Vienna Seminar in Mathematical Finance and Probability
This seminar is jointly organised by the following research units / departments:
Regular Time & Location:
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.
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.
Past Talks / Winter Term 2017
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.
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.
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.
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.
Past Talks / Summer Term 2017
Regular Time & Location:
2017-07-19, Wed., 11:00-12:00, seminar room DB gelb 04 ("Freihaus" building, yellow section, 4th floor):
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.
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.
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.
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)
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.
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.
2017-06-06, 16:30 in seminar room DA grün 06A (Freihaus, green section, 6th floor)
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)
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
Past Talks / Winter Term 2016/17
Regular Time & Location:
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.
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.
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).
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.
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),
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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:
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.
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).
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.
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.
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.
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.
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.
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:
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?
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.
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.
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.
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.)
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.
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.
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.
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:
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)
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Past Talks / Winter Term 2014/15
Regular Time & Location:
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.
Abstract: I consider fundamental questions of arbitrage pricing arising when the uncertainty model incorporates uncertainty about volatility.
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.
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.
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.
Abstract: In this talk we study the regularity of solutions to the SDE
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.
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
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.
2014-11-05, 17:15, seminar room SR11:
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.
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.
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.
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.
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:
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).
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.
2014-07-01 (Tuesday, 15:30, seminar room 8, 2nd floor, Oskar-Morgenstern-Platz 1, 1090 Wien):
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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