidean lattice in dimensions larger than or equal to three. It had previously been shown by Bricmont, Lebowitz, and Maes that the critical level is non-negative in any dimension and finite in dimension three. Rodriguez and Sznitman have extended this result to showing that it is finite in all dimensions, and positive in all large large enough dimensions.
We show that the critical parameter is positive in any dimension larger than or equal to three. In particular, this entails the percolation of sign clusters of the Gaussian free field.
This is based on joint work with A. Prévost (Köln) and P.-F. Rodriguez (Los Angeles).

2017-06-06, 16:30 in seminar room DA grün 06A (Freihaus, green section, 6th floor)
Marcel Nutz (Columbia University, USA)
A Mean-Field Competition

Abstract: We introduce a mean field game with rank-based reward: competing agents optimize their effort to achieve a goal, are ranked according to their completion time, and paid a reward based on their relative rank. On the one hand, we propose a tractable Poissonian model in which we can characterize the optimal efforts for a given reward scheme. On the other hand, we study the principal agent problem of designing an optimal reward scheme. A surprising, explicit solution is found to minimize the time until a given fraction of the population has reached the goal. (Work-in-progress with Yuchong Zhang)

2017-06-01, 16:30:
Giovanni Conforti (Université Lille 1, FR)
The bridges of the Langevin dynamics

Abstract: The Langevin dynamics is a basic model for a random system converging to an equilibrium state. Such convergence can be very precisely quantified when the underlying potential is convex. In this talk we look at the bridges of the Langevin dynamics and present a detailed quantitative study of their dynamics, including quantitative bounds for the distance from the invariant measure. The results rely on a new coupling between bridges with different end points, and they show how the key quantity which regulates the bridge dynamics is no longer the convexity of the potential, but rather its reciprocal characteristics.

2017-06-01, 17:15:
Daniel Lacker (Brown University, USA)
Mean field games of timing and models for bank runs

Abstract: The goal of the paper is to introduce a set of problems which we call mean field games of timing. We motivate the formulation by a dynamic model of bank run in a continuous-time setting. We briefly review the economic and game theoretic contributions at the root of our effort, and we develop a mathematical theory for stochastic games where the strategic decisions of the players are merely choices of times at which they leave the game, and the interaction between the strategic players is of a mean field nature. Based on joint work with Rene Carmona and Francois Delarue.

2017-05-18:
Ting-Kam Leonard Wong (University of Southern California, USA)
From optimal rebalancing to information geometry

Abstract: What is the optimal frequency to rebalance a portfolio? For the class of functionally generated portfolios in stochastic portfolio theory, we show that the answer is given in terms of a "dualistic" Pythagorean theorem. Along the way, we establish fascinating connections with optimal transport and information geometry - the differential geometry of probability distributions. A key role is played by the concept of L-divergence which generalizes the diversification return (aka excess growth rate) of a portfolio. Our results extend the classical information geometry of Bregman divergence developed by Amari and others. This is joint work with Soumik Pal.

2017-05-11:
Ales Cerny (Cass Business School, UK)
Convex duality and Orlicz spaces in expected utility maximization

Abstract: In this talk we report further progress towards a complete theory of expected utility maximization with semimartingale price processes for arbitrary utility function. Without any technical assumptions, we establish a surprising Fenchel duality result on conjugate Orlicz spaces, offering a fresh perspective on the classical papers of Kramkov and Schachermayer (1999, 2003). The analysis points to an intriguing interplay between no-arbitrage conditions and classical convex optimization, and motivates study of the Fundamental Theorem of Asset Pricing (FTAP) for Orlicz tame strategies.
Joint work with Sara Biagini, LUISS, Rome.

2017-05-04:
Zehra Eksi (WU Wien, AT)
Portfolio optimization: a pure jump model with unobservable characteristics and linear market impact

Abstract: We consider an investor faced with the utility maximization problem in which the stock price process has pure-jump dynamics affected by an unobservable continuous-time finite-state Markov chain, the intensity of which can also be controlled by actions of the investor. Using the classical filtering theory, we reduce this problem with partial information to one with complete information and solve it for logarithmic and power utility functions and characterize the optimal portfolio strategies. In particular, we apply control theory for piecewise deterministic Markov processes (PDMP) to our problem and derive the optimality equation for the value function and characterize the value function as unique viscosity solution of the associated dynamic programming equation. Finally, we provide a toy example, where the unobservable state process is driven by a two-state Markov chain, and discuss how investor's ability to control the intensity of the state process affects the optimal portfolio strategies as well as the optimal wealth under both partial and complete information cases.
This is a joint work with Sühan Altay (TU Wien) and Katia Colaneri (University of Perugia).

2017-04-27:
Katia Colaneri (University of Perugia, IT)
Unit-linked life insurance policies: optimal hedging in partially observable market models

Abstract: In this paper we investigate the hedging problem of a unit-linked life insurance contract via the local risk-minimization approach, when the insurer has a restricted information on the market. In particular, we consider an endowment insurance contract whose final value depends on the trend of a stock market where premia are invested.
We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor that also influences the mortality rate of the policyholder.
To allow for mutual dependence between the financial and the insurance markets, we use the progressive enlargement of filtration approach. We characterize the optimal hedging strategy in terms of the integrand in the Galtchouk-Kunita-Watanabe decomposition of the insurance claim with respect to the minimal martingale measure and the available information flow, and find a relation with the corresponding hedging strategy under full information. Finally, we discuss applications in a Markovian setting via filtering.
This is a joint work with Claudia Ceci and Alessandra Cretarola.

2017-04-06:
Todor Bilarev (HU Berlin, DE)
Superhedging with transient impact

Abstract: In this talk, we will first discuss modeling issues in a market model with a single risky asset and a large trader whose actions have impact on the asset's price in a transient way, i.e. the impact from a trade is decreasing in time. We postulate the evolution of the asset price process in a multiplicative way (multiplicative market impact model) that guarantees positivity of prices. At first, the gains from trading can be uniquely defined for continuous strategies of finite variation. We extend the model to general (cadlag) trading by continuously extending the gains functional in a suitable (non-standard) topology on the space of strategies (the Skorokhod M1 topology in probability).
Having specified our model for a general class of trading strategies/controls, we consider the problem of pricing European options by superreplication that we formulate as a stochastic target problem. When initial and terminal impact are taken into account, a version of the (geometric) dynamic programming principle holds (in special coordinates) and thus the minimal superreplication price can be characterized as the (discontinuous) viscosity solution of a non-linear PDE where the transient nature plays a key role. Changing slightly the problem formulation by omitting the initial and terminal impact (covered options) leads to a very different in nature pricing pde where gamma constraints are needed.

2017-03-30:
Daniel Bartl (University of Konstanz, DE)
Pointwise time-consistent convex expectations

Abstract: We study conditional convex expectation in discrete time without a reference measure or the assumption that an essential supremum exists. It is shown that a certain pointwise continuity condition is equivalent to the validity of a dual representation in terms of linear conditional expectations over sigma-additive probabilities minus a penalty function which enjoys a certain measurability. Moreover, we prove that a family of convex expectations is time-consistency if and only if the respective penalty functions have an additive structure.

2017-03-16:
Hadrien De March (CMAP, École Polytechnique, FR)
Structure of martingale transport plans in general dimensions

Abstract: Martingale transport plans on the line are known from Beiglböck & Juillet to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in R^d, d>=1. Our decomposition is a partition of R^d consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well-defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.

2017-03-09:
Johannes Heiny (Aarhus University, DK)
Limit theorems for the largest eigenvalues of the sample covariance matrix of a heavy-tailed time series

Abstract: We study the joint distributional convergence of the largest eigenvalues of the sample covariance matrix of a p-dimensional heavy-tailed time series when p converges to infinity together with the sample size n. We generalize the growth rates of p existing in the literature. Assuming a regular variation condition with tail index alpha<4, we employ a large deviations approach to show that the extreme eigenvalues are essentially determined by the extreme order statistics from an array of iid random variables. The asymptotic behavior of the extreme eigenvalues is then derived routinely from classical extreme value theory. The resulting approximations are strikingly simple considering the high dimension of the problem at hand.
We develop a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the case where rows and columns of the data are linearly dependent. Based on the weak convergence of this point process we derive the limit laws of various functionals of the eigenvalues.
This talk is based on a joint work with Richard Davis and Thomas Mikosch.

2017-03-02:
Denis Parganlija (TU Wien, AT)
A New Physics-Based Approach to Studies of Financial Markets based on the Linear Sigma Model

Abstract: Five decades ago, large quantities of experimental data in nuclear physics instigated the construction of various models based on the assumed correlations in the observed data. These models have a well-defined structure in terms of mathematics: for example, they describe data distribution in terms of a probability distribution developed by mathematical physicists Breit and Wigner and they are constructed based on rigorous use of symmetries under certain unitary transformations. Unlike many models in financial mathematics, they are not stochastic per construction - but they are still highly successful in describing the (essentially stochastic) nuclear processes.
The models have a clear mathematical basis; the input of physics is only given once the matching of the degrees of freedom to the physical ones is performed. However, this is no condicio sine qua non: model parameters and observables may in principle also be matched to those present in financial mathematics, or even econometrics. The main example in my talk will then be to demonstrate to which extent a particular physics approach (the so-called Linear Sigma Model) can be used to parametrise line-shapes observed in stock exchanges and to forecast their development. This may provide a new tool for hedging of stock options and calculation of derivative prices.

Past Talks / Winter Term 2016/17

Regular Time & Location:
Thursdays, 16:30-18:00, seminar room SR09 (Univ. Vienna),
Oskar-Morgenstern-Platz 1, 1090 Wien, 2nd floor.

2017-01-26:
Dario Trevisan (University of Pisa, Italy)
A PDE approach to a 2-dimensional matching problem

Abstract: We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by S. Caracciolo et al. (Phys. Rev. E, 90, 012118, 2014) that linearizes the Monge-Ampère equation. Moreover, it provides a new approach to classical bounds due to Ajtai et al. (Combinatorica, 4, 1984). Joint work with L. Ambrosio and F. Stra.

2017-01-24, 10:30–11:30:
Wendelin Werner (ETH Zürich, Switzerland)
Random cracks in space

Abstract: We will describe in non-technical terms some old and new ideas about what basic natural random objects and fields one can define in a given space with some geometric structure, and what one can do with them. This will probably include various joint recent and ongoing work with Jason Miller, Scott Sheffield, Qian Wei and Titus Lupu.

2017-01-19:
Arnulf Jentzen (ETH Zurich, Switzerland)
Stochastic algorithms for the approximative pricing of financial derivatives

Abstract: In this talk we present a few recent results on approximation algorithms for forward stochastic differential equations (SDEs) and forward-backward stochastic differential equations (FBSDEs) that appear in models for the approximative pricing of financial derivatives. In particular, we review strong convergence results for Cox-Ingersoll-Ross (CIR) processes, high dimensional nonlinear FBSDEs, and high dimensional nonlinear parabolic partial differential equations (PDEs). CIR processes appear in interest rates models and in the Heston equity derivative pricing model as instantaneous variance processes (squared volatility processes). High dimensional nonlinear FBSDEs and high dimensional nonlinear PDEs, respectively, are frequently employed as models for the value function linking the price of the underlying to the price of the financial derivative in pricing models incorporating nonlinear effects such as the default risk of the issuer and/or the holder of the financial derivative. The talk is based on joint works with Weinan E (Beijing University & Princeton University), Mario Hefter (University of Kaiserslautern), Martin Hairer (University of Warwick), Martin Hutzenthaler (University of Duisburg-Essen), and Thomas Kruse (University of Duisburg-Essen).

2017-01-12:
Martin Huesmann (TU Wien, Austria / Univ. Bonn, Germany)
Transport cost estimates for random measures in dimension one

Abstract: Motivated by matching and allocation problems we introduce the optimal transport problem between two invariant random measures. Since this is a transport problem between two infinite measures the total transport cost will always be infinite. It turns that the proper replacement is the transport cost per unit volume; assuming that the transport cost per unit volume is finite existence and uniqueness of optimal invariant couplings can be established.
After reviewing the essential parts of this theory I will show that in dimension one there is a sharp threshold for the transport cost between the Lebesgue measure and an invariant random measure to be finite. More precisely, we show that the L^1 cost is always infinite (provided the random measure is sufficiently random) and we establish sharp and easily checkable conditions for the L^p cost to be finite for 0<p<1.
If time permits, we end with some challenging open problems.

2017-01-12, 17:30:
Piet Porkert (TU Wien)
Upper bounds for the Wasserstein and Kolmogorov distances between random sums and their weak limits via Stein's method

Abstract: We discuss upper bounds for the Wasserstein and Kolmogorov distances between Poisson mixture sums and their related normal variance mixture distributions. To this end we use a conditional version of Stein's equation and utilize techniques established in the theory of Stein's method for the normal distribution. A non-central limit theorem follows as a byproduct.

2016-12-06: Sem.R. DA grün 06A (TU Wien),
Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, green section, 6th floor
Kais Hamza (Monash University, Australia)
Alternative models in Finance

Abstract: The Black-Scholes formula has been derived under the assumption of constant volatility in stocks. In spite of evidence that this parameter is not constant, this formula is widely used by the markets. It is therefore natural to ask whether a model for stock price exists such that the Black-Scholes formula holds while the volatility is non-constant. In this talk I will review a number of results on the existence of alternative models in option pricing and beyond. This is joint work with Fima Klebaner, Olivia Mah and Jie Yen Fan.

2016-12-01:
Christian Bayer (WIAS)
Smoothing the payoff for efficient computation of basket option

Abstract: We consider the problem of pricing basket options in a multivariate Black-Scholes or Variance-Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high-dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse-grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster than Monte Carlo or Quasi Monte Carlo methods in dimensions up to 25.
(Joint work with Markus Siebenmorgen und Raul Tempone)

2016-11-24:
Ludovic Tangpi (Universität Wien)
A convex duality approach to the FTAP

Abstract: In this talk, we will present an approach to the fundamental theorem of asset pricing that is based on a convex dual representation result of F. Delbaen as opposed to separation theorems. We will then discuss several examples where our theorem applies.

This is based on a joint work with M. Kupper.

2016-11-17:
Mladen Savov (Bulgarian Academy of Sciences, BG)
Bernstein-gamma functions and exponential functionals of Lévy processes

Abstract: We introduce briefly the class of Bernstein-gamma functions and compute with their help the Mellin transform of the exponential functional of Lévy processes. Then investigating in the detail the Mellin transform we announce a number of new results for the exponential functional of Lévy processes including asymptotics, smoothness and factorizations of its law. We also discuss how these results relate to the existing body of literature and in what aspect they can be useful in a couple of areas where the exponential functional plays a significant role. We emphasize that a weaker version of this talk has been delivered at this seminar in October 2014.

2016-11-10:
Martin Larsson (ETH Zurich, Switzerland)
Conditional infimum and recovery of monotone processes

Abstract: Monotone processes, just like martingales, can often be reconstructed from their final values. Examples include the running maximum of supermartingales, of fractional Brownian motion, and more generally, running maxima and local times of sticky processes. An interesting corollary is that any positive local martingale can be reconstructed from its final value and its global maximum. These results are derived from a simple no-arbitrage principle for monotone processes on certain complete lattices, analogous to the fundamental theorem of asset pricing in mathematical finance. The framework of complete lattices is sufficiently general to handle also more exotic examples, such as the process of convex hulls of multidimensional diffusions, and the process of sites visited by a random walk. The notion of conditional infimum is at the center of all these results.

2016-11-03:
Rémi Peyre (Universität Wien)
Fractional Brownian motion, financial mathematics and stopping times

Abstract: Fractional Brownian motion (fBm) [whose definition I will recall in the talk] is a quite natural model for stochastic evolutions. In the context of financial mathematics, modelling an asset's price by (geometric) fBm is certainly irrelevant within the classical setting, for fBm is no semimartingale [cf. Delbaen & Schachermayer]; however, if we restrict the authorised trading strategies, or if we introduce transaction costs, it becomes a nontrivial natural question whether an asset's price may be modelled by fBm while satisfying the no-arbitrage condition.
This kind of questions lead to look at what may or may not happen to the trajectory of a fBm just after an arbitrarily chosen stopping time. Obviously the behaviour of the trajectory may be very different from what it is typically, but to which extent? For example, can one find a stopping time after which the fBm would have nonzero probability to go on upwards or downwards? If not, fBm would be said to have the two-way crossing property, which has been beautifully shown to imply no-arbitrage results.
In this talk I will present a work of mine in which I proved the two-way crossing property for fBm, by studying sharp properties of its behaviour after a stopping time, in particular at the order of the local law of the iterated logarithm. Emphasis will be set of the ideas and techniques involved, which seem to me to be also interesting as such.

2016-10-27:
Alexander Schnurr (University of Siegen, Germany)
The Spectrum of Applications of the Probabilistic Symbol

Abstract: Levy processes, that is, processes with stationary and independent increments having cadlag paths, already form an interesting and important class of stochastic processes. However, from the point-of-view of the so called probabilistic symbol they are the most simple case, being homogeneous in space and time.
Starting from Levy processes, we will increase the class of processes which we are able to analyze step-by-step. The classes under consideration include solutions of Levy driven SDEs, Feller processes (with sufficiently rich domain) and certain kinds of semimartingales. The main tool we are using is the probabilistic symbol which is the right-hand side derivative (in time) of the characteristic functions of the process.
In the last part of the talk we sketch several applications of this symbol in the context of paths- and distributional- properties.

2016-10-20:
Friedrich Hubalek (TU Wien)
A binomial order book model and its Brownian limit

Abstract: We introduce a simple binomial order book model. The model provides an elementary and intuitive motivation for related work with Rheinlander and Kruhner on a Brownian order book model. We study the dynamics and the distribution of the order volume process, discrete time trading excursions, and trading sequences, which we call order avalanches. We obtain limit results that provide guidance for and correspond to results in the Brownian model, for example an SPDE for the order volume process. The methods come from the classical fluctuation theory for random walks, in particular discrete time path decompositions, combinatorial enumeration, and generating functions.

2016-10-13:
Torben Krüger (IST Austria)
Local eigenvalue statistics for random matrices with general short range correlations

Abstract: The statistics of eigenvalues of random matrix ensembles often exhibits universal behavior as the sizes of the matrices grow to infinity. By this we mean that statistical quantities (e.g. k-point correlation functions of eigenvalues, fluctuations of eigenvalues around their expected positions, distributions of gap sizes between neighboring eigenvalues, etc.) do not depend on most of the details of the model. We prove such a universality statement for non-centered random matrices H with general short range correlations. Our analysis shows that the resolvent G(z) = 1/(H-z) of of the random matrix H approaches a deterministic limit M(z) as long as the spectral smoothing Im[z] is larger than the typical eigenvalue spacing. The limit satisfies the Matrix-Dyson-Equation - 1/M = z - E[H] + E[(H-E[H])M(H-E[H])] which only depends on the second moments of the entries of H. The key novelty is a detailed stability analysis of this non-linear matrix valued equation in asymptotically infinite dimensions.

2016-10-06:
Alexander Cox (University of Bath)
Model-independent pricing with additional information: a Skorokhod embedding approach

Abstract: We analyze the pricing problem of an agent having additional (potentially insider) information on the market in a model-independent setup. Following Hobson's approach we reformulate this problem as a constrained Skorokhod embedding problem, and show a natural supperreplication result. Furthermore, we establish a monotonicity principle for the constrained SEP, giving a geometric characterisation of the support of the optimisers (in the spirit of Beiglboeck, Cox and Huesmann (2014)) which allows us to link the additional information with geometric properties of the optimizers to the constrained embedding problem. Surprisingly, for certain types of information the absence of arbitrage can be easily checked by considering only unconstrained solutions. We give some numerical evidence of the value of the informed agent's information, in terms of the change in price of variance options.
The talk is based on a joint work with Beatrice Acciaio and Martin Huesmann.

2016-09-29:
Tilmann Blümmel (TU Wien)
Understanding the structure of No Arbitrage

Abstract: The fundamental theorem of asset pricing (FTAP) relates the existence of an element in the set (EMM) of equivalent sigma-martingale measures to a no arbitrage condition, the "no free lunch with vanishing risk"-condition (NFLVR). The latter is equivalent to the classical "no arbitrage"-condition (NA) and the "no unbounded profit with bounded risk"-condition (NUPBR). For continuous semimartingales, (NUPBR) is equivalent to the "structure conditon" (SC) which allows for an explicit characterization of the elements in (EMM). But even more important, it provides a natural candidate for an equivalent sigma-martingale measure, the so-called minimal martingale measure (MMM). Unfortunately, for non-continuous semimartingales the (MMM) is, if it exists, in general only a signed measure. Hence, the following natural questions arise: Does there exist a natural candidate for an equivalent sigma-martingale measure if the semimartingale is not continuous? Does there exist a characterization of the elements in (EMM)? Moreover, is the natural candidate, as in the case of a continuous semimartingale, related to a particular structure condition on the underlying semimartingale? The aim of the talk is to answer these questions for quasi-left-continuous semimartingales in a rather basic/didactic way that could be part of a course on continuous time mathematical finance.

Past Talks / Summer Term 2016

Regular Time & Location:
Thursdays, 16:30-18:00, seminar room FH grün 04 (TU Wien),
Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, green section, 4th floor.

2016-06-16:
Mathias Pohl (University of Vienna)
An Applied Take on Dependence Uncertainty

Abstract: When it comes to the aggregation of risk, marginal risks are often modeled separately from their dependence structure, with respect to which there is high uncertainty. In the literature, this situation is known as dependence uncertainty and various ways to compute bounds for aggregated risks have be proposed. As these bounds are typically far apart, we restrict ourselves to a neighborhood of a pre-specified dependence structure instead of considering all possible scenarios. For this propose, we make use of copulas and introduce the Wasserstein distance as a meaningful distance measure between them. Novel results connecting copulas and optimal transport are presented. Bounds for the average value at risk of the sum of two standard uniform random variables serve as an illustrative example.

2016-06-02:
Stefan Ankirchner (University of Jena, Germany)
Play safe if ahead, take risk if behind

Abstract: Usually one expects that economic agents prefer low fluctuations to large ones: the less volatile a risk factor, the more predictable the future resources. However, there are situations where economic agents aim for high fluctuations. The talk will illustrate this within simple control models. One example comprises agents who aim at maximizing the occupation time of a state process in a certain region (winning region).

2016-05-12:
Roberto Renò (University of Verona, Italy)
The drift burst hypothesis

Abstract: The usual tenet that volatility dominates over the drift over short time intervals is not necessarily true when the drift term is locally explosive. The Drift Burst Hypothesis postulates the existence of such locally explosive drifts in the price dynamics. We provide theoretical and empirical support for the hypothesis. Theoretically, we show that feedback trading may imply the presence of endogenous drift bursts in the data generating process. Empirically, after providing suitable identification methods for drift explosion, we apply the detection methodology to high-frequency data and show that drift bursts can usually be associated to "flash crashes", that their occurrence rate is actually quite large, and that they are most typically followed by a price reversal.

2016-04-14:
David Belius (University of Zurich, Switzerland)
Some log-correlated random fields and their extrema

Abstract: Log-correlated random fields, and their extrema, show up in diverse settings, including the theory of cover times, random matrix theory and number theory. Often this can be explained by way of a multiscale decomposition which exhibits an approximate branching structure. I will recall the main ideas behind the analysis of the most basic model of the logcorrelated class, namely Branching Random Walk, where the branching structure is explicit, and explain how to adapt these to models where the branching structure is not immediately obvious.

2016-03-30, 15:00:
Michael Kupper (Universität Konstanz, Germany)
Duality formulas for robust pricing and hedging in discrete time

Abstract: We focus on robust super- and subhedging dualities for contingent claims that can depend on several underlying assets. In addition to strict super- and subhedging, we also consider relaxed versions which, instead of eliminating the shortfall risk completely, aim to reduce it to an acceptable level. This yields robust price bounds with tighter spreads. As applications we study strict super- and subhedging with general convex transaction costs and trading constraints as well as risk based hedging with respect to robust versions of the average value at risk and entropic risk measure. Our approach is based on representation results for increasing convex functionals and allows for general financial market structures. As a side result it yields a robust version of the fundamental theorem of asset pricing. The talk is based on joint work with Patrick Cheridito and Ludovic Tangpi.

2016-03-30, 14:00:
Martin Keller-Ressel (TU Dresden, Germany)
Implied Volatilities from Strict Local Martingales

Abstract: Several authors have proposed to model price bubbles in stock markets by specifying a strict local martingale for the risk-neutral stock price process. Such models are consistent with absence of arbitrage (in the NFLVR sense) while allowing fundamental prices to diverge from actual prices and thus modeling investors' exuberance during the appearance of a bubble. We show that the strict local martingale property as well as the "distance to a true martingale" can be detected from the asymptotic behavior of implied option volatilities for large strikes, thus providing a model-free asymptotic test for the strict local martingale property of the underlying. This talk is based on joint work with Antoine Jacquier.

2016-03-17:
Miklos Rasonyi (Renyi Institute, Hungarian Academy of Sciences)
Optimal investment in the APM of Ross

Abstract: We highlight the difficulties of treating optimal investment problems with an expected utility criterion in the context of large financial markets. Under appropriate assumptions, we solve these difficulties in the particular case of a well-known model of microeconomics: the Arbitrage Pricing Model proposed by S. A. Ross.

2016-02-18:
Tongseok Lim (University of British Columbia, Vancouver, Canada)
On the structure of underlying assets under marginal constraints, which maximize / minimize the price of an option

Abstract: It's a critical issue to deal with how to determine the price of an option from the information we could obtain from the financial market. Although it is in general not possible to determine the exact price of an option by the market information, it recently has been vigorously researched on how to calculate the range of the price's upper limit and the lowest limit. In particular, while many papers have been published on the case where the option depends only on one underlying asset, few results have been appeared on the various assets dependent case. We show that, in the latter case, the structure of the underlying assets which optimize the price of an option exhibit a certain extremal configuration. We also introduce open conjectures on this multi-dimensional optimization problem.

Past Talks / Winter Term 2015/16

Regular Time & Location:
Thursdays, 16:30-18:00, seminar room SR09 (Univ. Vienna),
Oskar-Morgenstern-Platz 1, 1090 Wien, 2nd floor.

2016-01-14:
Martin Herdegen (ETH Zurich, Switzerland)
Equilibrium models with small transaction costs

Abstract: After the financial crisis, it has been proposed in many countries to introduce a financial transaction tax on stocks and derivatives. But how would such a tax affect financial markets? Would it have the beneficial consequences hoped for by its supporters or rather the detrimental consequences feared by its opponents?
To answer questions of this kind, one needs to consider general equilibrium models, where prices are determined endogenously. Indeed, taxes change agents’ individual decision making, which in turn affects the market prices determined by their interactions. The new market environment then again alters the agents’ behaviour, leading to a notoriously intractable fixed point problem.
In this talk, we present an asymptotic approach to this problem. We show that for small proportional transaction costs, prices of stocks and bonds need not change at the leading order. As a consequence, the recent partial equilibrium results on small frictions including Soner/Touzi 2014 or Kallsen/Muhle-Karbe 2015 can be extended to a general equilibrium setup.
The talk is based on joint work in progress with Johannes Muhle-Karbe (University of Michigan).

2016-01-07:
Blanka Horvath (ETH Zurich, Switzerland)
Mass at Zero and small-strike implied volatility expansion in the SABR Model

Abstract: We study the probability mass at the origin in the SABR stochastic volatility model, and derive several tractable expressions for it, in particular when time becomes small or large. In the uncorrelated case, saddlepoint expansions allow for (semi) closed-form asymptotic formulae.
As an application–the original motivation for this paper–we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage free, allow us to quantify the impact of the mass at zero on currently used implied volatility expansions. In particular we discuss how much those expansions become erroneous.

2015-12-17:
Kevin Schnelli (ISTA)
Local law of addition of random matrices on optimal scale

Abstract: Describing the eigenvalue distribution of the sum of two general Hermitian matrices is basic question going back to Weyl. If the matrices have high dimensionality and are in general position in the sense that one of them is conjugated by a random Haar unitary matrix, the eigenvalue distribution of the sum is given by the free additive convolution of the respective spectral distributions. This result was obtained by Voiculescu on the macroscopic scale. In this talk, I show that it holds on the microscopic scale all the way down to the eigenvalue spacing. This shows a remarkable rigidity phenomenon for the eigenvalues.

2015-12-10:
I. Cetin Gülüm (FAM @ TU Wien)
A Variant of Strassen's Theorem

Abstract: Strassen's theorem asserts that a stochastic process is increasing in convex order if and only if there is a martingale with the same marginal distributions. Such processes, or families of measures, are nowadays known as peacocks. We extend this classical result in a novel direction, relaxing the requirement on the martingale. Instead of equal marginal laws, we just require them to be within closed balls, defined by some metric on the space of probability measures. In our main result, the metric is the infinity Wasserstein distance. Existence of a peacock within a prescribed distance is reduced to a countable collection of rather explicit conditions. We also discuss this problem when the underlying metric is the stop-loss distance, the Prokhorov distance and the Lévy distance.

2015-12-03:
Kim Weston (Carnegie Mellon University, Pittsburgh, US)
When is the dual optimizer a martingale?

Abstract: An unpleasant qualitative feature of the general theory of optimal investment is that the dual optimizer may not correspond to the density of a martingale measure. Using the probabilistic (Ap) condition from BMO spaces, I will provide sufficient conditions on the financial market and investor under which the dual optimizer is a martingale. Finally, I will construct a counterexample in which the dual optimizer is a strict local martingale showing that in some sense the (Ap) condition is necessary. (This work is joint with Dmitry Kramkov.)

2015-11-26, 17:30:
Christian Bayer (WIAS Berlin, Germany)
Pricing under rough volatility

Abstract: From an analysis of the time series of realized variance (RV) using recent high frequency data, Gatheral, Jaisson and Rosenbaum (2014) previously showed that log-RV behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated variance. We analyze in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.

2015-11-26, 16:30:
Ruodu Wang (University of Waterloo, Canada)
Recent advances in risk aggregation and dependence uncertainty

Abstract: Modeling inter-dependence among multiple risks often faces statistical as well as modeling challenges, with considerable uncertainty arising naturally. To deal with the uncertainty at the level of dependence in multivariate models, the field of risk aggregation with dependence uncertainty was developed in the past few years. The main object of interest is the set of possible distributions of risk aggregation with given marginal information and arbitrary dependence structure. A direct characterization of this set is unavailable at the moment, and many open questions are found around it. A few selected concrete mathematical problems will be discussed. Applications of the results in this field can be found in any field where uncertainty in multivariate models is of interest; this includes, for instance, risk measures, decision-making, model-independent pricing, scheduling, and optimal transportation.

2015-11-19:
Francesco Caravenna (University of Milano-Bicocca, Italy)
Multi-linear Central Limit Theorems and Scaling Limits of Disordered Systems

Abstract: I will first discuss Central Limit Theorems (CLTs) for multi-linear polynomials of independent random variables, generalizing the usual case of sums. The limit is often non Gaussian and can be characterized as a function of a Brownian motion. The core of the proof is a deep stability result (Lindeberg principle) which asserts that the distribution of a multi-linear polynomial is insensitive, in a quantitative way, to the details of the individual random variables.
I will then present applications of such multi-linear CLTs to the statistical mechanics of disordered systems. For a class of much studied lattice models, including the Ising model and some polymer models, we can prove the existence of a universal scaling limit in the weak coupling regime. No prerequisite on statistical mechanics will be assumed. If time permits, connections with the Stochastic Heat Equation will be described.
(Joint work with Rongfeng Sun and Nikos Zygouras)

2015-11-12:
Maren Schmeck (University of Cologne, DE)
Pricing options on forwards in energy markets: the role of mean reversion's speed

Abstract: Consider the problem of pricing options on forwards in energy markets. In our recent article we considered an underlying spot price which highly fluctuates but also quickly mean reverts to its original level. In such a case we found that fast mean reverting spikes do not matter in option pricing and that the Black 76 formula gives therefore a good approximation for options' prices. In this paper we study the impact of slowly mean reverting components in the spot price dynamics. We find both upper and lower error bounds for the option's price and we show that in this setting the Black 76 formula can misprice substantially.

Past Talks / Summer Term 2015

Regular Time & Location:
Thursdays, 16:30-18:00, seminar room FH grün 04 (TU Wien), former seminar room 101C,
Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, green section, 4th floor.

2015-07-30
Lars Rösler (WU Wien)
Pricing of Contingent Capital Notes in a Structural Credit Risk Model with Incomplete Information

Abstract: This talk deals with Contingent Convertible Notes (CoCos). These are corporate bonds that are equipped with a conversion feature, which is designed with the aim at strengthening the equity capital of the corporate (usually a bank) if it enters into financial distress. We will discuss pricing of various CoCos in a structural credit risk model with incomplete information on the asset value. In fact, thismodel is capable of handling of various types of CoCos. In contrast to many other approaches in the literature, we allow for the possibility that a default occurs before the trigger event.

2015-07-09, seminar room SR09 (Univ. of Vienna)
Sigrid Kallblad (École Polytechnique, Paris, FR)
Model-independent bounds for Asian options: a dynamic programming principle

Abstract: We consider the problem of finding model-independent bounds on the price of an Asian option, when the call prices at the maturity date of the option are known. Our methods differ from most approaches to model-independent pricing in that we consider the problem as a dynamic programming problem, where the controlled process is the conditional distribution of the asset at the maturity date. By formulating the problem in this manner, we are able to determine the model-independent price through a PDE formulation. Notably, this approach does not require specific constraints on the payoff function (e.g. convexity), and would appear to be generalisable to many related problems. This is joint work with A.M.G. Cox.

2015-06-25, 17:30:
Antoine Jacquier (Imperial College London, UK)
Variations on the Heston Theme

Abstract: The Heston model is one of the most popular stochastic volatility models used in mathematical finance, both in academia and by practitioners. Calibration on (Equity) implied volatility surfaces usually exhibit a good fit and the affine structure of the model makes it very amenable to option pricing. However, both the short-term smile and the VIX smile are notoriously mis-calibrated. We propose here variations of the Heston model, which we call "randomised (Heston) volatility models"; these variations, still with continuous paths, preserve the affine structure while allowing for better small-maturity asymptotic behaviour and are more consistent with the behaviour of the VIX smile.

2015-06-25, 16:30:
Umut Cetin (London School of Economics, UK)
Linear Inverse Problems and Market Microstructure

Abstract: In the setting of Kyle's model we discuss the connection between the existence of an equilibrium in a financial market with asymmetrically informed agents and the solutions to a class of linear inverse problems with kernels given by the transition functions of a Markov process. In particular we will observe that when the informed trader receives a continuous signal changing over time and the market makers are risk averse, existence of an equilibrium becomes related to an ill-posed inverse problem for a backward parabolic equation with a given initial condition. A necessary and sufficient condition will be given in order for the inverse problem to have a solution in some L² -space. For a transient diffusion this condition can be interpreted in terms of its last passage times.

2015-06-18, 17:30:
Rémi Lassalle (Instituto Superior Técnico, University of Lisbon, PT)
Some optimal transportation problems related to stochastic differential equations

Abstract: In this talk i will introduce some problems of optimal transport related to stochastic differential equations. The related transference plans between two probabilities are those which further satisfy a constraint on couplings introduced by Yamada and Watanabe in the proof or their celebrated result on SDE. These plans will be called causal since on path spaces they imbedd adapted processes. Since the constraints is directly meaningfull on two Polish spaces endowed with filtrations, i will first provide a very general result of existence for the Primal problem, under mild conditions on the filtrations and on the cost function. Then i will focus on causal transport of the Wiener measure for a particular cost function, in order to relate these problems to stochastic differential equations. Some open problems of stochastic calculus will be also pointed out.

2015-06-18, 16:30:
Caroline Hillairet (CMAP, École Polytechnique, FR)
Affine long term yield curves: an application of the Ramsey rule with progressive utility

Abstract: The purpose of this paper relies on the study of long term affine yield curves modeling. It is inspired by the Ramsey rule of the economic literature, that links discount rate and marginal utility of aggregate optimal consumption. For such a long maturity modelization, the possibility of adjusting preferences to new economic information is crucial, justifying the use of progressive utility. This paper studies, in a framework with affine factors, the yield curve given from the Ramsey rule. It first characterizes consistent progressive utility of investment and consumption, given the optimal wealth and consumption processes. A special attention is paid to utilities associated with linear optimal processes with respect to their initial conditions, which is for example the case of power progressive utilities. Those utilities are the basis point to construct other progressive utilities generating non linear optimal processes but leading yet to still tractable computations. This is of particular interest to study the impact of initial wealth on yield curves.

2015-06-11:
Pierre-Francois Rodriguez (ETH Zurich)
On near-critical level set-percolation for the Gaussian free field

Abstract: We investigate a correlated, non-planar percolation model, obtained by considering level sets of the massive free field above a given height h. The long-range dependence present in the model is a notorious impediment when trying to analyze the behavior near criticality. Alongside the critical threshold h_✶ for percolation, a second parameter h_✶✶ ≥ h_✶ characterizes a strongly subcritical regime. We prove that the relevant crossing probabilities satisfy an approximate 0-1 law around h_✶✶. This (firmly) suggests that the phase transition is sharp.

2015-05-21:
Artem Sapozhnikov (Universität Leipzig, DE)
Large-scale invariance in percolation models (with strong correlations)

Abstract: I will discuss recent progress in understanding supercritical percolation models on lattices, particularly in the presence of strong spatial correlations. This includes quenched Gaussian heat kernel bounds, Harnack inequalities, and local CLT for the random walk on infinite percolation clusters. The results apply to the random interlacements at all levels, the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of local uniqueness.

2015-05-07:
Ulrich Horst (Humboldt-Universität zu Berlin, DE)
A Functional Limit Theorem for Limit Order Books with State Dependent Price Dynamics.

Abstract: We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics. The talk is based on joint work with Christian Bayer and Jinniao Qiu.

2015-04-30:
Christophe Profeta (Université d'Evry Val d'Essonne, FR)
Peacocks and Associated Martingales

Abstract: We call peacock an integrable process which is increasing in the convex order. By a celebrated theorem of Kellerer, we may associate to any peacock (at least) one martingale which has the same one-dimensional marginals.
The general study of peacock originally comes from an example of Carr, Ewald and Xiao, who have proven that a normalized integrated geometric Brownian motion is a peacock. In other words, under the Black-Scholes assumption, the price of an Asian put or call option with fixed strike is increasing over time.
Starting from this example, we shall first present different classes of peacocks, and then give several methods to find associated martingales : via sheet methods, via time reversal, via Skorokhod embeddings... (This talk is based on a joint book with F. Hirsch, B. Roynette and M. Yor).

2015-04-16:
David Ruiz Baños (University of Oslo, NO)
Construction of higher order differentiable strong solutions of SDEs with discontinuous coefficients driven by fractional Brownian motion

Abstract: In this paper we present a new method for the construction of strong solutions of SDEs with merely integrable or bounded drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H < 1/2. Furthermore, we prove the rather surprising result of the higher order Fréchet differentiability of stochastic flows of such SDEs in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a "local time variational calculus". We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.

2015-03-19:
Jorge P. Zubelli (IMPA, Rio de Janeiro, Brazil)
Multiscale Models of Commodities and Derivatives on Futures

Abstract: We discuss a method for computing the first-order approximation of the price of derivatives on futures in the context of multiscale stochastic volatility. The central argument of our method could be applied to interest rate derivatives and compound derivatives as well. The model proposed here is well-suited for commodities since it incorporates mean reversion of the spot price and multiscale stochastic volatility. It allows an effective and straightforward calibration procedure of the group market parameters to implied volatilities. Furthermore, it only requires the first-order approximation of the underlying derivative. Our method was validated by calibrating the group market parameters to options on crude-oil futures, and displays a very good fit of the implied volatility.
This is joint work with J.-P. Fouque and Y. Saporito.

2015-03-19:
Giacomo Scandolo (University of Florence, IT)
Assessing financial model risk and an application to electricity prices

Abstract: Model risk has a huge impact on any risk measurement procedure and its quantification is therefore a crucial step. We introduce three quantitative measures of model risk when choosing a particular reference model within a given class: the absolute, the relative, and the local measure of model risk. Each of the measures has a specific purpose and so allows for flexibility. We illustrate the various notions by studying some simple, but relevant examples. Finally, we present an application of model risk quantification within the German electricity market.
(based on joint works with P. Barrieu and A. Gianfreda)

Past Talks / Winter Term 2014/15

Regular Time & Location:
Thursdays, 16:30-18:00, seminar room SR09 (Univ. Vienna),
Oskar-Morgenstern-Platz 1, 1090 Wien, 2nd floor.

2015-02-26:
Francesc Font Clos (Centre de Recerca Matemàtica, Spain)
Analysis of survival times for a thresholded birth-death process

Abstract: Thresholds are frequently used to define and differentiate events in the analysis of experimental data, either due to device limitations or to filter out small, unimportant events. The thresholding procedure is generally assumed to be harmless for the observables of interest, but a careful analysis usually lacks. To illustrate the perils of such a procedure, I will show how thresholding a simple birth-death process introduces a new scaling region in the distribution of survival times, with a scaling exponent unrelated to the true asymptotes of the process. The model can be solved analytically, and the solution is related to the area below the reciprocal of a Brownian excursion. If time allows, I will discuss how a data collapse could be used to detect threshold-induced effects on real experimental data.

2015-02-05:
Patrick Beißner (Bielefeld University, DE)
Microeconomic Theory of Financial Markets under Uncertain Volatility

Abstract: I consider fundamental questions of arbitrage pricing arising when the uncertainty model incorporates uncertainty about volatility.
1. Arrow-Debreu equilibria with linear price systems contain a natural limit of implementation through Radner equilbria with continuous-time trading. Only mean-ambiguous free claims satisfying equivalently the classical martingale representation property are elements of the marketed space.
2. Part 1. motivates a new principle of risk- and ambiguity-neutral valuation as an extension to Ross (1976). In the spirit of Harrison and Kreps (1979), I establish a microeconomic foundation of viability in which ambiguity-neutrality imposes a fair pricing principle via symmetric multiple-prior martingales. The resulting equivalent symmetric martingale measure set exists under Peng's G-Brownian motion.

2015-01-22:
Dan Hackmann (York University, CA)
Analytical methods for Lévy processes with applications to finance

Abstract: In a setting where the log-stock price is a Levy process, a variety of option prices can be calculated via integral transforms (Fourier, Laplace, Mellin). Transform methods lead us naturally to two theoretical objects: the Wiener-Hopf factors, and the exponential functional. Unfortunately, neither the Wiener-Hopf factors nor the distribution of the exponential functional are known explicitly for many important processes (e.g. VG and CGMY processes). In this talk I will demonstrate some ways in which we may approximate popular processes by analytically tractable processes for which the Wiener-Hopf factors and the distribution of the exponential functional are known. This includes a new, simple and efficient algorithm to approximate any process with completely monotone jumps (e.g. VG and CGMY process) by a hyper-exponential process. I will also give the results of a recent paper on finding the distribution of the exponential functional of a meromorphic process.

2015-01-15:
Florian Baumgartner (University of Innsbruck, AT)
Representations of infinite dimensional Lévy processes and subordination

Abstract: In this talk, we introduce a technique to handle Lévy processes with values in a large class of locally convex Suslin spaces. It enables one to treat the convergence of the interlacing procedure for the small jumps in a Banach subspace of the original space. This is used to prove a Lévy-Itô decomposition theorem and to get more insight in the structure of subordinated Lévy processes, in particular, subordinated Brownian motion.

2014-12-18:
Christoph Temmel (VU University Amsterdam, NL)
Disagreement percolation for simple point processes

Abstract: Markov point processes are a spatial generalisation of the memoryless property of Markov chains. They play a key role in statistical mechanics and spatial probability theory. A Markov random field is specified by a consistent family of conditional distributions on finite volume regions with boundary conditions. In the infinite volume limit, such a specification gives rise to one or more Gibbs measures. The existence of only a unique or several Gibbs measures lies behind the phase transitions in statistical mechanics. Disagreement percolation for lattice models by Maes and van den Berg is a technique to compare the competing influence of different boundary conditions with a product field. In other words, it relates the question of uniqueness of the Gibbs measure to percolation type problems. We generalise the technique to simple point processes and demonstrate the best possible improvement in the case of the hard-sphere model.

2014-12-11:
Paul Krühner (TU Dortmund University, DE)
Optimal density bounds for SDEs with discontinuous drift coefficients

Abstract: In this talk we study the regularity of solutions to the SDE
dX(t) = b(t,X(t))dt + a(t,X(t))dW(t)
in a finite dimensional space where b is only assumed to be measurable and bounded. Malliavin invented a great method to study the density properties - like boundedness of the density - of X(t) under smoothness assumptions. His approach has been generalised in various directions and to various different applications. Contrary to that method, we find sharp upper and lower bounds for the density without using Malliavin calculus or any other type of variational calculus. This talk is based on joint work with David Banos.

2014-12-04:
Michael Schmutz (University of Bern and Swiss Financial Market Supervisory Authority (FINMA), CH)
Risk based solvency frameworks and related challenges

Abstract: Risk-based solvency frameworks such as Solvency II to be introduced in the EU or the Swiss Solvency Test (SST) in force since 2011 in Switzerland seek to assess the financial health of insurance companies by quantifying the capital adequacy through calculating the solvency capital requirement (SCR). Companies can use their own economic capital models (internal models) for this calculation, provided the internal model is approved by the insurance supervisor. The Swiss supervisor has recently completed the first round of internal model approvals. This has provided the supervisor and the industry with many insights into the challenges of designing, assessing, and supervising such models and has shown that there is a considerable number of challenges, in particular modelling challenges, that have not yet been solved in a completely satisfactory way. Some of the most important challenges and problems will be discussed along with some approaches to solutions.

2014-11-20:
Soumik Pal (University of Washington, Seattle, USA)
The geometry of relative arbitrage

Abstract: Suppose we do not impose any stochastic models on how stock prices will evolve in the future. Is it possible, by active trading, to do better than a market index (say, S&P 500)? We will show the following surprising fact in both discrete and continuous time. If we restrict ourselves to portfolios that are functions of the current stock prices, there is exactly one class of trading strategies that achieves this goal. Remarkably, these strategies are produced as solutions of Monge-Kantorovich optimal transport problem on the multidimensional unit simplex with a cost function that can be described as the log partition function. These portfolios are essentially the Functionally Generated Portfolios discovered by Robert Fernholz in a continuous time semimartingale price set-up. Based on joint work with Leonard Wong

2014-11-13:
Johannes Ruf (Department of Mathematics, University College London, UK)
Convergence of local supermartingales and Novikov-type conditions for processes with jumps

Abstract: In the first part of the talk, we characterize the event of convergence of a local supermartingale. Conditions are given in terms of its predictable characteristics and jump measure. Furthermore, it is shown that L^1-boundedness of a related process is necessary and sufficient for convergence. The notion of extended local integrability plays a key role.
In the second part of the talk, we provide a novel proof for the sufficiency of Novikov-Kazamaki type conditions for the martingale property of nonnegative local martingales with jumps. The proof is based on explosion criteria for related processes under a possibly non-equivalent measure.
This is joint work with Martin Larsson.

2014-11-05, 17:15, seminar room SR11:
Matthias Erbar (University of Bonn, DE)
Ricci curvature for finite Markov chains

Abstract: In this talk I will present a new notion of Ricci curvature that applies to finite Markov chains and weighted graphs. It is defined using tools from optimal transport in terms of convexity properties of the Boltzmann entropy functional on the space of probability measures over the graph. I will discuss consequences of lower curvature bounds in terms of functional inequalities (such as modified log-Sobolev and isoperimetric inequalities) and show many examples of graphs and discrete interacting particle systems where explicit curvature bounds can be obtained.

2014-10-30:
Giovanni Puccetti (University of Firenze, Italy)
An Academic Response to Basel 3.5

Abstract: We review and discuss some of the most recent mathematical achievements in the field of Risk Aggregation and Model Uncertainty and we discuss their implications on the current Basel regulatory framework, with particular emphasis on VaR/ES risk measurement.

2014-10-23:
Johanna Penteker (Johannes Kepler Universität, Linz)
p-summing multiplication operators, dyadic Hardy spaces and atomic decomposition

Abstract: Absolutely summing multiplication operators as considered in this talk can be traced back to the work of Maurey-Pisier where they prove the equivalence of Gaussian and Bernoulli random variables in L^2(X) provided that the target space X is of non-trivial cotype. My talk starts with a generally accessible survey of Maurey-Pisier's classical argument. Then I continue by presenting our own work and consider multiplication operators from a C(K) space into a dyadic Hardy space H^p , 0<p ⇐ 2. Those operators are bounded and what is important to me, 2-summing. Pietsch's theorem guarantees therefore the existence of a Pietsch measure for these operators. The existence is guaranteed by a Hahn-Banach argument. Hence, the Pietsch measure is not determined constructively. I use the atomic decomposition property of the Hardy spaces to determine an explicit formula for the Pietsch measure of these multiplication operators.

2014-10-09:
Mladen Savov (University of Reading, UK)
Recent developments for exponential functionals and some possible implications for pricing Asian options

Abstract: The theory of exponential functionals of Levy processes has seen a big development in recent years. Many new and substantial results have been obtained and improved by a few groups of researchers. Despite these advancements the pricing of an Asian option under general Levy dynamics seems elusive. In this talk we will present a general discussion for these latest results and point to their implications for pricing Asian options and the numerous remaining difficulties.

2014-09-11:
Danila Zaev (National Research University, RU)
Monge-Kantorovich problem with additional constraints

Abstract: Monge-Kantorovich problem is a problem of transportation of one given distribution of mass to another in an optimal way. The theory around this problem studies existence, uniqueness and a form of such transfers. It appears that this theory is a cornerstone of the modern measure theory, and also it is very useful in various applications. In my talk I will speak about modifications of the Monge-Kantorovich problem, namely about problems where sets of admissible transport plans are restricted in some way. An example of such restriction is an invariance with respect to an action of some group, another one is a martingale property. Both examples can be seen as the particular cases of Monge-Kantorovich problem with additional linear constraint of the following general form: admissible measures should vanish on a given functional subspace. An important application of invariant Monge-Kantorovich problem is the possibility of a meaningful formulation for the problem on infinite-dimensional spaces. Some known results about properties of optimal transport maps in such cases will be also discussed.

Past Talks / Summer Term 2014

Regular Time & Location:
Thursdays, 16:30-18:00, seminar room 101C (TU Wien),
Wiedner Hauptstraße 8, 1040 Wien, "Freihaus" building, green section, 4th floor.

2014-07-31:
Elisa Alos Alcalde (Universitat Pompeu Fabra, Barcelona, Spain)
A general method to develop closed-form approximations formulas and to estimate their error bounds, with applications to the study of spread options

Abstract: Practicioners need easy-to-use, simple, not time-consuming, but also accurate and reliable techniques. It is really difficult to simultaneously obtain simplicity, accuracy and flexibility without the use of powerful mathematical tools. In this talk we present a methodology for short-time option pricing approximation, not depending on the specific model, nor on the specific option. This method is based on the classical Itô formula and on Malliavin calculus techniques, which allow us to obtain simple closed-form approximation formulas depending on the derivative operator. As an example, we apply this method to the study of spread options. The numerical analysis shows that these formulas are extremely accurate and improve some previous approaches on two-assets and three-assets spread options as Kirk.s formula or the decomposition mehod presented in Alòs, Eydeland and Laurence (2011).

2014-07-03:
Larry Goldstein (University of Southern California, Los Angeles, USA)
Applications of Stein Couplings for Concentration of Measure

Abstract: The existence of a bounded coupling of a non-negative random variable to one having the variable's size bias distribution implies concentration of measure with Poisson type tails. Applications of these types of concentration of measure results include the number of local maxima of a random function on a lattice, urn occupancy statistics in multinomial allocation models, and the volume contained in k-way intersections of n balls placed uniformly over a volume n subset of d dimensional space. The two final examples are members of a class of occupancy models with log concave marginals for which size bias couplings may be constructed more generally. Similarly, concentration bounds can be shown using the zero bias coupling, proving tail inequalities in Hoeffding's combinatorial central limit theorem under diverse assumptions on the permutation distribution. The bounds produced by these two couplings, which have their origin in Stein's method, offer improvements to those obtained by using other methods available in the literature.
This work is joint with Jay Bartroff, Subhankar Ghosh and Ümit Işlak.

2014-07-01 (Tuesday, 15:30, seminar room 8, 2nd floor, Oskar-Morgenstern-Platz 1, 1090 Wien):
Julio Backhoff (HU Berlin, Germany)
Sensitivity and robustness analysis of some stochastic optimization problems

Abstract: The robust approach to parameter uncertainty in stochastic optimization (S.O.) consists in hedging oneself against all reasonable parameters of the model at hand by taking a worst-case approach. In the case of robust utility maximization in financial market models one thus considers a family of reference probability measures (the 'uncertainty set') and seeks the best optimal strategy and the worst measure in such set. In this direction, and motivated by an application, we extend the existing convex analysis approach to the case when the uncertainty set is not compact but just weakly-closed in a pertinent 'modular space', and recover some of the existing results in the literature and provide new ones. The dual concept to robustness is that of sensitivity, whereby one computes first (or higher) order approximations to the value function of a S.O. problem w.r.t. its parameters. In this respect, we perform a first-order sensitivity analysis of general convex stochastic control problems and of specific non-convex variants. If time permits we shall discuss what a sensitivity analysis of utility maximization in financial market models yields.

2014-06-12:
Pingping Zeng (Hong Kong University of Science & Technology, Hong Kong, China)
Closed-form partial transform of triple joint density for pricing exotic options and variance derivatives under the 3/2 model

Abstract: Most of the empirical studies on stochastic volatility dynamics favor the 3/2 specification over the square-root (CIR) process in the Heston model. In the context of option pricing, the 3/2 stochastic volatility model is reported to be able to capture the volatility skew evolution better than the Heston model. In this article, we make a thorough investigation on the analytic tractability of the 3/2 stochastic volatility model by proposing a closed-form formula for the partial transform of the triple joint transition density (X,I,V) which stand for the log asset price, the quadratic variation (continuous realized variance) and the instantaneous variance, respectively. Two different approaches are presented for deriving the key result. In the first approach, we obtain the partial transform by utilizing the exponential affine structure of the pair (X,I) and solving the governing PDE that involves V only. The second approach is more probabilistic and it makes use of the change of measure and conditioning techniques. The closed-form partial transform enables us to deduce a variety of marginal transition density functions or characteristic functions that are crucial in pricing discretely sampled variance derivatives and exotic options that depend on both the asset price and quadratic variation. Various applications and numerical examples on pricing exotic derivatives with forward start or discrete monitoring features are given to demonstrate our unified pricing framework based on the closed-form partial transform under the 3/2 model.

2014-05-22:
Christos E. Kountzakis (University of the Aegean, GR, University of Vienna, AT)
The Order Form of the Fundamental Theorems of Asset Pricing

Abstract: In this article, we provide an order-form of the First and the Second Fundamental Theorem of Asset Pricing in the one -period market model. The space of the financial positions is supposed to be a Banach lattice. This form holds is relevant to any directed topological space.

2014-05-15:
Alexander Fribergh (Université Paul Sabatier, FR)
Biased random walk on supercritical percolation clusters

Abstract: We will present results on biased random walks on supercritical percolation clusters. This a natural model for observing trapping phenomena and anomalous long-term behaviors. We will explain why this model exhibits a phase transition from positive speed to zero speed as the bias increases. Furthermore, we shall discuss a subtle difficulty appearing when trying to rescale such a process to obtain scaling limits. This talk will be based on past and ongoing work of Alexander Fribergh and Alan Hammond.

2014-05-08:
Fabio Bellini (University of Milano-Bicocca, IT)
Elicitable risk measures and expectiles

Abstract: A statistical functional is elicitable if it can be defined as the minimizer of a suitable expected scoring function (see Gneiting (2011), Ziegel (2013) and the references therein). With financial applications in view, we suggest a slightly more restrictive definition than Gneiting (2011), and we derive several necessary conditions. For monetary risk measures, using the characterization results of Weber (2006), we show that elicitability leads to a subclass of the shortfall risk measures introduced by Föllmer and Schied (2002). In the coherent case the only example are the expectiles, that are becoming increasingly popular in the mathematical finance literature. We discuss some of their properties, with a particular emphasis on their tail asymptotic behaviour.

2014-04-24:
Jorge P. Zubelli (IMPA, Rio de Janeiro, Brazil)
Calibration of Stochastic Volatility Models with Applications to Commodities

Abstract: Local volatility models are extensively used and well-recognized for hedging and pricing in financial markets. They are frequently used, for instance, in the evaluation of exotic options so as to avoid arbitrage opportunities with respect to other instruments.
The PDE (inverse) problem consists in recovering the time and space varying diffusion coefficient in a parabolic equation from limited data. It is known that this corresponds to an ill-posed problem.
The ill-posed character of local volatility surface calibration from market prices requires the use of regularization techniques either implicitly or explicitly. Such regularization techniques have been widely studied for a while and are still a topic of intense research. We have employed convex regularization tools and recent inverse problem advances to deal with the local volatility calibration problem.
We describe a theoretical approach to calibrate the local volatility surface from quoted derivative prices, by introducing convex regularization techniques and a priori information. We investigate theoretical as well as practical consequences of our methods and illustrate our results both with data from commodity markets.
This work is part of ongoing collaboration with V. Albani (IMPA), A. De Cezaro (FURGS), and O. Scherzer (Vienna).

2014-04-03:
Gabor Pete (Technical University of Budapest, HU)
The scaling limit of the planar Minimal Spanning Tree

Abstract: In a joint work started long ago with Christophe Garban and Oded Schramm and completed only recently [arXiv:1309.0269], we prove that the Minimal Spanning Tree on a version of the triangular lattice in the complex plane has a unique scaling limit, which is invariant under rotations, scalings, and translations. However, it is not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works. I am planning to explain some of the key ideas in this project.

2014-03-20, 17:30:
Marius Hofert (TU Munich, DE)
An extreme value approach for modeling operational risk losses depending on covariates

Abstract: A general methodology for modeling loss data depending on covariates is developed. The parameters of the frequency and severity distributions of the losses may depend on covariates. The loss frequency over time is modeled via a non-homogeneous Poisson process with integrated rate function depending on the covariates. This corresponds to a generalized additive model which can be estimated with spline smoothing via penalized maximum likelihood estimation. The loss severity over time is modeled via a nonstationary generalized Pareto model depending on the covariates. Whereas spline smoothing can not be directly applied in this case, an efficient algorithm based on orthogonal parameters is suggested. The methodology is applied to a database of operational risk losses. Estimates, including confidence intervals, for Value-at-Risk (also depending on the covariates) as required by the Basel II/III framework are computed.

2014-03-20, 16:30:
Mathieu Rosenbaum (University Pierre and Marie Curie, Paris 6, FR)
Limit theorems for nearly unstable Hawkes processes

Abstract: Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the L1 norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox-Ingersoll-Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes based price model introduced by Bacry et al. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well known stylized facts of prices, both at the microstructure level and at the macroscopic scale.
(Joint work with Thibault Jaisson, Ecole Polytechnique Paris).

2014-03-13:
Claudio Fontana (University of Évry Val d'Essonne, FR)
Insider information, arbitrage profits and honest times

Abstract: In the context of semimartingale financial models, we study whether the addition of insider information can lead to arbitrage profits. In the first part of the talk, in the case of continuous semimartingale models, we consider the additional information associated to an honest time, which is shown to yield different arbitrage possibilities for an insider trader depending on the investment horizon. In the second part of the talk, we shall study the stability of absence of arbitrages of the first kind condition under progressive and initial enlargements of the original filtration.
(Based on joint work with B. Acciaio, M. Jeanblanc, K. Kardaras and S. Song)

2014-03-06:
Peter Markowich (University of Cambridge, UK)
Price Formation Modeling with PDE: From Boltzmann to Free Boundaries

Abstract: We present an analysis (uniqueness, existence and smoothness) of the Lasry-Lions price frormation free boundary model and a microscopic (kinetic) derivation as a scaling limit from a Boltzmann-type market-interaction model.

Vienna Seminar in Mathematical Finance and Probability

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