92 results found
Cont R, Kalinin A, 2019, On the support of solutions of stochastic differential equations with path-dependent coefficients
Given a stochastic differential equation with path-dependent coefficientsdriven by a multidimensional Wiener process, we show that the support of thelaw of the solution is given by the image of the Cameron-Martin space under theflow of the solutions of a system of path-dependent (ordinary) differentialequations. Our result extends the Stroock-Varadhan support theorem fordiffusion processes to the case of SDEs with path-dependent coefficients. Theproof is based on the Functional Ito calculus and interpolation estimates forstochastic processes in Holder norm.
Cont R, Perkowski N, 2019, Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, Transactions of the American Mathematical Society, Vol: 6, Pages: 161-186, ISSN: 0002-9947
We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of p-th variation along a sequence of time partitions. For paths with finite p-th variation along a sequence of time partitions, we derive a change of variable formula for p times continuously differentiable functions and show pointwise convergence of appropriately defined compensated Riemann sums. Results for functions are extended to regular path-dependent functionals using the concept of vertical derivative of a functional. We show that the pathwise integral satisfies an `isometry' formula in terms of p-th order variation and obtain a `signal plus noise' decomposition for regular functionals of paths with strictly increasing p-th variation. For less regular (Cp−1) functions we obtain a Tanaka-type change of variable formula using an appropriately defined notion of local time. These results extend to multidimensional paths and yield a natural higher-order extension of the concept of `reduced rough path'. We show that, while our integral coincides with a rough-path integral for a certain rough path, its construction is canonical and does not involve the specification of any rough-path superstructure.
Cont R, Sirignano J, 2018, Universal features of price formation in financial markets: perspectives from Deep Learning
Using a large-scale Deep Learning approach applied to a high-frequency database containing billions of electronic market quotes and transactions for US equities, we uncover nonparametric evidence for the existence of a universal and stationary price formation mechanism relating the dynamics of supply and demand for a stock, as revealed through the order book, to subsequent variations in its market price. We assess the model by testing its out-of-sample predictions for the direction of price moves given the history of price and order flow, across a wide range of stocks and time periods. The universal price formation model is shown to exhibit a remarkably stable out-of-sample prediction accuracy across time, for a wide range of stocks from different sectors. Interestingly, these results also hold for stocks which are not part of the training sample, showing that the relations captured by the model are universal and not asset-specific.
Chiu H, Cont R, 2018, On pathwise quadratic variation for càdlàg functions, Electronic Communications in Probability, Vol: 23, Pages: 1-12
© 2018, Institute of Mathematical Statistics. All rights reserved. We revisit Föllmer’s concept of quadratic variation of a càdlàg function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of càdlàg processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition which implies the Lebesgue decomposition of the pathwise quadratic variation as a result, rather than requiring it as an extra condition.
Cont R, Sirignano JA, 2018, Universal Features of Price Formation in Financial Markets: Perspectives From Deep Learning
Using a large-scale Deep Learning approach applied to a high-frequency database containing billions of electronic market quotes and transactions for US equities, we uncover nonparametric evidence for the existence of a universal and stationary price formation mechanism relating the dynamics of supply and demand for a stock, as revealed through the order book, to subsequent variations in its market price. We assess the model by testing its out-of-sample predictions for the direction of price moves given the history of price and order flow, across a wide range of stocks and time periods. The universal price formation model is shown to exhibit a remarkably stable out-of-sample prediction accuracy across time, for a wide range of stocks from different sectors. Interestingly, these results also hold for stocks which are not part of the training sample, showing that the relations captured by the model are universal and not asset-specific.The universal model --- trained on data from all stocks --- outperforms, in terms of out-of-sample prediction accuracy, asset-specific linear and nonlinear models trained on time series of any given stock, showing that the universal nature of price formation weighs in favour of pooling together financial data from various stocks, rather than designing asset or sector-specific models as commonly done. Standard data normalizations based on volatility, price level or average spread, or partitioning the training data into sectors or categories such as large/small tick stocks, do not improve training results. On the other hand, inclusion of price and order flow history over many past observations is shown to improve forecasting performance, showing evidence of path-dependence in price dynamics.
Cont R, Gordy M, 2017, Special Issue: Monitoring Systemic Risk: Data, Models and Metrics, Statistics and Risk Modeling, Vol: 34, ISSN: 2193-1402
Cont R, Ananova A, 2016, Pathwise integration with respect to paths of finite quadratic variation, Journal de Mathematiques Pures et Appliquees, Vol: 107, Pages: 737-757, ISSN: 0021-7824
We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands.We show that the integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero quadratic variation.
Cont R, Duffie D, Glasserman P, et al., 2016, Preface to the special issue on systemic risk: Models and mechanisms, Operations Research, Vol: 64, Pages: 1053-1055, ISSN: 0030-364X
Cont R, Kukanov A, 2016, Optimal order placement in limit order markets, Quantitative Finance, Vol: 17, Pages: 21-39, ISSN: 1469-7696
To execute a trade, participants in electronic equity markets may choose to submit limit orders or market orders across various exchanges where a stock is traded. This decision is influenced by the characteristics of the order flow and queue sizes in each limit order book, as well as the structure oftransaction fees and rebates across exchanges. We propose a quantitativeframework for studying this order placement problem by formulating it as a convex optimization problem. This formulation allows to study how the interplay between the state of order books, the fee structure, order flow properties and preferences of a trader determine the optimal placement decision. In the case of a single exchange, we derive an explicit solution for the optimal split between limit and market orders. For the general problem of order placement across multiple exchanges, we propose a stochastic algorithm for computing the optimal policy and study the sensitivity of the solution to various parameters using a numerical implementation of the algorithm.
Cont R, Wagalath L, 2016, Risk management for whales, Risk -London- Risk Magazine Limited-, ISSN: 0952-8776
We propose framework for modeling portfolio risk which integrates market risk with liquidation costs which may arise in stress scenarios. Our model provides a systematic method for computing liquidation-adjusted risk measures for a portfolio. Calculation of Liquidation-adjusted VaR (LVaR) for sample portfolios reveals a substantial impact of liquidation costs on portfolio risk for portfolios with large concentrated positions.
Amini H, Cont R, Minca A, 2016, Resilience to Contagion in Financial Networks, Mathematical Finance, Vol: 26, Pages: 329-365, ISSN: 0960-1627
Propagation of balance-sheet or cash-flow insolvency across financialinstitutions may be modeled as a cascade process on a network representingtheir mutual exposures. We derive rigorous asymptotic results for the magnitudeof contagion in a large financial network and give an analytical expression forthe asymptotic fraction of defaults, in terms of network characteristics. Ourresults extend previous studies on contagion in random graphs to inhomogeneousdirected graphs with a given degree sequence and arbitrary distribution ofweights. We introduce a criterion for the resilience of a large financialnetwork to the insolvency of a small group of financial institutions andquantify how contagion amplifies small shocks to the network. Our resultsemphasize the role played by "contagious links" and show that institutionswhich contribute most to network instability in case of default have both largeconnectivity and a large fraction of contagious links. The asymptotic resultsshow good agreement with simulations for networks with realistic sizes.
Cont R, Wagalath L, 2016, INSTITUTIONAL INVESTORS AND THE DEPENDENCESTRUCTURE OF ASSET RETURNS, International Journal of Theoretical & Applied Finance, Vol: 19, ISSN: 1793-6322
We propose a model of a financial market with multiple assets that takes into accountthe impact of a large institutional investor rebalancing its positions so as to maintaina fixed allocation in each asset. We show that feedback effects can lead to significantexcess realized correlation between asset returns and modify the principal componentstructure of the (realized) correlation matrix of returns. Our study naturally links, ina quantitative manner, the properties of the realized correlation matrix — correlationbetween assets, eigenvectors and eigenvalues — to the sizes and trading volumes oflarge institutional investors. In particular, we show that even starting with uncorrelated“fundamentals”, fund rebalancing endogenously generates a correlation matrix of returnswith a first eigenvector with positive components, which can be associated to the market,as observed empirically. Finally, we show that feedback effects flatten the differencesbetween the expected returns of assets and tend to align them with the returns of theinstitutional investor’s portfolio, making this benchmark fund more difficult to beat, notbecause of its strategy but precisely because of its size and market impact.
Cont R, Bally V, Caramellino L, 2016, Stochastic Integration by Parts and Functional Itô Calculus, Publisher: Birkhäuser, ISBN: 978-3-319-27128-6
This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012).The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes.Rama Cont's notes provide an introduction to the Functional Itô Calculus, a non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations.This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to practitioners in mathematical finance.
Cont R, LU Y, 2015, Weak approximation of martingale representations, Stochastic Processes and Their Applications, Vol: 126, Pages: 857-882, ISSN: 0304-4149
We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are obtained by computing a directional derivative of the weak Euler scheme and yield a consistent estimator for the integrand in the martingale representation formula for any square-integrable functional of the solution of an SDE with path-dependent coefficients. Explicit convergence rates are derived for functionals which are Lipschitz-continuous in the supremum norm. Our results require neither the Markov property, nor any differentiability conditions on the functional or the coefficients of the stochastic differential equations involved.
Cont R, 2015, The end of the waterfall: Default resources of central counterparties, Journal of Risk Management in Financial Institutions, Vol: 8, Pages: 365-389, ISSN: 1752-8887
Central counterparties (CCPs) have become pillars of the new global financial architecture following the financial crisis of 2008. The key role of CCPs in mitigating counterparty risk and contagion has in turn cast them as systemically important financial institutions whoseeventual failure may lead to potentially serious consequences for financial stability, andprompted discussions on CCP risk management standards and safeguards for recovery andresolutions of CCPs in case of failure. We contribute to the debate on CCP default resourcesby focusing on the incentives generated by the CCP loss allocation rules for the CCP and itsmembers and discussing how the design of loss allocation rules may be used to align theseincentives in favor of outcomes which benefit financial stability. After reviewing theingredients of the CCP loss waterfall and various proposals for loss recovery provisions forCCPs, we examine the risk management incentives created by different ingredients in theloss waterfall and discuss possible approaches for validating the design of the waterfall.We emphasize the importance of CCP stress tests and argue that such stress tests need toaccount for the interconnectedness of CCPs through common members and cross-marginagreements. A key proposal is that capital charges on assets held against CCP Default Fundsshould depend on the quality of the risk management of the CCP, as assessed throughindependent stress tests.
Cont R, Bentata A, 2015, Forward equations for option prices in semimartingale models, Finance and Stochastics, Pages: 617-651, ISSN: 1432-1122
We derive a forward partial integro-differential equation for prices of calloptions in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous- semimartingale. A uniquenesstheorem is given for the solutions of this equation. This result generalizesDupire's forward equation to a large class of non-Markovian models with jumps.
Cont R, Minca A, 2015, Credit default swaps and systemic risk, Annals of Operations Research, Vol: 247, Pages: 523-547, ISSN: 0254-5330
We present a network model for investigating the impact on systemic risk of central clearing of over the counter (OTC) credit default swaps (CDS). We model contingent cash flows resulting from CDS and other OTC derivatives by a multi-layered network with a core-periphery structure, which is flexible enough to reproduce the gross and net exposures as well as the heterogeneity of market shares of participating institutions. We analyze illiquidity cascades resulting from liquidity shocks and show that the contagion of illiquidity takes place along a sub-network constituted by links identified as ’critical receivables’. A key role is played by the long intermediation chains inherent to the structure of the OTC network, which may turn into chains of critical receivables. We calibrate our model to data representing net and gross OTC exposures of large dealer banks and use this model to investigate the impact of central clearing on network stability. We find that, when interest rate swaps are cleared, central clearing of credit default swaps through a well-capitalized CCP can reduce the probability and the magnitude of a systemic illiquidity spiral by reducing the length of the chains of critical receivables within the financial network. These benefits are reduced, however, if some large intermediaries are not included as clearing members.
Cont R, Schaanning E, 2014, Fire sales, indirect contagion and systemic stress-testing, Publisher: SSRN
We present a framework for modeling the phenomenon of fire sales in a network of financial institutions with common asset holdings, subject to leverage or capital constraints. Asset losses triggered by macro-shocks may interact with portfolio constraints, resulting in liquidation of assets, which in turn affects market prices, leading to contagion of losses when portfolios are marked to market. If mark-to-market losses are large, this may in turn lead to a new round of fire sales.In contrast to balance sheet contagion mechanisms based on direct linkages, this price-mediated contagion is transmitted through common asset holdings, which we quantify through liquidity-weighted overlaps across portfolios. Exposure to price-mediated contagion leads to the concept of indirect exposure to an asset class, as a consequence of which the risk of a portfolio depends on the matrix of asset holdings of other large and leveraged portfolios with similar assets. Our model provides an operational systemic stress testing method for quantifying the exposure of the financial system to these effects.Using data from the European Banking Authority, we apply this method to the examine the exposure of the EU banking system to price-mediated contagion.Our results indicate that, even with optimistic estimates of market depth, moderately large macro-shocks may trigger fire sales which then lead to substantial losses across bank portfolios, modifying the outcome of bank stress tests.Moreover, we show that price-mediated contagion leads to a heterogeneous cross-sectional loss distribution across banks, which cannot be replicated simply by applying a macro-shock to bank portfolios in absence of fire sales. We propose a bank-level indicator, based on the analysis of liquidity-weighted overlaps across bank portfolios, which is shown to be strongly correlated with bank losses due to fire sales and may be used to quantify the contribution of a financial institution to price-mediated contagion. Unlike m
Cont R, Wagalath L, 2014, Fire sales forensics: Measuring endogenous risk, Mathematical Finance, Vol: 26, Pages: 835-866, ISSN: 0960-1627
We propose a tractable framework for quantifying the impact of loss-triggered fire sales on portfolio risk, in a multi-asset setting. We derive analytical expressions for the impact of fire sales on the realized volatility and correlations of asset returns in a fire sales scenario and show that our results provide a quantitative explanation for the spikes in volatility and correlations observed during such deleveraging episodes. These results are then used to develop an econometric framework for the forensic analysis of fire sales episodes, using observations of market prices. We give conditions for the identifiability of model parameters from time series of asset prices, propose a statistical test for the presence of fire sales, and an estimator for the magnitude of fire sales in each asset class. Pathwise consistency and large sample properties of the estimator are studied in the high-frequency asymptotic regime. We illustrate our methodology by applying it to the forensic analysis of two recent deleveraging episodes: the Quant Crash of August 2007 and the Great Deleveraging following the default of Lehman Brothers in Fall 2008. © 2014 Wiley Periodicals, Inc.
Cont R, 2014, Central clearing of OTC derivatives: Bilateral vs multilateral netting, Statistics & Risk Modeling, Vol: 31, Pages: 3-22, ISSN: 2193-1402
We study the impact of central clearing of over-the-counter (OTC) transactions on counterparty exposures in a market with OTC transactions across several asset classes with heterogeneous characteristics. The impact of introducing a central counterparty (CCP) on expected interdealer exposure is determined by the tradeoff between multilateral netting across dealers on one hand and bilateral netting across asset classes on the other hand. We find this tradeoff to be sensitive to assumptions on heterogeneity of asset classes in terms of `riskyness' of the asset class as well as correlation of exposures across asset classes. In particular, while an analysis assuming independent, homogeneous exposures suggests that central clearing is efficient only if one has an unrealistically high number of participants, the opposite conclusion is reached if differences in riskyness and correlation across asset classes are realistically taken into account. We argue that empirically plausible specifications of model parameters lead to the conclusion that central clearing does reduce interdealer exposures: the gain from multilateral netting in a CCP overweighs the loss of netting across asset classes in bilateral netting agreements. When a CCP exists for interest rate derivatives, adding a CCP for credit derivatives is shown to decrease overall exposures. These findings are shown to be robust to the statistical assumptions of the model as well as the choice of risk measure used to quantify exposures.
Arulkumaran N, Rhodes A, 2013, Preface, Publisher: Elsevier BV, Pages: 789-789
Arulkumaran N, Rhodes A, 2013, Critical illness in obstetrics. Preface.
Cont R, Deguest R, 2013, EQUITY CORRELATIONS IMPLIED BY INDEX OPTIONS: ESTIMATION AND MODEL UNCERTAINTY ANALYSIS, Mathematical Finance, Vol: 23, Pages: 496-530, ISSN: 0960-1627
We propose a method for constructing an arbitrage-free multiasset pricing model which is consistent with a set of observed single- and multiasset derivative prices. The pricing model is constructed as a random mixture of N reference models, where the distribution of mixture weights is obtained by solving a well-posed convex optimization problem. Application of this method to equity and index options shows that, whereas multivariate diffusion models with constant correlation fail to match the prices of index and component options simultaneously, a jump-diffusion model with a common jump component affecting all stocks enables to do so. Furthermore, we show that even within a parametric model class, there is a wide range of correlation patterns compatible with observed prices of index options. Our method allows, as a by product, to quantify this model uncertainty with no further computational effort and propose static hedging strategies for reducing the exposure of multiasset derivatives to model uncertainty.
Cont R, Fournie D-A, 2013, Functional Ito calculus and stochastic integral representation of martingales, Annals of Probability, Vol: 41, Pages: 109-133, ISSN: 0091-1798
We develop a non-anticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependentfunctionals which possess certain directional derivatives. The construction isbased on a pathwise derivative, introduced by B Dupire, for functionals on thespace of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space ofsquare-integrable martingales. This extension defines a weak derivative whichis shown to be the inverse of the Ito integral and which may be viewed as a non-anticipative "lifting" of the Malliavin derivative. These results lead to a constructive martingale representation formula for Ito processes. By contrast with the Clark-Haussmann-Ocone formula, this representation only involves non-anticipative quantities which may be computed pathwise.
Cont R, Minca A, 2013, RECOVERING PORTFOLIO DEFAULT INTENSITIES IMPLIED BY CDO QUOTES, Mathematical Finance, Vol: 23, Pages: 94-121
We propose a stable nonparametric algorithm for the calibration of “top-down” pricing models for portfolio credit derivatives: given a set of observations of market spreads for collateralized debt obligation (CDO) tranches, we construct a risk-neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk-neutral loss process “closest” to a prior loss process, verifying the calibration constraints. We formalize the problem in terms of minimization of relative entropy with respect to the prior under calibration constraints and use convex duality methods to solve the problem: the dual problem is shown to be an intensity control problem, characterized in terms of a Hamilton–Jacobi system of differential equations, for which we present an analytical solution. Given a set of observed CDO tranche spreads, our method allows to construct a default intensity process which leads to tranche spreads consistent with the observations. We illustrate our method on ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the previous number of defaults, and offer quantitative evidence for contagion effects in the (risk-neutral) loss process.
Cont R, Kukanov A, Stoikov S, 2013, The Price Impact of Order Book Events, Journal of Financial Econometrics, Vol: 12, Pages: 47-88
We study the price impact of order book events - limit orders, market ordersand cancelations - using the NYSE TAQ data for 50 U.S. stocks. We show that,over short time intervals, price changes are mainly driven by the order flowimbalance, defined as the imbalance between supply and demand at the best bidand ask prices. Our study reveals a linear relation between order flowimbalance and price changes, with a slope inversely proportional to the marketdepth. These results are shown to be robust to seasonality effects, and stableacross time scales and across stocks. We argue that this linear price impactmodel, together with a scaling argument, implies the empirically observed"square-root" relation between price changes and trading volume. However, therelation between price changes and trade volume is found to be noisy and lessrobust than the one based on order flow imbalance.
Cont R, Moussa A, Santos EB, 2013, Network structure and systemic risk in banking systems., Handbook of Systemic Risk, Editors: Fouque, Langsam, Publisher: Cambridge University Press, Pages: 327-367, ISBN: 9781107023437
We present a quantitative methodology for analyzing the potential for contagion and systemic risk in a network of interlinked financial institutions, using a metric for the systemic importance of institutions: the Contagion Index. We apply this methodology to a data set of mutual exposures and capital levels of financial institutions in Brazil in 2007 and 2008, and analyze the role of balance sheet size and network structure in each institution's contribution to systemic risk. Our results emphasize the contribution of heterogeneity in network structure and concentration of counterparty exposures to a given institution in explaining its systemic importance. These observations plead for capital requirements which depend on exposures, rather than aggregate balance sheet size, and which target systemically important institutions.
Cont R, Kokholm T, 2013, A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES, Mathematical Finance, Vol: 23, Pages: 248-274, ISSN: 0960-1627
We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for volatility derivatives, such as VIX options, as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index.
Cont R, Wagalath L, 2013, RUNNING FOR THE EXIT: DISTRESSED SELLING AND ENDOGENOUS CORRELATION IN FINANCIAL MARKETS, Mathematical Finance, Vol: 23, Pages: 718-741
We propose a simple multiperiod model of price impact from trading in a market with multiple assets, which illustrates how feedback effects due to distressed selling and short selling lead to endogenous correlations between asset classes. We show that distressed selling by investors exiting a fund and short selling of the fund’s positions by traders may have nonnegligible impact on the realized correlations between returns of assets held by the fund. These feedback effects may lead to positive realized correlations between fundamentally uncorrelated assets, as well as an increase in correlations across all asset classes and in the fund’s volatility which is exacerbated in scenarios in which the fund undergoes large losses. By studying the diffusion limit of our discrete time model, we obtain analytical expressions for the realized covariance and show that the realized covariance may be decomposed as the sum of a fundamental covariance and a liquidity-dependent “excess” covariance. Finally, we examine the impact of these feedback effects on the volatility of other funds. Our results provide insight into the nature of spikes in correlation associated with the failure or liquidation of large funds.
Cont R, Deguest R, He X, 2013, Loss-Based Risk Measures, Statistics and Risk Modeling, Vol: 30, Pages: 133-167
Starting from the requirement that risk measures of financial portfolios should be based on their losses, not their gains, we define the notion of loss-based risk measure and study the properties of this class of risk measures. We characterize loss-based risk measures by a representation theorem and give examples of such risk measures. We then discuss the statistical robustness of estimators of loss-based risk measures: we provide a general criterion for qualitative robustness of risk estimators and compare thiscriterion with sensitivity analysis of estimators based on influence functions. Finally, we provide examples of statistically robust estimators for loss-based risk measures.
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