The estimation of loss distributions for dynamic portfolios requires the simulation of scenarios representing realistic joint dynamics of their components. We propose a novel data-driven approach for simulating realistic, high-dimensional multi-asset scenarios, focusing on accurately representing tail risk for a class of static and dynamic trading strategies. We exploit the joint elicitability property of Value-at-Risk (VaR) and Expected Shortfall (ES) to design a Generative Adversarial Network (GAN) that learns to simulate price scenarios preserving these tail risk features. We demonstrate the performance of our algorithm on synthetic and market data sets through detailed numerical experiments.

Question of the day: how to manage a large (or small) portfolio in low interest rate conditions, while equity markets bear significant draw-down risk? More generally, how to build an “antifragile” portfolio that can weather the most extreme market scenarios without impacting long-term performances? Do active strategies systematically create or increase already existing market instabilities?

Residual networks, or ResNets, are multilayer neural network architectures in which a skip connection is introduced at every layer. This allows very deep networks to be trained by circumventing vanishing and exploding gradients, mentioned in [3]. The increased depth in ResNets has lead to commensurate performance gains in applications ranging from speech recognition to computer vision.

The clearing of over-the-counter transactions through central counterparties (CCPs), one of the pillars of financial reform following the crisis of 2007-2008, has promoted CCPs as key elements of the new global financial architecture. Given the cost of implementing central clearing mandates and the associated collateral requirements, it is important to examine how these reforms have affected risks in the financial system and whether central clearing has attained the initial objective of the reform, which is to enhance financial stability and reduce systemic risk. We show that, rather than eliminating counterparty risk, central clearing transforms it into liquidity risk: margin calls transform accounting losses into realised losses which affect the liquidity buffers of clearing members. Accordingly, initial margin and default fund calculations should account for this liquidity risk in a realistic manner, especially for large positions. While recent discussions have centered on the solvency of CCPs, their capital and ‘skin-in-the-game’ and capital requirements for CCP exposures of banks, we argue that these issues are secondary and that the main focus of risk management and financial stability analysis should be on the liquidity of clearing members and the liquidity resources of CCPs. Clearing members should assess their exposure to CCPs in terms of liquidity, rather than counterparty risk. Stress tests involving CCPs should focus on liquidity stress testing and adequacy of liquidity resources.

The traditional approach to the stress testing of financial institutions focuses on capital adequacy and solvency. Liquidity stress tests have been applied in parallel to and independently from solvency stress tests, based on scenarios which may not be consistent with those used in solvency stress tests. We propose a structural framework for the joint stress testing of solvency and liquidity: our approach exploits the mechanisms underlying the solvency-liquidity nexus to derive relations between solvency shocks and liquidity shocks. These relations are then used to model liquidity and solvency risk in a coherent framework, involving external shocks to solvency and endogenous liquidity shocks arising from these solvency shocks. We define the concept of “Liquidity at Risk”, which quantifies the liquidity resources required for a financial institution facing a stress scenario. Finally, we show that the interaction of liquidity and solvency may lead to the amplification of equity losses due to funding costs which arise from liquidity needs.

Do option investors rationally exercise their options? Numerous studies report evidence of irrational behavior. In this paper, we pay careful attention to intraday option quotes and reach the opposite conclusion. An exercise boundary violation (EBV) occurs when the best bid price for an American option is below the option’s intrinsic value. Far from being unusual, we show that EBVs occur very frequently. Under these conditions, the rational response of an investor liquidating an option is to exercise the option rather than sell it. Empirically, we find that the likelihood of early exercise is strongly influenced by the existence and duration of EBVs. Not only do these results reverse standard theory on American option valuation and optimal exercise strategy, but they also suggest that the ability to avoid selling at an EBV price creates an additional source of value for American options that is unrelated, and in addition to, dividend payments. This additional value may help explain why American options appear overpriced relative to European options.

We provide a mathematical definition of fragility a semi-measure of dispersion and and antifragility as negative or positive sensitivity to volatility (a variant of negative or positive “vega”) and examine the link to nonlinear effects. We integrate model error (and biases) into the fragile or antifragile context. Unlike risk, which is linked to psychological notions such as subjective preferences (hence cannot apply to a coffee cup) we offer a measure that is universal and concerns any object that has a probability distribution (whether such distribution is known or, critically, unknown). We propose a detection of fragility, robustness, and antifragility using a single “fast-and-frugal”, model-free, probability free heuristic that also picks up exposure to model error.

We are able to explicitly compute the equilibrium strategies, in two steps. We first derive the optimal strategy of the high-frequency trader given any strategy adopted by the investor. Then, we solve the problem of the investor given the optimal strategy of the high-frequency trader, in terms of the resolvent of a Fredholm integral equation. Our results show that the high-frequency trader adopts a predatory strategy whenever the value of the trading signal is high, and follows a cooperative strategy otherwise. We also show that there is a net gain in performance for the investor from taking into account the order flow of the high-frequency trader. A U-shaped intraday pattern in trading volume is shown to arise endogenously as a result of the strategic behavior of the agents.

Column Generation (CG) is an iterative algorithm for solving linear programs (LPs) with an extremely large number of variables (columns). CG is the workhorse for tackling large-scale integer linear programs, which rely on CG to solve LP relaxations within a branch and price algorithm. Two canonical applications are the Cutting Stock Problem (CSP) and Vehicle Routing Problem with Time Windows (VRPTW). In VRPTW, for example, each binary variable represents the decision to include or exclude a route, of which there are exponentially many; CG incremen- tally grows the subset of columns being used, ultimately converging to an optimal solution. We propose RLCG, the first Reinforcement Learning (RL) approach for CG. Unlike typical column selection rules which myopically select a column based on local information at each iteration, we treat CG as a sequential decision-making problem: the column selected in a given iteration affects subsequent column selec- tions. This perspective lends itself to a Deep Reinforcement Learning approach that uses Graph Neural Networks (GNNs) to represent the variable-constraint structure in the LP of interest. We perform an extensive set of experiments using the publicly available BPPLIB benchmark for CSP and Solomon benchmark for VRPTW. RLCG converges faster and reduces the number of CG iterations by 22.4% for CSP and 40.9% for VRPTW on average compared to a commonly used greedy policy.

In highly connected financial networks, the failure of a single institution can cascade into additional bank failures. This systemic risk can be mitigated by adjusting the loans, holding shares, and other liabilities connecting institutions in a way that prevents cascading of failures. We are approaching the systemic risk problem by attempting to optimize the connections between the institutions. In order to provide a more realistic simulation environment, we have incorporated nonlinear/discontinuous losses in the value of the banks. To address scalability challenges, we have developed a two-stage algorithm where the networks are partitioned into modules of highly interconnected banks and then the modules are individually optimized. We developed new algorithms for classical and quantum partitioning for directed and weighted graphs (first stage) and a new methodology for solving Mixed Integer Linear Programming problems with constraints for the systemic risk context (second stage). We compare classical and quantum algorithms for the partitioning problem. Experimental results demonstrate that our two-stage optimization with quantum partitioning is more resilient to financial shocks, delays the cascade failure phase transition, and reduces the total number of failures at convergence under systemic risks with reduced time complexity.

Time series observations can be seen as realizations of an underlying dynamical system governed by rules that we typically do not know. For time series learning tasks we create our model using available data. Training on available realizations, where data is limited, often induces severe over-fitting thereby preventing generalization. To address this issue, we introduce a general recursive framework for time series augmentation, which we call the Recursive Interpolation Method (RIM). New augmented time series are generated using a recursive interpolation function from the original time series for use in training. We perform theoretical analysis to characterize the proposed RIM and to guarantee its performance under certain conditions. We apply RIM to diverse synthetic and real-world time series cases to achieve strong performance over non-augmented data on a variety of learning tasks. Our method is also computationally more efficient and leads to better performance when compared to state of the art time series data augmentation.

Reinforcement learning (RL) for dynamic asset allocation is an emerging field of study. Total return, the common performance metric, is useful for comparing algorithms but does not help us determine how close an RL algorithm is to an optimal solution. In real-world financial applications, a bad deci- sion could prove to be fatal. One of the key ideas of our work is to combine the two paradigms of the mean-variance optimization approach (Markowitz criteria) and the optimal capital growth approach (Kelly criteria) via the actor-critic approach. By using an actor-critic approach, we can balance optimization of risk and growth by configuring the actor to optimize the mean-variance while the critic is configured to maximize growth. We propose a Geometric Policy Score used by the critic to assess the quality of the actions taken by the actor. This could allow portfolio manager prac- titioners to better understand the investment RL policy. We present an extensive and in-depth study of RL algorithms for use in portfolio management (PM). We studied eight published policy-based RL algorithms which are preferred over value-based RL because they are better suited for contin- uous action spaces and are considered to be state of the art, Deterministic Policy Gradient (DPG), Stochastic Policy Gradients (SPG), Deep Deterministic Policy Gradient (DDPG), Trust Region Pol- icy Optimization (TRPO), Proximal Policy Optimization (PPO), Twin Delayed Deep Deterministic Policy Gradient (TD3), Soft Actor Critic (SAC), and Evolution Strategies (ES) for Policy Optimiza- tion. We implemented all eight and we were able to modify all of them for PM but our initial testing determined that their performance was not satisfactory. Most algorithms showed difficulty converg- ing during the training process due to the non-stationary and noisy nature of financial environments, along with other challenges. We selected the four most promising algorithms DPG, SPG, DDPG, PPO for further improvements. The modification of RL algorithms to finance required unconven- tional changes. We have developed a novel approach for encoding multi-type financial data in a way that is compatible with RL. We use a multi-channel convolutional neural network (CNN-RL) framework, where each channel corresponds to a specific type of data such as high-low-open-close prices and volumes. We also designed a reward function based on concepts such as alpha, beta, and diversification that are financially meaningful while still being learnable by RL. In addition, port- folio managers will typically use a blend of time series analysis and cross-sectional analysis before making a decision. We extend our approach to incorporate, for the first time, cross-sectional deep RL in addition to time series RL. Finally, we demonstrate the performance of the RL agents and benchmark them against commonly used passive and active trading strategies, such as the uniform buy-and-hold (UBAH) index and the dynamical multi-period Mean-Variance-Optimization (MVO) model.

Traditional model validation has become a staple of the finance industry, a well-understood and rigorously developed line of defense intended to help institutions avoid unnecessary financial losses caused by poorly performing models. Using tools such as logistic regression, time series modelling, and model benchmarking, the field has progressed steadily using traditional statistical validation techniques and tools. Many institutions by now have already improved their traditional model validation tools by incorporating machine learning (ML) and artificial intelligence (AI) tools to identify additional features used to better predict outcomes or disasters. The ML/AI algorithms use much larger datasets and uncover hidden relationships that often lead to big improvements in predictive power. However, the datasets used to develop ML/AI tools are also often static, even if they are updated at discrete intervals. Of course, one can also break any dataset into in-time and out-of-time samples, but even that can be considered a kind of static dataset. In real-life situations, one may need non-static models that go beyond static ML/AI. We are seeing a new wave in model validation in consumer credit that supplements the traditional and ML/AI algorithms by incorporating truly dynamic elements, intended to catch models as they might be failing, using real-time updated data and AI algorithms. These models can be sensitized to unexpected changes in data or predictions, triggering review or corrective action when necessary.

We will analyze the derivatives industry using modern concepts of strategy and industrial structure, focusing on five economic competitive forces with a discussion of the competitive forces of some major players. This economic framework will help us better analyze the economic structure of the industry and better describe the important changes that are happening. An understanding of the industry structure would allow derivatives players make more informed decisions about product positioning, cost positioning, and many other strategic business choices.

This textbook introduces students to the basics of stochastic calculus while inviting them to solve cash flow valuation problems. This provides a useful foundation for the further study of derivatives pricing. Numerous exercises are provided with solutions.

Reframing modern portfolio theory with Gaussian cash flows rather than percentage returns, the CFPM (cash flow portfolio model) sets a structural foundation for valuing both traditional capital assets and derivatives. Asset prices are shown to be decreasing functions of both cash flow covariances and variances. The usual single-period CAPM formulas can be derived, but the expected returns are determined endogenously. All risk is implicitly priced in expected returns, leading to reinterpreted rules for portfolio selection and capital budgeting. Derivatives obey the same total covariance-based pricing relationships as cash flows, except that they exist in zero net supply. Together with a single no-arbitrage convexity constraint, the Bachelier and Black-Scholes/Merton option pricing models are derived in a discrete time setting without continuous trading. The closed-form CFPM extends to multiple periods. The resulting model aspires to replace multiperiod discounting of cash flows at constant single-period CAPM discount rates.