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Emissions markets play a crucial role in reducing pollution by encouraging firms to minimize costs. However, their structure heavily relies on the decisions of policymakers, on the future economic activities, and on the availability of abatement technologies. This study examines how changes in regulatory standards, firms' abatement costs, and emissions levels affect allowance prices and firms' efforts to reduce emissions. This is done in a Radner equilibrium framework encompassing inter-temporal decision-making, uncertainty, and a comprehensive assessment of the market dynamics and outcomes. The results of this research have the potential to assist policymakers in enhancing the structure and efficiency of emissions trading systems, through an in-depth comprehension of the reactions of market stakeholders towards different market situations.

Cr\'epey, Frikha, and Louzi (2023) introduced a multilevel stochastic approximation scheme to compute the value-at-risk of a financial loss that is only simulatable by Monte Carlo. The optimal complexity of the scheme is in $O({\varepsilon}^{-5/2})$, ${\varepsilon} > 0$ being a prescribed accuracy, which is suboptimal when compared to the canonical multilevel Monte Carlo performance. This suboptimality stems from the discontinuity of the Heaviside function involved in the biased stochastic gradient that is recursively evaluated to derive the value-at-risk. To mitigate this issue, this paper proposes and analyzes a multilevel stochastic approximation algorithm that adaptively selects the number of inner samples at each level, and proves that its optimal complexity is in $O({\varepsilon}^{-2}|\ln {\varepsilon}|^{5/2})$. Our theoretical analysis is exemplified through numerical experiments., Comment: 43 pages, 6 tables, 4 figures

We present a unified framework for computing CVA sensitivities, hedging the CVA, and assessing CVA risk, using probabilistic machine learning meant as refined regression tools on simulated data, validatable by low-cost companion Monte Carlo procedures. Various notions of sensitivities are introduced and benchmarked numerically. We identify the sensitivities representing the best practical trade-offs in downstream tasks including CVA hedging and risk assessment., Comment: This is the long, preprint version of the eponymous paper forthcoming in Risk Magazine

Invariance times are stopping times $\tau$ such that local martingales with respect to some reduced filtration and an equivalently changed probability measure, stopped before $\tau$ , are local martingales with respect to the original model filtration and probability measure. They arise naturally for modeling the default time of a dealer bank, in the mathematical finance context of counterparty credit risk. Assuming an invariance time endowed with an intensity and a positive Az{\'e}ma supermartingale, this work establishes a dictionary relating the semimartingale calculi in the original and reduced stochastic bases, regarding in particular conditional expectations, martingales, stochastic integrals, random measure stochastic integrals, martingale representation properties, semimartingale characteristics, Markov properties, transition semigroups and infinitesimal generators, and solutions of backward stochastic differential equations.

We propose a distributional formulation of the spanning problem of a multi-asset payoff by vanilla basket options. This problem is shown to have a unique solution if and only if the payoff function is even and absolutely homogeneous, and we establish a Fourier-based formula to calculate the solution. Financial payoffs are typically piecewise linear, resulting in a solution that may be derived explicitly, yet may also be hard to numerically exploit. One-hidden-layer feedforward neural networks instead provide a natural and efficient numerical alternative for discrete spanning. We test this approach for a selection of archetypal payoffs and obtain better hedging results with vanilla basket options compared to industry-favored approaches based on single-asset vanilla hedges.

Motivated by the equations of cross valuation adjustments (XVAs) in the realistic case where capital is deemed fungible as a source of funding for variation margin, we introduce a simulation/regression scheme for a class of anticipated BSDEs, where the coefficient entails a conditional expected shortfall of the martingale part of the solution. The scheme is explicit in time and uses neural network least-squares and quantile regressions for the embedded conditional expectations and expected shortfall computations. An a posteriori Monte Carlo validation procedure allows assessing the regression error of the scheme at each time step. The superiority of this scheme with respect to Picard iterations is illustrated in a high-dimensional and hybrid market/default risks XVA use-case.

Based on supermodularity ordering properties, we show that convex risk measures of credit losses are nondecreasing w.r.t. credit-credit and, in a wrong-way risk setup, credit-market, covariances of elliptically distributed latent factors. These results support the use of such setups for computing credit provisions and economic capital or for conducting stress test exercises and risk management analysis.

Cr\'epey, Frikha, and Louzi (2023) introduced a nested stochastic approximation algorithm and its multilevel acceleration to compute the value-at-risk and expected shortfall of a random financial loss. We hereby establish central limit theorems for the renormalized estimation errors associated with both algorithms as well as their averaged versions. Our findings are substantiated through a numerical example., Comment: 56 pages, 1 figure, 4 tables

For vanilla derivatives that constitute the bulk of investment banks' hedging portfolios, central clearing through central counterparties (CCPs) has become hegemonic. A key mandate of a CCP is to provide an efficient and proper clearing member default resolution procedure. When a clearing member defaults, the CCP can hedge and auction or liquidate its positions. The counterparty credit risk cost of auctioning has been analyzed in terms of XVA metrics in Bastide, Cr{\'e}pey, Drapeau, and Tadese (2023). In this work we assess the costs of hedging or liquidating. This is done by comparing pre- and post-default market equilibria, using a Radner equilibrium approach for portfolio allocation and price discovery in each case. We show that the Radner equilibria uniquely exist and we provide both analytical and numerical solutions for the latter in elliptically distributed markets. Using such tools, a CCP could decide rationally on which market to hedge and auction or liquidate defaulted portfolios.

Darwinian model risk is the risk of mis-price-and-hedge biased toward short-to-medium systematic profits of a trader, which are only the compensator of long term losses becoming apparent under extreme scenarios where the bad model of the trader no longer calibrates to the market. The alpha leakages that characterize Darwinian model risk are undetectable by the usual market risk tools such as value-at-risk, expected shortfall, or stressed value-at-risk.Darwinian model risk can only be seen by simulating the hedging behavior of a bad model within a good model. In this paper we extend to callable assets the notion of hedging valuation adjustment introduced in previous work for quantifying and handling such risk. The mathematics of Darwinian model risk for callable assets are illustrated by exact numerics on a stylized callable range accrual example. Accounting for the wrong hedges and exercise decisions, the magnitude of the hedging valuation adjustment can be several times larger than the mere difference, customarily used in banks as a reserve against model risk, between the trader's price of a callable asset and its fair valuation.

We propose a multilevel stochastic approximation (MLSA) scheme for the computation of the value-at-risk (VaR) and expected shortfall (ES) of a financial loss, which can only be computed via simulations conditional on the realization of future risk factors. Thus, the problem of estimating its VaR and ES is nested in nature and can be viewed as an instance of stochastic approximation problems with biased innovations. In this framework, for a prescribed accuracy $\epsilon$, the optimal complexity of a nested stochastic approximation algorithm is shown to be of order $\epsilon$--3. To estimate the VaR, our MLSA algorithm attains an optimal complexity of order $\epsilon$--2--$\delta$ , where $\delta$ \< 1 is some parameter depending on the integrability degree of the loss, while to estimate the ES, it achieves an optimal complexity of order $\epsilon$--2 |ln $\epsilon$|2. Numerical studies of the joint evolution of the error rate and the execution time demonstrate how our MLSA algorithm regains a significant amount of the performance lost due to the nested nature of the problem.

We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finite dimensional Gaussian process (GP) regression approach under no-arbitrage constraints based on prices, and a neural net (NN) approach with penalization of arbitrages based on implied volatilities. We demonstrate the performance of these approaches relative to the SSVI industry standard. The GP approach is proven arbitrage-free, whereas arbitrages are only penalized under the SSVI and NN approaches. The GP approach obtains the best out-of-sample calibration error and provides uncertainty quantification.The NN approach yields a smoother local volatility and a better backtesting performance, as its training criterion incorporates a local volatility regularization term.

We consider the computation by simulation and neural net regression of conditional expectations, or more general elicitable statistics, of functionals of processes $(X, Y )$. Here an exogenous component $Y$ (Markov by itself) is time-consuming to simulate, while the endogenous component $X$ (jointly Markov with $Y$) is quick to simulate given $Y$, but is responsible for most of the variance of the simulated payoff. To address the related variance issue, we introduce a conditionally independent, hierarchical simulation scheme, where several paths of $X$ are simulated for each simulated path of $Y$. We analyze the statistical convergence of the regression learning scheme based on such block-dependent data. We derive heuristics on the number of paths of $Y$ and, for each of them, of $X$, that should be simulated. The resulting algorithm is implemented on a graphics processing unit (GPU) combining Python/CUDA and learning with PyTorch. A CVA case study with a nested Monte Carlo benchmark shows that the hierarchical simulation technique is key to the success of the learning approach., Comment: This article has been accepted for publication in Mathematical Finance, published by Wiley

A major concern when dealing with financial time series involving a wide variety ofmarket risk factors is the presence of anomalies. These induce a miscalibration of the models used toquantify and manage risk, resulting in potential erroneous risk measures. We propose an approachthat aims to improve anomaly detection in financial time series, overcoming most of the inherentdifficulties. Valuable features are extracted from the time series by compressing and reconstructingthe data through principal component analysis. We then define an anomaly score using a feedforwardneural network. A time series is considered to be contaminated when its anomaly score exceeds agiven cutoff value. This cutoff value is not a hand-set parameter but rather is calibrated as a neuralnetwork parameter throughout the minimization of a customized loss function. The efficiency of theproposed approach compared to several well-known anomaly detection algorithms is numericallydemonstrated on both synthetic and real data sets, with high and stable performance being achievedwith the PCA NN approach. We show that value-at-risk estimation errors are reduced when theproposed anomaly detection model is used with a basic imputation approach to correct the anomaly.

We propose a non-asymptotic convergence analysis of a two-step approach to learn a conditional value-at-risk (VaR) and a conditional expected shortfall (ES) using Rademacher bounds, in a non-parametric setup allowing for heavy-tails on the financial loss. Our approach for the VaR is extended to the problem of learning at once multiple VaRs corresponding to different quantile levels. This results in efficient learning schemes based on neural network quantile and least-squares regressions. An a posteriori Monte Carlo procedure is introduced to estimate distances to the ground-truth VaR and ES. This is illustrated by numerical experiments in a Student-$t$ toy model and a financial case study where the objective is to learn a dynamic initial margin.

In this paper we revisit Burnett (2021) \& Burnett and Williams (2021)'s notion of hedging valuation adjustment (HVA), originally intended to deal with dynamic hedging frictions such as transaction costs, in the direction of model risk. The corresponding HVA reconciles a global fair valuation model with the local models used by the different desks of the bank. Model risk and dynamic hedging frictions indeed deserve a reserve, but a risk-adjusted one, so not only an HVA, but also a contribution to the KVA of the bank. The orders of magnitude of the effects involved suggest that local models should not so much be managed via reserves, as excluded altogether.

We present a one-period XVA model encompassing bilateral and centrally cleared trading in a unified framework with explicit formulas for most quantities at hand. We illustrate possible uses of this framework for running stress test exercises on a financial network from a clearing member's perspective or for optimizing the porting of the portfolio of a defaulted clearing member.

We devise a neural network based compression/completion methodology for financial nowcasting. The latter is meant in a broad sense encompassing completion of gridded values, interpolation, or outlier detection, in the context of financial time series of curves or surfaces (also applicable in higher dimensions, at least in theory). In particular, we introduce an original architecture amenable to the treatment of data defined at variable grid nodes (by far the most common situation in financial nowcasting applications, so that PCA or classical autoencoder methods are not applicable). This is illustrated by three case studies on real data sets. First, we introduce our approach on repo curves data (with moving time-to-maturity as calendar time passes). Second, we show that our approach outperforms elementary interpolation benchmarks on an equity derivative surfaces data set (with moving time-to-maturity again). We also obtain a satisfying performance for outlier detection and surface completion. Third, we benchmark our approach against PCA on at-the-money swaption surfaces redefined at constant expiry/tenor grid nodes. Our approach is then shown to perform as well as (even if not obviously better than) the PCA which, however, is not be applicable to the native, raw data defined on a moving time-to-expiry grid)., Comment: his article has been accepted for publication in Journal of Computational Finance, publishedby Incisive Media Ltd

XVAs denote various counterparty risk related valuation adjustments that are applied to financial derivatives since the 2007--09 crisis. We root a cost-of-capital XVA strategy in a balance sheet perspective which is key in identifying the economic meaning of the XVA terms. Our approach is first detailed in a static setup that is solved explicitly. It is then plugged in the dynamic and trade incremental context of a real derivative banking portfolio. The corresponding cost-of-capital XVA strategy ensures to bank shareholders a submartingale equity process corresponding to a target hurdle rate on their capital at risk, consistently between and throughout deals. Set on a forward/backward SDE formulation, this strategy can be solved efficiently using GPU computing combined with deep learning regression methods in a whole bank balance sheet context. A numerical case study emphasizes the workability and added value of the ensuing pathwise XVA computations., Comment: This is an Accepted Manuscript of an article to be published by Taylor & Francis in Quantitative Finance, which will be available online (once published) at: https://doi.org/10.1080/14697688.2020.1817533

Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation of European vanilla option prices which jointly yields the full surface of local volatilities. We demonstrate the modification of the loss function or the feed forward network architecture to enforce (hard constraints approach) or favor (soft constraints approach) the no-arbitrage conditions and we specify the experimental design parameters that are needed for adequate performance. A novel component is the use of the Dupire formula to enforce bounds on the local volatility associated with option prices, during the network fitting. Our methodology is benchmarked numerically on real datasets of DAX vanilla options.

Modeling counterparty risk is computationally challenging because it requires the simultaneous evaluation of all the trades with each counterparty under both market and credit risk. We present a multi-Gaussian process regression approach, which is well suited for OTC derivative portfolio valuation involved in CVA computation. Our approach avoids nested simulation or simulation and regression of cash flows by learning a Gaussian metamodel for the mark-to-market cube of a derivative portfolio. We model the joint posterior of the derivatives as a Gaussian process over function space, with the spatial covariance structure imposed on the risk factors. Monte-Carlo simulation is then used to simulate the dynamics of the risk factors. The uncertainty in portfolio valuation arising from the Gaussian process approximation is quantified numerically. Numerical experiments demonstrate the accuracy and convergence properties of our approach for CVA computations, including a counterparty portfolio of interest rate swaps., Comment: 36 pages, 16 figures

We prove that the default times (or any of their minima) in the dynamic Gaussian copula model of Cr{\'e}pey, Jeanblanc, and Wu (2013) are invariance times in the sense of Cr{\'e}pey and Song (2017), with related invariance probability measures different from the pricing measure. This reflects a departure from the immersion property, whereby the default intensities of the surviving names and therefore the value of credit protection spike at default times. These properties are in line with the wrong-way risk feature of counterparty risk embedded in credit derivatives, i.e. the adverse dependence between the default risk of a counterparty and an underlying credit derivative exposure.

On a probability space $(\Omega,\mathcal{A},\mathbb{Q})$ we consider two filtrations $\mathbb{F}\subset \mathbb{G}$ and a $\mathbb{G}$ stopping time $\theta$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$ predictable processes on $(0,\theta]$. In this setup it is well-known that, for any $\mathbb{F}$ semimartingale $X$, the process $X^{\theta-}$ ($X$ stopped "right before $\theta$") is a $\mathbb{G}$ semimartingale.Given a positive constant $T$, we call $\theta$ an invariance time if there exists a probability measure $\mathbb{P}$ equivalent to $\mathbb{Q}$ on $\mathcal{F}\_T$ such that, for any $(\mathbb{F},\mathbb{P})$ local martingale $X$, $X^{\theta-}$ is a $(\mathbb{G},\mathbb{Q})$ local martingale. We characterize invariance times in terms of the $(\mathbb{F},\mathbb{Q})$ Az\'ema supermartingale of $\theta$ and we derive a mild and tractable invariance time sufficiency condition. We discuss invariance times in mathematical finance and BSDE applications.

In the aftermath of the 2007 global financial crisis, banks started reflecting into derivative pricing the cost of capital and collateral funding through XVA metrics. Here XVA is a catch-all acronym whereby X is replaced by a letter such as C for credit, D for debt, F for funding, K for capital and so on, and VA stands for valuation adjustment. This behaviour is at odds with economies where markets for contingent claims are complete, whereby trades clear at fair valuations and the costs for capital and collateral are both irrelevant to investment decisions. In this paper, we set forth a mathematical formalism for derivative portfolio management in incomplete markets for banks. A particular emphasis is given to the problem of finding optimal strategies for retained earnings which ensure a sustainable dividend policy.

The ongoing concern about systemic risk since the outburst of the global financial crisis has highlighted the need for risk measures at the level of sets of interconnected financial components, such as portfolios, institutions or members of clearing houses. The two main issues in systemic risk measurement are the computation of an overall reserve level and its allocation to the different components according to their systemic relevance. We develop here a pragmatic approach to systemic risk measurement and allocation based on multivariate shortfall risk measures, where acceptable allocations are first computed and then aggregated so as to minimize costs. We analyze the sensitivity of the risk allocations to various factors and highlight its relevance as an indicator of systemic risk. In particular, we study the interplay between the loss function and the dependence structure of the components. Moreover, we address the computational aspects of risk allocation. Finally, we apply this methodology to the allocation of the default fund of a CCP on real data., Comment: Code, results and figures can also be consulted at https://github.com/yarmenti/MSRA

This paper develops an XVA (costs) analysis of centrally cleared trading, parallel to the one that has been developed in the last years for bilateral transactions. We introduce a dynamic framework that incorporates the sequence of cash-flows involved in the waterfall of resources of a clearing house. The total cost of the clearance framework for a clearing member, called CCVA for central clearing valuation adjustment, is decomposed into a CVA corresponding to the cost of its losses on the default fund in case of defaults of other member, an MVA corresponding to the cost of funding its margins and a KVA corresponding to the cost of the regulatory capital and also of the capital at risk that the member implicitly provides to the CCP through its default fund contribution. In the end the structure of the XVA equations for bilateral and cleared portfolios is similar, but the input data to these equations are not the same, reflecting different financial network structures. The resulting XVA numbers differ, but, interestingly enough, they become comparable after scaling by a suitable netting ratio.

We develop a multi-curve term structure setup in which the modelling ingredients are expressed by rational functionals of Markov processes. We calibrate to LIBOR swaptions data and show that a rational two-factor lognormal multi-curve model is sufficient to match market data with accuracy. We elucidate the relationship between the models developed and calibrated under a risk-neutral measure Q and their consistent equivalence class under the real-world probability measure P. The consistent P-pricing models are applied to compute the risk exposures which may be required to comply with regulatory obligations. In order to compute counterparty-risk valuation adjustments, such as CVA, we show how positive default intensity processes with rational form can be derived. We flesh out our study by applying the results to a basis swap contract., Comment: 34 pages, 9 figures

The credit crisis and the ongoing European sovereign debt crisis have highlighted the native form of credit risk, namely the counterparty risk. The related Credit Valuation Adjustment, (CVA), Debt Valuation Adjustment (DVA), Liquidity Valuation Adjustment (LVA) and Replacement Cost (RC) issues, jointly referred to in this paper as Total Valuation Adjustment (TVA), have been thoroughly investigated in the theoretical papers [Cr12a] and [Cr12b]. The present work provides an executive summary and numerical companion to these papers, through which the TVA pricing problem can be reduced to Markovian pre-default TVA BSDEs. The first step consists in the counterparty clean valuation of a portfolio of contracts, which is the valuation in a hypothetical situation where the two parties would be risk-free and funded at a risk-free rate. In the second step, the TVA is obtained as the value of an option on the counterparty clean value process called Contingent Credit Default Swap (CCDS). Numerical results are presented for interest rate swaps in the Vasicek, as well as in the inverse Gaussian Hull-White short rate model, also allowing one to assess the related model risk issue., Comment: 29 pages, 9 figures

It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian set-ups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and one further needs a suitable comparison theorem. In this paper, we derive such estimates and comparison results. In the last section, applicability of the results is illustrated with a pricing problem in finance., Comment: Published in at http://dx.doi.org/10.1214/08-AAP517 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper, the authors develop a method of detecting correlations between epidemic patterns in different regions that are due to human movement and introduce a null model in which the travel-induced correlations are cancelled. They apply this method to the well-documented cases of seasonal influenza outbreaks in the United States and France. In the United States (using data for 1972-2002), the authors observed strong short-range correlations between several states and their immediate neighbors, as well as robust long-range spreading patterns resulting from large domestic air-traffic flows. The stability of these results over time allowed the authors to draw conclusions about the possible impact of travel restrictions on epidemic spread. The authors also applied this method to the case of France (1984-2004) and found that on the regional scale, there was no transportation mode that clearly dominated disease spread. The simplicity and robustness of this method suggest that it could be a useful tool for detecting transmission channels in the spread of epidemics., Comment: 8 pages, 7 figures, 3 tables

We study numerically the variability of the outbreak of diseases on complex networks. We use a SI model to simulate the disease spreading at short times, in homogeneous and in scale-free networks. In both cases, we study the effect of initial conditions on the epidemic's dynamics and its variability. The results display a time regime during which the prevalence exhibits a large sensitivity to noise. We also investigate the dependence of the infection time on nodes' degree and distance to the seed. In particular, we show that the infection time of hubs have large fluctuations which limit their reliability as early-detection stations. Finally, we discuss the effect of the multiplicity of shortest paths between two nodes on the infection time. Furthermore, we demonstrate that the existence of even longer paths reduces the average infection time. These different results could be of use for the design of time-dependent containment strategies.