Title: Markovian transition semigroups under model uncertainty
Abstract: When considering stochastic processes for the modelling of real world phenomena, a major issue is so-called model uncertainty or epistemic uncertainty. The latter refer to the impossibility of perfectly capturing information about the future in a single stochastic framework. In a dynamic setting, this leads to the task of constructing consistent families of nonlinear transition semigroups. In this talk, we present two ways to incorporate model uncertainty into a Markovian dynamics. One approach considers parameter uncertainty in the generator of a Markov process while the other considers perturbations of a reference model within a Wasserstein proximity. We show that, in typical situations, these two a priori different approaches lead to the same convex transition semigroup. We also discuss the role of Lipschitz sets and provide a comparison result for convex monotone semigroups. The talk is based on joint works with Jonas Blessing, Robert Denk, Sven Fuhrmann, and Max Nendel.
Abstract: In this talk, we introduce two problems of contract theory, in continuous-time, with a multitude of agents. First, we will study a model of optimal contracting in a hierarchy, which generalises the one-period framework of Sung (2015). The hierarchy is modelled by a series of interlinked principal-agent problems, leading to a sequence of Stackelberg equilibria. More precisely, the principal (she) can contract with a manager (he), to incentivise him to act in her best interest, despite only observing the net benefits of the total hierarchy. The manager in turn subcontracts the agents below him. We will see through a simple example that, while the agents only control the drift of their outcome, the manager controls the volatility of the Agents’ continuation utility. Therefore, even this relatively simple introductory example justifies the use of recent results on optimal contracting for drift and volatility control. We will also discuss some possible extensions of this model, in particular the elaboration of more general contracts, indexing the compensation of one worker on the result of the others. This constitutes a first approach to even more complex contracts, in the case, for example, of a continuum of workers with mean-field interactions.
This will lead us to introduce the second problem, namely optimal contracting for demand-response management, which consists in extending the model by Aïd, Possamaï, and Touzi (2018)  to a mean-field of consumers. More precisely, the principal (an electricity producer, or provider) contracts with a continuum of agents (the consumers), to incentivise them to decrease the mean and the volatility of their energy consumption during high peak demand. In addition, we introduce a common noise, impacting all consumption processes, to take into account the impact of weather conditions on the agents electricity consumption. This mean-field framework with common noise leads us to consider a more extensive class of contracts. In particular, we prove that the results of  can be improved by indexing the contracts on the consumption of one agent and aggregate consumption statistics from the distribution of the entire population of consumers. Finally, we will conclude by mentioning that this principal-agent approach with a multitude of agents can be used to address many situations, in particular in the current epidemic context.
A collection of joint works with R. Elie, T. Mastrolia, D. Possamaï, G. Turinici, and X. Warin.
Speaker: Hoi Ying Wong (Chinese University of Hong Kong)
Title: Primal return ambiguity and dual risk ambiguity
Abstract: Consider a robust consumption-investment problem for a risk- and ambiguity-averse investor who is concerned about return ambiguity in risky asset prices. When the investor aims to maximize the worst-case scenario of his/her consumption-investment objective, we propose a dual approach to the robust optimization problem in a dual economy with risk ambiguity. Using the G-expectation framework, we establish the duality theorem to bridge between the primal problem with return ambiguity and the dual problem with risk ambiguity, and hence characterize the robust strategy for a general class of utility functions subject to the non-negative consumption rate and wealth constraints. The risk ambiguity in the dual problem induces correlation ambiguity when the primal economy comprises multiple risky assets with return ambiguity. By analyzing the dual economy, we show that the robust investment strategy favors a sparse portfolio, in addition to its usual feature – having the least exposure to ambiguity risk. We also extend the analysis to robust consumption-investment with retirement option. Using the duality approach, the robust retirement timing is characterized through a G-stopping. (This talk is based on joint works with Kyunghyun Park.)
Title: Entropy Martingale Optimal Transport and Nonlinear Pricing-Hedging Duality
Abstract: We develop a duality between a novel Entropy Martingale Optimal Transport problem (A) and an associated optimization problem (B). In the (dual) problem (A) we follow the approach taken in the Entropy Optimal Transport (EOT) problem by Liero et al. “Optimal entropy-transport problems and a new Hellinger-Kantorovic distance between positive measures”, Invent. math. 2018, but we add the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as in Liero et al. The marginals are not any more fixed a priori, as in MOT, because we may not have sufficient information to detect them with enough accuracy, for example because there are not sufficiently many traded call and put options. So the infimum is taken over all martingale probability measures, but those that are far from some estimate are appropriately penalized via a divergence term D. This is a key difference with the approaches taken in the previous literature, where the addition of the entropic term is made without smoothing the strict marginal constraints, which are kept. Whereas in our problems we add uncertainty regarding the marginals themselves. In the (primal) problem (B) the objective functional, associated via Fenchel coniugacy to the terms D, is not any more linear, as in OT or in MOT. This leads to an optimization problem which also has a clear financial interpretation as a nonlinear subhedging value. Our theory allows us to establish a nonlinear robust pricing-hedging duality in financial mathematics, which covers a wide range of known robust results. We also focus on Wasserstein-induced penalization and we study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT. Joint work with Alessandro Doldi.
Speaker: Jianfeng Zhang (University of Southern California)
Title: Mean Field Game Master Equations with Monotonicity and Anti-monotonicity Conditions in Displacement Sense
Abstract: It is well known that certain monotonicity condition is crucial for the global well-posedness of mean field game master equations. In this talk we introduce a new methodology which would help us to find appropriate monotonicity conditions, in the spirit of the displacement monotonicity rather than the more popular Lasry-Lions monotonicity, for mean field game master equations with non-separable Hamiltonians. We then establish the global well-posedness for such equations, which is new in the literature. Moreover, we apply the same arguments to anti-monotonicity conditions, under which the global well-posedness is typically not expected. We show that the master equations can still be globally well-posed, provided that the coefficients are sufficiently anti-monotone in some sense. The talk is based on a joint work with Gangbo, Meszaros, Mou, and another ongoing work with Mou.
Abstract: Information is arguably one of the key drivers in financial markets. It determines on what basis market participants take their decisions and choose their actions. So the way we describe information flows and how we do optimal control in them is crucial for financial modelling and the aim of the talk is to illustrate some new possibilities in, for instance, optimal investment problems with jumps.
More precisely, in continuous-time financial modelling we typically use a filtration to specify the flow of information and the non-anticipativity of strategic actions is ensured by requiring them to be predictable with respect to this filtration. This is of little consequence in purely Brownian models where filtrations are continuous. Interest rate announcements by central banks on the macro-level or incoming large limit or market orders on the micro-level, though, are both examples of events where the flow of information exhibits a jump. Here, the restriction to predictable strategies may very well not be fully adequate as market participants work hard to device signals and act on them until the last moment to proactively position themselves for these “shocks”.
We will discuss how Meyer \(\sigma\)-fields can be used to conveniently incorporate such strategic signals in information flows and thus extend controls beyond the classical realm of predictable strategies. By way of illustration, we discuss the modelling possibilities in Merton’s classical problem of optimal investment and in the singular control problems with irreversible investment, answering in particular the question what value to assign an opportunity to become less predictable.
(This talk is based on joint work with David Besslich and Laura Körber.)
Abstract: Most integrated assessment models (IAM) for climate change, such as the Dynamic Integrated Climate-Economic (DICE) model popularized by Nobel laureate William Nordhaus, have at their core an economic module that is based on the mainstream macroeconomic paradigm of Dynamic Stochastic General Equilibrium (DSGE) models. These economic models have been the subject of intense criticism since the last financial crisis not only for their inability to predict or explain financial instabilities, but also for their adherence to “micro-foundations” that are at odds with observed behaviour of agents and lack of rigour in statistical validation. In this talk, I will review some recent work that proposes new integrated assessment models for climate change where the DSGE core is replaced by stock-flow consistent (SFC) macroeconomic models. These alternative models have much richer dynamic outcomes and allow the exploration of nonlinear feedback loops that are entirely absent from DICE models, in particular the crucial interaction between private debt, economic activity, and global temperature. On the other hand, the outcome of these models can be affected by both initial conditions and parameter uncertainty, so it is important to subject them to a thorough sensitivity analysis, which I’ll also discuss in the talk. Finally, I’ll present simulation results of the effects of several policy measures, including green quantitative easing and elements of the Green New Deal.
Abstract: This talk on financial market design addresses the costs (and sometimes the benefits) of fragmenting trade across multiple venues. Size discovery trading crosses buy and sell orders, with no bid-ask spread and no price impact, by exploiting the price determined on a separate exchange market. Although popular in practice, size discovery reduces the depth of exchange markets and, as modeled, worsens overall allocative efficiency. On the other hand, fragmenting trade in the same asset across multiple exchanges can improve allocative efficiency. This talk draws from research with Samuel Antill and Daniel Chen.
Speaker: Tomoyuki Ichiba (University of California Santa Barbara)
Title: Relative arbitrage among investors
Abstract: The relative arbitrage portfolio, formulated in Stochastic Portfolio Theory (SPT), outperforms a market portfolio over a given time-horizon with probability one under some conditions on the volatilities in the market, where the optimal relative arbitrage can be characterized by the smallest nonnegative continuous solution of a Cauchy problem. In this talk, we consider two regimes: finitely many investors and mean-field, and the corresponding Nash equilibrium of investors who compete with a benchmark determined by the market portfolio and other investors’ performance. With the market price of risk processes depending on the market portfolio and total volumes invested, we solve the multi-agent optimization problem under the framework of SPT. This is joint work with Tianjiao Yang.
Speaker: Bruno Bouchard (Université Paris Dauphine)
Title: Ito’s formula for concave or C1 path-dependent functions and applications in mathematical finance
Abstract: We will discuss several versions of the Ito’s formula in the case where the function is path-dependent and only concave or C1 in the sense of Dupire. In particular, we will show that it can be used to solve (super-) hedging problems in the context of market-impact or under volatility uncertainty.
Title: COVID-19: Modelling Another Global Systemic Phenomenon
Abstract: This talk will describe my efforts to comprehend the second great global crisis in our lifetime, based on what I learned from trying to model the Great Financial Crisis. I’ll try to convince you that insights into systemic risk made by financial mathematicians lead to network pandemic models that provide unified understanding not easy to discern from conventional SIR models. The Inhomogeneous Random Social Network (IRSN) framework, a direct offshoot of the Inhomogeneous Random Financial Network (IRFN) framework I developed in 2019, combines agent-based assumptions with a hierarchical network architecture for human society, with an aim to capture the daily dynamics of the spread of infection in a highly heterogenous population. The stochastic cascade dynamics can be simulated for networks with a moderate size N, or, under a certain convenient mathematical assumption, computed analytically in the large N limit. Since agent-based simulation experiments rapidly exceed available computer resources, I like to see what can be learned with the large N shortcut, which is comparable in computation time to conventional SIR methods. Three aspects of the framework illustrate some of the underlying mathematical ideas: the dose-response mechanism for infection, the role of superspreaders and how to capture frailty bias.
The paper introducing this model can be downloaded here: https://pubmed.ncbi.nlm.nih.gov/33313455/
Title: Improving Value-at-Risk prediction under model uncertainty
Abstract: Several well-established benchmark predictors exist for the Value-at-risk (VaR), a main instrument for financial risk management. Hybrid methods combining AR-GARCH filtering with skewed-t residuals and extreme value theory-based approach are particularly recommended. This study introduces yet another VaR predictor, the G-VaR, following an entirely novel methodology. Inspired by the recent mathematical theory of sublinear expectation, the G-VaR is built upon the concept of model uncertainty which, in the present case, signifies that the inherent volatility of financial returns cannot be characterized by a single but infinite many statistical distributions. By considering a worst scenario among these potential distributions, the G-VaR predictor is precisely identified.
Experiments on both the NASDAQ Composite Index and the S&P500 index demonstrate an excellent performance of the G-VaR predictor compared to most of benchmark VaR predictors. This talk is based on joint works with Shuzhen Yang and Jianfeng Yao.
Thursday, 25 February 2021, 12:00 (GMT +1) please note the different time – around lunchtime in central Europe!
Speaker: Alexander Schied (University of Waterloo)
Title: Robustness in risk measurement: the impact of incentives
Abstract: Statistical robustness is a desirable property for a regulatory risk measure. Previous research has stressed that Value at Risk is more robust than Expected Shortfall if both are applied to the same financial position. In reality, however, the regulatory choice of a particular risk measure imposes certain incentives, which impact the underlying position even before a particular risk measure is applied. Thus, one cannot decouple the technical properties of a risk measure from the incentives it creates. In this talk, we describe a first attempt of taking such incentives into account when assessing a risk measure’s robustness properties. To this end, we develop a general methodology which we call “robustness against optimization”. The new notion is studied for various classes of risk measures and expected utility and loss. In doing so, we arrive at conclusions, which are different from those of the previous literature and perhaps somewhat surprising. The talk is based on joint work with Paul Embrechts and Ruodu Wang.
Title: PDGM: a Neural Network Approach to Solve Path-Dependent Partial Differential Equations
Abstract: In this talk, we present a novel numerical method for Path-Dependent Partial Differential Equations (PPDEs). These equations firstly appeared in the seminal work of Dupire [QF, 2019, originally published in 2009], where the functional Itô calculus was developed to deal with path-dependent financial derivatives contracts. More specifically, we generalize the Deep Galerkin Method (DGM) of Sirignano and Spiliopoulos  to deal with these equations. The method, which we call Path-Dependent DGM (PDGM), consists of using a combination of feed-forward and Long Short-Term Memory architectures to model the solution of the PPDE. We then analyze several numerical examples from the Financial Mathematics literature that show the capabilities of the method under very different situations.
Title: Optional pricing in a non-linear incomplete market model with default: the European and American cases
Abstract: We study option pricing in an incomplete market consisting of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. We consider the case when the portfolio processes follow non-linear dynamics with a non-linear driver f.
We first study the superhedging prices and associated superhedging strategies for European options. By using a dynamic programming approach, we provide a dual formulation of the seller’s (superhedging) price involving a suitable set of equivalent probability measures, which we call f-martingale probability measures. We also establish a characterization of the seller’s price as the initial value of the minimal supersolution of a constrained Backward Stochastic Differential Equation with default.
We then study American options with irregular payoff in this market. Both points of view of the seller and of the buyer are analyzed. We give a dual representation of the seller’s (superhedging) price in terms of the value of a non-linear mixed control/stopping problem, and provide two infinitesimal characterizations of the seller’s price process in terms of the minimal supersolution of a constrained reflected BSDE and of an optional reflected BSDE. Under some regularity assumptions on the payoff, we also prove a duality result for the buyer’s price in terms of the value of a non-linear control/stopping game problem.
The talk is based on joint works with Miryana Grigorova, University of Leeds and Marie-Claire Quenez, LPSM, Université Paris Denis Diderot.