Imperial College London

ProfessorDamianoBrigo

Faculty of Natural SciencesDepartment of Mathematics

Chair in Mathematical Finance
 
 
 
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Contact

 

damiano.brigo

 
 
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Location

 

805Weeks BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

141 results found

Brigo D, 1996, New results on the Gaussian protection filter with small observation noise, SYSTEMS & CONTROL LETTERS, Vol: 28, Pages: 273-279, ISSN: 0167-6911

Journal article

Brigo D, 1995, On the nice behaviour of the Gaussian projection filter with small observation noise, SYSTEMS & CONTROL LETTERS, Vol: 26, Pages: 363-370, ISSN: 0167-6911

Journal article

Brigo D, Hanzon B, LeGland F, 1995, A differential geometric approach to nonlinear filtering: The projection filter, 34th IEEE Conference on Decision and Control, Publisher: I E E E, Pages: 4006-4011

Conference paper

El-Bachir N, Brigo D, An analytically tractable time-changed jump-diffusion default intensity model

We present a stochastic default intensity model where the intensity follows a tractable jump-diffusion process obtained by applying a deterministic change of time to a non mean-reverting square root jump-diffusion process. The model generates higher implied volatilities for default swaptions than mean-reverting versions, consistent with volatility levels observed on the market.

Scholarly edition

Graceffa F, Brigo D, Pallavicini A, On the consistency of jump-diffusion dynamics for FX rates under inversion

In this note we investigate the consistency under inversion of jump diffusionprocesses in the Foreign Exchange (FX) market. In other terms, if the EUR/USDFX rate follows a given type of dynamics, under which conditions will USD/EURfollow the same type of dynamics? In order to give a numerical description ofthis property, we first calibrate a Heston model and a SABR model to marketdata, plotting their smiles together with the smiles of the reciprocalprocesses. Secondly, we determine a suitable local volatility structureensuring consistency. We subsequently introduce jumps and analyze both constantjump size (Poisson process) and random jump size (compound Poisson process). Inthe first scenario, we find that consistency is automatically satisfied, forthe jump size of the inverted process is a constant as well. The second case ismore delicate, since we need to make sure that the distribution of jumps in thedomestic measure is the same as the distribution of jumps in the foreignmeasure. We determine a fairly general class of admissible densities for thejump size in the domestic measure satisfying the condition.

Working paper

Brigo D, Pallavicini A, Papatheodorou V, 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.

Working paper

Brigo D, Morini M, Tarenghi M, 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.

Scholarly edition

Brigo D, Pallavicini A, Torresetti R, 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.

Scholarly edition

Brigo D, Predescu M, Capponi A, Credit Default Swaps Liquidity modeling: A survey

We review different approaches for measuring the impact of liquidity on CDS prices. We start with reduced form models incorporating liquidity as an additional discount rate. We review Chen, Fabozzi and Sverdlove (2008) and Buhler and Trapp (2006, 2008), adopting different assumptions on how liquidity rates enter the CDS premium rate formula, about the dynamics of liquidity rate processes and about the credit-liquidity correlation. Buhler and Trapp (2008) provides the most general and realistic framework, incorporating correlation between liquidity and credit, liquidity spillover effects between bonds and CDS contracts and asymmetric liquidity effects on the Bid and Ask CDS premium rates. We then discuss the Bongaerts, De Jong and Driessen (2009) study which derives an equilibrium asset pricing model incorporating liquidity effects. Findings include that both expected illiquidity and liquidity risk have a statistically significant impact on expected CDS returns. We finalize our review with a discussion of Predescu et al (2009), which analyzes also data in-crisis. This is a statistical model that associates an ordinal liquidity score with each CDS reference entity and allows one to compare liquidity of over 2400 reference entities. This study points out that credit and illiquidity are correlated, with a smile pattern. All these studies highlight that CDS premium rates are not pure measures of credit risk. Further research is needed to measure liquidity premium at CDS contract level and to disentangle liquidity from credit effectively.

Scholarly edition

Brigo D, Buescu C, Morini M, 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.

Scholarly edition

Pallavicini A, Perini D, Brigo D, 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.

Working paper

Albanese C, Brigo D, Oertel F, 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.

Working paper

Brigo D, Capponi A, Pallavicini A, Papatheodorou Vet al., 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.

Working paper

Brigo D, Pistone G, Optimal approximations of the Fokker-Planck-Kolmogorov equation: projection, maximum likelihood eigenfunctions and Galerkin methods

We study optimal finite dimensional approximations of the generallyinfinite-dimensional Fokker-Planck-Kolmogorov (FPK) equation, finding the curvein a given finite-dimensional family that best approximates the exact solutionevolution. For a first local approximation we assign a manifold structure tothe family and a metric. We then project the vector field of the partialdifferential equation (PDE) onto the tangent space of the chosen family, thusobtaining an ordinary differential equation for the family parameter. A secondglobal approximation will be based on projecting directly the exact solutionfrom its infinite dimensional space to the chosen family using the nonlinearmetric projection. This will result in matching expectations with respect tothe exact and approximating densities for particular functions associated withthe chosen family, but this will require knowledge of the exact solution ofFPK. A first way around this is a localized version of the metric projectionbased on the assumed density approximation. While the localization will removeglobal optimality, we will show that the somewhat arbitrary assumed densityapproximation is equivalent to the mathematically rigorous vector fieldprojection. More interestingly we study the case where the approximating familyis defined based on a number of eigenfunctions of the exact equation. In thiscase we show that the local vector field projection provides also the globallyoptimal approximation in metric projection, and for some families thiscoincides with a Galerkin method.

Journal article

Brigo D, Jeanblanc M, Vrins F, SDEs with uniform distributions: Peacocks, Conic martingales and ergodic uniform diffusions

It is known since Kellerer (1972) that for any process that is increasing forthe convex order, or "peacock" as in Hirsch et al. 2011, there existmartingales with the same marginals laws. Nevertheless, there is no generalconstructive method for finding such martingales that yields diffusions. Weconsider the uniform peacock, namely the peacock with uniform law at all timeson a generic time-varying support [a(t); b(t)]. We derive explicitly thecorresponding SDEs and prove that, under certain "conic" conditions on a(t) andb(t), they admit a unique strong diffusive solution. To guess the candidate SDEwe resort to the approach of inverting the Fokker Planck equation. Dupire(1994) did this for volatility modeling. Here we tackle the inversion with thecaveats needed when dealing with uniform margins with conic boundaries. Thiswas done originally in the unpublished preprint by Brigo (1999). Independently,Madan and Yor (2002) obtained the result as a simple application of Dupire.Once the SDE is guessed, we analyze it rigorously, discussing the cases whereour approach adds strong uniqueness of the solution of the SDE and cases whereonly a weak solution is obtained. We further study the local time and activityof the solution. We then study the peacock with uniform law at all times on aconstant support [-1; 1] and derive the SDE of an associated mean-revertingdiffusion process with uniform margins that is not a martingale. For therelated SDE we prove existence of a solution. We derive the exact transitiondensities for both the mean reverting and the original conic martingale cases.We prove limit-laws and ergodic results: the SDE solution transition law tendsto be uniform after a long time. Finally, we provide a numerical studyconfirming the desired uniform behaviour. These results may be used to modelrandom probabilities, recovery rates or correlations.

Journal article

Armstrong J, Brigo D, Coordinate-free Stochastic Differential Equations as Jets

We explain how It\^o Stochastic Differential Equations (SDEs) on manifoldsmay be defined using 2-jets of smooth functions. We show how this relationshipcan be interpreted in terms of a convergent numerical scheme. We show how jetscan be used to derive graphical representations of It\^o SDEs. We show how jetscan be used to derive the differential operators associated with SDEs in acoordinate free manner. We relate jets to vector flows, giving a geometricinterpretation of the It\^o--Stratonovich transformation. We show howpercentiles can be used to give an alternative coordinate free interpretationof the coefficients of one dimensional SDEs. We relate this to the jetapproach. This allows us to interpret the coefficients of SDEs in terms of "fandiagrams". In particular the median of a SDE solution is associated to thedrift of the SDE in Stratonovich form for small times.

Journal article

Brigo D, Vrins F, Disentangling wrong-way risk: pricing CVA via change of measures and drift adjustment

A key driver of Credit Value Adjustment (CVA) is the possible dependencybetween exposure and counterparty credit risk, known as Wrong-Way Risk (WWR).At this time, addressing WWR in a both sound and tractable way remainschallenging: arbitrage-free setups have been proposed by academic researchthrough dynamic models but are computationally intensive and hard to use inpractice. Tractable alternatives based on resampling techniques have beenproposed by the industry, but they lack mathematical foundations. This probablyexplains why WWR is not explicitly handled in the Basel III regulatoryframework in spite of its acknowledged importance. The purpose of this paper isto propose a new method consisting of an appealing compromise: we start from astochastic intensity approach and end up with a pricing problem where WWR doesnot enter the picture explicitly. This result is achieved thanks to a set ofchanges of measure: the WWR effect is now embedded in the drift of theexposure, and this adjustment can be approximated by a deterministic functionwithout affecting the level of accuracy typically required for CVA figures. Theperformances of our approach are illustrated through an extensive comparison ofExpected Positive Exposure (EPE) profiles and CVA figures produced either by(i) the standard method relying on a full bivariate Monte Carlo framework and(ii) our drift-adjustment approximation. Given the uncertainty inherent to CVA,the proposed method is believed to provide a promising way to handle WWR in asound and tractable way.

Journal article

Brigo D, Pistone G, Projection based dimensionality reduction for measure valued evolution equations in statistical manifolds

We propose a dimensionality reduction method for infinite-dimensionalmeasure-valued evolution equations such as the Fokker-Planck partialdifferential equation or the Kushner-Stratonovich resp. Duncan-Mortensen-Zakaistochastic partial differential equations of nonlinear filtering, withpotential applications to signal processing, quantitative finance, heat flowsand quantum theory among many other areas. Our method is based on theprojection coming from a duality argument built in the exponential statisticalmanifold structure developed by G. Pistone and co-authors. The choice of thefinite dimensional manifold on which one should project the infinitedimensional equation is crucial, and we propose finite dimensional exponentialand mixture families. This same problem had been studied, especially in thecontext of nonlinear filtering, by D. Brigo and co-authors but the $L^2$structure on the space of square roots of densities or of densities themselveswas used, without taking an infinite dimensional manifold environment space forthe equation to be projected. Here we re-examine such works from theexponential statistical manifold point of view, which allows for a deepergeometric understanding of the manifold structures at play. We also show thatthe projection in the exponential manifold structure is consistent with theFisher Rao metric and, in case of finite dimensional exponential families, withthe assumed density approximation. Further, we show that if the sufficientstatistics of the finite dimensional exponential family are chosen among theeigenfunctions of the backward diffusion operator then the statistical-manifoldor Fisher-Rao projection provides the maximum likelihood estimator for theFokker Planck equation solution. We finally try to clarify how the finitedimensional and infinite dimensional terminology for exponential and mixturespaces are related.

Journal article

Brigo D, Pede N, Petrelli A, Multi Currency Credit Default Swaps Quanto effects and FX devaluation jumps

Credit Default Swaps (CDS) on a reference entity may be traded in multiplecurrencies, in that protection upon default may be offered either in thedomestic currency where the entity resides, or in a more liquid and globalforeign currency. In this situation currency fluctuations clearly introduce asource of risk on CDS spreads. For emerging markets, but in some cases even inwell developed markets, the risk of dramatic Foreign Exchange (FX) ratedevaluation in conjunction with default events is relevant. We address thisissue by proposing and implementing a model that considers the risk of foreigncurrency devaluation that is synchronous with default of the reference entity. Preliminary results indicate that perceived risks of devaluation can induce asignificant basis across domestic and foreign CDS quotes. For the Republic ofItaly, a USD CDS spread quote of 440 bps can translate into a EUR quote of 350bps in the middle of the Euro-debt crisis in the first week of May 2012. Morerecently, from June 2013, the basis spreads between the EUR quotes and the USDquotes are in the range around 40 bps. We explain in detail the sources for such discrepancies. Our modelingapproach is based on the reduced form framework for credit risk, where thedefault time is modeled in a Cox process setting with explicit diffusiondynamics for default intensity/hazard rate and exponential jump to default. Forthe FX part, we include an explicit default-driven jump in the FX dynamics. Asour results show, such a mechanism provides a further and more effective way tomodel credit / FX dependency than the instantaneous correlation that can beimposed among the driving Brownian motions of default intensity and FX rates,as it is not possible to explain the observed basis spreads during theEuro-debt crisis by using the latter mechanism alone.

Journal article

Brigo D, Durand C, An initial approach to Risk Management of Funding Costs

In this note we sketch an initial tentative approach to funding costsanalysis and management for contracts with bilateral counterparty risk in asimplified setting. We depart from the existing literature by analyzing theissue of funding costs and benefits under the assumption that the associatedrisks cannot be hedged properly. We also model the treasury funding spread bymeans of a stochastic Weighted Cost of Funding Spread (WCFS) which helpsdescribing more realistic financing policies of a financial institution. Weelaborate on some limitations in replication-based Funding / Credit ValuationAdjustments we worked on ourselves in the past, namely CVA, DVA, FVA andrelated quantities as generally discussed in the industry. We advocate as adifferent possibility, when replication is not possible, the analysis of thefunding profit and loss distribution and explain how long term funding spreads,wrong way risk and systemic risk are generally overlooked in most of thecurrent literature on risk measurement of funding costs. As a matter of initialillustration, we discuss in detail the funding management of interest rateswaps with bilateral counterparty risk in the simplified setup of our frameworkthrough numerical examples and via a few simplified assumptions.

Journal article

Brigo D, Rapisarda F, Sridi A, The arbitrage-free Multivariate Mixture Dynamics Model: Consistent single-assets and index volatility smiles

We introduce a multivariate diffusion model that is able to price derivativesecurities featuring multiple underlying assets. Each asset volatility smile ismodeled according to a density-mixture dynamical model while the same propertyholds for the multivariate process of all assets, whose density is a mixture ofmultivariate basic densities. This allows to reconcile single name andindex/basket volatility smiles in a consistent framework. Our approach could bedubbed a multidimensional local volatility approach with vector-state dependentdiffusion matrix. The model is quite tractable, leading to a complete marketand not requiring Fourier techniques for calibration and dependence measures,contrary to multivariate stochastic volatility models such as Wishart. We proveexistence and uniqueness of solutions for the model stochastic differentialequations, provide formulas for a number of basket options, and analyze thedependence structure of the model in detail by deriving a number of results oncovariances, its copula function and rank correlation measures andvolatilities-assets correlations. A comparison with sampling simply-correlatedsuitably discretized one-dimensional mixture dynamical paths is made, both interms of option pricing and of dependence, and first order expansionrelationships between the two models' local covariances are derived. We alsoshow existence of a multivariate uncertain volatility model of which ourmultivariate local volatilities model is a Markovian projection, highlightingthat the projected model is smoother and avoids a number of drawbacks of theuncertain volatility version. We also show a consistency result where theMarkovian projection of a geometric basket in the multivariate model is aunivariate mixture dynamics model. A few numerical examples on basket andspread options pricing conclude the paper.

Journal article

Sarais G, Brigo D, Inflation securities valuation with macroeconomic-based no-arbitrage dynamics

We develop a model to price inflation and interest rates derivatives usingcontinuous-time dynamics that have some links with macroeconomic monetary DSGEmodels equipped with a Taylor rule: in particular, the reaction function of thecentral bank, the bond market liquidity, inflation and growth expectations playan important role. The model can explain the effects of non-standard monetarypolicies (like quantitative easing or its tapering) and shed light on howcentral bank policy can affect the value of inflation and interest ratesderivatives. The model is built under standard no-arbitrage assumptions. Interestingly,the model yields short rate dynamics that are consistent with a time-varyingHull-White model, therefore making the calibration to the nominal interestcurve and options straightforward. Further, we obtain closed forms for bothzero-coupon and year-on-year inflation swap and options. The calibrationstrategy we propose is fully separable, which means that the calibration can becarried out in subsequent simple steps that do not require heavy computation. Amarket calibration example is provided. The advantages of such structural inflation modelling become apparent whenone starts doing risk analysis on an inflation derivatives book: because themodel explicitly takes into account economic variables, a trader can easilyassess the impact of a change in central bank policy on a complex book of fixedincome instruments, which is normally not straightforward if one is usingstandard inflation pricing models.

Journal article

Brigo D, Graziano GD, Optimal execution comparison across risks and dynamics, with solutions for displaced diffusions

We solve a version of the optimal trade execution problem when the mid assetprice follows a displaced diffusion. Optimal strategies in the adapted classunder various risk criteria, namely value-at-risk, expected shortfall and a newcriterion called "squared asset expectation" (SAE), related to a version of thecost variance measure, are derived and compared. It is well known thatdisplaced diffusions (DD) exhibit dynamics which are in-between arithmeticBrownian motions (ABM) and geometric Brownian motions (GBM) depending of thechoice of the shift parameter. Furthermore, DD allows for changes in thesupport of the mid asset price distribution, allowing one to include a minimumpermitted value for the mid price, either positive or negative. We study thedependence of the optimal solution on the choice of the risk aversioncriterion. Optimal solutions across criteria and asset dynamics are comparablealthough differences are not negligible for high levels of risk aversion andlow market impact assets. This is illustrated with numerical examples.

Journal article

Brigo D, Mai J-F, Scherer M, Consistent iterated simulation of multi-variate default times: a Markovian indicators characterization

We investigate under which conditions a single simulation of joint defaulttimes at a final time horizon can be decomposed into a set of simulations ofjoint defaults on subsequent adjacent sub-periods leading to that finalhorizon. Besides the theoretical interest, this is also a practical problem aspart of the industry has been working under the misleading assumption that thetwo approaches are equivalent for practical purposes. As a reasonable trade-offbetween realistic stylized facts, practical demands, and mathematicaltractability, we propose models leading to a Markovian multi-variatesurvival--indicator process, and we investigate two instances of static modelsfor the vector of default times from the statistical literature that fall intothis class. On the one hand, the "looping default" case is known to be equippedwith this property, and we point out that it coincides with the classical"Freund distribution" in the bivariate case. On the other hand, if allsub-vectors of the survival indicator process are Markovian, this constitutes anew characterization of the Marshall--Olkin distribution, and hence ofmulti-variate lack-of-memory. A paramount property of the resulting model isstability of the type of multi-variate distribution with respect to eliminationor insertion of a new marginal component with marginal distribution from thesame family. The practical implications of this "nested margining" property areenormous. To implement this distribution we present an efficient and unbiasedsimulation algorithm based on the L\'evy-frailty construction. We highlightdifferent pitfalls in the simulation of dependent default times and examine,within a numerical case study, the effect of inadequate simulation practices.

Journal article

Brigo D, Garcia J, Pede N, CoCo Bonds Valuation with Equity- and Credit-Calibrated First Passage Structural Models

After the beginning of the credit and liquidity crisis, financialinstitutions have been considering creating a convertible-bond type contractfocusing on Capital. Under the terms of this contract, a bond is converted intoequity if the authorities deem the institution to be under-capitalized. Thispaper discusses this Contingent Capital (or Coco) bond instrument and presentsa pricing methodology based on firm value models. The model is calibrated toreadily available market data. A stress test of model parameters is illustratedto account for potential model risk. Finally, a brief overview of how theinstrument performs is presented.

Journal article

Brigo D, The direct L2 geometric structure on a manifold of probability densities with applications to Filtering

In this paper we introduce a projection method for the space of probabilitydistributions based on the differential geometric approach to statistics. Thismethod is based on a direct L2 metric as opposed to the usual Hellingerdistance and the related Fisher Information metric. We explain how thisapparatus can be used for the nonlinear filtering problem, in relationship alsoto earlier projection methods based on the Fisher metric. Past projectionfilters focused on the Fisher metric and the exponential families that made thefilter correction step exact. In this work we introduce the mixture projectionfilter, namely the projection filter based on the direct L2 metric and based ona manifold given by a mixture of pre-assigned densities. The resultingprediction step in the filtering problem is described by a linear differentialequation, while the correction step can be made exact. We analyze therelationship of a specific class of L2 filters with the Galerkin basednonlinear filters, and highlight the differences with our approach, concerningparticularly the continuous--time observations filtering problems.

Journal article

Brigo D, Chourdakis K, Consistent single- and multi-step sampling of multivariate arrival times: A characterization of self-chaining copulas

This paper deals with dependence across marginally exponentially distributedarrival times, such as default times in financial modeling or inter-failuretimes in reliability theory. We explore the relationship between dependence andthe possibility to sample final multivariate survival in a long time-intervalas a sequence of iterations of local multivariate survivals along a partitionof the total time interval. We find that this is possible under a form ofmultivariate lack of memory that is linked to a property of the survival timescopula. This property defines a "self-chaining-copula", and we show that thiscoincides with the extreme value copulas characterization. The self-chainingcondition is satisfied by the Gumbel-Hougaard copula, a full characterizationof self chaining copulas in the Archimedean family, and by the Marshall-Olkincopula. The result has important practical implications for consistentsingle-step and multi-step simulation of multivariate arrival times in a waythat does not destroy dependency through iterations, as happens wheninconsistently iterating a Gaussian copula.

Journal article

Brigo D, Pistone G, Projecting the Fokker-Planck Equation onto a finite dimensional exponential family

In the present paper we discuss problems concerning evolutions of densitiesrelated to Ito diffusions in the framework of the statistical exponentialmanifold. We develop a rigorous approach to the problem, and we particularizeit to the orthogonal projection of the evolution of the density of a diffusionprocess onto a finite dimensional exponential manifold. It has been shown by D.Brigo (1996) that the projected evolution can always be interpreted as theevolution of the density of a different diffusion process. We give also acompactness result when the dimension of the exponential family increases, as afirst step towards a convergence result to be investigated in the future. Theinfinite dimensional exponential manifold structure introduced by G. Pistoneand C. Sempi is used and some examples are given.

Journal article

Brigo D, Francischello M, Pallavicini A, An indifference approach to the cost of capital constraints: KVA and beyond

The strengthening of capital requirements has induced banks and traders toconsider charging a so called capital valuation adjustment (KVA) to the clientsin OTC transactions. This roughly corresponds to charge the clients ex-ante theprofit requirement that is asked to the trading desk. In the following we tryto delineate a possible way to assess the impact of capital constraints in thevaluation of a deal. We resort to an optimisation stemming from an indifferencepricing approach, and we study both the linear problem from the point of viewof the whole bank and the non-linear problem given by the viewpoint ofshareholders. We also consider the case where one optimises the median ratherthan the mean statistics of the profit and loss distribution.

Journal article

Armstrong J, Brigo D, Optimizing S-shaped utility and implications for risk management

We consider market players with tail-risk-seeking behaviour as exemplified bythe S-shaped utility introduced by Kahneman and Tversky. We argue that riskmeasures such as value at risk (VaR) and expected shortfall (ES) areineffective in constraining such players. We show that, in many standard marketmodels, product design aimed at utility maximization is not constrained at allby VaR or ES bounds: the maximized utility corresponding to the optimal payoffis the same with or without ES constraints. By contrast we show that, inreasonable markets, risk management constraints based on a second moreconventional concave utility function can reduce the maximum S-shaped utilitythat can be achieved by the investor, even if the constraining utility functionis only rather modestly concave. It follows that product designs leading tounbounded S-shaped utilities will lead to unbounded negative expectedconstraining utilities when measured with such conventional utility functions.To prove these latter results we solve a general problem of optimizing aninvestor expected utility under risk management constraints where both investorand risk manager have conventional concave utility functions, but the investorhas limited liability. We illustrate our results throughout with the example ofthe Black--Scholes option market. These results are particularly importantgiven the historical role of VaR and that ES was endorsed by the Baselcommittee in 2012--2013.

Journal article

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