## Publications

141 results found

Armstrong J, Brigo D, Rossi Ferrucci E, 2019, Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections, *Proceedings of the London Mathematical Society*, Vol: 119, Pages: 176-213, ISSN: 1460-244X

We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Itô‐vector and Itô‐jet projections. This allows one to systematically develop low‐dimensional approximations to high‐dimensional SDEs using differential geometric techniques. The approach generalizes the notion of projecting a vector field onto a submanifold in order to derive approximations to ordinary differential equations, and improves the previous Stratonovich projection method by adding optimality analysis and results. Indeed, just as in the case of ordinary projection, our definitions of projection are based on optimality arguments and give in a well‐defined sense ‘optimal’ approximations to the original SDE in the mean‐square sense over small times. We also explain how the Stratonovich projection satisfies an optimality criterion that is more ad hoc and less appealing than the criteria satisfied by the Itô projections we introduce.As an application, we consider approximating the solution of the non‐linear filtering problem with a Gaussian distribution. We show how the newly introduced Itô projections lead to optimal approximations in the Gaussian family and briefly discuss the optimal approximation for more general families of distributions. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

Brigo D, Pede N, Petrelli A, 2019, Multi-currency credit default swaps, *International Journal of Theoretical and Applied Finance*, Vol: 22, ISSN: 0219-0249

Credit default swaps (CDS) on a reference entity may be traded in multiple currencies, in that, protection upon default may be offered either in the currency where the entity resides, or in a more liquid and global foreign currency. In this situation, currency fluctuations clearly introduce a source of risk on CDS spreads. For emerging markets, but in some cases even in well-developed markets, the risk of dramatic foreign exchange (FX)-rate devaluation in conjunction with default events is relevant. We address this issue by proposing and implementing a model that considers the risk of foreign currency devaluation that is synchronous with default of the reference entity. As a fundamental case, we consider the sovereign CDSs on Italy, quoted both in EUR and USD. Preliminary results indicate that perceived risks of devaluation can induce a significant basis across domestic and foreign CDS quotes. For the Republic of Italy, a USD CDS spread quote of 440 bps can translate into an EUR quote of 350bps in the middle of the Euro-debt crisis in the first week of May 2012. More recently, from June 2013, the basis spreads between the EUR quotes and the USD quotes are in the range around 40bps. We explain in detail the sources for such discrepancies. Our modeling approach is based on the reduced form framework for credit risk, where the default time is modeled in a Cox process setting with explicit diffusion dynamics for default intensity/hazard rate and exponential jump to default. For the FX part, we include an explicit default-driven jump in the FX dynamics. As our results show, such a mechanism provides a further and more effective way to model credit/FX dependency than the instantaneous correlation that can be imposed among the driving Brownian motions of default intensity and FX rates, as it is not possible to explain the observed basis spreads during the Euro-debt crisis by using the latter mechanism alone.

Brigo D, Pisani C, Rapisarda F, 2019, The multivariate mixture dynamics model: Shifted dynamics and correlation skew, *Annals of Operations Research*, ISSN: 0254-5330

The multi variate mixture dynamics model is a tractable, dynamical, arbitrage-free multivariate model characterized by transparency on the dependence structure, since closed form formulae for terminal correlations, average correlations and copula function are available. It also allows for complete decorrelation between assets and instantaneous variances. Each single asset is modelled according to a lognormal mixture dynamics model, and this univariate version is widely used in the industry due to its flexibility and accuracy. The same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allows for consistency of single asset and index/portfolio smile. In this paper, we generalize the MVMD model by introducing shifted dynamics and we propose a definition of implied correlation under this model. We investigate whether the model is able to consistently reproduce the implied volatility of FX cross rates once the single components are calibrated to univariate shifted lognormal mixture dynamics models. We consider in particular the case of the Chinese Renminbi FX rate, showing that the shifted MVMD model correctly recovers the CNY/EUR smile given the EUR/USD smile and the USD/CNY smile, thus highlighting that the model can also work as an arbitrage free volatility smile extrapolation tool for cross currencies that may not be liquid or fully observable. We compare the performance of the shifted MVMD model in terms of implied correlation with those of the shifted simply correlated mixture dynamics model where the dynamics of the single assets are connected naively by introducing correlation among their Brownian motions. Finally, we introduce a model with uncertain volatilities and correlation. The Markovian projection of this model is a generalization of the shifted MVMD model.

Brigo D, Francischello M, Pallavicini A, 2019, Nonlinear valuation under credit, funding, and margins: existence, uniqueness, invariance, and disentanglement, *European Journal of Operational Research*, Vol: 274, Pages: 788-805, ISSN: 0377-2217

Since the 2008 global financial crisis, the banking industry has been using valuation adjustments to account for default risk and funding costs. These adjustments are computed separately and added together by practitioners as if the valuation equations were linear. This assumption is too strong and does not allow to model market features such as different borrowing and lending rates and replacement default closeout. Hence we argue that the full valuation equations are nonlinear, and this paper is devoted to studying the nonlinear valuation equations introduced in Pallavicini et al (2011).We illustrate all the cash flows exchanged by the parties involved in a derivative contract, in presence of default risk, collateralisation with re-hypothecation and funding costs. Then we show how to obtain semi-linear PDEs or Forward Backward Stochastic Differential Equations (FBSDEs) from present-valuing said cash flows in an arbitrage-free setup, and we study the well-posedness of these PDEs and FBSDEs in a viscosity and classical sense.Moreover, from a financial perspective, we discuss cases where classical valuation adjustments (XVA) can be disentangled. We show how funding costs are offset by treasury valuation adjustments when one takes a whole-bank perspective in the valuation, while the same costs are not offset by such adjustments when taking a shareholder perspective. We show that although we use a risk-neutral valuation framework based on a locally risk-free bank account, our final valuation equations do not depend on the risk-free rate. Finally, we show how to consistently derive a netting set valuation from a portfolio level one.

Armstrong J, Brigo D, 2019, Risk managing tail-risk seekers: VaR and expected shortfall vs S-shaped utility, *Journal of Banking & Finance*, Vol: 101, Pages: 122-135, ISSN: 0378-4266

We consider market players with tail-risk-seeking behaviour modelled by S-shaped utility, as introduced by Kahneman and Tversky. We argue that risk measures such as value at risk (VaR) and expected shortfall (ES) are ineffective in constraining such players, as such measures cannot reduce the traders expected S-shaped utilities. Indeed, when designing payoffs aiming to maximize utility under a VaR or ES risk limit, the players will attain the same supremum of expected utility with or without VaR or ES limits. By contrast, we show that risk management constraints based on a second more conventional concave utility function can reduce the maximum S-shaped utility that can be achieved by the investor. Indeed, product designs leading to progressively larger S-shaped utilities will lead to progressively lower expected constraining conventional utilities, violating the related risk limit. These results hold in a variety of market models, including the Black Scholes options model, and are particularly relevant for risk managers given the historical role of VaR and the endorsement of ES by the Basel committee in 2012–2013.

Brigo D, Vrins F, 2018, Disentangling wrong-way risk: pricing credit valuation adjustment via change of measures, *European Journal of Operational Research*, Vol: 269, Pages: 1154-1164, ISSN: 0377-2217

In many financial contracts (and in particular when trading OTC derivatives), participantsare exposed to counterparty risk. The latter is typically rewarded by adjusting the “risk-freeprice” of derivatives; an adjustment known ascredit value adjustment(CVA). A key driverof CVA is the dependency between exposure and counterparty risk, known aswrong-way risk(WWR). In practice however, correctly addressing WWR is very challenging and calls forheavy numerical techniques. This might explain why WWR is not explicitly handled in theBasel III regulatory framework in spite of its acknowledged importance. In this paper wepropose a sound and tractable method to deal efficiently with WWR. Our approach consistsof embedding the WWR effect in the drift of the exposure dynamics. Even though thiscalls for infinite changes of measures, we end up with an appealing compromise betweentractability and mathematical rigor, preserving the level of accuracy typically required forCVA figures. The good performances of the method are discussed in a stochastic-intensitydefault setup based on extensive comparisons of Expected Positive Exposure (EPE) profilesand CVA figures produced (i) by a full bivariate Monte Carlo implementation of the initialmodel with (ii) our drift-adjustment technique.

Armstrong J, Brigo D, 2018, Rogue traders versus value-at-risk and expected shortfall, *Risk -London- Risk Magazine Limited-*, Pages: 63-63, ISSN: 0952-8776

We show that, in a Black and Scholes market, value at risk and ex-pected shortfall are irrelevant in limiting traders excessive tail-risk seekingbehaviour as modelled via Kahneman and Tversky’s S-shaped utility. Tohave effective constraints one can introduce a risk limit based on a secondbut concave utility function.

Brigo D, Hvolby T, Vrins F, Wrong-way risk adjusted exposure: analytical approximations for options in default intensity models, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Publishing Co.

We examine credit value adjustment (CVA) estimation under wrong-way risk(WWR) by computing the expected positive exposure (EPE) under an equiva-lent measure as suggested in [1], adjusting the drift of the underlying for defaultrisk. We apply this technique to European put and call options and derive theanalytic formulas for EPE under WWR obtained with various approximationsof the drift adjustment. We give the results of numerical experiments basedon 4 parameter sets, and supply figures of the CVA based on both of the sug-gested proxys, comparing with CVA based on a 2D-Monte Carlo scheme andGaussian Copula resampling. We also show the CVA obtained by the formulasfrom Basel III. We observe that the Basel III formula does not account forthe credit-market correlation, while the Gaussian Copula resampling methodestimates a too large impact of this correlation. The two proxies account forthe credit-market correlation, and give results that are mostly similar to the2D-Monte Carlo results.

Brigo D, Pede N, Petrelli A, Examples of wrong–way risk in CVA induced by devaluations on default, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Press

When calculatingCredit Valuation Adjustment(CVA), theinteraction between the portfolio’s exposure and the counter-party’s credit worthiness is referred to asWrong–Way Risk(WWR). Making the assumption that the Brownian mo-tions driving both the market (exposure) and the (counter-party) credit risk–factors dynamics are correlated representsthe simplest way of modelling the dependence structure be-tween these two components. For many practical applica-tions, however, such an approach may fail to account for theright amount of WWR, thus resulting in misestimates of theportfolio’s CVA. We present a modelling framework wherea further — and indeed stronger — source of market/creditdependence is introduced through devaluation jumps on themarket risk–factors’ dynamics. Such jumps happen upon thecounterparty’s default and are a particularly realistic featureto include in case of sovereign or systemically important coun-terparties. Moreover, we show that, in the special case wherethe focus is on FX/credit WWR, devaluation jumps provide an effective way of incorporating market information comingfrom quanto Credit Default Swap (CDS) basis spreads and wederive the corresponding CVA pricing equations as a systemof coupled PDEs.

Brigo D, Piat C, Static vs adapted optimal execution strategies in two benchmark trading models, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Publishing Co.

We consider the optimal solutions to the trade execution problem in the two different classes of i) fully adapted or adaptive and ii) deterministic or static strategies, comparing them. We do this in two different benchmark models. The first model is a discrete time framework with an information flow process, dealing with both permanent and temporary impact, minimizing the expected cost of the trade. The second model is a continuous time framework where the objective function is the sum of the expected cost and a value at risk (or expected shortfall) type risk criterion. Optimal adapted solutions are known in both frameworks from the original works of Bertsimas and Lo (1998) and Gatheral and Schied (2011). In this paper we derive the optimal static strategies for both benchmark models and we study quantitatively the improvement in optimality when moving from static strategies to fully adapted ones. We conclude that, in the benchmark models we study, the difference is not relevant, except for extreme unrealistic cases for the model or impact parameters. This indirectly confirms that in the similar framework of Almgren and Chriss (2000) one is fine deriving a static optimal solution, as done by those authors, as opposed to a fully adapted one, since the static solution happens to be tractable and known in closed form.

Armstrong J, Brigo D, 2018, Intrinsic stochastic differential equations as jets, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 474, ISSN: 1364-5021

We explain how Itô stochastic differential equations(SDEs) on manifolds may be defined using 2-jets ofsmooth functions. We show how this relationship canbe interpreted in terms of a convergent numericalscheme. We also show how jets can be used toderive graphical representations of Itô SDEs, and weshow how jets can be used to derive the differentialoperators associated with SDEs in a coordinatefreemanner. We relate jets to vector flows, givinga geometric interpretation of the Itô–Stratonovichtransformation. We show how percentiles can be usedto give an alternative coordinate-free interpretation ofthe coefficients of one-dimensional SDEs. We relatethis to the jet approach. This allows us to interpretthe coefficients of SDEs in terms of ‘fan diagrams’. Inparticular, the median of an SDE solution is associatedwith the drift of the SDE in Stratonovich form for smalltimes.

Brigo D, Mai, Jan, et al., Consistent iterated simulation of multivariate defaults: Markov indicators, lack of memory, extreme-value copulas, and the Marshall–Olkin distribution, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Publishing Co.

A current market-practice to incorporate multivariate defaults in global riskfactorsimulations is the iteration of (multiplicative) i.i.d. survival indicator incrementsalong a given time-grid, where the indicator distribution is based on acopula ansatz. The underlying assumption is that the behavior of the resultingiterated default distribution is similar to the one-shot distribution. It is shownthat in most cases this assumption is not fulfilled and furthermore numericalanalysis is presented that shows sizeable differences in probabilities assignedto both “survival-of-all” and “mixed default/survival” events. Moreover, theclasses of distributions for which probabilities from the “terminal one-shot”and “terminal iterated” distribution coincide are derived for problems considering“survival-of-all” events as well as “mixed default/survival” events. Forthe former problem, distributions must fulfill a lack-of-memory type property,which is, e.g., fulfilled by min-stable multivariate exponential distributions.These correspond in a copula-framework to exponential margins coupled viaextreme-value copulas. For the latter problem, while looping default inspiredmultivariate Freund distributions and more generally multivariate phase-type distributions could be a solution, under practically relevant and reasonableadditional assumptions on portfolio rebalancing and nested distributions, theunique solution is the Marshall–Olkin class.

Bormetti G, Brigo D, Francischello M, et al., 2018, Impact of multiple curve dynamics in credit valuation adjustments under collateralization, Publisher: ROUTLEDGE JOURNALS, TAYLOR & FRANCIS LTD

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Brigo D, Rapisarda F, Sridi A, 2017, The multivariate mixture dynamics: consistent no-arbitrage single-asset and index volatility smiles, *IISE Transactions*, Vol: 50, Pages: 27-44, ISSN: 0740-817X

We introduce a new arbitrage-free multivariate dynamic asset pricing model that allows us to reconcile single name and index/basket volatility smiles using a tractable and explicit dependence structure that goes beyond instantaneous correlation. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. After introducing the model, we derive tractable index option smile formulas resulting from the model and related closed form solutions for multivariate densities taking the form of multivariate mixtures. Using Markovian projection techniques, we relate our model to a multivariate uncertain volatility model and show a consistency result with geometric baskets with hints on possible uses in investigating triangular relationships between foreign exchange rates and the related smiles in practice. We also derive closed form solutions for a number of terminal statistics of dependence, and derive a precise relationship with a simpler but less tractable model based on a basic instantaneous correlation structure. Finally, closed form solutions for volatility/assets correlations illuminating the relationship with the uncertain volatility model are introduced. The model tractability makes it particularly suited for calibration and risk management applications, where speed of calculations and tractability are essential. A few numerical examples on basket and spread options pricing conclude the paper.

Brigo D, Buescu C, Rutkowski M, 2017, Funding, repo and credit inclusive valuation as modified option pricing, *Operations Research Letters*, Vol: 45, Pages: 665-670, ISSN: 0167-6377

We take the holistic approach of computing an OTC claim value that incorporates credit and funding liquidity risks and their interplays, instead of forcing individual price adjustments: CVA, DVA, FVA, KVA. The resulting nonlinear mathematical problem features semilinear PDEs and FBSDEs. We show that for the benchmark vulnerable claim there is an analytical solution, and we express it in terms of the Black–Scholes formula with dividends. This allows for a detailed valuation analysis, stress testing and risk analysis via sensitivities.

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

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

Armstrong J, Brigo D, 2017, Ito Stochastic Differential Equations as 2-Jets, 3rd International SEE Conference on Geometric Science of Information (GSI), Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 543-551, ISSN: 0302-9743

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

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

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

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

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

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

Brigo D, Fries C, Hull J, et al., 2016, FVA and electricity bill valuation adjustment - much of a difference?, Challenges in Derivatives Markets, Publisher: Springer, Pages: 147-168, ISSN: 2194-1009

Pricing counterparty credit risk, although being in the focus for almosta decade by now, is far from being resolved. It is highly controversial if any valuationadjustment besides the basic CVA should be taken into account, and ifso, for what purpose. Even today, the handling of CVA, DVA, FVA, ... differsbetween the regulatory, the accounting, and the economic point of view. Eventually,if an agreement is reached that CVA has to be taken into account, it remainsunclear if CVA can be modeled linearly, or if nonlinear models need tobe resorted to. Finally, industry practice and implementation differ in several aspects.Hence, a unified theory and treatment of FVA and alike is not yet tangible.The conference Challenges in Derivatives Markets, held at Technische Universitat¨Munchen in March/April 2015, featured a panel discussion with panelists repre- ¨senting different point of views: John Hull, who argues that FVA might not exist at all; in contrast to Christian Fries, who sees the need of all relevant costs to becovered within valuation but not within adjustments. Damiano Brigo emphasizesthe nonlinearity of (most) valuation adjustments and is concerned about overlappingadjustments and double-counting. Finally, Daniel Sommer puts the exit pricein the focus. The following (mildly edited) record of the panel discussion repeats themain arguments of the discussants – ultimately culminating in the awareness that ifeverybody charges an electricity bill valuation adjustment, it has to become part ofany quoted price.

Brigo D, Pistone G, Dimensionality reduction for measure valued evolution equations in statistical manifolds, Computational information geometry for image and signal processing, ISSN: 1860-4862

Armstrong J, Brigo D, Extrinsic projection of Ito SDEs on submanifolds with applications to nonlinear ltering, Computational Information Geometry for Image and Signal Processing, Publisher: Springer, ISSN: 1860-4862

We define the notion of the extrinsic Itˆo projection of astochastic differential equation (SDE) on a submanifold. This allows oneto systematically develop low dimensional approximations to high dimensionalSDEs in a differential geometric setting. We consider the exampleof approximating the non-linear filtering problem with a Gaussian distributionand show how the Itˆo projection leads to improved approximationsin the Gaussian family. We briefly discuss the approximations formore general families of distribution. We perform a numerical comparisonof our projection filters with the classical Extended Kalman Filterto demonstrate the efficacy of the approach.

Brigo D, Mai JF, Scherer M, 2016, Markov multi-variate survival indicators for default simulation as a new characterization of the Marshall-Olkin law, *Statistics & Probability Letters*, Vol: 114, Pages: 60-66, ISSN: 0167-7152

A new characterization of the Marshall–Olkin distribution is provided: all subvectorsof the associated survival indicators are continuous-time Markov chains.This property is crucial to overcome practical limitations for the modeling of highdimensionaldefault times (rebalancing, iterative simulation, consistent sub-portfolios).

Armstrong J, Brigo D, 2015, Nonlinear filtering via stochastic PDE projection on mixture manifolds in L^2 direct metric, *Mathematics of Control, Signals, and Systems*, Vol: 28, ISSN: 0932-4194

We examine some differential geometric approaches to finding approximatesolutions to the continuous time nonlinear filtering problem. Our primary focusis a new projection method for the optimal filter infinite-dimensional stochastic partial differential equation (SPDE), based on the direct L2 metric and on a family of normal mixtures. This results in a new finite-dimensional approximate filter based on the differential geometric approach to statistics. We compare this new filter to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and compare the L2 mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We discuss differences between projecting the SPDE for the normalized density, known as Kushner–Stratonovich equation, and the SPDE for the unnormalized density known as Zakai equation. We prove that for a simple choice of the mixture manifold the L2 mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence doesnot hold and the L2 mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples. We leverage an algebraic ring structure by proving that in presence of a given structure in the system coefficients the key integrations needed to implement the new filter equations can be executed offline.

Brigo D, Francischello M, Pallavicini A, 2015, Invariance, existence and uniqueness of solutions of nonlinear valuation PDEs and FBSDEs inclusive of credit risk, collateral and funding costs

We study conditions for existence, uniqueness and invariance of thecomprehensive nonlinear valuation equations first introduced in Pallavicini etal (2011). These equations take the form of semilinear PDEs andForward-Backward Stochastic Differential Equations (FBSDEs). After summarizingthe cash flows definitions allowing us to extend valuation to credit risk anddefault closeout, including collateral margining with possiblere-hypothecation, and treasury funding costs, we show how such cash flows, whenpresent-valued in an arbitrage free setting, lead to semi-linear PDEs or moregenerally to FBSDEs. We provide conditions for existence and uniqueness of suchsolutions in a viscosity and classical sense, discussing the role of thehedging strategy. We show an invariance theorem stating that even though westart from a risk-neutral valuation approach based on a locally risk-free bankaccount growing at a risk-free rate, our final valuation equations do notdepend on the risk free rate. Indeed, our final semilinear PDE or FBSDEs andtheir classical or viscosity solutions depend only on contractual, market ortreasury rates and we do not need to proxy the risk free rate with a realmarket rate, since it acts as an instrumental variable. The equationsderivations, their numerical solutions, the related XVA valuation adjustmentswith their overlap, and the invariance result had been analyzed numerically andextended to central clearing and multiple discount curves in a number ofprevious works, including Pallavicini et al (2011), Pallavicini et al (2012),Brigo et al (2013), Brigo and Pallavicini (2014), and Brigo et al (2014).

Brigo D, Garcia J, Pede N, 2015, COCO BONDS PRICING WITH CREDIT AND EQUITY CALIBRATED FIRST-PASSAGE FIRM VALUE MODELS, *INTERNATIONAL JOURNAL OF THEORETICAL AND APPLIED FINANCE*, Vol: 18, ISSN: 0219-0249

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Brigo D, Buescu C, Pallavicini A,
et al., 2015, A NOTE ON THE SELF-FINANCING CONDITION FOR FUNDING, COLLATERAL AND DISCOUNTING, *INTERNATIONAL JOURNAL OF THEORETICAL AND APPLIED FINANCE*, Vol: 18, ISSN: 0219-0249

Brigo D, Nordio C, 2015, A Random Holding Period Approach for Liquidity-Inclusive Risk Management, Conference on Risk Management Reloaded, Publisher: SPRINGER, Pages: 3-18, ISSN: 2194-1009

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Armstrong J, Brigo D, 2015, Stochastic PDE Projection on Manifolds: Assumed-Density and Galerkin Filters, 2nd International SEE Conference on Geometric Science of Information (GSI), Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 713-722, ISSN: 0302-9743

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