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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805Weeks BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

139 results found

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

Brigo D, Nordio C, 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.

Scholarly edition

Brigo D, Tarenghi M, 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.

Scholarly edition

Brigo D, Tarenghi M, 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.

Scholarly edition

Brigo D, El-Bachir N, An exact formula for default swaptions' pricing in the SSRJD stochastic intensity model

We develop and test a fast and accurate semi-analytical formula for single-name default swaptions in the context of a shifted square root jump diffusion (SSRJD) default intensity model. The model can be calibrated to the CDS term structure and a few default swaptions, to price and hedge other credit derivatives consistently. We show with numerical experiments that the model implies plausible volatility smiles.

Scholarly edition

Morini M, Brigo D, 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.

Scholarly edition

Brigo D, El-Bachir N, Credit Derivatives Pricing with a Smile-Extended Jump Stochastic Intensity Model

We present a two-factor stochastic default intensity and interest rate model for pricing single-name default swaptions. The specific positive square root processes considered fall in the relatively tractable class of affine jump diffusions while allowing for inclusion of stochastic volatility and jumps in default swap spreads. The parameters of the short rate dynamics are first calibrated to the interest rates markets, before calibrating separately the default intensity model to credit derivatives market data. A few variants of the model are calibrated in turn to market data, and different calibration procedures are compared. Numerical experiments show that the calibrated model can generate plausible volatility smiles. Hence, the model can be calibrated to a default swap term structure and few default swaptions, and the calibrated parameters can be used to value consistently other default swaptions (different strikes and maturities, or more complex structures) on the same credit reference name.

Scholarly edition

Brigo D, Mercurio F, 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.

Scholarly edition

Brigo D, Capponi A, Bilateral counterparty risk valuation with stochastic dynamical models and application to Credit Default Swaps

We introduce the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, including 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 value of the contract at the relevant default times. We allow for correlation between the default times of the investor, counterparty and underlying portfolio risk factors. We use arbitrage-free stochastic dynamical models. We then specialize our analysis to Credit Default Swaps (CDS) as underlying portfolio, generalizing the work of Brigo and Chourdakis (2008) [5] who deal with unilateral and asymmetric counterparty risk. We introduce stochastic intensity models and a trivariate copula function on the default times exponential variables to model default dependence. Similarly to [5], we find that both default correlation and credit spread volatilities have a relevant and structured impact on the adjustment. Differently from [5], the two parties will now agree on the credit valuation adjustment. We study a case involving British Airways, Lehman Brothers and Royal Dutch Shell, illustrating the bilateral adjustments in concrete crisis situations.

Scholarly edition

Brigo D, 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.

Scholarly edition

Brigo D, Pallavicini A, Torresetti R, 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

Working paper

Brigo D, 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.

Scholarly edition

Brigo D, 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.

Scholarly edition

Brigo D, Dalessandro A, Neugebauer M, Triki Fet al., 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.

Scholarly edition

Brigo D, Hanzon B, 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.

Scholarly edition

Brigo D, Morini M, 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.

Scholarly edition

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

Brigo D, Chourdakis K, Bakkar I, 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.

Scholarly edition

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

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