I'll give a general introduction to tensor-triangular geometry, the algebraic study of tensor-triangulated categories as they appear in topology, geometry and representation theory. Then I'll discuss an elementary idea, that of a "field" in this theory, and explain what we currently know about them.
This paper studies the spread of losses and defaults in financial networks with two important features: collateral requirements and alternative contract termination rules in bankruptcy. When collateral is committed to a firm’s counterparties, a solvent firm may default if it lacks sufficient liquid assets to meet its payment obligations. Collateral requirements can thus increase defaults and payment shortfalls. Moreover, one firm may benefit from the failure of another if the failure frees collateral committed by the surviving firm, giving it additional resources to make other payments. Contract termination at default may also improve the ability of other firms to meet their obligations. As a consequence of these features, the timing of payments and collateral liquidation must be carefully specified, and establishing the existence of payments that clear the network becomes more complex. Using this framework, we study the consequences of illiquid collateral for the spread of losses through fire sales; we compare networks with and without selective contract termination; and we analyze the impact of alternative bankruptcy stay rules that limit the seizure of collateral at default. Under an upper bound on derivatives leverage, full termination reduces payment shortfalls compared with selective termination.
In this talk, we shall propose a relaxed control regularization with general exploration rewards to design robust feedback controls for multi-dimensional continuous-time stochastic exit time problems. We establish that the regularized control problem admits a H\”{o}lder continuous feedback control, and demonstrate that both the value function and the feedback control of the regularized control problem are Lipschitz stable with respect to parameter perturbations. Moreover, we show that a pre-computed feedback relaxed control has a robust performance in a perturbed system, and derive a first-order sensitivity equation for both the value function and optimal feedback relaxed control. These stability results provide a theoretical justification for recent reinforcement learning heuristics that including an exploration reward in the optimization objective leads to more robust decision making. We finally prove first-order monotone convergence of the value functions for relaxed control problems with vanishing exploration parameters, which subsequently enables us to construct the pure exploitation strategy of the original control problem based on the feedback relaxed controls. This is joint work with Christoph Reisinger (available at https://arxiv.org/abs/2001.03148).
We adopt Deep Reinforcement Learning algorithms to design trading strategies for continuous futures contracts. Both discrete and continuous action spaces are considered and volatility scaling is incorporated to create reward functions which scale trade positions based on market volatility. We test our algorithms on the 50 most liquid futures contracts from 2011 to 2019, and investigate how performance varies across different asset classes including commodities, equity indices, fixed income and FX markets. We compare our algorithms against classical time series momentum strategies, and show that our method outperforms such baseline models, delivering positive profits despite heavy transaction costs. The experiments show that the proposed algorithms can follow large market trends without changing positions and can also scale down, or hold, through consolidation periods.
The full-length text is available at https://arxiv.org/abs/1911.10107.
In this talk we will discuss a data-driven approach based on neural networks (NN) for calibrating financial asset price models. Determining optimal values of the model parameters is formulated as training hidden neurons within a machine learning framework, based on available financial option prices. The framework consists of two parts: a forward pass in which we train the weights of the NN off-line, valuing options under many different asset model parameter settings; and a backward pass, in which we evaluate the trained NN-solver on-line, aiming to find the weights of the neurons in the input layer. We will show how the same data-driven approach can be used to estimate the Black-Scholes implied volatility and dividend yield for American options in a fast and robust way. We then discuss the complexity of the optimization problem through an analysis of the loss surface of the neural network. We finally will present some numerical examples which show that neural networks can be an efficient and reliable technique for the calibration of financial assets and the extraction of implied information.
Morse theory has a long history with many spectacular applications in different areas of mathematics. In this talk I will explain an extension of the main theorem of Morse theory that works for a large class of functions on singular spaces. The main example to keep in mind is that of moment maps on varieties, and I will present some applications to the topology of symplectic quotients of singular spaces.
It is well-known that the Teichmüller space of a compact surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R). Higher Teichmüller components are generalizations of this that exist for the moduli space of representations of the fundamental group into certain real simple Lie groups of higher rank. As for the usual Teichmüller space, these components consist entirely of discrete and faithful representations. Several cases have been identified over the years. First, the Hitchin components for split groups, then the maximal Toledo invariant components for Hermitian groups, and more recently certain components for SO(p,q). In this talk, I will describe a general construction of (still somewhat conjecturally) all possible Teichmüller components, and a parametrization of them using Higgs bundles.
We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.
The 2007-2008 financial crisis forced governments to choose between the unattractive alternatives of either bailing out a systemically important bank (SIBs) or having it fail in a disruptive manner. Bail-in has been put forward as the primary tool to resolve a failing bank, which would end the too-big-to-fail problem by letting stakeholders shoulder the losses, while minimising the calamitous systemic impact of a bank failure. Though the aptness of bail-in has been evinced in relatively minor idiosyncratic bank failures, its efficacy in maintaining stability in cases of large bank failures and episodes of system-wide crises remains to be practically tested. This paper investigates the financial stability implications of the bail-in design, in all these cases. We develop a multi-layered network model of the European financial system that captures the prevailing endogenous-amplification mechanisms: exposure loss contagion, overlapping portfolio contagion, funding contagion, bail-inable debt revaluations, and bail-inable debt runs. Our results reveal that financial stability hinges on a set of `primary' and `secondary' bail-in parameters, including the failure threshold, recapitalisation target, debt-to-equity conversion rate, loss absorption requirements, debt exclusions and bail-in-design certainty – and we uncover how. We also demonstrate that the systemic footprint of the bail-in design is not properly understood without the inclusion of multiple contagion mechanisms and non-banks. Our evidence fortunately suggests that the pivot for stability is in the hands of policymakers. It also suggests, however, that the current bail-in design might be in the regime of instability.
In this paper we extend the existing literature on xVA along three directions. First, we enhance current BSDE-based xVA frameworks to include initial margin by following the approach of Crépey (2015a) and Crépey (2015b). Next, we solve the consistency problem that arises when the front- office desk of the bank uses trade-specific discount curves that differ from the discount curve adopted by the xVA desk. Finally, we address the existence of multiple aggregation levels for contingent claims in the portfolio between the bank and the counterparty, providing suitable extensions of our proposed single-claim xVA framework.
This is a joint work with: Francesca Biagini and Immacolata Oliva
Preprint available at: https://arxiv.org/abs/1905.11328