## Publications

137 results found

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

Armstrong J, Brigo D, 2018, Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections

We define two new notions of projection of a stochastic differential equation(SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allowsone to systematically develop low dimensional approximations to highdimensional SDEs using differential geometric techniques. The approachgeneralizes the notion of projecting a vector field onto a submanifold in orderto derive approximations to ordinary differential equations, and improves theprevious Stratonovich projection method by adding optimality analysis andresults. Indeed, just as in the case of ordinary projection, our definitions ofprojection are based on optimality arguments and give in a well-defined sense"optimal" approximations to the original SDE in the mean-square sense. We alsoshow that the Stratonovich projection satisfies an optimality criterion that ismore ad hoc and less appealing than the criteria satisfied by the Itoprojections we introduce. As an application we consider approximating thesolution of the non-linear filtering problem with a Gaussian distribution andshow how the newly introduced Ito projections lead to optimal approximations inthe Gaussian family and briefly discuss the optimal approximation for moregeneral families of distribution. We perform a numerical comparison of ouroptimally approximated filter with the classical Extended Kalman Filter todemonstrate the efficacy of the approach.

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

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- Citations: 1

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 [1], 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

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- Citations: 1

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

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- Citations: 1

Armstrong J, Brigo D, Ito Stochastic Differential Equations as 2-Jets, Geometric Science of Information 2017, Publisher: Springer Verlag, ISSN: 0302-9743

We explain how Itˆo Stochastic Differential Equations on manifoldsmay be defined as 2-jets of curves and show how this relationshipcan be interpreted in terms of a convergent numerical scheme. We usejets as a natural language to express geometric properties of SDEs. Weexplain that the mainstream choice of Fisk-Stratonovich-McShane calculusfor stochastic differential geometry is not necessary. We give a newgeometric interpretation of the Itˆo–Stratonovich transformation in termsof the 2-jets of curves induced by consecutive vector flows. We discussthe forward Kolmogorov equation and the backward diffusion operatorin geometric terms. In the one-dimensional case we consider percentilesof the solutions of the SDE and their properties. In particular the medianof a SDE solution is associated to the drift of the SDE in Stratonovichform for small times.

Bormetti G, Brigo D, Francischello M, et al., 2016, Impact of Multiple Curve Dynamics in Credit Valuation Adjustments, Challenges in Derivatives Markets, Publisher: Springer, Pages: 251-266, ISSN: 2194-1009

We present a detailed analysis of interest rate derivatives valuation undercredit risk and collateral modeling. We show how the credit and collateral extendedvaluation framework in Pallavicini et al (2011) can be helpful in defining the keymarket rates underlying the multiple interest rate curves that characterize currentinterest rate markets. We introduce the collateralized valuation measures and formulatea consistent realistic dynamics for the rates emerging from our analysis. Wepoint out limitations of multiple curve models with deterministic basis consideringvaluation of particularly sensitive products such as basis swaps.

Brigo D, Liu Q, Pallavicini A, et al., 2016, Nonlinear Valuation Under Collateralization, Credit Risk, and Funding Costs, Challenges in Derivatives Markets, Publisher: Springer, Pages: 3-35, ISSN: 2194-1009

We develop a consistent, arbitrage-free framework for valuing derivativetrades with collateral, counterparty credit risk, and funding costs. Credit, debit, liquidity,and funding valuation adjustments (CVA, DVA, LVA, and FVA) are simplyintroduced as modifications to the payout cash-flows of the trade position.The framework is flexible enough to accommodate actual trading complexitiessuch as asymmetric collateral and funding rates, replacement close-out, and rehypothecationof posted collateral – all aspects which are often neglected. Thegeneralized valuation equation takes the form of a forward-backward SDE or semilinearPDE. Nevertheless, it may be recast as a set of iterative equations which can beefficiently solved by our proposed least-squares Monte Carlo algorithm. We implementnumerically the case of an equity option and show how its valuation changeswhen including the above effects.In the paper we also discuss the financial impact of the proposed valuation frameworkand of nonlinearity more generally. This is fourfold: Firstly, the valuationequation is only based on observable market rates, leaving the value of a derivativestransaction invariant to any theoretical risk-free rate. Secondly, the presenceof funding costs makes the valuation problem a highly recursive and nonlinear one.Thus, credit and funding risks are non-separable in general, and despite common practice in banks, CVA, DVA, and FVA cannot be treated as purely additive adjustmentswithout running the risk of double counting. To quantify the valuation errorthat can be attributed to double counting, we introduce a ’nonlinearity valuation adjustment’(NVA) and show that its magnitude can be significant under asymmetricfunding rates and replacement close-out at default. Thirdly, as trading parties cannotobserve each others’ liquidity policies nor their respective funding costs, the bilateralnature of a derivative price breaks down. The value of a trade to a counterpartywill not be j

Brigo D, Francischello M, Pallavicini A, 2016, Analysis Of Nonlinear Valuation Equations Under Credit And Funding Effects, Challenges in Derivatives Markets, Publisher: Springer, Pages: 37-52, ISSN: 2194-1009

We study conditions for existence, uniqueness and invariance of the comprehensivenonlinear valuation equations first introduced in Pallavicini et al (2011)[11]. These equations take the form of semi-linear PDEs and Forward-BackwardStochastic Differential Equations (FBSDEs). After summarizing the cash flows definitionsallowing us to extend valuation to credit risk and default closeout, includingcollateral margining with possible re-hypothecation, and treasury funding costs, weshow how such cash flows, when present-valued in an arbitrage free setting, leadto semi-linear PDEs or more generally to FBSDEs. We provide conditions for existenceand uniqueness of such solutions in a classical sense, discussing the role of thehedging strategy. We show an invariance theorem stating that even though we startfrom a risk-neutral valuation approach based on a locally risk-free bank accountgrowing at a risk-free rate, our final valuation equations do not depend on the riskfree rate. Indeed, our final semi-linear PDE or FBSDEs and their classical solutionsdepend only on contractual, market or treasury rates and we do not need to proxythe risk free rate with a real market rate, since it acts as an instrumental variable. Theequations derivations, their numerical solutions, the related XVA valuation adjustmentswith their overlap, and the invariance result had been analyzed numericallyand extended to central clearing and multiple discount curves in a number of previousworks, including [11], [12], [10], [6] and [4].

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

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- Citations: 1

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

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- Citations: 4

Brigo D, Fries CP, Hull J, et al., 2016, FVA and electricity bill valuation adjustment—much of a difference?, Pages: 147-168, ISSN: 2194-1009

© The Author(s) 2016. Pricing counterparty credit risk, although being in the focus for almost a decade by now, is far from being resolved. It is highly controversial if any valuation adjustment besides the basic CVA should be taken into account, and if so, for what purpose. Even today, the handling of CVA, DVA, FVA,… differs between the regulatory, the accounting, and the economic point of view. Eventually, if an agreement is reached that CVA has to be taken into account, it remains unclear if CVA can be modelled linearly, or if nonlinear models need to be resorted to. Finally, industry practice and implementation differ in several aspects. Hence, a unified theory and treatment of FVA and alike is not yet tangible. The conference Challenges in Derivatives Markets, held at Technische Universität München in March/April 2015, featured a panel discussion with panelists representing different points of view: John Hull, who argues that FVA might not exist at all; in contrast to Christian Fries, who sees the need of all relevant costs to be covered within valuation but not within adjustments. Damiano Brigo emphasises the nonlinearity of (most) valuation adjustments and is concerned about overlapping adjustments and double-counting. Finally, Daniel Sommer puts the exit price in the focus. The following (mildly edited) record of the panel discussion repeats the main arguments of the discussants—ultimately culminating in the awareness that if everybody charges an electricity bill valuation adjustment, it has to become part of any quoted price.

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

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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 INT 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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- Citations: 38

Brigo D, Pallavicini A, 2013, CCPs, Central Clearing, CSA, Credit Collateral and Funding Costs Valuation FAQ: Re-hypothecation, CVA, Closeout, Netting, WWR, Gap-Risk, Initial and Variation Margins, Multiple Discount Curves, FVA?

We present a dialogue on Funding Costs and Counterparty Credit Risk modeling,inclusive of collateral, wrong way risk, gap risk and possible Central Clearingimplementation through CCPs. This framework is important following the factthat derivatives valuation and risk analysis has moved from exotic derivativesmanaged on simple single asset classes to simple derivatives embedding the newor previously neglected types of complex and interconnected nonlinear risks weaddress here. This dialogue is the continuation of the "Counterparty Risk,Collateral and Funding FAQ" by Brigo (2011). In this dialogue we focus more onfunding costs for the hedging strategy of a portfolio of trades, on thenon-linearities emerging from assuming borrowing and lending rates to bedifferent, on the resulting aggregation-dependent valuation process and itsoperational challenges, on the implications of the onset of central clearing,on the macro and micro effects on valuation and risk of the onset of CCPs, oninitial and variation margins impact on valuation, and on multiple discountcurves. Through questions and answers (Q&A) between a senior expert and ajunior colleague, and by referring to the growing body of literature on thesubject, we present a unified view of valuation (and risk) that takes all suchaspects into account.

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