Publications
170 results found
Pallavicini A, Perini D, Brigo D, 2011, Funding Valuation Adjustment: a consistent framework including CVA, DVA, collateral,netting rules and re-hypothecation
In this paper we describe how to include funding and margining costs into arisk-neutral pricing framework for counterparty credit risk. We considerrealistic settings and we include in our models the common market practicessuggested by the ISDA documentation without assuming restrictive constraints onmargining procedures and close-out netting rules. In particular, we allow forasymmetric collateral and funding rates, and exogenous liquidity policies andhedging strategies. Re-hypothecation liquidity risk and close-out amountevaluation issues are also covered. We define a comprehensive pricing frameworkwhich allows us to derive earlier results on funding or counterparty risk. Somerelevant examples illustrate the non trivial settings needed to derive knownfacts about discounting curves by starting from a general framework and withoutresorting to ad hoc hypotheses. Our main result is a bilateral collateralizedcounterparty valuation adjusted pricing equation, which allows to price a dealwhile taking into account credit and debt valuation adjustments along withmargining and funding costs in a coherent way. We find that the equation has arecursive form, making the introduction of an additive funding valuationadjustment difficult. Yet, we can cast the pricing equation into a set ofiterative relationships which can be solved by means of standard least-squareMonte Carlo techniques.
Albanese C, Brigo D, Oertel F, 2011, Restructuring Counterparty Credit Risk
We introduce an innovative theoretical framework to model derivativetransactions between defaultable entities based on the principle of arbitragefreedom. Our framework extends the traditional formulations based on Credit andDebit Valuation Adjustments (CVA and DVA). Depending on how the defaultcontingency is accounted for, we list a total of ten different structuringstyles. These include bipartite structures between a bank and a counterparty,tri-partite structures with one margin lender in addition, quadri-partitestructures with two margin lenders and, most importantly, configurations whereall derivative transactions are cleared through a Central Counterparty (CCP).We compare the various structuring styles under a number of criteria includingconsistency from an accounting standpoint, counterparty risk hedgeability,numerical complexity, transaction portability upon default, induced behaviourand macro-economic impact of the implied wealth allocation.
Brigo D, 2011, Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Netting, Collateral, Re-hypothecation, WWR, Basel, Funding, CCDS and Margin Lending
We present a dialogue on Counterparty Credit Risk touching on Credit Value atRisk (Credit VaR), Potential Future Exposure (PFE), Expected Exposure (EE),Expected Positive Exposure (EPE), Credit Valuation Adjustment (CVA), DebitValuation Adjustment (DVA), DVA Hedging, Closeout conventions, Netting clauses,Collateral modeling, Gap Risk, Re-hypothecation, Wrong Way Risk, Basel III,inclusion of Funding costs, First to Default risk, Contingent Credit DefaultSwaps (CCDS) and CVA restructuring possibilities through margin lending. Thedialogue is in the form of a Q&A between a CVA expert and a newly hiredcolleague.
Brigo D, Buescu C, Morini M, 2011, Impact of the first to default time on Bilateral CVA
We compare two different bilateral counterparty valuation adjustment (BVA)formulas. The first formula is an approximation and is based on subtracting thetwo unilateral Credit Valuation Adjustment (CVA)'s formulas as seen from thetwo different parties in the transaction. This formula is only a simplifiedrepresentation of bilateral risk and ignores that upon the first defaultcloseout proceedings are ignited. As such, it involves double counting. Wecompare this formula with the fully specified bilateral risk formula, where thefirst to default time is taken into account. The latter correct formula dependson default dependence between the two parties, whereas the simplified one doesnot. We also analyze a candidate simplified formula in case the replacementcloseout is used upon default, following ISDA's recommendations, and we findthe simplified formula to be the same as in the risk free closeout case. Weanalyze the error that is encountered when using the simplified formula in acouple of simple products: a zero coupon bond, where the exposure isunidirectional, and an equity forward contract where exposure can go both ways.For the latter case we adopt a bivariate exponential distribution due to Gumbelto model the joint default risk of the two parties in the deal. We present anumber of realistic cases where the simplified formula differs considerablyfrom the correct one.
Bielecki T, Brigo D, Patras F, 2011, Credit Risk Frontiers, Publisher: Bloomberg Press, ISBN: 9781576603581
This book addresses these aspects of modeling and analysis of credit derivatives that have not been adequately studied and/or adequately understood in the past.
Brigo D, Capponi A, Pallavicini A, et al., 2011, Collateral Margining in Arbitrage-Free Counterparty Valuation Adjustment including Re-Hypotecation and Netting
This paper generalizes the framework for arbitrage-free valuation ofbilateral counterparty risk to the case where collateral is included, withpossible re-hypotecation. We analyze how the payout of claims is modified whencollateral margining is included in agreement with current ISDA documentation.We then specialize our analysis to interest-rate swaps as underlying portfolio,and allow for mutual dependences between the default times of the investor andthe counterparty and the underlying portfolio risk factors. We usearbitrage-free stochastic dynamical models, including also the effect ofinterest rate and credit spread volatilities. The impact of re-hypotecation, ofcollateral margining frequency and of dependencies on the bilateralcounterparty risk adjustment is illustrated with a numerical example.
BRIGO DAMIANO, PALLAVICINI ANDREA, PAPATHEODOROU VASILEIOS, 2011, ARBITRAGE-FREE VALUATION OF BILATERAL COUNTERPARTY RISK FOR INTEREST-RATE PRODUCTS: IMPACT OF VOLATILITIES AND CORRELATIONS, International Journal of Theoretical and Applied Finance, Vol: 14, Pages: 773-802
The purpose of this paper is introducing rigorous methods and formulas for bilateral counterparty risk credit valuation adjustment (CVA) on interest-rate portfolios. In doing so, we summarize the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, including the default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net present value of the contract at the relevant default times. We allow for correlation between the default times of the investor and counterparty, and for correlation of each with the underlying risk factor, namely interest rates. We also analyze the often neglected impact of credit spread volatility. We include close-out netting rules in our examples, although other agreements, such as periodic margining or collateral posting, are left for future work.
Brigo D, Morini M, 2011, NO-ARMAGEDDON MEASURE FOR ARBITRAGE-FREE PRICING OF INDEX OPTIONS IN A CREDIT CRISIS, Mathematical Finance, Pages: 573-593
In this work, we consider three problems of the standard market approach to credit index options pricing: the definition of the index spread is not valid in general, the considered payoff leads to a pricing which is not always defined, and the candidate numeraire for defining a pricing measure is not strictly positive, which leads to a nonequivalent pricing measure. We give a solution to the three problems, based on modeling the flow of information through a suitable subfiltration. With this we consistently take into account the possibility of default of all names in the portfolio, that is neglected in the standard market approach. We show on market inputs that, while the pricing difference can be negligible in normal market conditions, it can become highly relevant in stressed market conditions, like the situation caused by the credit crunch.
Brigo D, Morini M, 2010, Dangers of Bilateral Counterparty Risk: the fundamental impact of closeout conventions
We analyze the practical consequences of the bilateral counterparty riskadjustment. We point out that past literature assumes that, at the moment ofthe first default, a risk-free closeout amount will be used. We argue that thelegal (ISDA) documentation suggests in many points that a substitution closeoutshould be used. This would take into account the risk of default of thesurvived party. We show how the bilateral counterparty risk adjustment changesstrongly when a substitution closeout amount is considered. We model the twoextreme cases of default independence and co-monotonicity, which highlight prosand cons of both risk free and substitution closeout formulations, and allow usto interpret the outcomes as dramatic consequences on default contagion.Finally, we analyze the situation when collateral is present.
Brigo D, Nordio C, 2010, Liquidity-adjusted Market Risk Measures with Stochastic Holding Period
Within the context of risk integration, we introduce in risk measurementstochastic holding period (SHP) models. This is done in order to obtain a`liquidity-adjusted risk measure' characterized by the absence of a fixed timehorizon. The underlying assumption is that - due to changes on market liquidityconditions - one operates along an `operational time' to which the P&L processof liquidating a market portfolio is referred. This framework leads to amixture of distributions for the portfolio returns, potentially allowing forskewness, heavy tails and extreme scenarios. We analyze the impact of possibledistributional choices for the SHP. In a multivariate setting, we hint at thepossible introduction of dependent SHP processes, which potentially lead to nonlinear dependence among the P&L processes and therefore to tail dependenceacross assets in the portfolio, although this may require drastic choices onthe SHP distributions. We also find that increasing dependence as measured byKendall's tau through common SHP's appears to be unfeasible. We finally discusspotential developments following future availability of market data.
Brigo D, Pallavicini A, Torresetti R, 2010, Credit models and the crisis : a journey into CDOs, copulas, correlations and dynamic models., Publisher: Wiley, ISBN: 978-0-470-66566-4
Responding to the immediate need for clarity in the market and academic research environments, this book follows the development of credit derivatives and CDOs at a technical level, analyzing the impact, strengths and weaknesses of methods ...
Brigo D, El-Bachir N, 2010, An exact formula for default swaptions pricing in the SSRJD stochastic intensity model, Mathematical Finance, Vol: 20, Pages: 365-382
Brigo D, Pallavicini A, Torresetti R, 2009, Credit models and the crisis, or: how I learned to stop worrying and love the CDOs
We follow a long path for Credit Derivatives and Collateralized DebtObligations (CDOs) in particular, from the introduction of the Gaussian copulamodel and the related implied correlations to the introduction ofarbitrage-free dynamic loss models capable of calibrating all the tranches forall the maturities at the same time. En passant, we also illustrate the impliedcopula, a method that can consistently account for CDOs with differentattachment and detachment points but not for different maturities. Thediscussion is abundantly supported by market examples through history. Thedangers and critics we present to the use of the Gaussian copula and of impliedcorrelation had all been published by us, among others, in 2006, showing thatthe quantitative community was aware of the model limitations before thecrisis. We also explain why the Gaussian copula model is still used in its basecorrelation formulation, although under some possible extensions such as randomrecovery. Overall we conclude that the modeling effort in this area of thederivatives market is unfinished, partly for the lack of an operationallyattractive single-name consistent dynamic loss model, and partly because of thediminished investment in this research area.
Brigo D, Morini M, Tarenghi M, 2009, Credit Calibration with Structural Models: The Lehman case and Equity Swaps under Counterparty Risk
In this paper we develop structural first passage models (AT1P and SBTV) withtime-varying volatility and characterized by high tractability, moving from theoriginal work of Brigo and Tarenghi (2004, 2005) [19] [20] and Brigo and Morini(2006)[15]. The models can be calibrated exactly to credit spreads usingefficient closed-form formulas for default probabilities. Default events arecaused by the value of the firm assets hitting a safety threshold, whichdepends on the financial situation of the company and on market conditions. InAT1P this default barrier is deterministic. Instead SBTV assumes two possiblescenarios for the initial level of the default barrier, for taking into accountuncertainty on balance sheet information. While in [19] and [15] the models areanalyzed across Parmalat's history, here we apply the models to exactcalibration of Lehman Credit Default Swap (CDS) data during the monthspreceding default, as the crisis unfolds. The results we obtain with AT1P andSBTV have reasonable economic interpretation, and are particularly realisticwhen SBTV is considered. The pricing of counterparty risk in an Equity ReturnSwap is a convenient application we consider, also to illustrate theinteraction of our credit models with equity models in hybrid products context.
Brigo D, Tarenghi M, 2009, Credit Default Swap Calibration and Equity Swap Valuation under Counterparty Risk with a Tractable Structural Model
In this paper we develop a tractable structural model with analytical defaultprobabilities depending on some dynamics parameters, and we show how tocalibrate the model using a chosen number of Credit Default Swap (CDS) marketquotes. We essentially show how to use structural models with a calibrationcapability that is typical of the much more tractable credit-spread basedintensity models. We apply the structural model to a concrete calibration caseand observe what happens to the calibrated dynamics when the CDS-implied creditquality deteriorates as the firm approaches default. Finally we provide atypical example of a case where the calibrated structural model can be used forcredit pricing in a much more convenient way than a calibrated reduced formmodel: The pricing of counterparty risk in an equity swap.
Brigo D, Tarenghi M, 2009, Credit Default Swap Calibration and Counterparty Risk Valuation with a Scenario based First Passage Model
In this work we develop a tractable structural model with analytical defaultprobabilities depending on a random default barrier and possibly randomvolatility ideally associated with a scenario based underlying firm debt. Weshow how to calibrate this model using a chosen number of reference CreditDefault Swap (CDS) market quotes. In general this model can be seen as apossible extension of the time-varying AT1P model in Brigo and Tarenghi (2004).The calibration capability of the Scenario Volatility/Barrier model (SVBAT1P),when keeping time-constant volatility, appears inferior to the one of AT1P withtime-varying deterministic volatility. The SVBAT1P model, however, maintainsthe benefits of time-homogeneity and can lead to satisfactory calibrationresults, as we show in a case study where we compare different choices onscenarios and parameters. Similarly to AT1P, SVBAT1P is suited to pricinghybrid equity/credit derivatives and to evaluate counterparty risk in equitypayoffs, and more generally to evaluate hybrid credit/equity payoffs. Weconsider the equity return swap in Brigo and Tarenghi (2004) and show itsvaluation under SVBAT1P with the same CDS and equity calibration input usedearlier for AT1P, and further we hint at equity default swap valuation in theconclusions.
Brigo D, Pallavicini A, Papatheodorou V, 2009, Bilateral counterparty risk valuation for interest-rate products: impact of volatilities and correlations
The purpose of this paper is introducing rigorous methods and formulas forbilateral counterparty risk credit valuation adjustments (CVA's) oninterest-rate portfolios. In doing so, we summarize the general arbitrage-freevaluation framework for counterparty risk adjustments in presence of bilateraldefault risk, as developed more in detail in Brigo and Capponi (2008),including the default of the investor. We illustrate the symmetry in thevaluation and show that the adjustment involves a long position in a put optionplus a short position in a call option, both with zero strike and written onthe residual net present value of the contract at the relevant default times.We allow for correlation between the default times of the investor andcounterparty, and for correlation of each with the underlying risk factor,namely interest rates. We also analyze the often neglected impact of creditspread volatility. We include Netting in our examples, although otheragreements such as Margining and Collateral are left for future work.
Brigo D, Chourdakis K, Bakkar I, 2009, Counterparty risk valuation for Energy-Commodities swaps: Impact of volatilities and correlation
It is commonly accepted that Commodities futures and forward prices, inprinciple, agree under some simplifying assumptions. One of the most relevantassumptions is the absence of counterparty risk. Indeed, due to margining,futures have practically no counterparty risk. Forwards, instead, may bear thefull risk of default for the counterparty when traded with brokers or outsideclearing houses, or when embedded in other contracts such as swaps. In thispaper we focus on energy commodities and on Oil in particular. We use a hybridcommodities-credit model to asses impact of counterparty risk in pricingformulas, both in the gross effect of default probabilities and on the subtlereffects of credit spread volatility, commodities volatility andcredit-commodities correlation. We illustrate our general approach with a casestudy based on an oil swap, showing that an accurate valuation of counterpartyrisk depends on volatilities and correlation and cannot be accounted forprecisely through a pre-defined multiplier.
BRIGO DAMIANO, CHOURDAKIS KYRIAKOS, 2009, COUNTERPARTY RISK FOR CREDIT DEFAULT SWAPS: IMPACT OF SPREAD VOLATILITY AND DEFAULT CORRELATION, International Journal of Theoretical and Applied Finance, Vol: 12, Pages: 1007-1026
We consider counterparty risk for Credit Default Swaps (CDS) in presence of correlation between default of the counterparty and default of the CDS reference credit. Our approach is innovative in that, besides default correlation, which was taken into account in earlier approaches, we also model credit spread volatility. Stochastic intensity models are adopted for the default events, and defaults are connected through a copula function. We find that both default correlation and credit spread volatility have a relevant impact on the positive counterparty-risk credit valuation adjustment to be subtracted from the counterparty-risk free price. We analyze the pattern of such impacts as correlation and volatility change through some fundamental numerical examples, analyzing wrong-way risk in particular. Given the theoretical equivalence of the credit valuation adjustment with a contingent CDS, we are also proposing a methodology for valuation of contingent CDS on CDS.
Ben Ameur H, Brigo D, Errais E, 2009, Pricing Credit Default Swaps Bermudan Options: An Approximate Dynamic Programming Approach, Quantitative Finance
Ben-Ameur H, Brigo D, Errais E, 2009, A dynamic programming approach for pricing CDS and CDS options, QUANTITATIVE FINANCE, Vol: 9, Pages: 717-726, ISSN: 1469-7688
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Brigo D, 2008, Constant Maturity Credit Default Swap Pricing with Market Models
In this work we derive an approximated no-arbitrage market valuation formulafor Constant Maturity Credit Default Swaps (CMCDS). We move from the CDSoptions market model in Brigo (2004), and derive a formula for CMCDS that isthe analogous of the formula for constant maturity swaps in the default freeswap market under the LIBOR market model. A "convexity adjustment"-likecorrection is present in the related formula. Without such correction, or withzero correlations, the formula returns an obvious deterministic-credit-spreadexpression for the CMCDS price. To obtain the result we derive a joint dynamicsof forward CDS rates under a single pricing measure, as in Brigo (2004).Numerical examples of the "convexity adjustment" impact complete the paper.
Morini M, Brigo D, 2008, Arbitrage-free Pricing of Credit Index Options: The no-armageddon pricing measure and the role of correlation after the subprime crisis
In this work we consider three problems of the standard market approach topricing of credit index options: the definition of the index spread is notvalid in general, the usually considered payoff leads to a pricing which is notalways defined, and the candidate numeraire one would use to define a pricingmeasure is not strictly positive, which would lead to a non-equivalent pricingmeasure. We give a general mathematical solution to the three problems, based on anovel way of modeling the flow of information through the definition of a newsubfiltration. Using this subfiltration, we take into account consistently thepossibility of default of all names in the portfolio, that is neglected in thestandard market approach. We show that, while the related mispricing can benegligible for standard options in normal market conditions, it can becomehighly relevant for different options or in stressed market conditions. In particular, we show on 2007 market data that after the subprime creditcrisis the mispricing of the market formula compared to the no arbitrageformula we propose has become financially relevant even for the liquidCrossover Index Options.
Brigo D, Pallavicini A, Torresetti R, 2008, Default correlation, cluster dynamics and single names: The GPCL dynamical loss model
We extend the common Poisson shock framework reviewed for example in Lindskogand McNeil (2003) to a formulation avoiding repeated defaults, thus obtaining amodel that can account consistently for single name default dynamics, clusterdefault dynamics and default counting process. This approach allows one tointroduce significant dynamics, improving on the standard "bottom-up"approaches, and to achieve true consistency with single names, improving onmost "top-down" loss models. Furthermore, the resulting GPCL model hasimportant links with the previous GPL dynamical loss model in Brigo,Pallavicini and Torresetti (2006a,b), which we point out. Model extensionsallowing for more articulated spread and recovery dynamics are hinted at.Calibration to both DJi-TRAXX and CDX index and tranche data across attachmentsand maturities shows that the GPCL model has the same calibration power as theGPL model while allowing for consistency with single names
Brigo D, Dalessandro A, Neugebauer M, et al., 2008, A Stochastic Processes Toolkit for Risk Management
In risk management it is desirable to grasp the essential statisticalfeatures of a time series representing a risk factor. This tutorial aims tointroduce a number of different stochastic processes that can help in graspingthe essential features of risk factors describing different asset classes orbehaviors. This paper does not aim at being exhaustive, but gives examples anda feeling for practically implementable models allowing for stylised featuresin the data. The reader may also use these models as building blocks to buildmore complex models, although for a number of risk management applications themodels developed here suffice for the first step in the quantitative analysis.The broad qualitative features addressed here are {fat tails} and {meanreversion}. We give some orientation on the initial choice of a suitablestochastic process and then explain how the process parameters can be estimatedbased on historical data. Once the process has been calibrated, typicallythrough maximum likelihood estimation, one may simulate the risk factor andbuild future scenarios for the risky portfolio. On the terminal simulateddistribution of the portfolio one may then single out several risk measures,although here we focus on the stochastic processes estimation preceding thesimulation of the risk factors Finally, this first survey report focuses onsingle time series. Correlation or more generally dependence across riskfactors, leading to multivariate processes modeling, will be addressed infuture work.
Brigo D, Mercurio F, 2008, Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing
In the present paper we construct stock price processes with the samemarginal log-normal law as that of a geometric Brownian motion and also withthe same transition density (and returns' distributions) between any twoinstants in a given discrete-time grid. We then illustrate how option pricesbased on such processes differ from Black and Scholes', in that option pricescan be either arbitrarily close to the option intrinsic value or arbitrarilyclose to the underlying stock price. We also explain that this is due to theparticular way one models the stock-price process in between the grid timeinstants which are relevant for trading. The theoretical result concerningscalar stochastic differential equations with prescribed diffusion coefficientwhose densities evolve in a prescribed exponential family, on which part of thepaper is based, is presented in detail.
Brigo D, 2008, The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation
In the present paper, given an evolving mixture of probability densities, wedefine a candidate diffusion process whose marginal law follows the sameevolution. We derive as a particular case a stochastic differential equation(SDE) admitting a unique strong solution and whose density evolves as a mixtureof Gaussian densities. We present an interesting result on the comparisonbetween the instantaneous and the terminal correlation between the obtainedprocess and its squared diffusion coefficient. As an application tomathematical finance, we construct diffusion processes whose marginal densitiesare mixtures of lognormal densities. We explain how such processes can be usedto model the market smile phenomenon. We show that the lognormal mixturedynamics is the one-dimensional diffusion version of a suitable uncertainvolatility model, and suitably reinterpret the earlier correlation result. Weexplore numerically the relationship between the future smile structures ofboth the diffusion and the uncertain volatility versions.
Brigo D, Hanzon B, 2008, On three filtering problems arising in mathematical finance
Three situations in which filtering theory is used in mathematical financeare illustrated at different levels of detail. The three problems originatefrom the following different works: 1) On estimating the stochastic volatilitymodel from observed bilateral exchange rate news, by R. Mahieu, and P.Schotman; 2) A state space approach to estimate multi-factors CIR models of theterm structure of interest rates, by A.L.J. Geyer, and S. Pichler; 3)Risk-minimizing hedging strategies under partial observation in pricingfinancial derivatives, by P. Fischer, E. Platen, and W. J. Runggaldier; In thefirst problem we propose to use a recent nonlinear filtering technique based ongeometry to estimate the volatility time series from observed bilateralexchange rates. The model used here is the stochastic volatility model. Thefilters that we propose are known as projection filters, and a brief derivationof such filters is given. The second problem is introduced in detail, and apossible use of different filtering techniques is hinted at. In fact thefilters used for this problem in 2) and part of the literature can beinterpreted as projection filters and we will make some remarks on how moregeneral and possibly more suitable projection filters can be constructed. Thethird problem is only presented shortly.
Brigo D, Pallavicini A, 2008, Counterparty Risk and Contingent CDS under correlation, Risk Magazine
Brigo D, Pallavicini A, 2008, Counterparty Risk under Correlation between Default and Interest Rates, Numercial Methods for Finance, Editors: Miller J, Edelman D, Appleby J, Publisher: Chapman Hall
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