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Counterparty Risk and Funding

Book Description

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.

Table of Contents

  1. Preliminaries
  2. Preface
    1. Introduction
    2. Outline
    3. About the title of the book
    4. Standing Notation, Terminology and Assumptions
    5. Bibliographic Guidelines
    6. Acknowledgements
  3. Part I Financial Landscape
    1. Chapter 1 A Galilean Dialogue on Counterparty Risk, CVA, DVA, Multiple Curves, Collateral and Funding
      1. 1.1 To the Discerning Reader
      2. 1.2 The First Day
        1. 1.2.1 General Introduction, Size of Derivatives Markets, Exposures, Credit Var, Basel
      3. 1.3 The Second Day
        1. 1.3.1 CVA, DVA, Pricing, Arbitrage Free Theory, Closeout. And the Data? Ratings?
      4. 1.4 The Third Day
        1. 1.4.1 FVA, Hard Maths with No Data? CVA VaR, Basel III Problems, Collateral and Gap Risk
      5. 1.5 The Fourth Day
        1. 1.5.1 Counterparty Risk Restructuring. CCDS, Papillon, Floating Rate CVA and Margin Lending. Global Calibration. Global Valuation. Available CVA Books and Forthcoming CVA Books.
        1. Figure 1.1
        2. Figure 2.1
        3. Figure 2.2
        4. Figure 2.3
    2. Chapter 2 The Whys of the LOIS
      1. 2.1 Financial Setup
      2. 2.2 Indifference Valuation Model
        1. 2.2.1 Credit and Funding Costs Specification
      3. 2.3 LOIS Formula
      4. 2.4 Numerical Study
      5. Conclusion
  4. Part II Model-Free Developments
    1. Chapter 3 Pure Counterparty Risk
      1. 3.1 Cash Flows
        1. 3.1.1 Promised Dividend
        2. 3.1.2 Collateral
        3. 3.1.3 Closeout Cash Flow
      2. 3.2 Valuation and Hedging
        1. 3.2.1 Valuation of the Contract
        2. 3.2.2 Valuation of Counterparty Risk
        3. 3.2.3 Exposure at Default
        4. 3.2.4 TVA and CVA/DVA/RC
          1. 3.2.4.1 Expected Positive/Negative Exposures
          2. 3.2.4.2 Unilateral Counterparty Risk
        5. 3.2.5 Dynamic Hedging of Counterparty Risk
          1. 3.2.5.1 Min-Variance Hedging
      3. 3.3 CSA Specifications
        1. 3.3.1 Close Out Valuation Schemes
        2. 3.3.2 Collateralization Schemes
        3. 3.3.3 Cure Period
        4. 3.3.4 Rehypothecation Risk and Segregation
        5. 3.3.5 Haircuts
        6. 3.3.6 Centrally Cleared Trading
    2. Chapter 4 Bilateral Counterparty Risk under Funding Constraints
      1. 4.1 Introduction
        1. 4.1.1 Outline
      2. 4.2 Market Model
        1. 4.2.1 Hedging Assets
        2. 4.2.2 Funding Assets
      3. 4.3 Trading Strategies
        1. 4.3.1 Self-Financing Condition
        2. 4.3.2 General Price-and-Hedge
      4. 4.4 Martingale Pricing Approach
        1. 4.4.1 Primary Market
        2. 4.4.2 ℚ-Price-and-Hedge BSDE
        3. 4.4.3 Arbitrage, Replication and Computational Issues
      5. 4.5 TVA
        1. 4.5.1 Clean Price
        2. 4.5.2 CSA Close-Out Cash-Flow
        3. 4.5.3 TVA Representation
          1. 4.5.3.1 CCDS Static Hedging Interpretation
      6. 4.6 Example
        1. 4.6.1 Setup
        2. 4.6.2 Analysis of a Solution
          1. 4.6.2.1 TVA
          2. 4.6.2.2 CSA Close-Out Pricing Schemes
        3. 4.6.3 Comparison with the Results of Burgard and Kjaer
        1. Figure 4.1
  5. Part III Reduced-Form BSDE Modeling
    1. Chapter 5 A Reduced-Form TVA BSDE Approach to Counterparty Risk under Funding Constraints
      1. 5.1 Introduction
        1. 5.1.1 Outline
      2. 5.2 Pre-Default BSDE Modeling
        1. 5.2.1 Bilateral Reduced Form Setup
        2. 5.2.2 Reduction of Filtration
        3. 5.2.3 Modeling Assumption
        4. 5.2.4 Cost Processes Analysis
      3. 5.3 Markov Case
        1. 5.3.1 Factor Process
        2. 5.3.2 Min-Variance Hedging of Market Risk
          1. Hedging of the Contract as a Whole
        3. 5.3.3 Min-Variance Hedging Constrained to Perfect Hedging of Jump-to-Default Risk
          1. Hedging of the Contract as a Whole
        4. 5.3.4 Unilateral or Bilateral in the End?
    2. Chapter 6 The Four Wings of the TVA
      1. 6.1 Introduction
      2. 6.2 TVA Representations
        1. 6.2.1 Setup
        2. 6.2.2 BSDEs
          1. 6.2.2.1 Pre-default Markov Setup
        3. 6.2.3 CVA, DVA, LVA and RC
      3. 6.3 CSA Specifications
        1. 6.3.1 Clean CSA Recovery Scheme
        2. 6.3.2 Pre-Default CSA Recovery Scheme
        3. 6.3.3 Full Collateralization CSA
        4. 6.3.4 Pure Funding
        5. 6.3.5 Asymmetrical TVA Approach
      4. 6.4 Clean Valuations
        1. 6.4.1 Products
        2. 6.4.2 Gaussian Vasicek Short Rate Model
          1. 6.4.2.1 Caplet
        3. 6.4.3 Lévy Hull-White Short Rate Model
          1. 6.4.3.1 Caplet
        4. 6.4.4 Numerics
      5. 6.5 TVA Computations
        1. 6.5.1 TVA Equations
        2. 6.5.2 Numerics
      6. Conclusion
        1. Figure 6.1
        2. Figure 6.2
        3. Figure 6.3
        4. Figure 6.4
        5. Figure 6.5
        6. Figure 6.6
        7. Figure 6.7
        8. Figure 6.8
        1. Table 6.1
  6. Part IV Dynamic Copula Models
    1. Chapter 7 Dynamic Gaussian Copula Model
      1. 7.1 Introduction
      2. 7.2 Model
        1. 7.2.1 Gaussian Distributions
        2. 7.2.2 Model of Default Times
          1. 7.2.2.1 Conditional Survival Distribution
        3. 7.2.3 Fundamental Martingales
          1. 7.2.3.1 Univariate Case
          2. 7.2.3.2 Portfolio Case
      3. 7.3 Clean Valuation and Hedging of Credit Derivatives
        1. 7.3.1 Pricing of a CDS
        2. 7.3.2 Pricing of a CDO
        3. 7.3.3 Hedging CDO with CDS
      4. 7.4 Counterparty Risk
        1. 7.4.1 Numerics
          1. 7.4.1.1 Spread Volatilities
          2. 7.4.1.2 CVA
        1. Figure 7.1
        2. Figure 7.2
        3. Figure 7.3
        4. Figure 7.4
        5. Figure 7.5
        6. Figure 7.6
        7. Figure 7.7
    2. Chapter 8 Common-Shock Model
      1. 8.1 Introduction
      2. 8.2 Model of Default Times
        1. 8.2.1 Conditional Joint Survival Function
          1. 8.2.1.1 Conditional Common-Shock Model
        2. 8.2.2 Itô-Markov Formula
        3. 8.2.3 Intensity Structure
      3. 8.3 Clean Pricing, Calibration and Hedging
        1. 8.3.1 Pricing Equations
          1. 8.3.1.1 Rolling CDS
        2. 8.3.2 Min-Variance Hedging
          1. 8.3.2.1 Hedging of a CDO Tranche Using Rolling CDS Contracts
        3. 8.3.3 Convolution Recursion Pricing Schemes
        4. 8.3.4 Random Recoveries
      4. 8.4 Numerical Results
        1. 8.4.1 Calibration Methodology with Piecewise Constant Default Intensities and Constant Recoveries
        2. 8.4.2 Calibration Methodology with Piecewise Constant Default Intensities and Stochastic Recoveries
        3. 8.4.3 Calibration Results with Piecewise Constant Default Intensities
          1. 8.4.3.1 The Implied Loss Distribution
        4. 8.4.4 Calibration Methodology and Results with Stochastic Intensities
        5. 8.4.5 Min-Variance Hedging Deltas
      5. 8.5 CVA Pricing and Hedging
        1. Figure 8.1
        2. Figure 8.2
        3. Figure 8.3
        4. Figure 8.4
        5. Figure 8.5
        6. Figure 8.6
        7. Figure 8.7
        8. Figure 8.8
        9. Figure 8.9
        1. Table 8.1
        2. Table 8.2
        3. Table 8.3
        4. Table 8.4
        5. Table 8.5
    3. Chapter 9 CVA Computations for One CDS in the Common-Shock Model
      1. 9.1 Introduction
      2. 9.2 Generalities
        1. 9.2.1 Specification-Free Results in a Common-Shock Setup
          1. 9.2.1.1 Clean Valuation
          2. 9.2.1.2 Min-Variance Hedging of the CVA Jump-To-Counterparty-Default Exposure
      3. 9.3 Common-Shock Model with Deterministic Intensities
        1. 9.3.1 Implementation
          1. 9.3.1.1 Linear Intensities
          2. 9.3.1.2 Calibration Issues
          3. 9.3.1.3 Constant Intensities
      4. 9.4 Numerical Results with Deterministic Intensities
      5. 9.5 Common-Shock Model with Stochastic Intensities
        1. 9.5.1 CIR++ Intensities
          1. 9.5.1.1 Calibration Methodology
        2. 9.5.2 Extended CIR Intensities
          1. 9.5.2.1 Implementation
      6. 9.6 Numerics
        1. 9.6.1 Calibration Results
        2. 9.6.2 CVA Stylized Features
        3. 9.6.3 Case of a Low-Risk Reference Entity
        4. 9.6.4 CDS Options-Implied Volatilities
        5. 9.6.5 Contribution of the Joint Default
      7. Conclusions
        1. Figure 9.1
        2. Figure 9.2
        3. Figure 9.3
        4. Figure 9.4
        5. Figure 9.5
        6. Figure 9.6
        7. Figure 9.7
        8. Figure 9.8
        9. Figure 9.9
        10. Figure 9.10
        1. Table 9.1
        2. Table 9.2
        3. Table 9.3
        4. Table 9.4
        5. Table 9.5
        6. Table 9.6
        7. Table 9.7
        8. Table 9.8
        9. Table 9.9
        10. Table 9.10
        11. Table 9.11
        12. Table 9.12
        13. Table 9.13
    4. Chapter 10 CVA Computations for Credit Portfolios in the Common-Shock Model
      1. 10.1 Portfolio of CDS
        1. 10.1.1 Common-Shock Model Specification
        2. 10.1.2 Numerical Results
      2. 10.2 CDO Tranches
        1. 10.2.1 Numerical Results
        1. Figure 10.1
        1. Table 10.1
        2. Table 10.2
        3. Table 10.3
        4. Table 10.4
  7. Part V Further Developments
    1. Chapter 11 Rating Triggers and Credit Migrations
      1. 11.1 Introduction
      2. 11.2 Credit Value Adjustment and Collateralization under Rating Triggers
        1. 11.2.1 Pricing Bilateral Counterparty Risk with Rating Triggers
        2. 11.2.2 Dynamic Collateralization
      3. 11.3 Markov Copula Approach for Rating-Based Pricing
      4. 11.4 Applications
        1. 11.4.1 Interest Rate Swap with Rating Triggers
        2. 11.4.2 CDS with Rating Triggers
        1. Figure 11.1
        2. Figure 11.2
        3. Figure 11.3
        4. Figure 11.4
        5. Figure 11.5
        6. Figure 11.6
        7. Figure 11.7
        8. Figure 11.8
        9. Figure 11.9
        10. Figure 11.10
        11. Figure 11.11
        12. Figure 11.12
        1. Table 11.1
        2. Table 11.2
        3. Table 11.3
        4. Table 11.4
        5. Table 11.5
        6. Table 11.6
        7. Table 11.7
        8. Table 11.8
        9. Table 11.9
        10. Table 11.10
        11. Table 11.11
        12. Table 11.12
        13. Table 11.13
        14. Table 11.14
        15. Table 11.15
    2. Chapter 12 A Unified Perspective
      1. 12.1 Introduction
      2. 12.2 Marked Default Time Reduced-Form Modeling
        1. 12.2.1 Pre-default Setup
      3. 12.3 Dynamic Gaussian Copula TVA Model
        1. 12.3.1 Model of Default Times
        2. 12.3.2 Pre-default TVA Model
      4. 12.4 Dynamic Marshall-Olkin Copula TVA Model
        1. 12.4.1 Model of Default Times
        2. 12.4.2 TVA Model
        3. 12.4.3 Reduced-Form TVA Approach
      5. Conclusion
  8. Part VI Mathematical Appendix
    1. Chapter 13 Stochastic Analysis Prerequisites
      1. Setup
      2. 13.1 Stochastic Integration
        1. 13.1.1 Semimartingales
        2. 13.1.2 Random Measures Integration Theory
      3. 13.2 Itô Processes
        1. 13.2.1 Finite Variation Jumps
        2. 13.2.2 General Case
          1. 13.2.2.1 Brackets
      4. 13.3 Jump-Diffusions
      5. 13.4 Feynman-Kac Formula
        1. 13.4.1 An Affine Formula
      6. 13.5 Backward Stochastic Differential Equations
      7. 13.6 Measure Changes and Random Intensity of Jumps
      8. 13.7 Reduction of Filtration and Hazard Intensity Pre-Default Credit Risk Modeling
        1. 13.7.1 Portfolio Credit Risk
    2. Chapter 14 Markov Consistency and Markov Copulas
      1. 14.1 Introduction
      2. 14.2 Consistent Markov Processes
      3. 14.3 Markov Copulas
      4. 14.4 Examples
        1. 14.4.1 Diffusions
        2. 14.4.2 Jump-Diffusions
          1. 14.4.2.1 Finite Markov Chains
        3. 14.4.3 Diffusion Modulated Markov Jump Processes.
  9. Bibliography