139 results found
Armstrong J, Brigo D, Rossi Ferrucci E, 2019, Optimal approximation of SDEs on submanifolds: the Itô‐vector and Itô‐jet projections, Proceedings of the London Mathematical Society, Vol: 119, Pages: 176-213, ISSN: 0024-6115
Brigo D, Pisani C, Rapisarda F, The multivariate mixture dynamics model: shifted dynamics and correlation skew, Annals of Operations Research, ISSN: 0254-5330
Brigo D, Francischello M, Pallavicini A, 2019, Nonlinear valuation under credit, funding, and margins: Existence, uniqueness, invariance, and disentanglement, EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, Vol: 274, Pages: 788-805, ISSN: 0377-2217
Armstrong J, Brigo D, 2019, Risk managing tail-risk seekers: VaR and expected shortfall vs S-shaped utility, Journal of Banking & Finance, Vol: 101, Pages: 122-135, ISSN: 0378-4266
We consider market players with tail-risk-seeking behaviour modelled by S-shaped utility, as introduced by Kahneman and Tversky. We argue that risk measures such as value at risk (VaR) and expected shortfall (ES) are ineffective in constraining such players, as such measures cannot reduce the traders expected S-shaped utilities. Indeed, when designing payoffs aiming to maximize utility under a VaR or ES risk limit, the players will attain the same supremum of expected utility with or without VaR or ES limits. By contrast, we show that risk management constraints based on a second more conventional concave utility function can reduce the maximum S-shaped utility that can be achieved by the investor. Indeed, product designs leading to progressively larger S-shaped utilities will lead to progressively lower expected constraining conventional utilities, violating the related risk limit. These results hold in a variety of market models, including the Black Scholes options model, and are particularly relevant for risk managers given the historical role of VaR and the endorsement of ES by the Basel committee in 2012–2013.
Brigo D, Vrins F, 2018, Disentangling wrong-way risk: pricing credit valuation adjustment via change of measures, EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, Vol: 269, Pages: 1154-1164, ISSN: 0377-2217
Armstrong J, Brigo D, 2018, Rogue traders versus value-at-risk and expected shortfall, Risk -London- Risk Magazine Limited-, Pages: 63-63, ISSN: 0952-8776
We show that, in a Black and Scholes market, value at risk and ex-pected shortfall are irrelevant in limiting traders excessive tail-risk seekingbehaviour as modelled via Kahneman and Tversky’s S-shaped utility. Tohave effective constraints one can introduce a risk limit based on a secondbut concave utility function.
Brigo D, Hvolby T, Vrins F, Wrong-way risk adjusted exposure: analytical approximations for options in default intensity models, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Publishing Co.
We examine credit value adjustment (CVA) estimation under wrong-way risk(WWR) by computing the expected positive exposure (EPE) under an equiva-lent measure as suggested in , adjusting the drift of the underlying for defaultrisk. We apply this technique to European put and call options and derive theanalytic formulas for EPE under WWR obtained with various approximationsof the drift adjustment. We give the results of numerical experiments basedon 4 parameter sets, and supply figures of the CVA based on both of the sug-gested proxys, comparing with CVA based on a 2D-Monte Carlo scheme andGaussian Copula resampling. We also show the CVA obtained by the formulasfrom Basel III. We observe that the Basel III formula does not account forthe credit-market correlation, while the Gaussian Copula resampling methodestimates a too large impact of this correlation. The two proxies account forthe credit-market correlation, and give results that are mostly similar to the2D-Monte Carlo results.
Brigo D, Pede N, Petrelli A, Examples of wrong–way risk in CVA induced by devaluations on default, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Press
When calculatingCredit Valuation Adjustment(CVA), theinteraction between the portfolio’s exposure and the counter-party’s credit worthiness is referred to asWrong–Way Risk(WWR). Making the assumption that the Brownian mo-tions driving both the market (exposure) and the (counter-party) credit risk–factors dynamics are correlated representsthe simplest way of modelling the dependence structure be-tween these two components. For many practical applica-tions, however, such an approach may fail to account for theright amount of WWR, thus resulting in misestimates of theportfolio’s CVA. We present a modelling framework wherea further — and indeed stronger — source of market/creditdependence is introduced through devaluation jumps on themarket risk–factors’ dynamics. Such jumps happen upon thecounterparty’s default and are a particularly realistic featureto include in case of sovereign or systemically important coun-terparties. Moreover, we show that, in the special case wherethe focus is on FX/credit WWR, devaluation jumps provide an effective way of incorporating market information comingfrom quanto Credit Default Swap (CDS) basis spreads and wederive the corresponding CVA pricing equations as a systemof coupled PDEs.
Brigo D, Piat C, Static vs adapted optimal execution strategies in two benchmark trading models
We consider the optimal solutions to the trade execution problem in the twodifferent classes of i) fully adapted or adaptive and ii) deterministic orstatic strategies, comparing them. We do this in two different benchmarkmodels. The first model is a discrete time framework with an information flowprocess, dealing with both permanent and temporary impact, minimizing theexpected cost of the trade. The second model is a continuous time frameworkwhere the objective function is the sum of the expected cost and a value atrisk (or expected shortfall) type risk criterion. Optimal adapted solutions areknown in both frameworks from the original works of Bertsimas and Lo (1998) andGatheral and Schied (2011). In this paper we derive the optimal staticstrategies for both benchmark models and we study quantitatively theimprovement in optimality when moving from static strategies to fully adaptedones. We conclude that, in the benchmark models we study, the difference is notrelevant, except for extreme unrealistic cases for the model or impactparameters. This indirectly confirms that in the similar framework of Almgrenand Chriss (2000) one is fine deriving a static optimal solution, as done bythose authors, as opposed to a fully adapted one, since the static solutionhappens to be tractable and known in closed form.
Armstrong J, Brigo D, 2018, Intrinsic stochastic differential equations as jets, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 474, ISSN: 1364-5021
Brigo D, Mai, Jan, et al., Consistent iterated simulation of multivariate defaults: Markov indicators, lack of memory, extreme-value copulas, and the Marshall–Olkin distribution, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Publishing Co.
A current market-practice to incorporate multivariate defaults in global riskfactorsimulations is the iteration of (multiplicative) i.i.d. survival indicator incrementsalong a given time-grid, where the indicator distribution is based on acopula ansatz. The underlying assumption is that the behavior of the resultingiterated default distribution is similar to the one-shot distribution. It is shownthat in most cases this assumption is not fulfilled and furthermore numericalanalysis is presented that shows sizeable differences in probabilities assignedto both “survival-of-all” and “mixed default/survival” events. Moreover, theclasses of distributions for which probabilities from the “terminal one-shot”and “terminal iterated” distribution coincide are derived for problems considering“survival-of-all” events as well as “mixed default/survival” events. Forthe former problem, distributions must fulfill a lack-of-memory type property,which is, e.g., fulfilled by min-stable multivariate exponential distributions.These correspond in a copula-framework to exponential margins coupled viaextreme-value copulas. For the latter problem, while looping default inspiredmultivariate Freund distributions and more generally multivariate phase-type distributions could be a solution, under practically relevant and reasonableadditional assumptions on portfolio rebalancing and nested distributions, theunique solution is the Marshall–Olkin class.
Brigo D, Rapisarda F, Sridi A, 2018, The multivariate mixture dynamics: Consistent no-arbitrage single-asset and index volatility smiles, IISE TRANSACTIONS, Vol: 50, Pages: 27-44, ISSN: 2472-5854
Bormetti G, Brigo D, Francischello M, et al., 2018, Impact of multiple curve dynamics in credit valuation adjustments under collateralization, Publisher: ROUTLEDGE JOURNALS, TAYLOR & FRANCIS LTD
Brigo D, Buescu C, Rutkowski M, 2017, Funding, repo and credit inclusive valuation as modified option pricing, OPERATIONS RESEARCH LETTERS, Vol: 45, Pages: 665-670, ISSN: 0167-6377
, 2017, Geometric Science of Information, Publisher: Springer International Publishing, ISSN: 0302-9743
Armstrong J, Brigo D, Extrinsic projection of Ito SDEs on submanifolds with applications to nonlinear ltering, Computational Information Geometry for Image and Signal Processing, Publisher: Springer, ISSN: 1860-4862
We define the notion of the extrinsic Itˆo projection of astochastic differential equation (SDE) on a submanifold. This allows oneto systematically develop low dimensional approximations to high dimensionalSDEs in a differential geometric setting. We consider the exampleof approximating the non-linear filtering problem with a Gaussian distributionand show how the Itˆo projection leads to improved approximationsin the Gaussian family. We briefly discuss the approximations formore general families of distribution. We perform a numerical comparisonof our projection filters with the classical Extended Kalman Filterto demonstrate the efficacy of the approach.
Brigo D, Pistone G, Dimensionality reduction for measure valued evolution equations in statistical manifolds, Computational information geometry for image and signal processing, ISSN: 1860-4862
Brigo D, Mai J-F, Scherer M, 2016, Markov multi-variate survival indicators for default simulation as a new characterization of the Marshall-Olkin law, STATISTICS & PROBABILITY LETTERS, Vol: 114, Pages: 60-66, ISSN: 0167-7152
Armstrong J, Brigo D, 2016, Nonlinear filtering via stochastic PDE projection on mixture manifolds in L-2 direct metric, MATHEMATICS OF CONTROL SIGNALS AND SYSTEMS, Vol: 28, ISSN: 0932-4194
Brigo D, Fries CP, Hull J, et al., 2016, FVA and Electricity Bill Valuation Adjustment-Much of a Difference?, Conference on Innovations in Derivatives Markets - Fixed Income Modeling, Valuation Adjustments, Risk Management, and Regulation, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 147-168, ISSN: 2194-1009
Brigo D, Liu QD, Pallavicini A, et al., 2016, Nonlinear Valuation Under Collateralization, Credit Risk, and Funding Costs, Conference on Innovations in Derivatives Markets - Fixed Income Modeling, Valuation Adjustments, Risk Management, and Regulation, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 3-35, ISSN: 2194-1009
Brigo D, Francischello M, Pallavicini A, 2016, Analysis of Nonlinear Valuation Equations Under Credit and Funding Effects, Conference on Innovations in Derivatives Markets - Fixed Income Modeling, Valuation Adjustments, Risk Management, and Regulation, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 37-52, ISSN: 2194-1009
Bormetti G, Brigo D, Francischello M, et al., 2016, Impact of Multiple-Curve Dynamics in Credit Valuation Adjustments, Conference on Innovations in Derivatives Markets - Fixed Income Modeling, Valuation Adjustments, Risk Management, and Regulation, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 251-266, ISSN: 2194-1009
Brigo D, Francischello M, Pallavicini A, 2015, Invariance, existence and uniqueness of solutions of nonlinear valuation PDEs and FBSDEs inclusive of credit risk, collateral and funding costs
We study conditions for existence, uniqueness and invariance of thecomprehensive nonlinear valuation equations first introduced in Pallavicini etal (2011). These equations take the form of semilinear PDEs andForward-Backward Stochastic Differential Equations (FBSDEs). After summarizingthe cash flows definitions allowing us to extend valuation to credit risk anddefault closeout, including collateral margining with possiblere-hypothecation, and treasury funding costs, we show how such cash flows, whenpresent-valued in an arbitrage free setting, lead to semi-linear PDEs or moregenerally to FBSDEs. We provide conditions for existence and uniqueness of suchsolutions in a viscosity and classical sense, discussing the role of thehedging strategy. We show an invariance theorem stating that even though westart from a risk-neutral valuation approach based on a locally risk-free bankaccount growing at a risk-free rate, our final valuation equations do notdepend on the risk free rate. Indeed, our final semilinear PDE or FBSDEs andtheir classical or viscosity solutions depend only on contractual, market ortreasury rates and we do not need to proxy the risk free rate with a realmarket rate, since it acts as an instrumental variable. The equationsderivations, their numerical solutions, the related XVA valuation adjustmentswith their overlap, and the invariance result had been analyzed numerically andextended to central clearing and multiple discount curves in a number ofprevious works, including Pallavicini et al (2011), Pallavicini et al (2012),Brigo et al (2013), Brigo and Pallavicini (2014), and Brigo et al (2014).
Brigo D, Garcia J, Pede N, 2015, COCO BONDS PRICING WITH CREDIT AND EQUITY CALIBRATED FIRST-PASSAGE FIRM VALUE MODELS, INTERNATIONAL JOURNAL OF THEORETICAL AND APPLIED FINANCE, Vol: 18, ISSN: 0219-0249
Brigo D, Buescu C, Pallavicini A, et al., 2015, A NOTE ON THE SELF-FINANCING CONDITION FOR FUNDING, COLLATERAL AND DISCOUNTING, INTERNATIONAL JOURNAL OF THEORETICAL AND APPLIED FINANCE, Vol: 18, ISSN: 0219-0249
Brigo D, Nordio C, 2015, A Random Holding Period Approach for Liquidity-Inclusive Risk Management, Conference on Risk Management Reloaded, Publisher: SPRINGER, Pages: 3-18, ISSN: 2194-1009
Armstrong J, Brigo D, 2015, Stochastic PDE Projection on Manifolds: Assumed-Density and Galerkin Filters, 2nd International SEE Conference on Geometric Science of Information (GSI), Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 713-722, ISSN: 0302-9743
Brigo D, Liu Q, Pallavicini A, et al., 2014, Nonlinear Valuation under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes
We develop an arbitrage-free framework for consistent valuation of derivativetrades with collateralization, counterparty credit gap risk, and funding costs,following the approach first proposed by Pallavicini and co-authors in 2011.Based on the risk-neutral pricing principle, we derive a general pricingequation where Credit, Debit, Liquidity and Funding Valuation Adjustments (CVA,DVA, LVA and FVA) are introduced by simply modifying the payout cash-flows ofthe deal. Funding costs and specific close-out procedures at default break thebilateral nature of the deal price and render the valuation problem anon-linear and recursive one. CVA and FVA are in general not really additiveadjustments, and the risk for double counting is concrete. We introduce a newadjustment, called a Non-linearity Valuation Adjustment (NVA), to addressdouble-counting. The theoretical risk free rate disappears from our finalequations. The framework can be tailored also to CCP trading under initial andvariation margins, as explained in detail in Brigo and Pallavicini (2014). Inparticular, we allow for asymmetric collateral and funding rates, replacementclose-out and re-hypothecation. The valuation equation takes the form of abackward stochastic differential equation or semi-linear partial differentialequation, and can be cast as a set of iterative equations that can be solved byleast-squares Monte Carlo. We propose such a simulation algorithm in a casestudy involving a generalization of the benchmark model of Black and Scholesfor option pricing. Our numerical results confirm that funding risk has anon-trivial impact on the deal price, and that double counting matters too. Weconclude the article with an analysis of large scale implications ofnon-linearity of the pricing equations.
Brigo D, Capponi A, Pallavicini A, 2014, ARBITRAGE-FREE BILATERAL COUNTERPARTY RISK VALUATION UNDER COLLATERALIZATION AND APPLICATION TO CREDIT DEFAULT SWAPS, MATHEMATICAL FINANCE, Vol: 24, Pages: 125-146, ISSN: 0960-1627
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