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2017 | Book

Introduction and Context Read chapter Read first chapter

Credit Correlation

Theory and Practice

Author: Youssef Elouerkhaoui

Publisher: Springer International Publishing

Book Series : Applied Quantitative Finance

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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About this book

This book provides an advanced guide to correlation modelling for credit portfolios, providing both theoretical underpinnings and practical implementation guidance. The book picks up where pre-crisis credit books left off, offering guidance for quants on the latest tools and techniques for credit portfolio modelling in the presence of CVA (Credit Value Adjustments). Written at an advanced level, it assumes that readers are familiar with the fundamentals of credit modelling covered, for example, in the market leading books by Schonbucher (2003) and O’Kane (2008). Coverage will include the latest default correlation approaches; correlation modelling in the ‘Marshall-Olkin’ contagion framework, in the context of CVA; numerical implementation; and pricing, calibration and risk challenges.

The explosive growth of credit derivatives markets in the early-to-mid 000’s was bought to a close by the 2007 financial crisis, where these instruments were held largely to blame for the economic downturn. However, in the wake of increased regulation across all financial instruments and the challenge of buying and selling bonds in large amounts, credit derivatives have once again been found to be the answer and the market has grown significantly.

Written by a practitioner for practitioners, this book will also interest researchers in mathematical finance who want to understand how things happen and work ‘on the floor’. Building the reader’s knowledge from the ground up, and with numerous real life examples used throughout, this book will prove a popular reference for anyone with a mathematical mind interested credit markets.

Table of Contents

Frontmatter
Chapter 1. Introduction and Context
Abstract
To set the context, we start this introduction with a presentation of the main (portfolio) credit derivative contracts that we are interested in. When we talk about portfolio credit derivative valuations, the first thing that we need to do is to generate a set of loss (or default) distributions, at different time horizons, from the single-name curves and some “correlation” assumptions.
Youssef Elouerkhaoui

Theoretical Tools

Frontmatter
Chapter 2. Mathematical Fundamentals
Abstract
In this chapter, we present the essential mathematical tools needed in the modelling of portfolio credit derivative products. This includes: doubly-stochastic Poisson processes, also known as Cox processes; point processes and their intensities, on some given filtration; and copula functions.
Youssef Elouerkhaoui
Chapter 3. Expectations in the Enlarged Filtration
Abstract
In this chapter, we derive a formula of the conditional expectation with respect to the enlarged filtration. This is a generalization of the Dellacherie formula. We shall use this key result to compute the expectations that we encounter in the conditional jump diffusion framework.
Youssef Elouerkhaoui
Chapter 4. Copulas and Conditional Jump Diffusions
Abstract
Enlarging the economic state-variables’ filtration by observing the default process of all available credits has some profound implications on the dynamics of intensities.
Youssef Elouerkhaoui

Correlation Models: Practical Implementation

Frontmatter
Chapter 5. Correlation Demystified: A General Overview
Abstract
This chapter gives a broad overview of default correlation modelling in the context of pricing and risk managing a correlation trading book. We cover both theoretical and practical market aspects, as well as numerical performance issues.
Youssef Elouerkhaoui
Chapter 6. Correlation Skew: A Black-Scholes Approach
Abstract
In this chapter, we view the valuation of CDO tranches as an option pricing problem. The payoff of a CDO tranche is a call-spread on the loss variable. By specifying the distribution of the loss variable at each time horizon, one would be able to value tranches. The standard way of defining this distribution is the base correlation approach. Here, we use a Black-Scholes analogy and we define an implied volatility for each tranche. Then, given a Black volatility surface, we parameterize the loss distribution with a Stochastic CEV model. We show that this parametric form gives a very good fit to the market tranche quotes. In addition, we give an application of the correlation skew Black approach to risk management and hedging.
Youssef Elouerkhaoui
Chapter 7. An Introduction to the Marshall-Olkin Copula
Abstract
In this chapter, we present the Marshall-Olkin copula model where the correlation profile is constructed via a set of common shocks, which can trigger joint defaults in the basket.
Youssef Elouerkhaoui
Chapter 8. Numerical Tools: Basket Expansions
Abstract
In the next few chapters, we study some efficient numerical methods for the valuation of large basket credit derivatives. While the approaches are presented in the Marshall-Olkin copula model, most of the numerical techniques are generic and could be used with other copulas as well. The methods presented span a large spectrum of applied mathematics: Fourier transforms, changes of probability measure, numerical stable schemes, high-dimensional Sobol integration, recursive convolution algorithms.
Youssef Elouerkhaoui
Chapter 9. Static Replication
Abstract
In principle, the direct (pricing) method requires, for each time step, \(2^{n+1}\) values, corresponding to the set of all possible default combinations; as the size of the underlying basket increases, the number of default configurations explodes exponentially. This significant limitation restricts the applicability of the method to baskets under 10 or 11 credits. As an alternative, we develop a different approach, which is based on a static replication idea. In this chapter, we describe how this static FTD replication is done: first, we show the relationship between kth-to-default and \((k-1)\)th-to-default swaps; then, we apply this recursion step-by-step until we arrive at the complete FTD expansion.
Youssef Elouerkhaoui
Chapter 10. The Homogeneous Transformation
Abstract
In general, the number of sub-FTDs in the replication formula is a function of n, the size of the basket, and k, the order of the basket default swap. The most time-consuming step in the evaluation is the generation of the sub-FTDs, for all possible combinations. If we had a homogeneous basket, then, for a given subset size l, all the FTD instruments would have exactly the same value; and the pricing equation would simplify substantially. In particular, the number of sub-FTDs to compute, would reduce to one evaluation per l-subset, hence a total of \(N\left( k,n\right) =k\) FTD evaluations for the whole \(k{\text {th}}\)-to-default swap. The first (natural) approximation that we consider is to transform the original non-homogeneous basket to a homogeneous one while preserving some properties of the aggregate default distribution. In the approach described here, for each default order, we use the corresponding percentile of the aggregate default distribution, and we require that this quantity remains invariant with respect to the homogeneous approximation. We shall see that this transformation is exact for an FTD swap, and that, for higher-order defaults, the approximation gives very good results.
Youssef Elouerkhaoui
Chapter 11. The Asymptotic Homogeneous Expansion
Abstract
The transformation, described in the previous chapter, produces a homogeneous portfolio, which mimics some properties of the aggregate default distribution, and can be used pari-pasu for the purposes of basket default swap valuation. By using this homogeneous portfolio, the numerical burden that comes with the pricing of large baskets is eased, and the valuation algorithm is significantly speeded up.
Youssef Elouerkhaoui
Chapter 12. The Asymptotic Expansion
Abstract
In this chapter, we relax the homogeneous portfolio assumption, and we derive an asymptotic series expansion of the \(k{\text {th}}\)-to-default Q-factor in the non-homogeneous case. We also show how to compute the conditional aggregate default distributions that appear in the expansion using the convolution recursion algorithm.
Youssef Elouerkhaoui
Chapter 13. CDO-Squared: Correlation of Correlation
Abstract
In this chapter, we analyze the “correlation of correlation” risk in the Marshall-Olkin copula framework. The valuation of higher-order correlation products such as “CDOs of CDOs” (also known as “CDO-Squared”) is mainly driven by correlation of correlation effects. First, We extend the first-to-default replication method to baskets of basket products.
Youssef Elouerkhaoui
Chapter 14. Second Generation Models: From Flat to Correlation Skew
Abstract
In this chapter, we review some popular correlation skew models. We give a brief description of each model and discuss the advantages and limitations of each modelling framework.
Youssef Elouerkhaoui
Chapter 15. Third Generation Models: From Static to Dynamic Models
Abstract
In this chapter, we review some of the most important dynamic credit models in the literature. We give a brief description of each model and discuss the advantages and limitations of each modelling framework. We also comment on the usefulness of each model for a given family of correlation products.
Youssef Elouerkhaoui

Advanced Topics in Pricing and Risk Management

Frontmatter
Chapter 16. Pricing Path-Dependent Credit Products
Abstract
This chapter addresses the problem of pricing (soft) path-dependent portfolio credit derivatives whose payoff depends on the loss variable at different time horizons. We review the general theory of copulas and Markov processes, and we establish the link between the copula approach and the Markov-Functional paradigm used in interest rates modelling. Equipped with these theoretical foundations, we show how one can construct a dynamic credit model, which matches the correlation skew at each tenor, by construction, and follows an exogenously specified choice of dynamics. Finally, we discuss the details of the numerical implementation and we give some pricing examples in this framework.
Youssef Elouerkhaoui
Chapter 17. Hedging in Incomplete Markets
Abstract
In this chapter, we present a methodology for hedging basket credit derivatives with single name instruments. Because of the market incompleteness due to the residual correlation risk, perfect replication cannot be achieved. We allow for mean self-financing strategies and use a risk-minimization criterion to find the hedge. Managing credit risk is always a fine balance between hedging the Jump-to-Default exposure (JTD) or the credit spread exposure (CR01). Recently, this credit hedging dilemma (JTD vs CR01) is becoming very topical in the context of managing counterparty credit risk for large derivatives books.
Youssef Elouerkhaoui
Chapter 18. Min-Variance Hedging with Carry
Abstract
In this chapter, we present the construction of the Min-Variance Hedging Delta operator used for basket products. Because of the market incompleteness—i.e., we cannot replicate a basket product with its underlying default swaps—min-variance hedging is the best thing that we can hope for. There will always be a residual correlation risk orthogonal to the sub-space of hedging instruments. We also present an extension of the standard MVH optimization to take into account the drift mismatch between the basket and the hedging portfolio. This defines the “Min-Variance Hedging Deltas with Carry”.
Youssef Elouerkhaoui
Chapter 19. Correlation Calibration with Stochastic Recovery
Abstract
In this chapter, we expand the base correlation framework by enriching it with Stochastic Recovery modelling as a way to address the model limitations observed in a distressed credit environment. We introduce the general class of Conditional-Functional recovery models, which specify the recovery rate as a function of the common conditioning factor of the Gaussian copula. Then, we review some of the most popular ones, such as: the Conditional Discrete model of Krekel (2008), the Conditional Gaussian of Andersen and Sidenius (2005) and the Conditional Mark-Down of Amraoui and Hitier (2008). We also look at stochastic recovery from an aggregate portfolio perspective and present a top-down specification of the problem. By establishing the equivalence between these two approaches, we show that the latter can provide a useful tool for analyzing the structure of various stochastic recovery model assumptions.
Youssef Elouerkhaoui

The Next Challenge

Frontmatter
Chapter 20. New Frontiers in Credit Modelling: The CVA Challenge
Abstract
In this chapter, we present a general framework for evaluating the (counterparty) Credit Valuation Adjustment for CDO tranches.
Youssef Elouerkhaoui
Backmatter
Metadata
Title
Credit Correlation
Author
Youssef Elouerkhaoui
Copyright Year
2017
Electronic ISBN
978-3-319-60973-7
Print ISBN
978-3-319-60972-0
DOI
https://doi.org/10.1007/978-3-319-60973-7