Counterparty Risk and Funding: A Tale of Two Puzzles, 1st Edition (Hardback) book cover

Counterparty Risk and Funding

A Tale of Two Puzzles, 1st Edition

By Stéphane Crépey, Tomasz R. Bielecki, Damiano Brigo

Chapman and Hall/CRC

388 pages | 51 B/W Illus.

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pub: 2014-06-23
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Solve the DVA/FVA Overlap Issue and Effectively Manage Portfolio Credit Risk

Counterparty Risk and Funding: A Tale of Two Puzzles explains how to study risk embedded in financial transactions between the bank and its counterparty. The authors provide an analytical basis for the quantitative methodology of dynamic valuation, mitigation, and hedging of bilateral counterparty risk on over-the-counter (OTC) derivative contracts under funding constraints. They explore credit, debt, funding, liquidity, and rating valuation adjustment (CVA, DVA, FVA, LVA, and RVA) as well as replacement cost (RC), wrong-way risk, multiple funding curves, and collateral.

The first part of the book assesses today’s financial landscape, including the current multi-curve reality of financial markets. In mathematical but model-free terms, the second part describes all the basic elements of the pricing and hedging framework. Taking a more practical slant, the third part introduces a reduced-form modeling approach in which the risk of default of the two parties only shows up through their default intensities. The fourth part addresses counterparty risk on credit derivatives through dynamic copula models. In the fifth part, the authors present a credit migrations model that allows you to account for rating-dependent credit support annex (CSA) clauses. They also touch on nonlinear FVA computations in credit portfolio models. The final part covers classical tools from stochastic analysis and gives a brief introduction to the theory of Markov copulas.

The credit crisis and ongoing European sovereign debt crisis have shown the importance of the proper assessment and management of counterparty risk. This book focuses on the interaction and possible overlap between DVA and FVA terms. It also explores the particularly challenging issue of counterparty risk in portfolio credit modeling. Primarily for researchers and graduate students in financial mathematics, the book is also suitable for financial quants, managers in banks, CVA desks, and members of supervisory bodies.


"… a fresh take on mitigation of counterparty risk … [the book] gives a ground-up approach for analysis and managing of risks associated with non-payment of promised cash flows due to the default by a party in an over-the-counter derivative transaction. It should be of value to researchers, graduate students, financial quants, managers in banks, CVA desks, and members of supervisory bodies."

—, July 2014

"The landscape of the rates and credit markets has changed so drastically since the 2008 crisis that older textbooks are barely relevant and, from an analytic perspective, appropriate methods have to be rethought from scratch. The present volume is one of the best contributions in this direction, featuring a clear description of the various ‘value adjustments,’ new models for portfolio credit risk, a unified analytic framework based on BSDEs, and detailed treatment of numerical methods."

—Mark Davis, Imperial College London

"Understanding the subtle interconnections between credit and funding is key to a modern valuation of derivatives. This timely contribution, written by world-class academics who are also well-recognized experts in the field, offers a rigorous and comprehensive treatment of the main theories underpinning the new valuation principles. Numerical examples are also provided to help the reader grasp key concepts and ideas of the advanced models and techniques here presented. Overall, an excellent textbook. Brigo’s dialogue is the icing on the cake."

—Fabio Mercurio, Head of Derivatives Research, Bloomberg LP

"A big hooray for this book on CVA, DVA, FVA/LVA, RVA, TVA, and other three letter acronyms (TLA!)."

—Peter Carr, PhD, Managing Director, Morgan Stanley, and Executive Director, NYU Courant Master of Science Program in Mathematics in Finance

Table of Contents

Financial Landscape

A Galilean Dialogue on Counterparty Risk, CVA, DVA, Multiple Curves, Collateral, and Funding

To the Discerning Reader

The First Day

The Second Day

The Third Day

The Fourth Day

The Whys of the LOIS

Financial Setup

Indifference Valuation Model

LOIS Formula

Numerical Study

Model-Free Developments

Pure Counterparty Risk

Cash Flows

Valuation and Hedging

CSA Specifications

Bilateral Counterparty Risk under Funding Constraints


Market Model

Trading Strategies

Martingale Pricing Approach



Reduced-Form BSDE Modeling

A Reduced-Form TVA BSDE Approach to Counterparty Risk under Funding Constraints


Pre-Default BSDE Modeling

Markov Case

The Four Wings of the TVA


TVA Representations

CSA Specifications

Clean Valuations

TVA Computations

Dynamic Copula Models

Dynamic Gaussian Copula Model



Clean Valuation and Hedging of Credit Derivatives

Counterparty Risk

Common-Shock Model


Model of Default Times

Clean Pricing, Calibration and Hedging

Numerical Results

CVA Pricing and Hedging

CVA Computations for one CDS in the Common-Shock Model



Common-Shock Model with Deterministic Intensities

Numerical Results with Deterministic Intensities

Common-Shock Model with Stochastic Intensities


CVA Computations for Credit Portfolios in the Common-Shock Model

Portfolio of CDS

CDO Tranches

Further Developments

Rating Triggers and Credit Migrations


Credit Value Adjustment and Collateralization under Rating Triggers

Markov Copula Approach for Rating-Based Pricing


A Unified Perspective


Marked Default Time Reduced-Form Modeling

Dynamic Gaussian Copula TVA Model

Dynamic Marshall-Olkin Copula TVA Model

Mathematical Appendix

Stochastic Analysis Prerequisites

Stochastic Integration

Itô Processes


Feynman-Kac Formula

Backward Stochastic Differential Equations

Measure Changes and Random Intensity of Jumps

Reduction of Filtration and Hazard Intensity Pre-Default Credit Risk Modeling

Markov Consistency and Markov Copulas


Consistent Markov Processes

Markov Copulas



About the Series

Chapman and Hall/CRC Financial Mathematics Series

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General