Title: Pricing options on flow forwards by neural networks in Hilbert space
Abstract: We propose a new methodology for pricing options on flow forwards by applying infinite-dimensional neural networks. We recast the pricing problem as an optimization problem in a Hilbert space of real-valued function on the positive real line, which is the state space for the term structure dynamics. This optimization problem is solved by facilitating a novel feedforward neural network architecture designed for approximating continuous functions on the state space. The proposed neural net is built upon the basis of the Hilbert space. We provide a case study that shows excellent numerical efficiency, with superior performance over that of a classical neural net trained on sampling the term structure curves.
Title: PCF-GAN: generating sequential data via the characteristic function of measures on the path space
Abstract: Implicit Generative Models (IGMs) have demonstrated the superior capacity in generating high-fidelity samples from the high dimensional space, especially for static image data. However, these methods struggle to capture the temporal dependence of joint probability distributions induced by time-series data. To tackle this issue, we directly compare the path distributions via the characteristic function of measures on the path space (PCF) from rough path theory, which uniquely characterises the law of stochastic processes. The distance metric via PCF enjoyed theoretical properties, including boundedness and differentiability with respect to generator parameters. The PCF can also be thought as a variant of the MMD loss on the path space, which enjoys linear time complexity in the sample size, in contrast with the quadratic-time Maximum Mean Discrepancy (MMD). Furthermore, the PCF loss can be optimised based on the path distribution by learning the optimal unitary representation of PCF, which avoids the need for manual kernel selection, and leads to an improvement in test power relative to the original PCF. Numerical results demonstrate that the proposed PCF-GAN consistently outperforms state-of-the-art baselines on several benchmarking datasets (e.g, rough volatility data, google stock data and air-quality data) in terms of various test metrics.
Title: Systemic Risk in Markets with Multiple Central Counterparties
Abstract: We provide a framework for modelling risk and quantifying payment shortfalls in cleared markets with multiple central counterparties (CCPs). Building on the stylised fact that clearing membership is shared among CCPs, we show that stress in this shared membership can transmit across markets through multiple CCPs. We provide stylised examples to lay out how such stress transmission can take place, as well as empirical evidence to illustrate that the mechanisms we study could be relevant in practice. Furthermore, we show how stress mitigation mechanisms such as variation margin gains haircutting by one CCP can have spillover effects on other CCPs. The framework can be used to enhance CCP stress-testing, which currently relies on the “Cover 2” standard requiring CCPs to be able to withstand the default of their two largest clearing members. We show that who these two clearing members are can be significantly affected by higher-order effects arising from interconnectedness through shared clearing membership.
This is joint work with Iñaki Aldasoro (Bank for International Settlements).
Speaker: Charles-Albert Lehalle (Abu Dhabi Investment Authority (ADIA) and Visiting Professor at Imperial College London)
Title: Mathematics Of Data Curation For Financial Applications
Abstract: The emergence of alternative data (satellite images, texts, credit cards, vessels positions, etc) opened the door to nowcasting a few years ago. Nowcasting targets to understand the current state of the physical and economic world, collecting information everywhere it is available. Hence, when used properly, these datasets can shed light on investment decisions and on the valuation of tradable instruments.
I am working with such datasets for more than six years now, managing teams of researchers using such them to feed investment strategies. It appears to me that dealing with such datasets for financial applications is very subtle, and that it requires to carefully identify their bias a way that is not needed for other applications. Besides, an important challenge is to be able to combine these datasets, or datasets of different origins describing the same phenomenon. For instance, the supply chain for companies is partially known by observing factories, or by picking the right information in the quarterly reports of the companies, and by observing the positions of vessels and ships on the oceans.
More recently I started to collect formal knowledge on the process of Data Curation, around concepts like Post-Stratification, Active Learning, Covariate Shift, Causality and Semi-Supervised Learning. During this talk I will share my understanding of these fields and how I see them providing solutions for nowcasting.
Speaker: Lisa Goldberg (University of California, Berkeley)
Title: James Stein for eigenvectors
Abstract: Estimated covariance matrices are widely used to construct portfolios with variance-minimizing optimization, yet the embedded sampling error produces portfolios with systematically underestimated variance. This effect is especially severe when the number of securities greatly exceeds the number of observations. In this high dimension low sample size (HL) regime, we show that a dispersion bias in the leading eigenvector of the estimated covariance matrix is a material source of distortion in the minimum variance portfolio. We correct the bias with the data-driven GPS (Global Positioning System) shrinkage estimator, which improves with the size of the market, and which is structurally identical to the James Stein estimator for a collection of averages. We illustrate the power of the GPS estimator with a numerical example, and conclude with open problems that have emerged from our research.
Title: Term structure modeling with overnight rates beyond stochastic continuity
Abstract: In the current reform of interest rate benchmarks, a central role is played by risk-free rates (RFRs), such as SOFR (secured overnight financing rate) in the US. A key feature of RFRs is the presence of jumps and spikes at periodic time intervals as a result of regulatory and liquidity constraints. This corresponds to stochastic discontinuities (i.e., jumps occurring at predetermined dates) in the dynamics of RFRs. In this work, we propose a general modelling framework where RFRs and term rates can have stochastic discontinuities and characterize absence of arbitrage in an extended HJM setup. When the term rate is generated by the RFR itself, we show that it solves a BSDE, whose driver is determined by the HJM drift restrictions. We develop a tractable specification driven by affine semimartingales, also extending the classical short rate approach to the case of stochastic discontinuities. In this context, we show that a simple specification allows to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Finally, we study hedging in the sense of local risk-minimization when the underlying term structures have stochastic discontinuities.
Based on joint work with Zorana Grbac and Thorsten Schmidt.
Title: A unifying approach to viability and arbitrage
Abstract: In this talk we address fundamental questions of both mathematical finance and financial economics by means of the concept of viability. We briefly present the ideas of Harrison and Kreps and how they posed the basis for the classical mathematical finance literature built on the existence of a reference probability measure. We discuss the need for an extension of such a theory, even in the probabilistic case, and construct a unifying framework based on a common order able to encompass classical probabilistic models as well as modern non probabilistic models of Knightian Uncertainty. We derive the equivalence of economic viability of asset prices and absence of arbitrage, under suitably revised notions, and a version of the fundamental theorem of asset pricing using the notion of sublinear pricing measures. Different versions of the Efficient Market Hypothesis can be formulated within our framework and we show that they are related to the assumptions one is willing to impose on the common order.
This talk is mainly based on a joint work with F. Riedel and H.M. Soner.
Speaker: Johannes Muhle-Karbe (Imperial College London)
Title: Liquidity Risk and Asset Prices
Abstract: How do asset prices depend on liquidity, that is, the ease with which the assets can be traded? And what is the impact of liquidity risk, that is, unpredictable changes of liquidity over time? To address such questions, we study equilibrium asset pricing models with transaction costs. In this context, the market clearing equilibrium prices and the corresponding optimal trading strategies can be generally characterised by systems of forward-backward stochastic differential equations. Even though these systems are multidimensional and fully coupled, surprisingly explicit results can still be obtained by approximating the solution for small transaction costs. This leads to a range of explicitly solvable settings that allow to study the interplay of liquidity risk, trading needs, and asset prices.
(Joint work in progress with Agostino Capponi and Xiaofei Shi.)