Talks (titles and abstracts)
Matteo Burzoni: Pathwise arbitrage theory
We provide a pathwise approach to arbitrage theory and superhedging duality in discrete time. We introduce the possibility of restricting the set of possible scenarios which the modeler deems to be feasible. This feature allows to study some models under Knightian Uncertainty as well as the classical probabilistic models for a particular choice of the set of scenarios. This is a joint work with M. Frittelli, Z. Hou, M. Maggis and J. Obloj.
Thomas Cass: A Stratonovich-Skorohod formula for Volterra Gaussian rough paths
Lyons’ theory of rough paths allows us to solve stochastic differential equations driven by a Gaussian processes X under certain conditions on the covariance function. The rough integral of these solutions against X again exist, and a natural question is to find a closed-form correction formula between this rough integral and the Skorohod integral of the solution. This is particularly useful in applications, in which typically one wants to compute (or estimate) the expectation of the rough integral. In the case of Brownian motion our formula reduced to the classical Stratonovich-Ito conversion formula. Previous works in the literature assumes the integrand to be the gradient of a smooth function of \(X_{t}\); our formula again recovers these results as special cases.
Joint work with Nengli Lim (Imperial College London and National University of Singapore).
Patrick Cheridito: Duality formulas for robust pricing and hedging in discrete time
In this paper we derive 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.
Rama Cont: Pathwise stochastic integration and functional calculus
We will discuss some new results on a pathwise approach to 'stochastic' integration with respect to paths with finite quadratic variation, building on Hans Föllmer's (1981) seminal paper on pathwise Ito calculus. This approach is shown to extend to path-dependent integrands and allows for integrands and integrators which include paths of Brownian motions and path-dependent functionals of such paths. We prove an isometry formula for this integral, prove a continuity property for the pathwise integral and derive a pathwise 'Doob-Meyer' decomposition is obtained for functionals of paths with strictly increasing quadratic variation. Finally, we discuss the relation with the notion of controlled rough path introduced by Gubinelli.
Dan Crisan: High-order discretizations to the filtering problem
The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. First order discretisations of these functionals have been introduced and analized by Picard in the eighties. We introduce a class of high-order time discretisation of these functionals corresponding to a chosen partition of the time interval and analyze the convergence rate of discretisation.
This is joint work with Salvador-Ortiz Latorre (Oslo).
Ibrahim Ekren: Hörmander condition for delayed diffusions
In this talk we present some results that extends the Hormander condition to delayed diffusions. We will state a sufficient condition for the existence of density for the marginals of a delayed diffusion. This condition takes into account and quantifies the additional noise introduced to the system through the delay. The talk is based on joint work with Reda Chhaibi.
Tolulope Rhoda Fadina: Credit risk and Good-deal bounds with ambiguity on the default intensity
In this talk, we discuss the concept of no-arbitrage in a credit risk market under ambiguity. We consider an intensity-based framework where we assume that the default intensity is strictly positive. This assumption is economically intuitive, as it is equivalent to an approach where at every time \(s\) credit risk is present and not negligible. However, we consider the realistic case where the intensity is not precisely known, but there is ambiguity on the intensity. In addition, we discuss the Good-deal bound pricing with application to credit risk under ambiguity.
Lukas Gonon: Skorokhod embedding via time-changed Lévy processes
Using a suitable interpolation of two given marginal distributions, we obtain explicit examples of discontinuous martingales matching the interpolating marginals. The construction is based on time-change arguments and uniqueness results for Fokker-Planck equations. As applications, our results yield a solution to the Skorokhod embedding problem for recurrent Lévy processes processes starting from non-trivial distributions.
The talk is based on joint work with Leif Döring, David Prömel and Oleg Reichmann.
Alex Hening: Killed Brownian Motion, PDE and models of default
We show that if \(B\) is a Brownian motion, \(\zeta\) a nonnegative random variable, \(\lambda>0\) is a killing rate parameter and \(\psi:\mathbb{R}\to [0,1]\) is a suitably smooth nonincreasing approximation to the indicator of the set \((-\infty,0]\) then, under certain assumptions, there exists a unique continuously differentiable function \(b:\mathbb{R}_+\to\mathbb{R}\) such that \(\mathbb{E}\left[-\lambda \int_0^t \psi(B_s-b(s))\,ds\right] = \mathbb{P}\{\zeta>t\}, t\geq 0\).
We also look at the case when \(\psi\) is the indicator of the set \((-\infty,0]\). This presents difficulties because \(\psi\) is discontinuous.We associate a partial differential equation (PDE) coupled with an ordinary differential equation (ODE) to this version of the inverse first passage time problem. We show the existence and uniqueness of this PDE/ODE system.
This is joint work with Boris Ettinger, Steve Evans and Tak Kwong Wong.
Calypso Herrera: Parallel American Monte Carlo
Introduction of a new algorithm for American Monte Carlo that can be used either for American-style options, callable structured products or for computing counterparty credit risk (e.g. CVA or PFE computation). Leveraging least squares regressions, the main novel feature of this algorithm is that it can be fully parallelized. Moreover, there is no need to store the paths and the payoff computation can be done forwards: this allows to price structured products with complex path and exercise dependencies. The key idea of this algorithm is to split the set of paths in several subsets which are used iteratively. I will give the convergence rate of the algorithm. I will illustrate this method on an American put option and compare the results with the Longstaff-Schwartz algorithm.
Blanka Horvath: Asymptotics for fractional volatility models
Martin Larsson: Conditional infimum and recovery of monotone processes
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 of these results.
Chong Liu: Compactness criterion for semimartingale laws and semimartingale optimal transport
We extend the results regarding the semimartingale optimal transport obtained by Tan and Touzi from the continuous paths space to the Skorokhod space. In the presence of jumps, the main obstruction for the control set \(\mathfrak{P}_\Theta\) being weakly compact is that the laws of purely discontinuous local martingales may converge weakly to a nontrivial diffusion process, which will make \(\mathfrak{P}_\Theta\) fail to be compact even if the parameter set \(\Theta\) itself is convex and compact.
In this talk we will give a sufficient and necessary condition for the compactness of \(\mathfrak{P}_\Theta\) which also identifies the convergence behavior of laws of purely discontinuous local martingales. As an application we use this argument to show that under some mild assumptions of the parameter set \(\Theta\), there exists an element in the control set \(\mathfrak{P}_\Theta\) which minimizes the cost of transportation determined by the characteristics of general semimartingales taking values in \(\Theta\). We also show that under such conditions, the duality result obtained in also holds true in our general setting and the value function is the unique viscosity solution of some nonlinear partial integro-differential equation.
Peng Luo: Portfolio Optimization under Probability and Discounting Uncertainty
We consider the portfolio optimization problem with random endowment under a general concave non-linear expectation given by a maximal subsolution of a BSDE. We provide a characterization of the optimal strategy in terms of fully coupled FBSDE, where we have an auxiliary BSDE dealing with the discounting process. We will present several concrete examples in which we construct the solutions explicitly.
This is joint work with Samuel Drapeau and Dewen Xiong.
Marvin Müller: SPDE Models for Limit Order Books
Motivated by observations in high frequency markets we introduce a model for
the order book density based on parabolic stochastic partial differential
equations. While the celebrated free boundary model for price formation of
Lasry-Lions (2007) was starting point for a wide range of price-
time continuous models for limit order books, tractability is one of the main
issues when working with infinite dimensional systems. We discuss existence of
finite dimensional realizations which reduce the complexity of the model
drastically. Following empirical observations by Cont, Kukanov and Stoikov
(2014), the so called order flow imbalance induces naturally a model for short
term price dynamics which becomes explicit in the particular framework.
Ariel Neufeld: Robust Utility Maximization with Lévy Processes
We present a tractable framework for Knightian uncertainty, the so-called nonlinear Lévy processes, and use it to formulate and solve problems of robust utility maximization for an investor with logarithmic or power utility.
The uncertainty is specified by a set of possible Lévy triplets; that is, possible instantaneous drift, volatility and jump characteristics of the price process.
We show that an optimal investment strategy exists and compute it in semi-closed form. Moreover, we provide a saddle point analysis describing a worst-case model.
This talk is based on joint work with Marcel Nutz.
Eyal Neuman: Optimal portfolio liquidation in target zone models
We study optimal buying and selling strategies in target zone models. In these models the price is modeled by a diffusion process which is reflected at one or more barriers. Such models arise for example when a currency exchange rate is kept above a certain threshold due to central bank intervention. We consider the optimal portfolio liquidation problem for an investor for whom prices are optimal at the barrier and who creates temporary price impact. This problem will be formulated as the minimization of a cost-risk functional over strategies that only trade when the price process is located at the barrier. We solve the corresponding to singular stochastic control problem by means of a scaling limit of critical branching particle systems, which is known as a catalytic superprocess. We also study the central bank favourable strategies in target zone markets. When the investors additionally create permanent price impact, the central bank has to keep the exchange rate above a random moving barrier. We derive the optimal strategy of the central bank, which protects the exchange rate from crushing, by means of the Skorokhod problem.
This is joint work with Alexander Schied.
David Prömel: Paracontrolled convolution equations
Based on the notion of paracontrolled distributions, existence and uniqueness results are presented for convolution equations. In particular, this wide class of equations includes rough/stochastic differential equations with possible delay, stochastic Volterra equations, and convolutions equations driven by Levy processes.
Juan Sagredo: Existence of Sequential Competitive Equilibrium in Krusell–Smith Type Economies
In this paper, we give a proof of existence of sequential competitive equilibrium in Krusell-Smith type economies with different kinds of agents, taxes, unemployment insurance, and endogenous labor choice. We also provide a formal construction of a family of shock processes exhibiting conditional-no-aggregate-uncertainty.
Eric Schaanning: Fire sales, price-mediated contagion and systemic risk
Regulatory stress tests for the banking and financial sectors do currently not include any feedback mechanisms. The Basel Committee on Banking Supervision has identified this as a key shortcoming. We develop a new framework to model fire sales with this goal in mind. A key concept that arises in our model is that of indirect exposures: Due to the potential for losses via price-mediated contagion, banks have larger exposures to asset classes or risk factors than is apparent from their balance sheet. Using a dataset of 90 European Banks by the European Banking Authority, we estimate these indirect exposures and find them to be of significant economic size.
Indirect exposures thus play an important role in quantifying systemic risk and they may change the outcome of a regulatory macro stress test: When applying our model to the European banking system, around 10 % of the banks pass the individual stress test, but fail the systemic stress test, which incorporates losses through fire sales and price-mediated contagion.
Finally, we compare the performance of our model to a previously used fire sales model of the ECB and the NY Fed and find ours to outperform the other one in several key aspects.
Mario Sikic: Robust Utility Maximization in Discrete-Time Markets with Friction
We study a robust stochastic optimization problem in the quasi-sure setting in discrete time. We show that under a lineality-type condition the problem admits a maximizer. This condition is implied by the no-arbitrage condition in models of financial markets. As a corollary we obtain existence of utility maximizer in frictionless market model, markets with proportional transaction costs and also more general convex costs, like in the case of market impact.
Pietro Siorpaes: The Martingale Polar Sets
Martingale optimal transport (MOT) is a variant of the classical optimal transport problem where a martingale constraint is imposed on the coupling. In a recent paper, Beiglböck, Nutz and Touzi show that in dimension one there is no duality gap and that the dual problem admits an optimizer. A key step towards this achievement is the characterization of the polar sets of the family of all martingale couplings. Here we extend this characterization to arbitrary finite dimension through a deeper study of the convex order.
Joint work with Jan Obłoj.
Josef Teichmann: Affine processes and non-linear PDEs
Martin Weidner: Hörmander's theorem for rough differential equations on manifolds
We introduce a new definition for solutions~\(Y\) to rough differential equations (RDEs) of the form \(\mathrm{d} Y_{t} =V\left (Y_{t}\right )\mathrm{d}\mathbf{X}_{t} ,Y_{0} =y_{0}\text{}\). By using the Grossman-Larson Hopf algebra on labelled rooted trees, we prove equivalence with the classical definition of a solution advanced by Davie when the state space~\(E\) for \(Y\) is a finite-dimensional vector space.~ The~notion of solution we propose, however, works~when \(E\) is~any smooth manifold~\(\mathcal{M}\) and is therefore equally well-suited for use as an intrinsic defintion of an \(\mathcal{M}\)-valued RDE solution. This enables us to prove existence, uniqueness and coordinate-invariant theorems for RDEs on \(\mathcal{M}\) bypassing the need to define a rough path on \(\mathcal{M}\). Using this framework, we generalise result of Cass, Hairer, Litterer and Tindel proving the smoothness of the density of \(\mathcal{M}\)-valued RDEs driven by non-degenerate Gaussian rough paths under Hörmander's bracket condition. In doing so, we reinterpret some of the foundational results of the Malliavin calculus to make them appropriate to the study of \(\mathcal{M}\)-valued Wiener functionals.
Harry Zheng: Turnpike Property for an Investment and Consumption Model
It is well known that the optimal portfolio strategy for optimal investment problems can be approximated by a wealth and time independent strategy if the planning horizon is distant and the utility of the terminal wealth behaves asymptotically like a power utility. This is called the turnpike property for investment. There is little literature on the turnpike property for optimal investment and consumption problems. In this paper we discuss the turnpike property for general utilities and the convergence rate of the approximation of optimal strategies. We find there exists a threshold value for the turnpike property of investment to hold and the threshold value only depends on the Sharpe ratio, the riskless interest rate and the discount rate, but not utility functions. There is no turnpike property for consumption in general. If consumption utility is power utility, then the turnpike property for investment always holds except for one special case.
This is a joint work with Baojun Bian of Tongji University, Shanghai.