Personal page of Professor Damiano Brigo at Imperial College London, Dept. of Mathematics
Professor (Chair), Stochastic Analysis Group
& co-Head of Mathematical Finance,
Imperial College London
For Citation Data, H-index, Social networks support, etc, please click here
short CV
Research PapersSeminars and Public Lectures
MSc lectures and London Graduate School PhD lectures
Randomness, Dynamics and Risk (video)
(Financial
Modeling, Systems Theory, Probability, Statistics)
,
Counterparty Credit Risk
___ Welcome! ___
Welcome!
Welcome to my web site. As you can easily see, this page does not conform to the flashing, singing, animated, pyrotechnic web-pages that you can find nowadays. No need to "skip intro" here, but in case you may "skip intro" all the same.
Notwithstanding this page somberness and lack of special effects, here you can find:
A short CV
including the description of my current and past tasks as a
mathematician working in financial modeling and signal processing and also my
academic appointments and activities;Thus I hope that if you are interested either in mathematical finance or in stochastic differential equations in connection with exponential families and the nonlinear-filtering problem, you will find this page to be at least a little helpful. The themes treated here concern indeed financial modelling, probability, systems theory and stochastic geometry from A to Zzzzzzzzzzzzzzz (where have you heard this one before), so bear with me....
This page is also an opportunity to convey an image of mathematicians and financial engineers different from the stereotype some people have in mind: not all mathematicians and physicists working in actual or academic finance take themselves too seriously, rather than seriously enough.
Anyway, enjoy your stay at this little "professional corner" of mine in the cyberspace.
You can contact me by following this link, and please be patient if I reply after a while or do not reply, the number of messages I receive is out of control, and goes on top of the huge amount of emails I receive in my professional accounts.
Arrivederci!
NEW and RECENT RESEARCH PAPERS
For a complete list of research papers click here
Coordinate-free stochastic differential equations as jets
Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections
SDEs with uniform distributions: Peacocks, Conic martingales and mean reverting uniform diffusions
Projection based dimensionality reduction for measure valued evolution equations in statistical manifolds
Optimal approximations of the Fokker-Planck-Kolmogorov equation: projection, maximum likelihood eigenfunctions and Galerkin methods
An indifference approach to the cost of capital constraints: KVA and beyond Static vs adapted optimal execution strategies in two benchmark trading models Disentangling wrong-way risk: pricing CVA via change of measures and drift adjustment Impact of Multiple Curve Dynamics in Credit Valuation Adjustments under Collateralization Invariance, existence and uniqueness of solutions of nonlinear valuation PDEs and FBSDEs inclusive of credit risk, collateral and funding costs Funding, Repo and Credit Inclusion in Option Pricing via Dividends Nonlinear Valuation under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes Multi Currency Credit Default Swaps: Quanto Effects and FX Devaluation Jumps The Multivariate Mixture Dynamics Model: Shifted dynamics and correlation skew Impact of Robotics, RPA and AI on the insurance industry: challenges and opportunities ALL DOWNLOADABLE SCIENTIFIC / ACADEMIC RESEARCH PAPERS
Research papers in stochastic nonlinear filtering, probability, statistics, information geometry, and in several areas of quantitative finance. Most papers are either directly downloadable or downloadable via SSRN/arXiv
Probability, Statistics, Nonlinear Filtering, Stochastic Differential and Information Geometry
Coordinate-free stochastic differential equations as jets
Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections
SDEs with uniform distributions: Peacocks, Conic martingales and mean reverting uniform diffusions
Projection based dimensionality reduction for measure valued evolution equations in statistical manifolds
Maximum likelihood eigenfunctions of the Fokker Planck equation and Hellinger projectionBack to scientific/academic works
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Counterparty Credit Risk, Collateral, Funding, CVA/DVA/FVA, multiple curves.
Disentangling wrong-way risk: pricing CVA via change of measures and drift adjustment Impact of Multiple Curve Dynamics in Credit Valuation Adjustments under Collateralization Invariance, existence and uniqueness of solutions of nonlinear valuation PDEs and FBSDEs inclusive of credit risk, collateral and funding costs Funding, Repo and Credit Inclusion in Option Pricing via Dividends Nonlinear Valuation under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes An indifference approach to the cost of capital constraints: KVA and beyond Back to scientific/academic works
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Credit Derivatives Papers: CDS, CDS Options, CDS Liquidity, CDOs, etc.
Multi Currency Credit Default Swaps: Quanto Effects and FX Devaluation Jumps Back to scientific/academic works
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Algorithmic trading and optimal execution
Static vs adapted optimal execution strategies in two benchmark trading models Back to scientific/academic works
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Volatility smile modeling
The Multivariate Mixture Dynamics Model: Shifted dynamics and correlation skew
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Interest-rate derivatives modeling, interest rate models with credit and liquidity effects, multiple curves
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Basket options
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General option-pricing theory
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Risk Management
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Probability, Statistics, Nonlinear Filtering, Stochastic Differential and Information Geometry.
(Click here to download a PDF file with the paper).
In this paper we introduce a projection method for the space of probability distributions based on the differential geometric approach to statistics. This method is based on a direct $L^2$ metric as opposed to the usual Hellinger distance and the related Fisher Information metric. We explain how this apparatus can be used for the nonlinear filtering problem, in relationship also to earlier projection methods based on the Fisher metric. Past projection filters focused on the Fisher metric and the exponential families that made the filter correction step exact. In this work we introduce the mixture projection filter, namely the projection filter based on the direct $L^2$ metric and based on a manifold given by a mixture of pre-assigned densities.
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Click here to download a PDF file with the paper from arXiv.
We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a projection method using the direct L2 metric onto a family of normal mixtures. We compare this method to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and we compare the L2 mixture projection filter with a particle method with the same number of parameters. We study particular systems that may illustrate the advantages of this 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.
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Part of this paper has been published in "Statistics and Probability Letters" 49 (2000), pp. 127-134. (Click here to download a PDF file with the paper).
In this paper we consider the continuous--time nonlinear filtering problem, which has an infinite--dimensional solution in general, as proved by Chaleyat--Maurel and Michel. There are few examples of nonlinear systems for which the optimal filter is finite dimensional, in particular Kalman's, Benes', and Daum's filters. In the present paper, we construct new classes of scalar nonlinear filtering problems admitting finite--dimensional filters. We consider a given (nonlinear) diffusion coefficient for the state equation, a given (nonlinear) observation function, and a given finite--dimensional exponential family of probability densities. We construct a drift for the state equation such that the resulting nonlinear filtering problem admits a finite--dimensional filter evolving in the prescribed exponential family augmented by the observaton function and its square.
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(Click here to download a PDF file with the paper).
We explain how Ito Stochastic Differential Equations on manifolds may be defined as 2-jets of curves and show how this relationship can be interpreted in terms of a convergent numerical scheme. We use jets as a natural language to express geometric properties of SDEs. We show how jets can lead to intuitive representations of It\^o SDEs, including three different types of drawings. We explain that the mainstream choice of Fisk-Stratonovich-McShane calculus for stochastic differential geometry is not necessary. We give a new geometric interpretation of the It\^o--Stratonovich transformation in terms of the 2-jets of curves induced by consecutive vector flows. We discuss the forward Kolmogorov equation and the backward diffusion operator in geometric terms. In the one-dimensional case we consider percentiles of the solutions of the SDE and their properties. This allows us to interpret the coefficients of SDEs in terms of "fan diagrams". In particular the median of a SDE solution is associated to the drift of the SDE in Stratonovich form for small times.
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(Click here to download a PDF file with the paper).
We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Ito-vector and Ito-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. As an application we consider approximating the solution of the non-linear filtering problem with a Gaussian distribution and show how the newly introduced Ito projections lead to optimal approximations in the Gaussian family and briefly discuss the optimal approximation for more general families of distribution. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.
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(Click here to download a PDF file with the paper).
We introduce a way to design Stochastic Differential Equations of diffusion type admitting a unique strong solution distributed as a uniform law with conic time-boundaries. We connect this general result to some special cases that where previously found in the peacock processes literature, and with the square root of time boundary case in particular. We introduce a special case with linear time boundary. We further introduce general mean-reverting diffusion processes having a constant uniform law at all times. This may be used to model random probabilities, random recovery rates or random correlations. We study local time and activity of such processes and verify via an Euler scheme simulation that they have the desired uniform behaviour.
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Working paper at a preliminary stage. (Click here to download a PDF file with the paper).
In the present paper we discuss problems concerning evolutions of densities related to Ito diffusions in the framework of the statistical exponential manifold. We develop a rigorous approach to the problem, and we particularize it to the orthogonal projection of the evolution of the density of a diffusion process onto a finite dimensional exponential manifold. It has been shown by D. Brigo (1996) that the projected evolution can always be interpreted as the evolution of the density of a different diffusion process. We give also a compactness result when the dimension of the exponential family increases, as a first step towards a convergence result to be investigated in the future. The infinite dimensional exponential manifold structure introduced by G. Pistone and C. Sempi is used and some examples are given.
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(Click here to download a PDF file with the paper).
We propose a dimensionality reduction method for infinite-dimensional measure-valued evolution equations such as the Fokker-Planck partial differential equation or the Kushner-Stratonovich resp. Duncan-Mortensen-Zakai stochastic partial differential equations of nonlinear filtering, with potential applications to signal processing, quantitative finance, heat flows and quantum theory among many other areas. Our method is based on the projection coming from a duality argument built in the exponential statistical manifold structure developed by G. Pistone and co-authors. The choice of the finite dimensional manifold on which one should project the infinite dimensional equation is crucial, and we propose finite dimensional exponential and mixture families. This same problem had been studied, especially in the context of nonlinear filtering, by D. Brigo and co-authors but the L2 structure on the space of square roots of densities or of densities themselves was used, without taking an infinite dimensional manifold environment space for the equation to be projected. Here we re-examine such works from the exponential statistical manifold point of view, which allows for a deeper geometric understanding of the manifold structures at play. We also show that the projection in the exponential manifold structure is consistent with the Fisher Rao metric and, in case of finite dimensional exponential families, with the assumed density approximation. Further, we show that if the sufficient statistics of the finite dimensional exponential family are chosen among the eigenfunctions of the backward diffusion operator then the statistical-manifold or Fisher-Rao projection provides the maximum likelihood estimator for the Fokker Planck equation solution. We finally try to clarify how the finite dimensional and infinite dimensional terminology for exponential and mixture spaces are related.
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(Click here to download a PDF file with the paper).
We study optimal finite dimensional approximations of the generally infinite-dimensional Fokker-Planck-Kolmogorov (FPK) equation, finding the curve in a given finite-dimensional family that best approximates the exact solution evolution. For a first local approximation we assign a manifold structure to the family and a metric. We then project the vector field of the partial differential equation (PDE) onto the tangent space of the chosen family, thus obtaining an ordinary differential equation for the family parameter. A second global approximation will be based on projecting directly the exact solution from its infinite dimensional space to the chosen family using the nonlinear metric projection. This will result in matching expectations with respect to the exact and approximating densities for particular functions associated with the chosen family, but this will require knowledge of the exact solution of FPK. A first way around this is a localized version of the metric projection based on the assumed density approximation. While the localization will remove global optimality, we will show that the somewhat arbitrary assumed density approximation is equivalent to the mathematically rigorous vector field projection. More interestingly we study the case where the approximating family is defined based on a number of eigenfunctions of the exact equation. In this case we show that the local vector field projection provides also the globally optimal approximation in metric projection, and for some families this coincides with a Galerkin method.
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PhD Thesis, Free University of Amsterdam, 1996. Several parts of this document have been published in journals such as "IEEE Transactions on Automatic Control", "Systems and Control Letters", "Bernoulli". (Click here to download a PDF file with the thesis, or click here to download a zipped PS file with the thesis from Francois Le Gland's page at IRISA).
This is the synthesis of three years of research work on the projection filter.
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Updated version published in "Communications in Statistics: Theory and Methods", Vol 34, issue 7, 2005. Cermics report 2003-250. Click here to download a PDF version of the related paper from the CERMICS web site.
[This paper is listed also in the Mathematical Finance papers area below]. Although there exists a large variety of copula functions, only a few are practically manageable, and often the choice in dependence modeling falls on the Gaussian copula. Further, most copulas are exchangeable, thus implying symmetric dependence. We introduce a way to construct copulas based on periodic functions. We study the two-dimensional case based on one dependence parameter and then provide a way to extend the construction to the n-dimensional framework. We can thus construct families of copulas in dimension n and parameterized by n - 1 parameters, implying possibly asymmetric relations. Such “periodic” copulas can be simulated easily.
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download a PDF version of the related paper from arXiv.org
[This paper is listed also in the Mathematical Finance papers area below] This paper deals with dependence across marginally exponentially distributed arrival times, such as default times in financial modeling or inter-failure times in reliability theory. We explore the relationship between dependence and the possibility to sample final multivariate survival in a long time-interval as a sequence of iterations of local multivariate survivals along a partition of the total time interval. We find that this is possible under a form of multivariate lack of memory that is linked to a property of the survival times copula. This property defines a "self-chaining-copula", and we show that this coincides with the extreme value copulas characterization. The self-chaining condition is satisfied by the Gumbel-Hougaard copula, a full characterization of self chaining copulas in the Archimedean family, and by the Marshall-Olkin copula. The result has important practical implications for consistent single-step and multi-step simulation of multivariate arrival times in a way that does not destroy dependency through iterations, as happens when inconsistently iterating a Gaussian copula.
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Counterparty Credit Risk, Collateral and Funding; CVA / DVA / FVA. Credit Derivatives Modeling.
Click here to download a PDF version of this paper from arXiv, or download it from SSRN here, or directly here here
We present a dialogue on Counterparty Credit Risk touching on Credit Value at Risk (Credit VaR), Potential Future Exposure (PFE), Expected Exposure (EE), Expected Positive Exposure (EPE), Credit Valuation Adjustment (CVA), Debit Valuation 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 Default Swaps (CCDS) and CVA restructuring possibilities through margin lending. The dialogue is in the form of a Q&A between a CVA expert and a newly hired colleague.
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Click here to download a PDF version of this paper from SSRN. The paper is available also on arXiv.org here
We present a dialogue on Funding Costs and Counterparty Credit Risk modeling, inclusive of collateral, wrong way risk, gap risk and possible Central Clearing implementation through CCPs. This framework is important following the fact that derivatives valuation and risk analysis has moved from exotic derivatives managed on simple single asset classes to simple derivatives embedding the new or previously neglected types of complex and interconnected nonlinear risks we address here. This dialogue is the continuation of the "Counterparty Risk, Collateral and Funding FAQ" by Brigo (2011). In this dialogue we focus more on funding costs for the hedging strategy of a portfolio of trades, on the non-linearities emerging from assuming borrowing and lending rates to be different, on the resulting aggregation-dependent valuation process and its operational challenges, on the implications of the onset of central clearing, on the macro and micro effects on valuation and risk of the onset of CCPs, on initial and variation margins impact on valuation, and on multiple discount curves. Through questions and answers (Q&A) between a senior expert and a junior colleague, and by referring to the growing body of literature on the subject, we present a unified view of valuation (and risk) that takes all such aspects into account.
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Click here to download a PDF version of this paper from arXiv.
A key driver of Credit Value Adjustment (CVA) is the possible dependency between exposure and counterparty credit risk, known as Wrong-Way Risk (WWR). At this time, addressing WWR in a both sound and tractable way remains challenging: arbitrage-free setups have been proposed by academic research through dynamic models but are computationally intensive and hard to use in practice. Tractable alternatives based on resampling techniques have been proposed by the industry, but they lack mathematical foundations. This probably explains why WWR is not explicitly handled in the Basel III regulatory framework in spite of its acknowledged importance. The purpose of this paper is to propose a new method consisting of an appealing compromise: we start from a stochastic intensity approach and end up with a pricing problem where WWR does not enter the picture explicitly. This result is achieved thanks to a set of changes of measure: the WWR effect is now embedded in the drift of the exposure, and this adjustment can be approximated by a deterministic function without affecting the level of accuracy typically required for CVA figures. The performances of our approach are illustrated through an extensive comparison of Expected 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 a sound and tractable way.
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Click here to download a PDF version of this paper from SSRN. The paper is available also on arXiv.org here
The introduction of CCPs in most derivative transactions will dramatically change the landscape of derivatives pricing, hedging and risk management, and, according to the TABB group, will lead to an overall liquidity impact about 2 USD trillions. In this article we develop for the first time a comprehensive approach for pricing under CCP clearing, including variation and initial margins, gap credit risk and collateralization, showing concrete examples for interest rate swaps. Mathematically, the inclusion of asymmetric borrowing and lending rates in the hedge of a claim lead to nonlinearities showing up in claim dependent pricing measures, aggregation dependent prices, nonlinear PDEs and BSDEs. This still holds in presence of CCPs and CSA. We introduce a modeling approach that allows us to enforce rigorous separation of the interconnected nonlinear risks into different valuation adjustments where the key pricing nonlinearities are confined to a funding costs component that is analyzed through numerical schemes for BSDEs. We present a numerical case study for Interest Rate Swaps that highlights the relative size of the different valuation adjustments and the quantitative role of initial and variation margins, of liquidity bases, of credit risk, of the margin period of risk and of wrong way risk correlations.
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Click here to download a PDF version of this paper from SSRN. The paper is available also on arXiv.org here
This paper specializes a number of earlier contributions to the theory of valuation of financial products in presence of credit risk, repurchase agreements and funding costs. Earlier works, including our own, pointed to the need of tools such as Backward Stochastic Differential Equations (BSDEs) or semi-linear Partial Differential Equations (PDEs), which in practice translate to ad-hoc numerical methods that are time-consuming and which render the full valuation and risk analysis difficult. We specialize here the valuation framework to benchmark derivatives and we show that, under a number of simplifying assumptions, the valuation paradigm can be recast as a Black-Scholes model with dividends. In turn, this allows for a detailed valuation analysis, stress testing and risk analysis via sensitivities. We refer to the full paper for a more complete mathematical treatment.
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Click here to download a PDF version of this paper from SSRN. The paper is available also on arXiv.org here
We study conditions for existence, uniqueness and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al (2011). These equations take the form of semilinear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions allowing us to extend valuation to credit risk and default closeout, including collateral margining with possible re-hypothecation, and treasury funding costs, we show how such cash flows, when present-valued in an arbitrage free setting, lead to semi-linear PDEs or more generally to FBSDEs. We provide conditions for existence and uniqueness of such solutions in a viscosity and classical sense, discussing the role of the hedging strategy. We show an invariance theorem stating that even though we start from a risk-neutral valuation approach based on a locally risk-free bank account growing at a risk-free rate, our final valuation equations do not depend on the risk free rate. Indeed, our final semilinear PDE or FBSDEs and their classical or viscosity solutions depend only on contractual, market or treasury rates and we do not need to proxy the risk free rate with a real market rate, since it acts as an instrumental variable. The equations derivations, their numerical solutions, the related XVA valuation adjustments with their overlap, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works, including Pallavicini et al (2011), Pallavicini et al (2012), Brigo et al (2013), Brigo and Pallavicini (2014), and Brigo et al (2014).
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Click here to download a PDF version of this paper from SSRN. The paper is available also on arXiv.org here
We develop an arbitrage-free framework for consistent valuation of derivative trades with collateralization, counterparty credit gap risk, and funding costs, following the approach first proposed by Pallavicini and co-authors in 2011. Based on the risk-neutral pricing principle, we derive a general pricing equation where Credit, Debit, Liquidity and Funding Valuation Adjustments (CVA, DVA, LVA and FVA) are introduced by simply modifying the payout cash-flows of the deal. Funding costs and specific close-out procedures at default break the bilateral nature of the deal price and render the valuation problem a non-linear and recursive one. CVA and FVA are in general not really additive adjustments, and the risk for double counting is concrete. We introduce a new adjustment, called a Non-linearity Valuation Adjustment (NVA), to address double-counting. Our framework is based on real market rates, since the theoretical risk free rate disappears from our final equations. The framework addresses common market practices of ISDA governed deals without restrictive assumptions on collateral margin payments and close-out netting rules, and can be tailored also to CCP trading under initial and variation margins, as explained in detail in Brigo and Pallavicini (2014). In particular, we allow for asymmetric collateral and funding rates, replacement close-out and re-hypothecation. The valuation equation takes the form of a backward stochastic differential equation or semi-linear partial differential equation, and can be cast as a set of iterative equations that can be solved by least-squares Monte Carlo. We propose such a simulation algorithm in a case study involving a generalization of the benchmark model of Black and Scholes for option pricing. Our numerical results confirm that funding risk has a non-trivial impact on the deal price, and that double counting matters too. We conclude the article with an analysis of large scale implications of non-linearity of the pricing equations: non-separability of risks, aggregation dependence in valuation, and local pricing measures as opposed to universal ones. This prompts a debate and a comparison between the notions of price and value, and will impact the operational structure of banks. This paper is an evolution, in particular, of the work by allavicini et al. (2011, 2012), Pallavicini and Brigo (2013), and Sloth (2013).
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Click here to download a PDF version of this paper from SSRN. The paper is available also on arXiv.org here
We present a detailed analysis of interest rate derivatives valuation under credit risk and collateral modeling. We show how the credit and collateral extended valuation framework in Pallavicini et al (2011), and the related collateralized valuation measure, can be helpful in defining the key market rates underlying the multiple interest rate curves that characterize current interest rate markets. A key point is that spot Libor rates are to be treated as market primitives rather than being defined by no-arbitrage relationships. We formulate a consistent realistic dynamics for the different rates emerging from our analysis and compare the resulting model performances to simpler models used in the industry. We include the often neglected margin period of risk, showing how this feature may increase the impact of different rates dynamics on valuation. We point out limitations of multiple curve models with deterministic basis considering valuation of particularly sensitive products such as basis swaps. We stress that a proper wrong way risk analysis for such products requires a model with a stochastic basis and we show numerical results confirming this fact.
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Click here to download a PDF version of this paper from SSRN. The paper is available also on arXiv.org here
The main result of this paper is a collateralized counterparty valuation adjusted pricing equation, which allows to price a deal while taking into account credit and debit valuation adjustments (CVA, DVA) along with margining and funding costs, all in a consistent way. Funding risk breaks the bilateral nature of the valuation formula. We find that the equation has a recursive form, making the introduction of a purely additive funding valuation adjustment (FVA) difficult. Yet, we can cast the pricing equation into a set of iterative relationships which can be solved by means of standard least-square Monte Carlo techniques. As a consequence, we find that identifying funding costs and debit valuation adjustments is not tenable in general, contrary to what has been suggested in the literature in simple cases. The assumptions under which funding costs vanish are a very special case of the more general theory. We define a comprehensive framework that allows us to derive earlier results on funding or counterparty risk as a special case, although our framework is more than the sum of such special cases. We derive the general pricing equation by resorting to a risk-neutral approach where the new types of risks are included by modifying the payout cash flows. We consider realistic settings and include in our models the common market practices suggested by ISDA documentation, without assuming restrictive constraints on margining procedures and close-out netting rules. In particular, we allow for asymmetric collateral and funding rates, and exogenous liquidity policies and hedging strategies. Re-hypothecation liquidity risk and close-out amount evaluation issues are also covered. Finally, relevant examples of non-trivial settings illustrate how to derive known facts about discounting curves from a robust general framework and without resorting to ad hoc hypotheses.
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Click here to download a PDF version of this paper. The paper is available also on ssrn.com and arXiv.org
In this paper we describe how to include funding and margining costs into a risk-neutral pricing framework for counter-party credit risk. We consider realistic settings and we include in our models the common market practices suggested by the ISDA documentation without assuming restrictive constraints on margining procedures and close-out netting rules. In particular, we allow for asymmetric collateral and funding rates, and exogenous liquidity policies and hedging strategies. Re-hypothecation liquidity risk and close-out amount evaluation issues are also covered. We define a comprehensive pricing framework which allows us to derive earlier results funding or counter-party risk. Some relevant examples illustrate the non trivial settings needed to derive known facts about discounting curves by starting from a general framework and without resorting to ad hoc hypotheses. Our main result is a bilateral collateralized counter-party valuation adjusted pricing equation, which allows to price a deal while taking into account credit and debt valuation adjustments along with margining and funding costs in a coherent way. We find that the equation has a recursive form, making the introduction of an additive funding valuation adjustment difficult. Yet, we can cast the pricing equation into a set of iterative relationships which can be solved by means of standard least-square Monte Carlo techniques.
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Click here to download a PDF version of this paper. The paper is available also on ssrn.com
In this paper we describe how to include funding and margining costs into a risk-neutral pricing framework for counter-party credit risk. We consider realistic settings and we include in our models the common market practices suggested by the ISDA documentation without assuming restrictive constraints on margining procedures and close-out netting rules. In particular, we allow for asymmetric collateral and funding rates, and exogenous liquidity policies and hedging strategies. Re-hypothecation liquidity risk and close-out amount evaluation issues are also covered. We define a comprehensive pricing framework which allows us to derive earlier results funding or counter-party risk. Some relevant examples illustrate the non trivial settings needed to derive known facts about discounting curves by starting from a general framework and without resorting to ad hoc hypotheses. Our main result is a bilateral collateralized counter-party valuation adjusted pricing equation, which allows to price a deal while taking into account credit and debt valuation adjustments along with margining and funding costs in a coherent way. We find that the equation has a recursive form, making the introduction of an additive funding valuation adjustment difficult. Yet, we can cast the pricing equation into a set of iterative relationships which can be solved by means of standard least-square Monte Carlo techniques.
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Click here to download a PDF version of this paper This paper has been submitted to the Bundesbank paper research series.
We introduce an innovative theoretical framework for the valuation and replication of derivative transactions between defaultable entities based on the principle of arbitrage free- dom. Our framework extends the traditional formulations based on Credit and Debit Valuation Adjustments (CVA and DVA). Depending on how the default contingency is accounted for, we list a total of ten di er- ent structuring styles. These include bi-partite structures between a bank and a counterparty, tri-partite structures with one margin lender in addition, quadri-partite structures with two mar- gin lenders and, most importantly, con gurations where all derivative transactions are cleared through a Central Counterparty Clearing House (CCP). We compare the various structuring styles under a number of criteria including consistency from an accounting standpoint, counterparty risk hedgeability, numerical complexity, transaction portability upon default, induced behaviour and macro-economic impact of the implied wealth allocation.
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Click here to download a PDF version of this paper.
We illustrate a problem in the self-financing condition used in the papers "Funding beyond discounting: collateral agreements and derivatives pricing" (Risk Magazine, February 2010) and "Partial Differential Equation Representations of Derivatives with Counterparty Risk and Funding Costs" (The Journal of Credit Risk, 2011). These papers state an erroneous self-financing condition. In the first paper, this is equivalent to assuming that the equity position is self-financing on its own and without including the cash position. In the second paper, this is equivalent to assuming that a subportfolio is self-financing on its own, rather than the whole portfolio. The error in the first paper is avoided when clearly distinguishing between price processes, dividend processes and gain processes. We present an outline of the derivation that yields the correct statement of the self-financing condition, clarifying the structure of the relevant funding accounts, and show that the final result in "Funding beyond discounting" is correct, even if the self-financing condition stated is not.
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Click here to download a PDF version of this paper from arXiv, or download it from SSRN here
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 domestic 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 example 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 a EUR quote of 350 bps 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 40 bps. 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.
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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
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After the beginning of the credit and liquidity crisis, financial institutions have been considering creating a convertible-bond type contract focusing on Capital. Under the terms of this contract, a bond is converted into equity if the authorities deem the institution to be under-capitalized. This paper discusses this Contingent Capital (or Coco) bond instrument and presents a pricing methodology based on firm value models. The model is calibrated to readily available market data. A stress test of model parameters is illustrated to account for potential model risk. Finally, a brief overview of how the instrument performs is presented.
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In
this work we consider three problems of the standard market approach to pricing
of credit index options: the definition of the index spread is not valid in
general, the usually considered payoff leads to a pricing which is not always
defined, and the candidate numeraire one would use to define a pricing measure
is not strictly positive, which would lead to a non-equivalent pricing measure.
We give a general mathematical solution to the three problems, based on a novel
way of modeling the flow of information through the definition of a new
subfiltration. Using this subfiltration, we take into account consistently the
possibility of default of all names in the portfolio, that is neglected in the
standard market approach. We show that, while the related mispricing can be
negligible for standard options in normal market conditions, it can become
highly relevant for different options or in stressed market conditions.
In particular, we show on 2007 market data that after the subprime credit
crisis the mispricing of the market formula compared to the no arbitrage
formula we propose has become financially relevant even for the liquid
Crossover Index Options.
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We follow a long path for Credit Derivatives and Collateralized Debt Obligations (CDOs) in particular, from the introduction of the Gaussian copula model and the related implied correlations to the introduction of arbitrage-free dynamic loss models capable of calibrating all the tranches for all the maturities at the same time. En passant, we also illustrate the implied copula, a method that can consistently account for CDOs with different attachment and detachment points but not for different maturities. The discussion is abundantly supported by market examples through history. The dangers and critics we present to the use of the Gaussian copula and of implied correlation had all been published by us, among others, in 2006, showing that the quantitative community was aware of the model limitations before the crisis. We also explain why the Gaussian copula model is still used in its base correlation formulation, although under some possible extensions such as random recovery. Overall we conclude that the modeling effort in this area of the derivatives market is unfinished, partly for the lack of an operationally attractive single-name consistent dynamic loss model, and partly because of the diminished investment in this research area.
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We extend the common Poisson shock framework reviewed for example in Lindskog and McNeil (2003) to a formulation avoiding repeated defaults, thus obtaining a model that can account consistently for single name default dynamics, cluster default dynamics and default counting process. This approach allows one to introduce significant dynamics, improving on the standard ``bottom-up" approaches, and to achieve true consistency with single names, improving on most ``top-down" loss models. Furthermore, the resulting GPCL model has important links with the previous GPL dynamical loss model in Brigo, Pallavicini and Torresetti (2006a,b), which we point out. Model extensions allowing for more articulated spread and recovery dynamics are hinted at. Calibration to both DJi-TRAXX and CDX index and tranche data across attachments and maturities shows that the GPCL model has the same calibration power as the GPL model while allowing for consistency with single names.
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In the first part we consider a dynamical model for the number of defaults of a pool of names. The model is based on the notion of generalized Poisson process, allowing for more than one default in small time intervals, contrary to many alternative approaches to loss modeling. We illustrate how to define the pool default intensity and discuss recovery assumptions. The models are tractable, pricing and simulation are straightforward, and consistent calibration to quoted index CDO tranches and tranchelets for several maturities is feasible, as we illustrate with numerical examples. In the second part we model directly the pool loss and we introduce extensions based on piecewise gamma, scenario-based or CIR random intensities, leading to richer spread dynamics, investigating calibration improvements and stability.
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We consider the risk neutral loss distribution as implied by index CDO tranche quotes through a “scenario default rate” model as opposed to the objective measure loss distribution based on historical analysis. The risk neutral loss distribution turns out to privilege large realizations of the loss with respect to the objective distribution, thus implying the well known presence of a risk premium. We quantify this difference numerically by pricing CDO tranches and indices under the two distributions. En passant we analyze the implied risk neutral default rate distributions calibrated from April-2004 throughout April-2006, pointing out its distinctive “bump feature” in the tail.
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We explain how the payoffs of credit indices and tranches are valued in terms of expected tranched losses (ETL). ETL are natural quantities to imply from market data. No-arbitrage constraints on ETL's as attachment points and maturities change are introduced briefy. As an alternative to the temporally inconsistent notion of implied correlation we consider the ETL surface, built directly from market quotes given minimal interpolation assumptions. We check that the kind of interpolation does not interfere excessively. Instruments bid/asks enter our analysis, contrary to Walker's (2006) earlier work on the ETL implied surface. By doing so we find less and very few violations of the no-arbitrage conditions. The ETL implied surface can be used to value tranches with nonstandard attachments and maturities as an alternative to implied correlation.
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We illustrate the two main types of implied correlation one may obtain from market CDO tranche spreads. Compound correlation is more consistent at single tranche level but for some market CDO tranche spreads cannot be implied. Base correlation is less consistent but more flexible and can be implied for a much wider set of CDO tranche market spreads. Furthermore, base correlation is more easily interpolated and leads to the possibility to price non-standard detachments. Even so, Base correlation may lead to negative expected tranche losses, thus violating basic no-arbitrage conditions. We illustrate these features with numerical examples.
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Following the recent introduction of new forms of Credit Default Swap (CDS) contracts expressed as upfront payments plus a fixed coupon, this note examines the methodology suggested by Barclays Capital, Goldman Sachs, JPMorgan, Markit (BGJM)/ISDA (2009), for conversion of CDS quotes between upfront and running. The proposed flat hazard rate (FHR) conversion method is to be understood as a rule-of-thumb single-contract quoting mechanism rather than as a modelling device. For example, an hypothetical investor who would put the FHR converted running spreads into her old running CDS library would strip wrong hazard rates, inconsistent with those coming directly from the quoted term structure of upfronts. This new methodology appears mostly as a device to transit the market towards adoption of the new upfront CDS as direct trading products while maintaining a semblance of running quotes for investors who may be suffering the transition. We caution though that i) the conversion done with proper hazard rates consistent across term would produce different results; ii) the quantities involved in the conversion should not be used as modelling tools anywhere; and iii) for highly distressed names with a high upfront paid by the protection buyer, the conversion to running spreads fails unless, as we propose, a third recovery scenario of 0% is added to the suggested 20% and 40%. This paper is not meant as a criticism of the proposed standardization of the conversion method but as a warning on the confusion this may generate when the method is not used carefully.
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Paper presented at the conference RISK EUROPE, April 8-9, 2003, Paris, and at the 6-th Columbia=JAFEE International Conference, Tokyo, March 15-16, 2003. Reduced version in Finance and Stochastics. Click here to download a PDF version of this paper.
In the present paper
we introduce a two-dimensional shifted square-root diffusion (SSRD) model for
interest rate derivatives and single-name credit derivatives, in a stochastic
intensity framework. The SSRD is the unique model, to the best of our
knowledge, allowing for an automatic calibration of the term structure of
interest rates and of credit default swaps (CDS's). Moreover, the model retains
free dynamics parameters that can be used to calibrate option data, such as
caps for the interest rate market and options on CDS's in the credit market.
The calibrations to the interest-rate market and to the credit market can be
kept separate, thus realizing a superposition that is of practical value.
We discuss the impact of interest-rate and
default-intensity correlation on calibration and pricing, and test it by means
of Monte Carlo simulation. We use a variant of Jamshidian's decomposition to
derive an analytical formula for CDS options under CIR++ stochastic intensity.
Finally, we develop an analytical approximation based on a Gaussian dependence
mapping for some basic credit derivatives terms involving correlated CIR processes.
Presented at the Third Bachelier Conference on
Mathematical Finance, Chicago, 2004. Click here to
download a PDF file containing this paper, from Laurent Cousot's web site at
NYU. Updated version to appear on the "International Journal of Theoretical and Applied
Finance".
In this paper we investigate implied volatility patterns in the Shifted Square Root Diffusion (SSRD) model as functions of the model parameters. We begin by recalling the Credit Default Swap (CDS) options market model that is consistent with a market Black-like formula, thus introducing a notion of implied volatility for CDS options. We examine implied volatilies coming from SSRD prices and characterize the qualitative behavior of implied volatilities as functions of the SSRD model parameters. We introduce an analytical approximation for the SSRD implied volatility that follows the same patterns in the model parameters and that can be used to have a first rough estimate of the implied volatility following a calibration. We compute numerically the CDS-rate volatility smile for the adopted SSRD model. We find a decreasing pattern of SSRD implied volatilities in the interest-rate/intensity correlation. We check whether it is possible to assume zero correlation after the option maturity in computing the option price and provide an upper bound for the Monte Carlo standard error in cases where this is not possible.
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Updated version published in Quantitative Finance. Click here to download a copy from Eymen Errais' web site at Stanford.
We propose a general setting for pricing single-name knock-out credit derivatives. Examples include Credit Default Swaps (CDS), European and Bermudan CDS options. The default of the underlying reference entity is modeled within a doubly stochastic framework where the default intensity follows a CIR++ process. We estimate the model parameters through a combination of a cross sectional calibration based method and a historical estimation approach. We propose a numerical procedure based on dynamic programming and a piecewise linear approximation to price American-style knock-out credit options. Our numerical investigation shows consistency, convergence and efficiency. We find that American-style CDS options can complete the credit derivatives market by allowing the investor to focus on spread movements rather than on the default event.
Click here to download a PDF version of this paper from the ICMA Centre.
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. Separable calibration to interest-rate and credit products is feasible,
as we illustrate with examples on the basic model and its variants.
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.
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We
develop and test a fast and accurate semi-analytical formula for single-name
default swaptions in the context of the shifted square root jump diffusion
(SSRJD) default intensity model. The formula consists of a decomposition of an
option on a summation of survival probabilities in a summation of options on
the underlying survival probabilities, where the strike for each option is
adjusted.
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In: Proceedings of the 4-th ICS Conference on
Statistical Finance, Tokyo, March 18-19, 2004.
Click here to download a PDF version of this
paper
We
consider the standard Credit Default Swap (CDS) payoff and some alternative
approximated versions, stemming from different conventions on the premium and
protection legs. We consider standard running CDS (RCDS), upfront CDS and
postponed-payments running CDS (PRCDS). Each different definition implies a
different definition of forward CDS rate, which we consider with some detail.
We introduce defaultable floating rate notes (FRN)'s. We point out which kind
of CDS payoff produces a forward CDS rate that is equal to the fair spread in
the considered defaultable FRN. An approximated equivalence between CDS's and
defaultable FRN's is established, which allows to view CDS options as
structurally similar to the optional component in defaultable callable notes.
We briefly investigate the possibility to express
forward CDS rates in terms of some basic rates and discuss a possible analogy
with the LIBOR and swap default-free models. Finally, we discuss the change of
numeraire approach for deriving a Black-like formula for CDS options or, equivalently,
defaultable callable FRN's. We also introduce an analytical formula for CDS
option prices under the CDS-calibrated SSRD stochastic-intensity model, and
discuss the impact of the different CIR++ dynamics parameters on the related
CDS options implied volatilities. Hints on possible methods for smile modeling
of CDS options are given for possible future developments of the CDS option
market.
To download a PDF copy of this paper go to
the SSRN website clicking here, or
download the paper directly from here.
In this work we derive an approximated no-arbitrage market valuation formula for Constant Maturity Credit Default Swaps (CMCDS). We move from the CDS options market model in Brigo (2004), and derive a formula for CMCDS that is the analogous of the formula for constant maturity swaps in the default free swap market under the LIBOR market model. A ``convexity adjustment"-like correction is present in the related formula. Without such correction, or with zero correlations, the formula returns an obvious deterministic-credit-spread expression for the CMCDS price. To obtain the result we derive a joint dynamics of forward CDS rates under a single pricing measure, as in Brigo (2004). Numerical examples of the ``convexity adjustment" impact complete the paper.
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In this work we analyze market payoffs of Credit Default Swaps (CDS) and we derive rigorous standard market formulas for pricing options on CDS. Formulas are based on modelling CDS spreads which are consistent with simple market payoffs, and we introduce a subfiltration structure allowing all measures to be equivalent to the risk neutral measure. Then we investigate market CDS spreads through change of measure and consider possible choices of rates for modelling a complete term structure of CDS spreads. We also consider approximations and apply them to pricing of specific market contracts. Results are derived in a probabilistic framework similar to that of Jamshidian (2004).
Presented at the IASTED conference at MIT, November
2004. To download a PDF copy of this paper go to the SSRN website
clicking here,
or download the paper directly from here.
In this paper we develop a tractable structural model with analytical default probabilities depending on some dynamics parameters, and we show how to calibrate the model using a chosen number of Credit Default Swap (CDS) market quotes. We essentially show how to use structural models with a calibration capability that is typical of the much more tractable credit-spread based intensity models. We apply the resulting AT1P structural model to a concrete calibration case and observe what happens to the calibrated dynamics when the CDS-implied credit quality deteriorates as the firm approaches default. Finally we provide a typical example of a case where the calibrated structural model can be used for credit pricing in a much more convenient way than a calibrated reduced form model: The pricing of counterparty risk in an equity swap.
Presented at the WBS workshop on Hybrid Equity/Credit
Products, London, March 14, 2005, and to be presented at Global Derivatives 2005,
Paris, May 23-26. To download a PDF copy of this paper go to the SSRN
website clicking here, or
download the paper directly from here.
In this work we develop a tractable structural model with analytical default probabilities depending on a random default barrier and possibly random volatility ideally associated with a scenario based underlying firm debt. We show how to calibrate this model using a chosen number of reference Credit Default Swap (CDS) market quotes. In general this model can be seen as a possible extension of the time-varying AT1P model in Brigo (2004). The calibration capability of the Scenario Volatility/Barrier model (SVBAT1P), when keeping time-constant volatility, appears inferior to the one of AT1P with time-varying deterministic volatility. The SVBAT1P model, however, maintains the benefits of time-homogeneity and can lead to satisfactory calibration results, as we show in a case study where we compare different choices on scenarios and parameters.
Similarly to AT1P, SVBAT1P is suited to pricing hybrid equity/credit derivatives and to evaluate counterparty risk in equity payoffs, and more generally to evaluate hybrid credit/equity payoffs. We consider the equity return swap in Brigo (2004) and show its valuation under SVBAT1P with the same CDS and equity calibration input used earlier for AT1P.
Updated version published in Risk Magazine (2006). Download here.
In this paper we develop structural first passage models (AT1P and SBTV) with time-varying volatility and characterized by high tractability. The models can be calibrated exactly to credit spreads using efficient closed-form formulas for default probabilities. In these models default events are caused by the value of the firm assets hitting a safety threshold, which depends on the financial situation of the company and on market conditions. In AT1P this default barrier is deterministic. Instead SBTV assumes two possible scenarios for the initial level of the default barrier, for taking into account uncertainty on balance sheet information and in particular risk of fraud. We apply the models to exact calibration of Parmalat Credit Default Swap (CDS) data during the months preceding default. In some cases these models show more calibration capability than a reduced-form model. The results we obtain with AT1P and SBTV have reasonable economic interpretation, and are particularly realistic when SBTV is considered. These results are analyzed in relation with the progressive unfolding of news on Parmalat crisis, and compared to the results we obtain for a company with higher credit quality.
An extended and updated version of this paper with the title
"Credit Calibration with Structural Models and Equity Return Swap valuation under Counterparty Risk"
will appear in: Bielecki, T., Brigo, D., and Patras, F. (Editors), Recent advancements in theory
and practice of credit derivatives, Bloomberg Press, 2010. Cermics report 2003-250. Click here to
download a PDF version of this paper.
In this paper we develop structural first passage models (AT1P and SBTV) with time-varying volatility and characterized by high tractability, moving from the original work of Brigo and Tarenghi (2004, 2005) and Brigo and Morini (2006). The models can be calibrated exactly to credit spreads using efficient closed-form formulas for default probabilities. Default events are caused by the value of the firm assets hitting a safety threshold, which depends on the financial situation of the company and on market conditions. In AT1P this default barrier is deterministic. Instead SBTV assumes two possible scenarios for the initial level of the default barrier, for taking into account uncertainty on balance sheet information. While in Brigo and Tarenghi (2004) and Brigo and Morini (2006) the models are analyzed across Parmalat's history, here we apply the models to exact calibration of Lehman Credit Default Swap (CDS) data during the months preceding default, as the crisis unfolds. The results we obtain with AT1P and SBTV have reasonable economic interpretation, and are particularly realistic when SBTV is considered. The pricing of counterparty risk in an Equity Return Swap is a convenient application we consider, also to illustrate the interaction of our credit models with equity models in hybrid products context.
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Updated version published in
"Communications in Statistics: Theory and Methods", Vol 34, issue 7,
2005. Cermics report 2003-250. Click here to
download a PDF version of the related paper from the CERMICS web
site. This version does not contain applications to finance yet.
Although there exists a large variety of copula functions, only a few are practically manageable, and often the choice in dependence modeling falls on the Gaussian copula. Further, most copulas are exchangeable, thus implying symmetric dependence. We introduce a way to construct copulas based on periodic functions. We study the two-dimensional case based on one dependence parameter and then provide a way to extend the construction to the n-dimensional framework. We can thus construct families of copulas in dimension n and parameterized by n - 1 parameters, implying possibly asymmetric relations. Such “periodic” copulas can be simulated easily.
Click here to
download a PDF version of the related paper from arXiv.org.
This paper deals with dependence across marginally exponentially distributed arrival times, such as default times in financial modeling or inter-failure times in reliability theory. We explore the relationship between dependence and the possibility to sample final multivariate survival in a long time-interval as a sequence of iterations of local multivariate survivals along a partition of the total time interval. We find that this is possible under a form of multivariate lack of memory that is linked to a property of the survival times copula. This property defines a "self-chaining-copula", and we show that this coincides with the extreme value copulas characterization. The self-chaining condition is satisfied by the Gumbel-Hougaard copula, a full characterization of self chaining copulas in the Archimedean family, and by the Marshall-Olkin copula. The result has important practical implications for consistent single-step and multi-step simulation of multivariate arrival times in a way that does not destroy dependency through iterations, as happens when inconsistently iterating a Gaussian copula.
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We question the industry practice of economic scenario generation involving statistically dependent default times. In particular, we investigate under which conditions a single simulation of joint default times at a final time horizon can be decomposed in a set of simulations of joint defaults on subsequent adjacent sub-periods leading to that final horizon. As a reasonable trade-off between realistic stylized facts, practical demands, and mathematical tractability, we propose models leading to a Markovian multi-variate default-indicator process. The well-known "looping default" case is shown to be equipped with this property, to be linked to the classical "Freund distribution", and to allow for a new construction with immediate multi-variate extensions. If, additionally, all sub-vectors of the default indicator process are also Markovian, this constitutes a new characterization of the Marshall-Olkin distribution, and hence of multi-variate lack-of-memory. A paramount property of the resulting model is stability of the type of multi-variate distribution with respect to elimination or insertion of a new marginal component with marginal distribution from the same family. The practical implications of this "nested margining" property are enormous. To implement this distribution we present an efficient and unbiased simulation algorithm based on the Levy-frailty construction. We highlight different pitfalls in the simulation of dependent default times and examine, within a numerical case study, the effect of inadequate simulation practices.
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download a PDF version of the related paper from the SSRN web site.
In this document we show how to handle counterparty risk for Interest Rate Swaps (IRS). First we establish a general formula, showing that counterparty risk adds one level of optionality to the contract. Then we introduce the default probabilities using a deterministic intensity model where the default time is modeled as the ¯rst jump of a time-inhomogeneous Poisson process. We consider Credit Default Swaps as liquid sources of market default probabilities. We then apply the general formula to a single IRS. The IRS price under counterparty risk turns out to be the sum of swaption prices with different maturities, each weighted with the probability of defaulting around that maturity. Then we consider a portfolio of IRS's in presence of a netting agreement. The related option cannot be valued as a standard swaption, and we resort both to Monte Carlo simulation and to two analytical approximations, investigating them by Monte Carlo simulation under several configurations of market inputs. We find a good agreement between the formula and the simulations in most cases. The approximated formula is well suited to risk management, where the computational time under each risk factors scenario is crucial and an analytical approximation contains it.
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Click here
to
download a PDF version of this paper. A refined version is "Risk
Neutral Pricing of Counterparty Risk, in: Pykhtin, M. (Editor), Counterparty
Credit Risk Modeling: Risk Management, Pricing and Regulation. Risk Books, London", 2005.
From the introduction: In this chapter we show how to handle counterparty risk when pricing some basic financial products. In particular we are analyzing in detail counterparty-risk (or Default-risk) Interest Rate Swaps and counterparty-risk equity return swaps. The reason to introduce counterparty risk when evaluating a contract is linked to the fact that many financial contracts are traded over the counter (OTC), so that the credit quality of the counterparty can be important. This is particularly appropriated when thinking of the different defaults experienced by some important companies during the last years. Also, regulatory issues related to the Basel II framework encourage the inclusion of counterparty risk into valuation.
We face the problem from the viewpoint of a safe (default-free) counterparty entering a financial contract with another counterparty that has a positive probability of defaulting before the maturity of the contract itself. We are assuming there are no guarantees in place (such as for example collateral). When investing in default risky assets we require a risk premium as a reward for assuming the default risk. If we think, for example, of a corporate bond, we know that the yield is higher than the corresponding yield of an equivalent treasury bond, and this difference is usually called credit spread. The (positive) credit spread implies a lower price for the bond when compared to default free bonds. This is a typical feature of every asset: The value of a generic claim traded with a counterparty subject to default risk is always smaller than the value of the same claim traded with a counterparty having a null default probability.
In the paper we focus on the following points in particular:
We assume absence of guarantees such as collateral;
Illustrate how the inclusion of counterparty risk in the valuation can make a payoff model dependent by adding one level of optionality;
Use the risk neutral default probability for the counterparty by extracting it from Credit Default Swap (CDS) data;
Because of the previous point, the chosen default model will have to be calibrated to CDS data;
When possible (and we will do so in Part III), take into account the correlation between the underlying of the contract and the default of the counterparty.
An updated version of this paper has been accepted for
publication in the International Journal of Theoretical and Applied Finance. Click here to download a PDF
version of this paper.
We consider counterparty risk for Credit Default Swaps (CDS) in presence of correlation between default of the counterparty and default of the CDS reference credit. Our approach is innovative in that, besides default correlation, which was taken into account in earlier approaches, we also model credit spread volatility. Stochastic intensity models are adopted for the default events, and defaults are connected through a copula function. We find that both default correlation and credit spread volatility have a relevant impact on the positive counterparty-risk credit valuation adjustment to be subtracted from the counterparty-risk free price. We analyze the pattern of such impacts as correlation and volatility change through some fundamental numerical examples, analyzing wrong-way risk in particular. Given the theoretical equivalence of the credit valuation adjustment with a contingent CDS, we are also proposing a methodology for valuation of contingent CDS on CDS.
An updated version of this paper has been published in Risk Magazine and also as a book
chapter. Click here to download a PDF
version of this paper.
We consider counterparty risk for interest rate payoffs in presence of
correlation between the default event and interest rates. The previous analysis
of Brigo and Masetti (2006), assuming independence, is further extended to
interest rate payoffs different from simple swap portfolios. A stochastic
intensity model is adopted for the default event. We find that correlation
between interest-rates and default has a relevant impact on the positive
adjustment to be subtracted from the default free price to take into account
counterparty risk. We analyze the pattern of such impacts as product
characteristics and tenor structures change through some fundamental numerical
examples. We find the counterparty risk adjustment to decrease with the
correlation for receiver payoffs, while the analogous adjustment for payer
payoffs increases. The impact of correlation decreases when the default
probability increases. Finally,
our analysis applies naturally also to Contingent Credit Default Swaps.
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The aim of this work is to
develop a pricing model for a kind of contract that we term "inflation
indexed credit default swaps (IICDS)". IICDS' payoffs are linked to
inflation, in that one of the legs of the swap is tied to the inflation rate.
In particular, the structure exchanges consumer price index (CPI) growth rate
plus a fixed spread minus the relevant libor rate for a protection payment in
case of early default of the reference credit. This is inspired by a real
market payoff we manged in our work. The method we introduce will be applied to
our case but is in fact much more general and may be envisaged in situations
involving inflation / credit / interest rate hybrids. The term IICDS itself can
be associated to quite different structures.
Many variables enter our IICDS
valuation. We have the CPI, the nominal and real interest rates, and the
default modeling variables. For our pricing purposes we need to choose a way of
modeling such variables in a convenient and practical fashion. Our choice fell
on the familiar short rate model setting, although frameworks based on recent
market models for credit and inflation could be attempted in principle, for
example by combining ideas on Credit Default Swap Market Models (Schoenbucher
2004, Brigo 2005) with ideas on Inflation Market Models (Belgrade, Benhamou and
Koehler 2004, Mercurio 2005).
We discuss numerical methods such as Euler discretization and Monte Carlo
simulation for our pricing procedure based on gaussian and CIR short rate
models for rates and default intensity. We analyze the numerical results
in details and discuss the impact of correlation between the different rates on
the valuation.
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Click here to download a PDF version of this paper.
This paper generalizes the framework for arbitrage-free valuation of bilateral counterparty risk to the case where collateral is included, with possible re-hypotecation. We analyze how the payout of claims is modified when collateral 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 and the counterparty and the underlying portfolio risk factors. We use arbitrage-free stochastic dynamical models, including also the effect of interest rate and credit spread volatilities. The impact of re-hypotecation, of collateral margining frequency and of dependencies on the bilateral counterparty risk adjustment is illustrated with a numerical example.
Click here to download a PDF version of this paper.
We analyze the practical consequences of the bilateral counterparty risk adjustment. We point out that past literature assumes that, at the moment of the first default, a risk-free closeout amount will be used. We argue that the legal (ISDA) documentation suggests in many points that a substitution closeout should be used. This would take into account the risk of default of the survived party. We show how the bilateral counterparty risk adjustment changes strongly when a substitution closeout amount is considered. We model the two extreme cases of default independence and co-monotonicity, which highlight pros and cons of both risk free and substitution closeout formulations, and allow us to interpret the outcomes as dramatic consequences on default contagion. Finally, we analyze the situation when collateral is present.
Click here to download a PDF version of this paper.
The purpose of this paper is introducing rigorous methods and formulas for bilateral counterparty risk credit valuation adjustments (CVA's) on interest-rate portfolios. In doing so, we summarize the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, as developed more in detail in Brigo and Capponi (2008), including the default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net present value of the contract at the relevant default times. We allow for correlation between the default times of the investor and counterparty, and for correlation of each with the underlying risk factor, namely interest rates. We also analyze the often neglected impact of credit spread volatility. We include Netting in our examples, although other agreements such as Margining and Collateral are left for future work.
Click here to download a PDF version of this paper.
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.
An updated version of this paper has
been published in Energy Risk. Click here to download a PDF version of this paper.
It is commonly accepted that Commodities futures and forward prices, in principle, agree under some simplifying assumptions. One of the most relevant assumptions is the absence of counterparty risk. Indeed, due to margining, futures have practically no counterparty risk. Forwards, instead, may bear the full risk of default for the counterparty when traded with brokers or outside clearing houses, or when embedded in other contracts such as swaps. In this paper we focus on energy commodities and on Oil in particular. We use a hybrid commodities-credit model to asses impact of counterparty risk in pricing formulas, both in the gross effect of default probabilities and on the subtler effects of credit spread volatility, commodities volatility and credit-commodities correlation. We illustrate our general approach with a case study based on an oil swap, showing that an accurate valuation of counterparty risk depends on volatilities and correlation and cannot be accounted for precisely through a pre-defined multiplier.
Volatility
smile modeling
Draft of a working paper. Click here to download the paper, or click here to download a PDF version of this paper from the SSRN web site.
In the present paper, given an evolving mixture of probability densities, we define a candidate diffusion process whose marginal law follows the same evolution. We derive as a particular case a stochastic differential equation (SDE) admitting a unique strong solution and whose density evolves as a mixture of Gaussian densities. We present an interesting result on the comparison between the instantaneous and the terminal correlation between the obtained process and its squared diffusion coefficient. As an application to mathematical finance, we construct diffusion processes whose marginal densities are mixtures of lognormal densities. We explain how such processes can be used to model the market smile phenomenon. We show that the lognormal mixture dynamics is the one-dimensional diffusion version of a suitable uncertain volatility model, and suitably reinterpret the earlier correlation result. We explore numerically the relationship between the future smile structures of both the diffusion and the uncertain volatility versions.
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A short version of this paper by Brigo and Mercurio has been published in the September 2000 issue of Risk Magazine, and a related paper appeared in the International Journal of Theoretical and Applied Finance. This version features a whole surface calibration example due to Rapisarda. Click here to download a PDF file containing this paper.
We introduce a general class of analytically tractable models for the dynamics of an asset price based on the assumption that the asset-price density is given by the mixture of known basic densities. We consider the lognormal-mixture model as a fundamental example, deriving explicit dynamics, closed form formulas for option prices and analytical approximations for the implied volatility function. We then introduce the asset-price model that is obtained by shifting the previous lognormal-mixture dynamics and investigate its analytical tractability. We finally consider a specific example of calibration to real market option data. The general mixture framework with a result on the relationship between the asset and its volatility is given in the paper above.
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Click here to download a PDF file containing this paper from SSRN, or here to download it from arXiv.
We introduce a multivariate diffusion model that is able to price derivative securities featuring multiple underlying assets. 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. This allows to reconcile single name and index/basket volatility smiles in a consistent framework. Our approach could be dubbed a multidimensional local volatility approach with vector-state dependent diffusion matrix. The model is quite tractable, leading to a complete market and not requiring Fourier techniques for calibration and dependence measures, contrary to multivariate stochastic volatility models such as Wishart. We prove existence and uniqueness of solutions for the model stochastic differential equations, provide formulas for a number of basket options, and analyze the dependence structure of the model in detail by deriving a number of results on covariances, its copula function and rank correlation measures and volatilities-assets correlations. A comparison with sampling simply-correlated suitably discretized one-dimensional mixture dynamical paths is made, both in terms of option pricing and of dependence, and first order expansion relationships between the two models' local covariances are derived. We also show existence of a multivariate uncertain volatility model of which our multivariate local volatilities model is a Markovian projection, highlighting that the projected model is smoother and avoids a number of drawbacks of the uncertain volatility version. We also show a consistency result where the Markovian projection of a geometric basket in the multivariate model is a univariate mixture dynamics model. A few numerical examples on basket and spread options pricing conclude the paper.
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Click here to download a PDF file containing this paper from SSRN, or here to download it from arXiv.
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 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.
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Interest-rate derivatives modeling, interest rate models with credit and liquidity effects and multiple curves
Paper available here from SSRN, and here from arXiv.
The market practice of extrapolating different term structures from different instruments lacks a rigorous justification in terms of cash flows structure and market observables. In this paper, we integrate our previous consistent theory for pricing under credit, collateral and funding risks into term structure modelling, integrating the origination of different term structures with such effects. Under a number of assumptions on collateralization, wrong-way risk, gap risk, credit valuation adjustments and funding effects, including the treasury operational model, and via an immersion hypothesis, we are able to derive a synthetic master equation for the multiple term structure dynamics that integrates multiple curves with credit/funding adjustments.
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Paper presented at the 2nd world congress of the
Bachelier Finance Society. Updated version
published in Quantitative Finance. Click here to download
a PDF version of this paper.
In this paper we are
concerned with the distributional difference of forward swap rates between the
lognormal forward--Libor model
(LFM) or ``Libor market model" and the
lognormal forward-swap model (LSM) or ``swap market model", the two
fundamental models for interest-rate derivatives. To measure this
distributional difference, we resort to a ``metric" in the space of
distributions, the well known Kullback-Leibler information (KLI). We explain
how the KLI can be used to measure the distance of a given distribution from
the lognormal (exponential) family of densities, and then apply this to our
models' comparison. The volatility of the projection of the LFM swap-rate
distribution onto the lognormal family is compared to a industry synthetic swap
volatility approximation obtained via ``freezing the drift" techniques in
the LFM. Finally, for some instantaneous covariance parameterizations of the
LFM we analyze how the above distance changes according to the parameter values
and to the parameterizations themselves, in an attempt to characterize the
situations where LFM and LSM are really distributionally close, as is assumed
by the market.
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An earlier version of this paper, co-authored by Cristina Capitani, has been presented at the 2001 Annual Conference in Financial Risk held in Budapest, at the 2001 "Quantitative Methods in Finance" conference in Sydney, and at the Bachelier Society second world congress in Crete in 2002. This paper is based on Chapters 6, 7 and 8 of "Interest-Rate Models: Theory and Practice" by Brigo and Mercurio. An extended version has been published in the European Journal of Operations Research. NEW DEVELOPMENTS ARE FOUND IN THE NEXT PAPER BELOW. .
In this paper we consider several parametric assumptions for the instantaneous covariance structure of the Libor market model, whose role in the modern interest-rate derivatives theory is becoming more and more central. We examine the impact of each different parameterization on the evolution of the term structure of volatilities in time, on terminal correlations and on the joint calibration to the caps and swaptions markets. We present a number of cases of calibration in the Euro market. In particular, we consider calibration via a parameterization establishing a controllable one to one correspondence between instantaneous covariance parameters and swaptions volatilities, and assess the benefits of smoothing the input swaption matrix before calibrating.
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A new version of the LIBOR calibration paper has been presented at the Third Bachelier World Congress in Chicago (2004) and can be downloaded from the SSRN network here.. For the earlier paper presented at the IV Italian Workshop on Mathematical Finance, ICER, Turin, January 30-31, 2003, click here .
This work focuses on
the swaptions automatic cascade calibration algorithm (CCA) for the LIBOR
Market Model (LMM) first appeared in Brigo and Mercurio (2001). This method
induces a direct analytical correspondence between market swaption volatilities
and LMM parameters, and allows for a perfect recovery of market quoted swaption
volatilities if a common industry swaptions approximation is used.
We present explicitly an extension of the CCA to calibrate the entire swaption
matrix rather than its upper triangular part. Then, while previous tests on
earlier data showed the appearance of numerical problems, we present here
different calibration cases leading to acceptable results. We analyze the
characteristics of the configurations used and concentrate on the effects of
different exogenous instantaneous historical or parametric correlation
matrices.
We also investigate the influence of manipulations in input swaptions data for
missing quotes, and devise a new algorithm maintaining all the positive
characteristics of the CCA while relying only on directly quoted market data.
Empirical results on a larger range of market situations and instantaneous
covariance assumptions show this algorithm to be more robust and efficient than
the previous version. Calibrated parameters are in general regular and
financially satisfactory, as confirmed by the analysis of various diagnostics
implied structures.
Finally we Monte Carlo investigate the reliability of the underlying LMM
swaption analytical approximation in the new context, and present some possibilities
to include information coming from the semi-annual tenor cap market.
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A reduced version of this paper has appeared in "Finance and Stochastics". Click here to download a PDF version of this paper.
In the present paper we show how to extend any time-homogeneous short-rate model and analytically tractable short-rate model (such as Vasicek (1977), Cox-Ingersoll-Ross (1985), Dothan (1978)) to a model which can reproduce any observed yield curve, through a procedure that preserves the possible analytical tractability of the original model. In the case of the Vasicek (1977) model, our extension is equivalent to that of Hull and White (1990), whereas in the case of the Cox-Ingersoll-Ross (1985) (CIR) model, our extension is more analytically tractable and avoids problems concerning the use of numerical solutions. Our approach can also be applied to the Dothan (1978) or Rendleman and Bartter (1980) model, thus yielding a "quasi" lognormal short-rate model which fits any given yield curve and for which there exist analytical formulae for prices of zero coupon bonds. We also consider the extension of time-homogeneous models without analytical formulae but whose tree-construction procedures are particularly appealing, such as the exponential Vasicek's. We explain why the CIR++ extended CIR model is the more interesting model obtained through our procedure. We also give explicit analytical formulae for bond options, hence swaptions, caps and floors, and we explain how the model can be used for Monte Carlo evaluation of European path-dependent interest-rate derivatives.We finally hint at the same extension for multifactor models and explain its strong points for concrete applications.
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This
note is meant to be an expansion of Section 6.9 of the first edition of
``Interest-Rate Models: Theory and Practice",
Springer, 2001, by Brigo and Mercurio. This is a preliminary draft.
Comments are welcome.
Click here to download a PDF
file containing this paper
We review both full-rank and reduced-rank parameterizations for correlation matrices. In particular, the full-rank parameterizations of Rebonato (1999d) and Schoenmakers and Coffey (2000), the reduced-rank angle parameterization of Rebonato (1999d) and Rebonato and Jackel (1999), and the well known zeroed-eigenvalues reduced-rank approximation are described in detail. Numerical examples are provided for the reduced-rank parameterizations and results are compared.
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Basket options
A short version of this paper has been presented at the 2001 annual meeting of the European Financial Management Association. Click here to download a PDF version of this paper. .
The aim of this paper is to present two moment-matching procedures for basket-options pricing and to test their distributional approximations via distances on the space of probability densities, the Kullback-Leibler information (KLI) and the Hellinger distance (HD). We are interested in measuring the KLI and the HD between the real simulated basket terminal distribution and the distributions used for the approximation, both in the lognormal and shifted-lognormal families. We isolate influences of instantaneous volatilities and instantaneous correlations, in order to assess which configurations of these variables have a major impact on the KLI and HD and therefore on the quality of the approximation. A number of numerical investigations is carried out.
General option-pricing theory
A short version of this paper by the same authors has appeared in "Finance and Stochastics". The related theoretical part has been published in "Statistics and Probability letters" 49 (2000), pp. 127--134. Click here to download a PDF version of this paper.
In the present paper we construct stock price processes with the same marginal log-normal law as that of a geometricBrownian motion and also with the same transition density (and returns' distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes', in that option prices can be either arbitrarily close to the option intrinsic value or arbitrarily close to the underlying stock price. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants which are relevant for trading. The theoretical result concerning scalar stochastic differential equations with prescribed diffusion coefficient whose densities evolve in a prescribed exponential family, on which part of the paper is based, is presented in detail.
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A related paper
appeared later on in "Insurance. Mathematics and Economics", 22
(1) (1998) pp. 53-64.
Click here to download a
Postscript file containing this paper
Three situations in
which filtering theory is used in mathematical finance are illustrated at
different levels of detail. The three problems originate from the following
different works:
1) On estimating the stochastic volatility model
from observed bilateral exchange rate news, by R. Mahieu, and P. Schotman;
2) A state space approach to estimate multi-factors
CIR models of the term structure of interest rates, by A.L.J. Geyer, and S.
Pichler;
3) Risk-minimizing hedging strategies under partial
observation in pricing financial derivatives, by P. Fischer, E. Platen, and W.
J. Runggaldier;
In the first problem we propose to use a recent
nonlinear filtering technique based on geometry to estimate the volatility time
series from observed bilateral exchange rates. The model used here is the
stochastic volatility model. The filters that we propose are known as
projection filters, and a brief derivation of such filters is given.
The second problem is introduced in detail, and a
possible use of different filtering techniques is hinted at. In fact the
filters used for this problem in 2) and part of the literature can be
interpreted as projection filters and we will make some remarks on how more
general and possibly more suitable projection filters can be constructed. The
third problem is only presented shortly.
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Risk Management
An updated version of this paper has been published in the Journal of Risk Management in Financial Institutions. Click here to download a PDF version of this paper, or download it from the SSRN web site here.
In risk management it is desirable to grasp the essential statistical features of a time series representing a risk factor. This tutorial aims to introduce a number of different stochastic processes that can help in grasping the essential features of risk factors describing different asset classes or behaviors. This paper does not aim at being exhaustive, but gives examples and a feeling for practically implementable models allowing for stylised features in the data. The reader may also use these models as building blocks to build more complex models, although for a number of risk management applications the models developed here suffice for the first step in the quantitative analysis. The broad qualitative features addressed here are { fat tails} and { mean reversion}. We give some orientation on the initial choice of a suitable stochastic process and then explain how the process parameters can be estimated based on historical data. Once the process has been calibrated, typically through maximum likelihood estimation, one may simulate the risk factor and build future scenarios for the risky portfolio. On the terminal simulated distribution of the portfolio one may then single out several risk measures, although here we focus on the stochastic processes estimation preceding the simulation of the risk factors Finally, this first survey report focuses on single time series. Correlation or more generally dependence across risk factors, leading to multivariate processes modeling, will be addressed in future work.
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Click here to download a PDF version of this paper, or download it from the SSRN web site here. The report is also available on arxiv.org
Within the context of risk integration, we introduce in risk measurement stochastic holding period (SHP) models. This is done in order to obtain a `liquidity-adjusted risk measure' characterized by the absence of a fixed time horizon. The underlying assumption is that - due to changes on market liquidity conditions - one operates along an `operational time' to which the P\&L process of liquidating a market portfolio is referred. This framework leads to a mixture of distributions for the portfolio returns, potentially allowing for skewness, heavy tails and extreme scenarios. We analyze the impact of possible distributional choices for the SHP. In a multivariate setting, we hint at the possible introduction of dependent SHP processes, which potentially lead to non linear dependence among the P\&L processes and therefore to tail dependence across assets in the portfolio, although this may require drastic choices on the SHP distributions. We finally discuss potential developments following future availability of market data.
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This paper appeared in the Journal of Financial Perspectives, Volume 4, issue 1, pp. 8-20. Click here to download a PDF version of this paper.
We consider the current challenges and opportunities in applications of Robotics to financial services and to insurance in particular. Combinations of Robot Process Automation (RPA) with digitization have been considered by the industry, with important benefits in cost reduction and efficiency. We highlight the general benefits of RPA and the related implementation challenges in detail. We discuss more advanced Artificial Intelligence (AI) applications, arguing that such applications depend on the general advancements of AI, where human level interaction is not yet available. We discuss the great potential for AI applications in the near future and consider some initial examples. We also briefly discuss the hard problems of AI in relation to intelligence and consciousness in the introductory part, and briefly look at the implications AI and robots could have for human society and employment.
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Algorithmic trading and optimal execution
Click here to download a PDF version of this paper from SSRN, or here to download it from arXiv.
We solve the optimal trade execution problem in the Almgren and Chirss framework with the Value-at-risk / Expected Shortfall based criterion of Gatheral and Schied when the underlying unaffected stock price follows a displaced diffusion model. The displaced diffusion model can conveniently model at the same time situations where either an arithmetic Brownian motion (ABM) or a geometric Browinan motion (GBM) type dynamics may prevail, thus serving as a bridge between the ABM and GBM frameworks. We introduce alternative risk criteria and we notice that the optimal trade execution strategy little depends on the specific risk criterion we adopt. In most situations the solution is close to the simple Volume Weighted Average Price (VWAP) solution regardless of the specific diffusion dynamics or risk criterion that is chosen, especially on realistic trading horizons of a few days or hours. This suggests that more general dynamics need to be considered, and possibly more extreme risk criteria, in order to find a relevant impact on the optimal strategy.
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Click here to download a PDF version of this paper from SSRN, or here to download it from arXiv.
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.
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A SUMMARY OF THE FILTERING PROBLEM AND OF MY PhD STUDIES
(Stochastic
differential equations, exponential families, stochastic nonlinear filtering, differential geometry)
My doctoral thesis treats the finite--dimensional approximation of distributions obtained via differential--geometric methods and exponential families. The key ingredients in the theory developed here are: Stochastic differential equations (SDE's), the filtering problem, the differential geometric approach to statistics, and the theory of exponential families.
SDE's are roughly an extension of ordinary
differential equations (ODE's) to the case where the evolution of the system is
afflicted by randomness. This evolution then needs to be described by a
mathematical object called SDE, since ODE's do not incorporate randomness.
The filtering problem consists of estimating the
state of a stochastic system from noise perturbed observations. One has a
system whose state evolves according to a SDE, and one observes a related
process which is generally a function of the state process plus some new
randomness. The filtering problem consists of estimating the signal at any time
instant from the history of the observation process up to the same instant.
If the evolution of the state and the observations
are described by linear equations, the solution of the problem is the
well known Kalman Filter (KF). This filter consists of a finite set
of recursive equations which permit to update the estimates including at
each time instant the new observations. In this case the filter is said to be
finite dimensional. The more general nonlinear filtering problem is far more
complicated because the resulting nonlinear filter is not finite
dimensional in general. Finite dimensionality of a filter is loosely defined as
a filter consisting of a finite set of recursive equations which update the
optimal estimate of the state based on the past observations. In general there
is no such set of equations for the nonlinear filtering problem. The
solution of the filtering problem in continuous time is the probability
distribution of the state given the past and current observations. This
solution is described by a mathematical object called a stochastic
partial--differential equation. This is in general an
infinite--dimensional equation, in the sense that its solution cannot be
characterized by the solution of a finite set of (stochastic) differential
equations. The past remedies to this infinite dimensionality (assumed--density
filter and extended KF) were based on heuristic considerations and not much is
known on the quality of their performances. In this thesis we present a new
method to obtain a finite set of SDEs which approximate the
infinite--dimensional equation for the optimal filter. We introduce the
projection filter (PF), which is a finite--dimensional nonlinear filter
based on the differential--geometric approach to statistics. The
projection filter is obtained by projecting the infinite--dimensional equation
for the optimal filter onto a finite--dimensional manifold. This projection is
mathematically well defined. Moreover, there is ample choice about what
finite--dimensional space one can project upon. In the thesis we use this
geometric framework to define and study in detail the projection filter for
exponential families of probability densities. We present results describing
the advantages of choosing exponential families: Simple filter equation,
possibility of defining the total projection residual measuring the local
approximation involved in the projection around each time instant, equivalence
with the (previously heuristics--based) assumed--density filters, perfect
update step in the case of discrete--time observations. Moreover we present
simulation results for the exponential projection filter applied to a
particular system called cubic sensor. Finally, some results on the nice
asymptotic behaviour of the Gaussian projection filter with small
observation noise are given. This treats roughly the case where the randomness
afflicting the observations becomes small. The Gaussian densities are a
particular case of exponential densities.
The framework of an exponential family of
densities with parameters described by SDE's and with the
differential--geometric structure developed for the filtering problem is
useful also for other applications. In the thesis we have solved several
problems related to existence of stochastic differential equations. These
results are related to areas such as stochastic realization theory,
mathematical finance, and existence of finite--dimensional optimal filters, as
we show in the final chapter.
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DOWNLOADABLE SLIDES OF TALKS, SEMINARS AND LECTURES
QUANT 2019 Conference, Venice, Feb 22, 2019.
To download the slides of this presentation as a PDF
document click here.
Beyond the Boundaries conference, Leeds,
5 May 2021;
Quant Minds, May 16 2019, Vienna.
To download the slides of this presentation as a PDF
document click here.
Options: 45 Years after the Publication of the Black Scholes Model conference,
Gershon Center, Hebrew University, Jerusalem, Dec 4-5, 2018.
To download the slides of this presentation as a PDF
document click here.
Quantitative Finance Seminar, March 15, 2016, Imperial College London;
University of Oxford, Quantitative Finance Seminar, 26 November 2015;
Fields Institute Seminar, Toronto, October 31th 2014;(video)
Plenary 30th anniversary AFFI Conference, Lyon, 28-31 May 2013;
2014 International Conference on Financial Engineering &
Innovation, Tongji University,Shanghai, March 14-16;
Cambridge, Isaac Newton Institute for Mathematical Sciences, March 28, 2013.
To download the slides of this presentation as a PDF
document click here.
London-Paris Bachelier Workshop 2017, September 21--22, 2017;
To download the slides of this presentation as a PDF
document click here.
This talk is similar to the previous one but less technical;
QFRA 2017, Corfu, June 15--16, 2017;
Oxford-Princeton Workshop in Math Finance, May 25--26, 2017;
Global Derivatives Plenary 2017, Barcelona, May 11;
To download the slides of this presentation as a PDF
document click here.
Institut de Recherche Mathematique Avancees, Strasbourg, March 17, 2017.
Scuola Normale Superiore, Pisa, October 10, 2016.
2016 Risk & Stochastics Conference, LSE, April 21, 2016.
To download the slides of this presentation as a PDF
document click here.
Innovations in Insurance, Risk & Asset Management, Technical University of Munich, 6 April 2017;
Bachelier Seminar, Institut Henri Poincare, Paris, April 8, 2016;
RiskMinds, Amsterdam, Dec 9, 2015;
Nomura, Internal Risk Seminar, London, Aug 12, 2015;
Quant Congress Europe, London, April 15, 2015;
Imperial-ETH worskshop, London, March 6, 2015.
To download the slides of this presentation as a PDF
document click here.
Oxford
data assimilation conference, 24-28 september 2012;
AHOI Workshop on Ambit Stochastics and Applications
at Imperial College London, 25-27 March 2013
To download the slides of this presentation as a PDF
document click here.
Presentation at the European Stability Mechanism, Luxembourg, May 12, 2016.
Presented at the Nomura Risk Seminar, London, Nomura, September 19, 2016.
To download the slides of this presentation as a PDF
document click here.
London Graduate School in Mathematical Finance - MF6 Course
London, Nov 6,7,13,14, 2012, Imperial College London;
To download the slides of this presentation as a PDF
document click here.
For material related to this course click here.
Inaugural Meeting of the Scottish Risk Academy, November 4, 2010;
Second Conference on the Mathematics of Credit Risk, Princeton, May 23-24 2008;
To download the slides of this presentation as a PDF
document click here.
Winter School in Mathematical Finance (Kasteel Oud Poelgeest,
Oestgeest, Amsterdam, December 16-18, 2002);
Quantitative Finance (Risk Conference, London, November 25-26, 2002),
Hitotsubashi University (Tokyo, March 2002)
University of Padua (October 26, 2001). To download the slides of this presentation as a PDF
document click here.
Conference on Credit Risk, Stevanovich Center, University of Chicago, 19-20 Oct 2007. To download the slides of this presentation as a PDF document click here.
Seminar at University of Chicago, October 26, 2006. To download the slides of this presentation as a PDF document click here.
Center for Financial Engineering, Columbia University, New York. May 11, 2006. To download the slides of this presentation as a PDF document click here.
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PUBLICATIONS, PRESS, IMPACT and NETWORKING |
BOOKS and peer-reviewed ARTICLES
A complete list of books and more than 130 peer reviewed publications in journals, volumes and conference proceedings is available here.
The books include "Interest rate models: theory and practice", that became a field reference book in interest rate modeling for derivatives. It has more than 3000 citations in Google Scholar and has been used across the industry in trading desks and in many universities for postgraduate courses. Books on credit risk, the credit crisis and valuation adjustments are also listed.
The peer review articles are publications in probability and publications in quantitative finance.
The publications in probability concern stochastic filtering and signal processing, differential geometry and statistics, the geometry of the Fokker Planck equation, a new characterization of the Marshall-Olkin law, and Stochastic and Rough Differential Equations on Manifolds. These publications appeared in top journals such as Proceedings of the Royal Society A, Proceedings of the London Mathematical Society, IEEE Transactions on Automatic Control, Bernoulli, Stochastic Processes and Their Applications and others.
The publications in quantitative finance concern fundamentals of option pricing, with and without probability (rough paths theory), volatility smile modeling, both univariate and multivariate, term structure modeling and interest rates, risk measures, credit risk, credit derivatives (credit default swaps and options, dynamic loss models for CDOs, papers on the credit crisis), valuation adjustments and XVA, nonlinear valuation via BSDEs and semi-linear PDEs, optimal execution, price impact, machine learning methods in credit risk and non-performing loans and interpretability for deep learning in finance. These publications appeared in top academic journals such as European Journal of Operational Research, Mathematical Finance, Finance and Stochastics, Journal of Banking and Finance, Quantitative Finance, Insurance: Mathematics and Economics, and many others.
A number of publications appeared also in the industry-influential Risk Magazine.
Ph.D. THESIS
D. Brigo, Filtering by Projection on the Manifold of Exponential Densities, Free University of Amsterdam, 1996. Zipped PostScript file available here.
PRESS INTERVIEWS
Short interview with Il Mondo (Corriere della Sera) in the "Jobs and Careers" reportage "I Matematici: Testa tra le nuvole ma piedi per terra [Mathematicians: Head in the clouds but feet on the ground]", Issue 14, April 13, 2001, pp.136-144.
Profile interview in Risk Magazine. Damiano has been interviewed for a Profile published in Risk Magazine with the title ``The risk-free myth", March 2011 issue
Interview in The Banker (the Financial Times magazine). Damiano has been interviewed for the article "`Has Basel got its numbers wrong?", June 21 2011.
IMPACT
Provided Impact to society. For example, the above publication with M. Morini and M. Tarenghi has been quoted as a technical support in a sentence in the court of law in Novara, Italy
NETWORKING
Invited at a Dinner at the Royal Society, London, Oct 27 2011, organized by Terry Lyons (Oxford), for a discussion on the future of Risk Management and of methodology in the Financial Industry, with executives and officials from Morgan Stanley, Nomura, Barcap, JP Morgan, Lloyds TSB, FSA, HM Treasury, Bank of England.
Invited at an industry dinner on CVA organized by SunGard at the Royal Stock Exchange, London, Nov 9 2011, together with Executives from BNP Paribas, Royal Bank of Scotland, Barcap, UBS, Nomura, Unicredit, Commerzbank, WestLB.
Invited at an industry dinner organized by NumeriX, Chicago, Nov 15, 2011, together with Executives from Morgan Stanley, NumeriX, Bloomberg.
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Arrivederci!