## Publications

141 results found

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, 2013, ARBITRAGE-FREE BILATERAL COUNTERPARTY RISK VALUATION UNDER COLLATERALIZATION AND APPLICATION TO CREDIT DEFAULT SWAPS, *Mathematical Finance*, Vol: 24, Pages: 1252146-1252146, ISSN: 0960-1627

We develop an arbitrage-free valuation framework for bilateral counterparty risk, where collateral is included with possible rehypothecation. We show that the adjustment is given by the sum of two option payoff terms, where each term depends on the netted exposure, i.e., the difference between the on-default exposure and the predefault collateral account. We then specialize our analysis to credit default swaps (CDS) as underlying portfolios, and construct a numerical scheme to evaluate the adjustment under a doubly stochastic default framework. In particular, we show that for CDS contracts a perfect collateralization cannot be achieved, even under continuous collateralization, if the reference entity’s and counterparty’s default times are dependent. The impact of rehypothecation, collateral margining frequency, and default correlation-induced contagion is illustrated with numerical examples.

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.

Brigo D, Armstrong J, 2013, Stochastic filtering by projection: the example of the cubic sensor, Geometric Science of Information: First International Conference, Pages: 685-692

The “projection method” is an approach to finding numerical approximations to the optimal filter for non linear stochastic filtering problems. One uses a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic differential equation given by the filtering problem onto a finite dimensional manifold inside the space of densities. This reduces the problem to finite dimensional stochastic differential equation.Previously, the projection method has only been considered for the Hilbert space structure associated with the Hellinger metric. We show through the numerical example of the quadratic sensor that the approach also works well when one projects using the direct L 2 metric.Previous implementations of projection methods have been limited to solving a single problem. We indicate how one can build a computational framework for applying the projection method more generally.

Brigo D, Morini M, Pallavicini A, 2013, Counterparty Credit Risk, Collateral and Funding: with Pricing Cases for all Asset Classes, Publisher: Wiley, ISBN: 978-0-470-74846-6

The book’s content is focused on rigorous and advanced quantitative methods for the pricing and hedging of counterparty credit and funding risk. The new general theory that is required for this methodology is developed from scratch, leading to a consistent and comprehensive framework for counterparty credit and funding risk, inclusive of collateral, netting rules, possible debit valuation adjustments, re-hypothecation and closeout rules. The book however also looks at quite practical problems, linking particular models to particular ‘concrete’ financial situations across asset classes, including interest rates, FX, commodities, equity, credit itself, and the emerging asset class of longevity. The authors also aim to help quantitative analysts, traders, and anyone else needing to frame and price counterparty credit and funding risk, to develop a ‘feel’ for applying sophisticated mathematics and stochastic calculus to solve practical problems. The main models are illustrated from theoretical formulation to final implementation with calibration to market data, always keeping in mind the concrete questions being dealt with. The authors stress that each model is suited to different situations and products, pointing out that there does not exist a single model which is uniformly better than all the others, although the problems originated by counterparty credit and funding risk point in the direction of global valuation. Finally, proposals for restructuring counterparty credit risk, ranging from contingent credit default swaps to margin lending, are considered.

BRIGO D, CAPPONI A, PALLAVICINI A,
et al., 2013, PRICING COUNTERPARTY RISK INCLUDING COLLATERALIZATION, NETTING RULES, RE-HYPOTHECATION AND WRONG-WAY RISK, *International Journal of Theoretical and Applied Finance*, Vol: 16, Pages: 1350007-1350007, ISSN: 0219-0249

<jats:p> This article is concerned with the arbitrage-free valuation of bilateral counterparty risk through stochastic dynamical models when collateral is included, with possible rehypothecation. The payout of claims is modified to account for collateral margining in agreement with International Swap and Derivatives Association (ISDA) documentation. The analysis is specialized to interest-rate and credit derivatives. In particular, credit default swaps are considered to show that a perfect collateralization cannot be achieved under default correlation. Interest rate and credit spread volatilities are fully accounted for, as is the impact of re-hypothecation, collateral margining frequency, and dependencies. </jats:p>

Pallavicini A, Perini D, Brigo D, 2012, Funding, Collateral and Hedging: uncovering the mechanics and the subtleties of funding valuation adjustments

The main result of this paper is a collateralized counterparty valuationadjusted pricing equation, which allows to price a deal while taking intoaccount credit and debit valuation adjustments (CVA, DVA) along with marginingand funding costs, all in a consistent way. Funding risk breaks the bilateralnature of the valuation formula. We find that the equation has a recursiveform, making the introduction of a purely additive funding valuation adjustment(FVA) difficult. Yet, we can cast the pricing equation into a set of iterativerelationships which can be solved by means of standard least-square Monte Carlotechniques. As a consequence, we find that identifying funding costs and debitvaluation adjustments is not tenable in general, contrary to what has beensuggested in the literature in simple cases. The assumptions under whichfunding costs vanish are a very special case of the more general theory. Wedefine a comprehensive framework that allows us to derive earlier results onfunding or counterparty risk as a special case, although our framework is morethan the sum of such special cases. We derive the general pricing equation byresorting to a risk-neutral approach where the new types of risks are includedby modifying the payout cash flows. We consider realistic settings and includein our models the common market practices suggested by ISDA documentation,without assuming restrictive constraints on margining procedures and close-outnetting rules. In particular, we allow for asymmetric collateral and fundingrates, and exogenous liquidity policies and hedging strategies.Re-hypothecation liquidity risk and close-out amount evaluation issues are alsocovered. Finally, relevant examples of non-trivial settings illustrate how toderive known facts about discounting curves from a robust general framework andwithout resorting to ad hoc hypotheses.

BRIGO D, BUESCU C, MORINI M, 2012, COUNTERPARTY RISK PRICING: IMPACT OF CLOSEOUT AND FIRST-TO-DEFAULT TIMES, *International Journal of Theoretical and Applied Finance*, Vol: 15, Pages: 1250039-1250039, ISSN: 0219-0249

<jats:p> In the absence of a universally accepted procedure for the credit valuation adjustment (CVA) calculation, we compare a number of different bilateral counterparty valuation adjustment (BVA) formulas. First we investigate the impact of the choice of the closeout convention used in the formulas. Important consequences on default contagion manifest themselves in a rather different way depending on which closeout formulation is used (risk-free or replacement), and on default dependence between the two entities in the deal. Second we compare the full bilateral formula with an approximation that is based on subtracting two unilateral credit valuation adjustment (UCVA) formulas. Although the latter might be attractive for its instantaneous implementation once one has a unilateral CVA system, it ignores the impact of the first-to-default time, when closeout procedures are ignited. We illustrate in a number of realistic cases both the contagion effect due to the closeout convention, and the CVA pricing error due to ignoring the first-to-default time. </jats:p>

Brigo D, Buescu C, Pallavicini A, et al., 2012, Illustrating a problem in the self-financing condition in two 2010-2011 papers on funding, collateral and discounting

We illustrate a problem in the self-financing condition used in the papers"Funding beyond discounting: collateral agreements and derivatives pricing"(Risk Magazine, February 2010) and "Partial Differential EquationRepresentations of Derivatives with Counterparty Risk and Funding Costs" (TheJournal of Credit Risk, 2011). These papers state an erroneous self-financingcondition. In the first paper, this is equivalent to assuming that the equityposition is self-financing on its own and without including the cash position.In the second paper, this is equivalent to assuming that a subportfolio isself-financing on its own, rather than the whole portfolio. The error in thefirst paper is avoided when clearly distinguishing between price processes,dividend processes and gain processes. We present an outline of the derivationthat yields the correct statement of the self-financing condition, clarifyingthe structure of the relevant funding accounts, and show that the final resultin "Funding beyond discounting" is correct, even if the self-financingcondition stated is not.

Brigo D, Capproni A, 2012, Bilateral Credit Valuation Adjustment with Application to Credit Default Swaps, Managing and Measuring Capital: For Banks and Financial Institutions, Editors: Ong, Publisher: Risk Books

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, 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 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, 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

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 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*

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

Brigo D, Pallavicini A, 2008, Counterparty Risk and Contingent CDS under correlation, *Risk Magazine*

Brigo D, 2008, CDS Options through Candidate Market Models and the CDS-Calibrated CIR++ Stochastic Intensity Model, Credit Risk: Models, Derivatives and Management, Editors: Wagner, Publisher: Taylor & Francis

Brigo D, Pallavicini A, Torresetti R, 2007, Cluster-based extension of the generalized poisson loss dynamics and consistency with single names, *International Journal of Theoretical and Applied Finance*, Vol: 10, ISSN: 0219-0249

We extend the common Poisson shock framework reviewed for example in Lindskog and McNeil [15] to a formulation avoiding repeated defaults, thus obtaining a model that can account consistently for single name default dynamics, cluster default dynamics and default counting process. This approach allows one to introduce significant dynamics, improving on the standard "bottom-up" approaches, and to achieve true consistency with single names, improving on most "top-down" loss models. Furthermore, the resulting GPCL model has important links with the previous GPL dynamical loss model in Brigo et al. [6], which we point out. Model extensions allowing for more articulated spread and recovery dynamics are hinted at. Calibration to both DJi-TRAXX and CDX index and tranche data across attachments and maturities shows that the GPCL model has the same calibration power as the GPL model while allowing for consistency with single names.

Brigo D, Pallavicini A, Torresetti R, 2007, CDO calibration with the dynamical Generalized Poisson Loss model, *Risk Magazine*

Brigo D, Pallavicini A, Torresetti R, 2007, Cluster-based extension of the generalized poisson loss dynamics and consistency with single names, Credit Correlation - Life After Copulas, Editors: Lipton A, Rennie A, Publisher: World Scientific

Brigo D, 2006, Constant Maturity CDS valuation with market models, *Risk Magazine*

Brigo D, Masetti M, 2006, Risk Neutral Pricing of Counterparty Risk, Counterparty Credit Risk Modeling: Risk Management, Pricing and Regulation, Editors: Pykhtin, Publisher: Risk Books

BRIGO DAMIANO, COUSOT LAURENT, 2006, THE STOCHASTIC INTENSITY SSRD MODEL IMPLIED VOLATILITY PATTERNS FOR CREDIT DEFAULT SWAP OPTIONS AND THE IMPACT OF CORRELATION, *International Journal of Theoretical and Applied Finance*, Vol: 09, Pages: 315-339

In this paper we investigate implied volatility patterns in the Shifted Square Root Diffusion (SSRD) model as functions of the model parameters. We begin by recalling the Credit Default Swap (CDS) options market model that is consistent with a market Black-like formula, thus introducing a notion of implied volatility for CDS options. We examine implied volatilities coming from SSRD prices and characterize the qualitative behavior of implied volatilities as functions of the SSRD model parameters. We introduce an analytical approximation for the SSRD implied volatility that follows the same patterns in the model parameters and that can be used to have a first rough estimate of the implied volatility following a calibration. We compute numerically the CDS-rate volatility smile for the adopted SSRD model. We find a decreasing pattern of SSRD implied volatilities in the interest-rate/intensity correlation. We check whether it is possible to assume zero correlation after the option maturity in computing the option price.

Brigo D, Mercurio F, 2006, Interest Rate Models: Theory and Practice - with Smile, Inflation and Credit, Heidelberg, Publisher: Springer Verlag

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