CREPEY Stephane

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This thesis deals with the computation of X-value adjustments, where X includes C for credit, F for financing, M for margin and K for capital. We study different approaches based on nested simulation and implemented on graphics processing units (GPUs). We first consider the problem, for an insurance company or a bank, of numerically computing its economic capital in the form of a value-at-risk or an expected shortfall over a given time horizon. Using a stochastic approximation approach on the value-at-risk or the expected shortfall we establish the convergence of the resulting patterns of the economic capital simulation. Then, we present a nested Monte Carlo (NMC) approach for the computation of XVA. We show that the overall computation of XVAs involves five layers of dependence. The highest layers are run first and trigger nested simulations on the fly if needed to compute an element from a lower layer. Finally, we present a single-layer nested Monte Carlo (1NMC) based algorithm to simulate the U-functions of a Markov process X. The main originality of the proposed method comes from the fact that it provides a recipe for simulating U_{t>=s} conditionally on X_s. The generality, the stability and the iterative character of this algorithm, even in high dimension, make its strength.

Stochastic Approximation is an iterative procedure for computing a zero θ of a function that is not explicit but defined as an expectation. It is, for example, a numerical tool for computing maximum likelihood in "regular" latent data models. If the definition of the statistical model is tainted with an uncertainty τ , of which only an a priori dπ(τ ) is known, then the zeros depend on τ and the natural question is to explore their distribution when τ ∼ dπ. In this paper, we propose an iterative algorithm based on a Stochastic Approximation scheme that,in the limit, computes θ (τ) for any τ and produces a characterization of its distribution. and weenounce sufficient conditions for the convergence of this algorithm.

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We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja (2010) or by regression à la Broadie, Du, and Moallemi (2015). Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in Gordy-Juneja (2010) and Broadie-Du-Moallemi (2015). Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking par- lance. This is illustrated numerically by a KVA case study implemented on GPUs.

We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja (2010) or by regression à la Broadie, Du, and Moallemi (2015). Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in Gordy-Juneja (2010) and Broadie-Du-Moallemi (2015). Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking par- lance. This is illustrated numerically by a KVA case study implemented on GPUs.

We begin this thesis report by evaluating the expected exposure, which represents one of the major components of XVA adjustments. Under the assumption of independence between exposure and financing and credit costs, we derive in Chapter 3 a new representation of expected exposure as the solution of an ordinary differential equation with respect to the time of default observation. For the one-dimensional case, we rely on arguments similar to those for Dupire's local volatility. And for the multidimensional case, we refer to the Co-aire formula. This representation allows us to explain the impact of volatility on the expected exposure: this time value involves the volatility of the underlyings as well as the first-order sensitivity of the price, evaluated on a finite set of points. Despite numerical limitations, this method is an accurate and fast approach for valuing unit XVA in dimension 1 and 2.The following chapters are dedicated to the risk aspects of correlations between exposure envelopes and XVA costs. We present a model of the general correlation risk through a multivariate stochastic diffusion, including both the underlying assets of the derivatives and the default intensities. In this framework, we present a new approach to valuation by asymptotic developments, such that the price of an XVA adjustment corresponds to the price of the zero-correlation adjustment, plus an explicit sum of corrective terms. Chapter 4 is devoted to the technical derivation and study of the numerical error in the context of the valuation of default contingent derivatives. The quality of the numerical approximations depends solely on the regularity of the credit intensity diffusion process, and is independent of the regularity of the payoff function. The valuation formulas for CVA and FVA are presented in Chapter 5. A generalization of the asymptotic developments for the bilateral default framework is addressed in Chapter 6.We conclude this dissertation by addressing a case of the specific correlation risk related to rating migration contracts. Beyond the valuation formulas, our contribution consists in presenting a robust approach for the construction and calibration of a rating transition model consistent with market implied default probabilities.

We present a nested Monte Carlo (NMC) approach implemented on graphics processing units (GPU) to X-valuation adjustments (XVA), where X ranges over C for credit, F for funding, M for margin, and K for capital. The overall XVA suite involves five compound layers of dependence. Higher layers are launched first and trigger nested simulations on-the-fly whenever required in order to compute an item from a lower layer. If the user is only interested in some of the XVA components, then only the sub-tree corresponding to the most outer XVA needs be processed computationally. Inner layers only need a square root number of simulation with respect to the most outer layer. Some of the layers exhibit a smaller variance. As a result, with GPUs at least, error controlled NMC XVA computations are doable. But, although NMC is naively suited to parallelization, a GPU implementation of NMC XVA computations requires various optimizations. This is illustrated on XVA computations involving equities, interest rate, and credit derivatives, for both bilateral and central clearing XVA metrics.

The ongoing concern about systemic risk since the outburst of the global financial crisis has highlighted the need for risk measures at the level of sets of interconnected financial components, such as portfolios, institutions or members of clearing houses. The two main issues in systemic risk measurement are the computation of an overall reserve level and its allocation to the different components according to their systemic relevance. We develop here a pragmatic approach to systemic risk measurement and allocation based on multivariate shortfall risk measures, where acceptable allocations are first computed and then aggregated so as to minimize costs. We analyze the sensitivity of the risk allocations to various factors and highlight its relevance as an indicator of systemic risk. In particular, we study the interplay between the loss function and the dependence structure of the components. Moreover, we address the computational aspects of risk allocation. Finally, we apply this methodology to the allocation of the default fund of a CCP on real data.

Two nonlinear Monte Carlo schemes, namely, the linear Monte Carlo expansion with randomization of Fujii and Takahashi (2012a,2012b) and the marked branching diffusion scheme of Henry-Labordère (2012), are compared in terms of applicability and numerical behavior regarding counterparty risk computations on credit derivatives. This is done in two dynamic copula models of portfolio credit risk: the dynamic Gaussian copula model and the model in which default dependence stems from joint defaults. For such high-dimensional and nonlinear pricing problems, more standard deterministic or simulation/regression schemes are ruled out by Bellman's " curse of dimensionality " and only purely forward Monte Carlo schemes can be used.

Banking operations are being rewired around XVA metrics quantifying market incompleteness. This paper focuses on the cost of funding of variation margin and the cost of capital, i.e. FVA and KVA. The two metrics are intertwined since economic capital is itself a source of funding. Accurate valuations require simulations of capital and funding costs. Motivated by Basel Pillar II, Solvency II and IFRS 4 Phase II, we propose a principled approach to accounting regulatory treatments for FVA and KVA.

In Crépey [9], a basic reduced-form counterparty risk modelling approach was introduced under a standard immersion hypothesis between a reference filtration and the filtration progressively enlarged by the default times of the two parties. This basic setup, with a related continuity assumption on some of the data at the first default time of the two parties, is too restrictive for wrong-way and gap risk applications, such as counterparty risk on credit derivatives. This paper introduces an extension of the basic approach, implements it through marked default times and applies it to counterparty risk on credit derivatives.

In incomplete markets, a basic Black-Scholes perspective has to be complemented by the valuation of market imperfections. Otherwise this results in Black-Scholes Ponzi schemes, such as the ones at the core of the last global financial crisis, where always more derivatives need to be issued for remunerating the capital attracted by the already opened positions. In this paper we consider the sustainable Black-Scholes equations that arise for a portfolio of options if one adds to their trade additive Black-Scholes price, on top of a nonlinear funding cost, the cost of remunerating at a hurdle rate the residual risk left by imperfect hedging. We assess the impact of model uncertainty in this setup.

We present a nested Monte Carlo (NMC) approach implemented on graphics processing units (GPU) to X-valuation adjustments (XVA), where X ranges over C for credit, F for funding, M for margin, and K for capital. The overall XVA suite involves five compound layers of dependence. Higher layers are launched first and trigger nested simulations on-the-fly whenever required in order to compute an item from a lower layer. If the user is only interested in some of the XVA components, then only the sub-tree corresponding to the most outer XVA needs be processed computationally. Inner layers only need a square root number of simulation with respect to the most outer layer. Some of the layers exhibit a smaller variance. As a result, with GPUs at least, error controlled NMC XVA computations are doable. But, although NMC is naively suited to parallelization, a GPU implementation of NMC XVA computations requires various optimizations. This is illustrated on XVA computations involving equities, interest rate, and credit derivatives, for both bilateral and central clearing XVA metrics.

This paper develops an XVA (costs) analysis of centrally cleared trading, parallel to the one that has been developed in the last years for bilateral transactions. We introduce a dynamic framework that incorporates the sequence of cash-flows involved in the waterfall of resources of a clearing house. The total cost of the clearance framework for a clearing member, called CCVA for central clearing valuation adjustment, is decomposed into a CVA corresponding to the cost of its losses on the default fund in case of defaults of other member, an MVA corresponding to the cost of funding its margins and a KVA corresponding to the cost of the regulatory capital and also of the capital at risk that the member implicitly provides to the CCP through its default fund contribution. In the end the structure of the XVA equations for bilateral and cleared portfolios is similar, but the input data to these equations are not the same, reflecting different financial network structures. The resulting XVA numbers differ, but, interestingly enough, they become comparable after scaling by a suitable netting ratio.

We prove that the default times (or any of their minima) in the dynamic Gaussian copula model of Crépey, Jeanblanc, and Wu (2013) are invariance times in the sense of Crépey and Song (2017), with related invariance probability measures different from the pricing measure. This reflects a departure from the immersion property, whereby the default intensities of the surviving names and therefore the value of credit protection spike at default times. These properties are in line with the wrong-way risk feature of counterparty risk embedded in credit derivatives, i.e. the adverse dependence between the default risk of a counterparty and an underlying credit derivative exposure.

We prove that the default times (or any of their minima) in the dynamic Gaussian copula model of Crépey, Jeanblanc, and Wu (2013) are invariance times in the sense of Crépey and Song (2017), with related invariance probability measures different from the pricing measure. This reflects a departure from the immersion property, whereby the default intensities of the surviving names and therefore the value of credit protection spike at default times. These properties are in line with the wrong-way risk feature of counterparty risk embedded in credit derivatives, i.e. the adverse dependence between the default risk of a counterparty and an underlying credit derivative exposure.

The uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm is analyzed. In our setup, this limit $\targetfn$ is defined as a zero of an intractable function and is modeled as uncertain through a parameter $\param$. We aim at deriving the function $\targetfn$, as well as the probabilistic distribution of $\targetfn(\param)$ given a probability distribution $\pi$ for $\param$. We introduce the so-called Uncertainty Quantification for SA (UQSA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of $\param \mapsto \targetfn(\param)$ on an orthogonal basis of a suitable Hilbert space. UQSA, run with a finite number of iterations $K$, returns a finite set of coefficients, providing an approximation $\widehat{\targetfn_K}(\cdot)$ of $\targetfn(\cdot)$. We establish the almost-sure and $L^p$-convergences in the Hilbert space of the sequence of functions $\widehat{\targetfn_K}(\cdot)$ when the number of iterations $K$ tends to infinity. This is done under mild, tractable conditions, uncovered by the existing literature for convergence analysis of infinite dimensional SA algorithms. For a suitable choice of the Hilbert basis, the algorithm also provides an approximation of the expectation, of the variance-covariance matrix and of higher order moments of the quantity $\widehat{\targetfn_K}(\param)$ when $\param$ is random with distribution $\pi$. UQSA is illustrated and the role of its design parameters is discussed numerically.

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In incomplete markets, a basic Black-Scholes perspective has to be complemented by the valuation of market imperfections. Otherwise this results in Black-Scholes Ponzi schemes, such as the ones at the core of the last global financial crisis, where always more derivatives need to be issued for remunerating the capital attracted by the already opened positions. In this paper we consider the sustainable Black-Scholes equations that arise for a portfolio of options if one adds to their trade additive Black-Scholes price, on top of a nonlinear funding cost, the cost of remunerating at a hurdle rate the residual risk left by imperfect hedging. We assess the impact of model uncertainty in this setup.

On a probability space $(\Omega,\mathcal{A},\mathbb{Q})$ we consider two filtrations $\mathbb{F}\subset \mathbb{G}$ and a $\mathbb{G}$ stopping time $\theta$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$ predictable processes on $(0,\theta]$. In this setup it is well-known that, for any $\mathbb{F}$ semimartingale $X$, the process $X^{\theta-}$ ($X$ stopped ``right before $\theta$'') is a $\mathbb{G}$ semimartingale. Given a positive constant $T$, we call $\theta$ an invariance time if there exists a probability measure $\mathbb{P}$ equivalent to $\mathbb{Q}$ on $\mathcal{F}_T$ such that, for any $(\mathbb{F},\mathbb{P})$ local martingale $X$, $X^{\theta-}$ is a $(\mathbb{G},\mathbb{Q})$ local martingale. We characterize invariance times in terms of the $(\mathbb{F},\mathbb{Q})$ Az\'ema supermartingale of $\theta$ and we derive a mild and tractable invariance time sufficiency condition. We discuss invariance times in mathematical finance and BSDE applications.

This thesis addresses various issues related to collateral management in the context of centralized trading through clearing houses. First, we present the notions of capital cost and funding cost for a bank, by placing them in an elementary Black-Scholes framework where the payoff of a standard call takes the place of a counterparty default exposure. We assume that the bank only imperfectly hedges this call and faces a funding cost higher than the risk-free rate, hence the FVA and KVA pricing corrections with respect to the Black-Scholes price. We then focus on the costs that a bank faces when trading in a CCP. To this end, we transpose the XVA framework of bilateral trading to centralized trading. The total cost for a member to trade through a CCP is thus decomposed into a CVA corresponding to the cost for the member to replenish its contribution to the guarantee fund in case of losses due to defaults by other members, an MVA corresponding to the cost of financing its initial margin and a KVA corresponding to the cost of capital put at risk by the member in the form of its contribution to the guarantee fund. We then question the regulatory assumptions previously used, looking at alternatives in which members would use a third party for their initial margin, who would post the margin in the member's place in exchange for a fee. We also consider a method of calculating the guarantee fund and its allocation that takes into account the risk of the chamber in the sense of the fluctuations of its P&L over the following year, as it results from the combination of the market risk and the default risk of the members. Finally, we propose the application of multivariate risk measure methodologies for the calculation of members' margins and/or guarantee funds. We introduce a notion of systemic risk measures in the sense that they are sensitive not only to the marginal risks of the components of a financial system (e.g., but not necessarily the positions of the members of a CCP), but also to their dependence.

In finance, model risk is the risk of financial loss resulting from the use of models. It is a complex risk to apprehend and covers several very different situations, especially the estimation risk (a model generally uses an estimated parameter) and the model specification error risk (which consists in using an inadequate model). This thesis focuses on the quantification of model risk in the construction of rate or credit curves and on the study of the compatibility of Sobol indices with the theory of stochastic orders. It is divided into three chapters. Chapter 1 deals with the study of model risk in the construction of rate or credit curves. In particular, we analyze the uncertainty associated with the construction of rate or credit curves. In this context, we have obtained no-arbitrage bounds associated with implied default or rate curves that are perfectly compatible with the quotations of the associated reference products. In Chapter 2 of the thesis, we make the link between global sensitivity analysis and stochastic order theory. In particular, we analyze how the Sobol indexes transform following an increase in the uncertainty of a parameter in the sense of the stochastic dispersive order or excess wealth. Chapter 3 of the thesis focuses on the quantile contrast index. We first make the link between this index and the CTE risk measure, and then we analyze the extent to which an increase in the uncertainty of a parameter in the sense of stochastic dispersive order or excess wealth leads to an increase in the quantile contrast index. Finally, we propose a method for estimating this index. We show, under appropriate assumptions, that the estimator we propose is consistent and asymptotically normal.

In the aftermath of the 2007 global financial crisis, banks started reflecting into derivative pricing the cost of capital and collateral funding through XVA metrics. Here XVA is a catch-all acronym whereby X is replaced by a letter such as C for credit, D for debt, F for funding, K for capital and so on, and VA stands for valuation adjustment. This behaviour is at odds with economies where markets for contingent claims are complete, whereby trades clear at fair valuations and the costs for capital and collateral are both irrelevant to investment decisions. In this paper, we set forth a mathematical formalism for derivative portfolio management in incomplete markets for banks. A particular emphasis is given to the problem of finding optimal strategies for retained earnings which ensure a sustainable dividend policy.

In Crépey [9], a basic reduced-form counterparty risk modelling approach was introduced under a standard immersion hypothesis between a reference filtration and the filtration progressively enlarged by the default times of the two parties. This basic setup, with a related continuity assumption on some of the data at the first default time of the two parties, is too restrictive for wrong-way and gap risk applications, such as counterparty risk on credit derivatives. This paper introduces an extension of the basic approach, implements it through marked default times and applies it to counterparty risk on credit derivatives.

We consider the problem of valuation of interest rate derivatives in the post-crisis setup. We develop a multiple-curve model, set in the HJM framework and driven by a L ́evy process. We proceed with joint calibration to OTM swaptions and co-terminal ATM swaptions of different tenors, the calibration to OTM swaptions guaranteeing that the model correctly captures volatility smile effects and the calibration to co-terminal ATM swaptions ensuring an appropriate term structure of the volatility in the model. To account for counterparty risk and funding issues, we use the calibrated multiple- curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counterparty risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration we study the case of a basis swap and a related swaption, for which we compute the counterparty risk and funding adjustments.

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This thesis deals with two areas of stochastic analysis and financial mathematics: the Malliavin calculus for Markov chains (Part I) and counterparty risk (Part II). Part I aims at studying the Malliavin calculus for Markov chains in continuous time. Two points are presented: proving the existence of the density for the solutions of a stochastic differential equation and computing the sensitivities of derivatives. Part II deals with current topics in the field of market risk, namely XVA (price adjustments) and multi-curve modeling.

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This thesis studies some stochastic control problems in a context of liquidity risk and impact on asset prices. The thesis is composed of four chapters.In the second chapter, we propose a modeling of a market making problem in a liquidity risk context in the presence of inventory constraints and regime shifts. This formulation can be considered as an extension of previous studies on this subject. The main result of this part is the characterization of the value function as a unique solution, in the sense of viscosity, of a system of Hamilton-Jacobi-Bellman equations . In the third chapter, we propose a numerical approximation scheme to solve a portfolio optimization problem in a context of liquidity risk and impact on asset prices. We show that the value function can be obtained as a limit of an iterative procedure where each iteration represents an optimal stopping problem and we use a numerical algorithm, based on optimal quantization, to compute the value function and the control policy. The convergence of the numerical scheme is obtained via monotonicity, stability and consistency criteria.In the fourth chapter, we focus on a coupled problem of singular control and impulse control in an illiquidity context. We propose a mathematical formulation to model the dividend distribution and the investment policy of a firm subject to liquidity constraints. We show that, under transaction costs and an impact on the price of illiquid assets, the firm's value function is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. An iterative numerical method is also proposed to compute the optimal buy, sell and dividend strategy.

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This thesis deals with several topics in financial mathematics: credit risk, portfolio optimization and interest rate modeling. Chapter 1 consists of three studies in the area of credit risk. The most innovative is the first one in which we build a model such that the immersion property is not verified under any equivalent martingale measure. Chapter 2 studies the problem of maximizing the sum of a terminal wealth utility and a consumption utility. Chapter 3 studies the valuation of interest rate derivatives in a multi-curve framework, which takes into account the difference between a risk-free rate curve and Libor rate curves of different tenors.

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The 2007 subprime crisis has induced a persistent disconnection between the Libor derivative markets of different tenors and the OIS market. Commonly proposed explanations for the corresponding spreads are a combination of credit risk and liquidity risk. However in the literature the meaning of liquidity is either not precisely stated, or it is simply defined as a residual spread after removal of a credit component. In this paper we propose a stylized equilibrium model in which a Libor-OIS spread (LOIS) emerges as a consequence of a credit component determined by the skew of the CDS curve of a representative Libor panelist (playing the role of the “borrower” in an interbank loan) and a liquidity component corresponding to a volatility of the spread between the refinancing (or funding) rate of a representative Libor panelist (playing the role of the “lender”) and the overnight interbank rate. The credit component is thus in fact a credit skew component, whilst the relevant notion of liquidity appears as the optionality, valued by the aforementioned volatility, of dynamically adjusting through time the amount of a rolling overnight loan, as opposed to lending a fixed amount up to the tenor horizon on Libor. “At-the-money” when the funding rate of the lender and the overnight interbank rate match on average, this results, under diffusive features, in a square root term structure of the LOIS, with a square root coefficient given by the above-mentioned volatility. Empirical observations reveal a square root term structure of the LOIS consistent with this theoretical analysis, with, on the EUR market studied in this paper on the period half-2007 half-2012, LOIS explained in a balanced way by credit and liquidity until the beginning of 2009 and dominantly explained by liquidity since then.

We study a BSDE with random terminal time that appears in the modeling of coun-terparty risk in finance. We proceed by reduction of the original BSDE into a simpler BSDE posed with respect to a smaller filtration and a changed probability measure. We relax the basic immersion conditions of the classical reduced-form modeling approach in credit risk by modeling the default time as an invariant time, i.e. a time such that local martingales with respect to a reduced filtration and a possibly changed probability measure, once stopped right before that time, stay local martingales with respect to the original model filtration and probability measure. Using an Azéma supermartingale characterization of invariant times, we establish the equivalence between the original and the reduced BSDE.

From a broad perspective, this work deals with the question of reduction of filtration, i.e., given a stopping time θ relative to a full model filtration G, when and how to separate the information that comes from θ from a reference filtration in order to simplify the computations. Toward this aim, some kind of local martingale invariance property is required, but under minimal assumptions, so that the model stays as flexible as possible in view of applications (to, in particular, counterparty and credit risk). Specifically, we define an invariant time as a stopping time with respect to the full model filtration such that local martingales with respect to a smaller filtration and a possibly changed probability measure, once stopped right before that time, are local martingales with respect to the original model filtration and probability measure. The possibility to change the measure provides an additional degree of freedom with respect to other classes of random times such as Cox times or pseudo-stopping times that are commonly used to model default times. We provide an Azéma supermartingale characterization of invariant times and we characterize the positivity of the stochastic exponential involved in a tentative measure change. We study the avoidance properties of invariant times and their connections with pseudo-stopping times.

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A basic reduced-form counterparty risk modeling approach hinges on a standard immersion hypothesis between a reference filtration and the filtration progressively enlarged by the default times of the two parties, also involving the continuity of some of the data at default time. This basic approach is too restrictive for application to credit derivatives, which are characterized by strong wrong-way risk, i.e. adverse dependence between the exposure and the credit riskiness of the counterparties, and gap risk, i.e. slippage between the portfolio and its collateral during the so called cure period that separates default from liquidation. This paper shows how a suitable extension of the basic approach can be devised so that it can be applied in dynamic copula models of counterparty risk on credit derivatives. More generally, this method is applicable in any marked default times intensity setup satisfying a suitable integrability condition. The integrability condition expresses that no mass is lost in a related measure change. The changed probability measure is not needed algorithmically. All one needs in practice is an explicit expression for the intensities of the marked default times.

We consider the problem of valuation of interest rate derivatives in the post-crisis set-up. We develop a multiple-curve model, set in the HJM framework and driven by a Levy process. We proceed with joint calibration to OTM swaptions and co-terminal ATM swaptions of different tenors, the calibration to OTM swaptions guaranteeing that the model correctly captures volatility smile effects and the calibration to co-terminal ATM swaptions ensuring an appropriate term structure of the volatility in the model. To account for counterparty risk and funding issues, we use the calibrated multiple-curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counterparty risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration, we study the case of a basis swap and a related swaption, for which we compute the counterparty risk and funding adjustments.

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We devise a bottom-up dynamic model of portfolio credit risk where instantaneous contagion is represented by the possibility of simultaneous defaults. Due to a Markovian copula nature of the model, calibration of marginals and dependence parameters can be performed separately using a two-step procedure, much like in a standard static copula setup. In this sense this solves the bottom-up top-down puzzle which the CDO industry had been trying to do for a long time. This model can be used for any dynamic portfolio credit risk issue, such as dynamic hedging of CDOs by CDSs, or CVA computations on credit portfolios.

In this paper, we prove that the conditional dependence structure of default times in the Markov model of "A Bottom-Up Dynamic Model of Portfolio Credit Risk. Part I: Markov Copula Perspective" belongs to the class of Marshall-Olkin copulas. This allows us to derive a factor representation in terms of "common-shocks", the latter being able to trigger simultaneous defaults in some prespecified groups of obligors. This representation depends on the current default state of the credit portfolio so that fast convolution pricing schemes can be exploited for pricing and hedging credit portfolio derivatives. As emphasized in "A Bottom-Up Dynamic Model of Portfolio Credit Risk: Part I: Markov Copula Perspective," the innovative breakthrough of this dynamic bottom-up model is a suitable decoupling property between the dependence structure and the default marginals as in "Dynamic Modeling of Dependence in Finance via Copulae Between Stochastic Processes" (like in static copula models but here in a full-flesh dynamic "Markov copula" model). Given the fast deterministic pricing schemes of the present paper, the model can then be jointly calibrated to single-name and portfolio data in two steps, as opposed to a global joint optimization procedures involving all the model parameters at the same time which would be untractable numerically. We illustrate this numerically by results of calibration against market data from CDO tranches as well as individual CDS spreads. We also discuss hedging sensitivities computed in the models thus calibrated.

Since the 2007 subprime crisis, OIS and Libor markets (Eonia and Euribor in the EUR market) diverged suddenly (See Fig.1 and 2). In this note we show how, by optimizing their lending between Libor and OIS markets, banks are led to apply a spread (LOIS) over the OIS rate when lending at Libor.

The spread between Libor and overnight index swap rates used to be negligible – until the crisis. Its behaviour since can be explained theoretically and empirically by a model driven by typical lenders’ liquidity and typical borrowers’ credit risk.

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This thesis deals with density models for default times and their application to credit and counterparty risk. The first part is a theoretical contribution to the study of projections on different filtrations of the Radon-Nikodym density, in the form of Doleans-Dade exponential, occurring during measurement changes. The main result is the characterization of the measurement changes that preserve the immersion, obtained by applying our projection formulas. The second part aims at an informational dynamization of the static Gaussian copula model applied to a credit portfolio, which can be seen as a density model allowing to deal with CDO hedging by CDS or counterparty risk on credit derivatives. The main contributions are the introduction of the dynamic perspective, which gives a theoretical justification to the Gaussian copula bump-sensitivities used by practitioners, and the application to CVA calculations on a CDS.

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This thesis deals with the modeling of credit derivatives and consists of two parts: The first part concerns the density model, recently proposed by El Karoui et al. where we make the assumption that the conditional law of default time knowing reference filtration is equivalent to its (unconditional) law. Under this assumption, we give different (and simpler) proofs to the already existing results in the theory of initial and progressive coarsening of filtrations. In addition, we present new results such as the predictable representation theorem for progressively coarsened filtration in the multidimensional case. We then propose several methods for constructing density models in both the one-dimensional and multi-dimensional cases. Finally, we show that the density model is an efficient approach for dynamic hedging of multi-name credit derivatives. In the second part, in order to study the counterparty risk in a CDS contract, we propose a Markov model in which simultaneous defaults are possible. The wrong-way risk is thus represented by the fact that, at the time of the counterparty default, there is a strictly positive probability that the reference entity will also default. We begin by considering a Markov chain with four states corresponding to two names. In this simple case, we obtain semi-explicit formulas for most important quantities, such as price, CVA, EPE, or hedge ratios. We then generalize this framework to account for spread risk by introducing stochastic factors. We treat a Markovian copula model with stochastic intensities. We also address the issue of dynamic CVA hedging with a written CDS on the counterparty. For the implementation of the model, we specify the intensities by affine processes, which given the dynamic copula property of the model, makes the calibration of this model efficient. Numerical results are presented to show the relevance of the CVA behavior in the model with the stylized market facts.

This thesis deals with the optimization of asset portfolios subject to default risk. The current crisis has allowed us to understand that it is important to take into account the risk of default to be able to give the real value of its portfolio. Indeed, due to the different exchanges of the financial market actors, the financial system has become a network of several connections which it is essential to identify in order to evaluate the risk of investing in a financial asset. In this thesis, we define a financial system with a finite number of connections and we propose a model of the dynamics of an asset in such a system by taking into account the connections between the different assets. The measurement of the correlation will be done through the jump intensity of the processes. Using Stochastic Differential Backward Equations (SDGE), we will derive the price of a contingent asset and take into account the model risk in order to better evaluate the optimal consumption and wealth if one invests in such a market.

This thesis consists of five independent parts dedicated to the modeling and study of the problems related to the risk of default, in partial information. The first part constitutes the Introduction. The second part is dedicated to the calculation of the survival probability of a firm, conditional on the information available to the investor, in a structural model with partial information. We use a hybrid numerical technique based on the Monte Carlo method and optimal quantization. In the third part we treat, with the Dynamic Programming approach, a discrete time problem of maximizing the utility of terminal wealth, in a market where securities subject to default risk are traded. The risk of contagion between defaults is modeled, as well as the possible uncertainty of the model. In the fourth part, we address the problem of uncertainty related to the investment time horizon. In a complete market subject to the risk of default, we solve, either with the martingale method or with Dynamic Programming, three problems of maximizing the utility of consumption: when the time horizon is fixed, finite but uncertain and infinite. Finally, in the fifth part we deal with a purely theoretical problem. In the context of the coarsening of filtrations, our goal is to re-demonstrate, in a specific framework, the already known results on the characterization of martingales, the decomposition of martingales with respect to the reference filtration as semimartingales in progressively and initially coarsened filtrations and the Predictable Representation Theorem.

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This thesis presents a combination of partial differential equation and probability techniques for various applied mathematical problems. Regularization methods are used to approach these problems numerically. The first part, differential games, aims at the numerical determination of the (discontinuous) barriers of the evasion pursuit games, via the introduction of auxiliary games in minimum distance. We propose a numerical scheme on a destructured mesh for the solutions of such games. The numerical approach is then validated on an analytically solved game. Finally, we establish a convergence result by probabilistic methods a la Kushner for differential game problems. The second part, financial mathematics, deals with numerical methods by partial differential equations in finance - direct methods, including the study of transparent edge conditions or of an American put on arithmetic mean, and inverse methods, with the study of a problem of calibration of the dynamics of an underlying asset from the prices of derivative products observed on the markets. If the pure calibration problem is underdetermined, it becomes on the other hand well posed by harmonic regularization, in a tikhonov type approach, as it follows from qualitative properties of call prices and bounds on their sensitivities in a black-scholes model with local volatility.