COUSIN Areski

The objective of this thesis is to study certain problems related to credit risk. These issues are divided into two main themes, namely the monotonicity of transition matrices, and the modeling of interdependence in credit risk. The first theme is motivated by the idealization of empirical transition matrices practiced by banks. We propose in this thesis an optimal solution that allows to approximate an empirical matrix by a monotonic matrix and thus to realize an idealization of the whole matrix. We also prove some theoretical results on the stability of monotonicity under two types of transformations. The second theme of the thesis concerns interdependence in credit risk in general, and we study contagion and its propagation as a special case. The idea is to see a credit portfolio as a network whose nodes are the entities of the portfolio connected via links. We therefore build a graphical Markov field model able to take into account both exogenous factors and interactions between entities. Under the formalism of this model, we study several aspects of interdependence in credit risk, including the occurrence of critical phenomena, the effect of network topology on risk factors and risk propagation. We have been able to make theoretical contributions on these topics by proving theorems and properties that are quite interesting and allow us to predict the behavior of the portfolio under certain conditions. On the other hand, we also propose several ways to solve the computational and calibration problems that made this type of model difficult to use in practice.

Analyzing the effect of business cycle on rating transitions has been a subject of great interest these last fifteen years, particularly due to the increasing pressure coming from regulators for stress testing. In this paper, we consider that the dynamics of rating migrations is governed by an unobserved latent factor. Under a point process filtering framework, we explain how the current state of the hidden factor can be efficiently inferred from observations of rating histories. We then adapt the classical Baum-Welsh algorithm to our setting and show how to estimate the latent factor parameters. Once calibrated, we may reveal and detect economic changes affecting the dynamics of rating migration, in real-time. To this end we adapt a filtering formula which can then be used for predicting future transition probabilities according to economic regimes without using any external covariates. We propose two filtering frameworks: a discrete and a continuous version. We demonstrate and compare the efficiency of both approaches on fictive data and on a corporate credit rating database. The methods could also be applied to retail credit loans.

This paper investigates the optimal asset allocation of a financial institution whose customers are free to withdraw their capital-guaranteed financial contracts at any time. Accounting for asset-liability mismatch risk of the institution, we present a general utility optimization problem in discrete time setting and provide a dynamic programming principle for the optimal investment strategies. Furthermore, we consider an explicit context, including liquidity risk, interest rate and credit intensity fluctuations, and show, by numerical results, that the optimal strategy improves the solvency and the asset returns of the institution compared to the baseline asset allocation.

Implied volatility surface is of crucial interest for risk management and exotic option pricing models. Its construction is usually carried out in accordance with the arbitrage-free principle. This condition leads to shape restrictions on the option prices such as monotonicity with respect to maturities and convexity with respect to strike prices. In this paper, we propose a new arbitrage-free construction method that extends classical spline techniques by additionally allowing for quantification of uncertainty. The proposed method extends the constrained kriging techniques developed in [MB16] and [CMR16] to the context of volatility surface construction. Assuming a Gaussian process prior, the posterior price surface becomes a truncated Gaussian field given shape constraints and market observations. Prices of illiquid instruments can also be incorporated when considered as noisy observations. Starting from a suitable finite-dimensional approximation of the Gaussian process prior, the no-arbitrage condition on the entire input domain is characterized by a finite number of linear inequality constraints. We define the most likely response surface and the most-likely noise values as the solution of a quadratic optimization problem. We use Hamiltonian Monte Carlo technics to simulate the posterior truncated Gaussian surface and build pointwise confidence bands. The Gaussian process hyper-parameters are estimated using maximum likelihood. The method is illustrated on Euro Stoxx 50 option prices by building no-arbitrage volatility surfaces and their corresponding confidence bands.

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In the past decade, Sobol's variance decomposition have been used as a tool - among others - in risk management. We show some links between global sensitivity analysis and stochastic ordering theories. This gives an argument in favor of using Sobol's indices in uncertainty quantification, as one indicator among others.

In this paper we propose a new methodology for solving an uncertain stochastic Marko-vian control problem in discrete time. We call the proposed methodology the adaptive robust control. We demonstrate that the uncertain control problem under consideration can be solved in terms of associated adaptive robust Bellman equation. The success of our approach is to the great extend owed to the recursive methodology for construction of relevant confidence regions. We illustrate our methodology by considering an optimal portfolio allocation problem, and we compare results obtained using the adaptive robust control method with some other existing methods.

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This doctoral thesis starts from the observation that a credit portfolio is subject to several risks that come mainly from the credit quality of the borrower and its drawing and prepayment behavior on its credit lines. It turns out that the observed risks are dynamic and depend on various factors, both micro and macro-economic. We wanted to understand the articulation of these risks to have an effective management of them in the present, but also a prospective vision if the economic conditions change, for a proactive management. To address this issue, we have structured our research around three axes that have resulted in three chapters in the form of articles.(i) Analysis of changes in credit ratings as a function of risk factors.The use of multi-factor migration models has allowed us to reproduce stylized facts cited in the literature and to identify others. We also reconstruct the business cycle between 2006 and 2014 which manages to capture the 2008 and 2012 crises.(ii) Design of a cash flow model that accounts for the evolution of borrowers' behaviors under the influence of their micro and macroeconomic environments.We prove the influence of credit rating, business cycle, estimated recovery rate and short term interest rate on utilization rates. This model also allows us to obtain risk measures such as Cash Flow-at-Risk and Stressed Cash Flow-at-Risk on credit portfolios through Monte Carlo simulations.(iii) Reflecting on the Disposition-to-Pay (DTP) of an ambiguity-neutral decision maker to reduce risk in the presence of uncertainty on probabilities. We show that the presence of multiple (possibly correlated) sources of ambiguity changes the welfare of a risk-averse decision maker even though the decision maker is ambiguity neutral.

The ORSA Own Risk Solvency and Assessment is a set of rules defined by the European Solvency II directive. It is intended to serve as a decision support tool and strategic risk analysis. In the context of the ORSA, insurance companies must assess their future solvency, on an ongoing and prospective basis. In order to do so, they must obtain projections of their balance sheet (assets and liabilities) over a certain time horizon. In this thesis, we focus mainly on the aspect of predicting future asset values. More precisely, we deal with the yield curve, its construction and extrapolation at a given date, and its predictions envisaged in the future. In the text, we refer to the "yield curve", but it is in fact the construction of discount factor curves. Counterparty default risk is not explicitly addressed, but techniques similar to those developed can be adapted to the construction of rate curves incorporating counterparty default risk.

This thesis deals with the study of stochastic order book modeling, and two stochastic control problems in a context of liquidity risk and impact on asset prices. The thesis is composed of two distinct parts.In the first part, we treat, under different aspects, a Markovian order book model.In particular, in chapter 2, we introduce a cumulative depth representation model. We consider different types of event arrivals with a dependence on the current state.Chapter 3 deals with the model stability problem through a semi-martingale approach for the classification of a countable Markov chain. We give, for each classification problem, a calibration of the model from empirical facts such as the average order book density profile.Chapter 4 is devoted to the estimation and calibration of our model from market data streams. Thus, we compare our model and the data using stylized facts and empirical facts. We give a concrete calibration to the different classification problems.Then, in chapter 5, we treat the optimal liquidation problem within the framework of the cumulative depth representation model.In the second part, we propose a modeling of an investor's optimal liquidation problem with stochastic resilience. We reduce the problem to a singular stochastic control problem. We show that the associated value function is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. Moreover, we use an iterative numerical method to compute the optimal strategy. The convergence of this numerical scheme is obtained via monotonicity, stability and consistency criteria.

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In this paper we propose a new methodology for solving an uncertain stochastic Marko-vian control problem in discrete time. We call the proposed methodology the adaptive robust control. We demonstrate that the uncertain control problem under consideration can be solved in terms of associated adaptive robust Bellman equation. The success of our approach is to the great extend owed to the recursive methodology for construction of relevant confidence regions. We illustrate our methodology by considering an optimal portfolio allocation problem, and we compare results obtained using the adaptive robust control method with some other existing methods.

No summary available.

Due to the lack of reliable market information, building financial term-structures may be associated with a significant degree of uncertainty. In this paper, we propose a new term-structure interpolation method that extends classical spline techniques by additionally allowing for quantification of uncertainty. The proposed method is based on a generalization of kriging models with linear equality constraints (market-fit conditions) and shape-preserving conditions such as monotonicity or positivity (no-arbitrage conditions). We define the most likely curve and show how to build confidence bands. The Gaussian process covariance hyper-parameters under the construction constraints are estimated using cross-validation techniques. Based on observed market quotes at different dates, we demonstrate the efficiency of the method by building curves together with confidence intervals for term-structures of OIS discount rates, of zero-coupon swaps rates and of CDS implied default probabilities. We also show how to construct interest-rate surfaces or default probability surfaces by considering time (quotation dates) as an additional dimension.

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.

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The construction of term structures is at the heart of financial evaluation and risk management see e.g. [1], [2], [3], [4] and [5]. A term structure is a curve that describes the evolution of an economic or financial quantity as a function of the maturity or time horizon. Typical examples are the term structure of risk-free interest rates, the term structure of bonds, the term structure of default probabilities and the term structure of implied volatilities of financial asset returns. In practice, market quotes of the underlying financial products are used and provide partial information on the term structures considered. Moreover, this information is more or less reliable depending on the liquidity of the maturity of the markets in question. The goal is to obtain a continuous maturity curve from this information.

In this paper, we study the statistical estimation of some factor migration models. This class of models is based on the assumption that rating migrations are driven by a set of common factors representing the business cycle evolution. In particular, we compare the estimation of the ordered Probit model as described for instance in Gagliardini and Gourieroux (2005) and of the multi-state latent factor intensity model used in Koopman et al. (2008). For these two approaches, we also distinguish the case where the underlying factors are observable and the case where they are assumed to be unobservable. The paper is supplied with an empirical study where the estimation is made on historical Standard & Poor's rating data on the period [01/2006 − 01/2014]. We find that the intensity model with observable factors is the one that best fits empirical transition probabilities. In line with Kavvathas (2001), this study shows that short migrations of investment grade firms are significantly correlated to the business cycle whereas, because of lack of observations, it is not possible to state any relation between long migrations (more than two grades) and the business cycle. Concerning non investment grade firms, downgrade migrations are negatively related to business cycle whatever the amplitude of the migration.

In this paper, we study the statistical estimation of some factor credit migration models, that is, multivariate migration models for which the transition matrix of each obligor is driven by the same dynamic factors. In particular, we compare the statistical estimation of the ordered Probit model as described for instance in Gagliardini and Gourieroux (2005) and of the multi-state latent factor intensity model used in Koopman et al. (2008). For these two approaches, we also distinguish the case where the underlying factors are observable and the case where they are assumed to be unobservable. The paper is supplied with an empirical study where the estimation is made on a set of historical Standard & Poor’s rating data on the period [01/2006 − 01/2014]. We find that the intensity model with observable factors is the one that has the best fit with respect to empirical transition probabilities. In line with Kavvathas (2001), this study shows that short migrations of investment grade firms are significantly correlated to the business cycle whereas, because of lack of observations, it is not possible to state any relation between long migrations (more than two grades) and the business cycle. Concerning non-investment grade firms, downgrade migrations are negatively related to business cycle whatever the amplitude of the migration.

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In this paper, we introduce two alternative extensions of the classical univariate Conditional-Tail-Expectation (CTE) in a multivariate setting. Contrary to allocation measures or systemic risk measures, these measures are also suitable for multivariate risk problems where risks are heterogenous in nature and cannot be aggregated together.

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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.

In this paper, we analyze the diversity of term structure functions (e.g., yield curves, swap curves, credit curves) constructed in a process which complies with some admissible properties: arbitrage-freeness, ability to fit market quotes and a certain degree of smoothness. When present values of building instruments are expressed as linear combinations of some primary quantities such as zero-coupon bonds, discount factor, or survival probabilities, arbitrage-free bounds can be derived for those quantities at the most liquid maturities. As a matter of example, we present an iterative procedure that allows to compute model-free bounds for OIS-implied discount rates and CDS-implied default probabilities. We then show how mean-reverting term structure models can be used as generators of admissible curves. This framework is based on a particular specification of the mean-reverting level which allows to perfectly reproduce market quotes of standard vanilla interest-rate and default-risky securities while preserving a certain degree of smoothness. The numerical results suggest that, for both OIS discounting curves and CDS credit curves, the operational task of term-structure construction may be associated with a significant degree of uncertainty.

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In this paper, we introduce two alternative extensions of the classical univariate Conditional-Tail-Expectation (CTE) in a multivariate setting. Contrary to allocation measures or systemic risk measures, these measures are also suitable for multivariate risk problems where risks are heterogenous in nature and cannot be aggregated together.

In this paper, we introduce two alternative extensions of the classical univariate Value-at-Risk (VaR) in a multivariate setting. The two proposed multivariate VaR are vector-valued measures with the same dimension as the underlying risk portfolio. The lower-orthant VaR is constructed from level sets of multivariate distribution functions whereas the upper-orthant VaR is constructed from level sets of multivariate survival functions. Several properties have been derived. In particular, we show that these risk measures both satisfy the positive homogeneity and the translation invariance property. Comparison between univariate risk measures and components of multivariate VaR are provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Illustrations are given in the class of Archimedean copulas.

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.

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The present paper provides a multi-period contagion model in the credit risk field. Our model is an extension of Davis and Lo's infectious default model. We consider an economy of n firms which may default directly or may be infected by other defaulting firms (a domino effect being also possible). The spontaneous default without external influence and the infections are described by not necessarily independent Bernoulli-type random variables. Moreover, several contaminations could be required to infect another firm. In this paper we compute the probability distribution function of the total number of defaults in a dependency context. We also give a simple recursive algorithm to compute this distribution in an exchangeability context. Numerical applications illustrate the impact of exchangeability among direct defaults and among contaminations, on different indicators calculated from the law of the total number of defaults. We then examine the calibration of the model on iTraxx data before and during the crisis. The dynamic feature together with the contagion effect seem to have a significant impact on the model performance, especially during the recent distressed period.

In this paper, we introduce two alternative extensions of the classical univariate Value-at-Risk (VaR) in a multivariate setting. The two proposed multivariate VaR are vector-valued measures with the same dimension as the underlying risk portfolio. The lower-orthant VaR is constructed from level sets of multivariate distribution functions whereas the upper-orthant VaR is constructed from level sets of multivariate survival functions. Several properties have been derived. In particular, we show that these risk measures both satisfy the positive homogeneity and the translation invariance property. Comparison between univariate risk measures and components of multivariate VaR are provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Illustrations are given in the class of Archimedean copulas.

The purpose of this thesis is to shed light on some aspects of risk management of synthetic CDO tranches. The first part deals with the risk analysis of CDO tranches in the class of factor models which includes a large number of popular approaches such as copula-based models, multivariate structural models, multivariate Poisson models or affine intensity models. Moreover, when considering a homogeneous credit portfolio, the assumption of a dependence structure based on a factor representation is no longer restrictive thanks to De Finetti's theorem. The law of conditional probability of default plays an important role in the evaluation of CDO tranches but also in the risk analysis of credit portfolios. Indeed, it is possible to compare different factor models by simply comparing their conditional probability of default. The second part addresses the hedging problem in the context of Markov contagion models for which the prices of assets contingent on default risk can be perfectly replicated under the assumption of no simultaneous defaults. Moreover, when the portfolio is homogeneous and when default intensities depend only on the current state of the number of defaults, the aggregate loss process is simply a Markov chain. In this case, it is possible to calibrate the aggregate loss intensities on a loss distribution and to efficiently compute dynamic hedging strategies using a recombining binomial tree in which the payoff of the CDO tranches can be perfectly duplicated thanks to the index and the risk-free asset.