Publications
170 results found
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.
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, Hvolby T, Vrins F, 2018, 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.
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
- Author Web Link
- Open Access Link
- Cite
- Citations: 4
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, 2017, 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.
Armstrong J, Brigo D, 2016, 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, Pistone G, 2016, Dimensionality reduction for measure valued evolution equations in statistical manifolds, Computational information geometry for image and signal processing, ISSN: 1860-4862
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, 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
- Author Web Link
- Cite
- Citations: 8
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
- Author Web Link
- Cite
- Citations: 2
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
- Author Web Link
- Cite
- Citations: 2
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
Crepey S, Bielecki TR, Brigo D, 2014, Stochastic Analysis Prerequisites, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: 311-333, ISBN: 978-1-4665-1645-8
Crepey S, Bielecki TR, Brigo D, 2014, Counterparty Risk and Funding A Tale of Two Puzzles <i>Preface</i>, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: XV-+, ISBN: 978-1-4665-1645-8
Crepey S, Bielecki TR, Brigo D, 2014, The Four Wings of the TVA, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: 135-163, ISBN: 978-1-4665-1645-8
Crepey S, Bielecki TR, Brigo D, 2014, CVA Computations for Credit Portfolios in the Common-Shock Model, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: 267-275, ISBN: 978-1-4665-1645-8
Crepey S, Bielecki TR, Brigo D, 2014, Common-Shock Model, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: 193-237, ISBN: 978-1-4665-1645-8
Crepey S, Bielecki TR, Brigo D, 2014, Bilateral Counterparty Risk under Funding Constraints, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: 85-112, ISBN: 978-1-4665-1645-8
Crepey S, Bielecki TR, Brigo D, 2014, A Reduced-Form TVA BSDE Approach to Counterparty Risk under Funding Constraints, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: 115-134, ISBN: 978-1-4665-1645-8
Crepey S, Bielecki TR, Brigo D, 2014, Markov Consistency and Markov Copulas, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: 335-341, ISBN: 978-1-4665-1645-8
Crepey S, Bielecki TR, Brigo D, 2014, The Whys of the LOIS, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: 47-59, ISBN: 978-1-4665-1645-8
Crepey S, Bielecki TR, Brigo D, 2014, Pure Counterparty Risk, COUNTERPARTY RISK AND FUNDING: A TALE OF TWO PUZZLES, Publisher: CRC PRESS-TAYLOR & FRANCIS GROUP, Pages: 65-83, ISBN: 978-1-4665-1645-8
This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters.