Partial Differential Equation Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs

Christoph Burgard and Mats Kjaer

Contents

Introduction

Preface to Chapter 1

1.

Being Two-Faced over Counterparty Credit Risk

2.

Risky Funding: A Unified Framework for Counterparty and Liquidity Charges

3.

DVA for Assets

4.

Pricing CDSs’ Capital Relief

5.

The FVA Debate

6.

The FVA Debate: Reloaded

7.

Regulatory Costs Break Risk Neutrality

8.

Risk Neutrality Stays

9.

Regulatory Costs Remain

10.

Funding beyond Discounting: Collateral Agreements and Derivatives Pricing

11.

Cooking with Collateral

12.

Options for Collateral Options

13.

Partial Differential Equation Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs

14.

In the Balance

15.

Funding Strategies, Funding Costs

16.

The Funding Invariance Principle

17.

Regulatory-Optimal Funding

18.

Close-Out Convention Tensions

19.

Funding, Collateral and Hedging: Arbitrage-Free Pricing with Credit, Collateral and Funding Costs

20.

Bilateral Counterparty Risk with Application to Credit Default Swaps

21.

KVA: Capital Valuation Adjustment by Replication

22.

From FVA to KVA: Including Cost of Capital in Derivatives Pricing

23.

Warehousing Credit Risk: Pricing, Capital and Tax

24.

MVA by Replication and Regression

25.

Smoking Adjoints: Fast Evaluation of Monte Carlo Greeks

26.

Adjoint Greeks Made Easy

27.

Bounding Wrong-Way Risk in Measuring Counterparty Risk

28.

Wrong-Way Risk the Right Way: Accounting for Joint Defaults in CVA

29.

Backward Induction for Future Values

30.

A Non-Linear PDE for XVA by Forward Monte Carlo

31.

Efficient XVA Management: Pricing, Hedging and Allocation

32.

Accounting for KVA under IFRS 13

33.

FVA Accounting, Risk Management and Collateral Trading

34.

Derivatives Funding, Netting and Accounting

35.

Managing XVA in the Ring-Fenced Bank

36.

XVA: A Banking Supervisory Perspective

37.

An Annotated Bibliography of XVA

Given the market conditions since the global financial crisis, counterparty credit risk implicitly embedded in derivative contracts has become increasingly relevant. This kind of risk represents the possibility that a counterparty will default while owing money under the terms of a derivative contract, or, more precisely, if the mark-to-market value of the derivative is positive to the seller at the time of default of the counterparty. While for exchange-traded contracts the counterparty credit risk is mitigated by the exchange’s presence as intermediary, this is not the case for over-the-counter products. For these, a number of different techniques are used to mitigate counterparty risk, most commonly by means of netting agreements and collateral mechanisms. The details of these agreements are specified, for example, by the International Swaps and Derivatives Association (ISDA) 2002 Master Agreement. However, the counterparty faces the similar risk of the seller defaulting when the mark-to-market value is positive to the counterparty. Taking into account the credit risk of both parties is commonly referred to as considering bilateral counterparty risk. When doing so, the value of

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