Landmarks in XVA
First published:
ISBN: 9781782722939
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A Non-Linear PDE for XVA by Forward Monte Carlo
Introduction
Preface to Chapter 1
Being Two-Faced over Counterparty Credit Risk
Risky Funding: A Unified Framework for Counterparty and Liquidity Charges
DVA for Assets
Pricing CDSs’ Capital Relief
The FVA Debate
The FVA Debate: Reloaded
Regulatory Costs Break Risk Neutrality
Risk Neutrality Stays
Regulatory Costs Remain
Funding beyond Discounting: Collateral Agreements and Derivatives Pricing
Cooking with Collateral
Options for Collateral Options
Partial Differential Equation Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs
In the Balance
Funding Strategies, Funding Costs
The Funding Invariance Principle
Regulatory-Optimal Funding
Close-Out Convention Tensions
Funding, Collateral and Hedging: Arbitrage-Free Pricing with Credit, Collateral and Funding Costs
Bilateral Counterparty Risk with Application to Credit Default Swaps
KVA: Capital Valuation Adjustment by Replication
From FVA to KVA: Including Cost of Capital in Derivatives Pricing
Warehousing Credit Risk: Pricing, Capital and Tax
MVA by Replication and Regression
Smoking Adjoints: Fast Evaluation of Monte Carlo Greeks
Adjoint Greeks Made Easy
Bounding Wrong-Way Risk in Measuring Counterparty Risk
Wrong-Way Risk the Right Way: Accounting for Joint Defaults in CVA
Backward Induction for Future Values
A Non-Linear PDE for XVA by Forward Monte Carlo
Efficient XVA Management: Pricing, Hedging and Allocation
Accounting for KVA under IFRS 13
FVA Accounting, Risk Management and Collateral Trading
Derivatives Funding, Netting and Accounting
Managing XVA in the Ring-Fenced Bank
XVA: A Banking Supervisory Perspective
An Annotated Bibliography of XVA
We consider the following semi-linear partial differentiation equation (PDE)
| (30.1) |
where V(t, x) is a function of time t, t ∈ [0, T] and space x (one- or multi-dimensional), L is an infinitesimal generator of some diffusion and r ≥ 0. The equation is combined with the terminal condition V(T, x) = ψ(x) and is solved to obtain V(0, x).
Having efficient numerical methods for this PDE is important, as the equation (or its close cousins) appears in a number of related but distinct applications. These include pricing derivatives contracts with a one-way Credit Support Annex (CSA) (Piterbarg 2010), risky close-outs for credit value adjustments (CVAs) (Burgard and Kjaer 2011), derivatives pricing with differential borrowing and lending rates (Mercurio 2014) and accounting-consistent valuation (Albanese et al 2015; Burgard and Kjaer 2015). And, as far back as 2005, a similar non-linear PDE appeared in Andreasen (2005) in the context of fixed physical cash supply and negative rates. More details can be found in Piterbarg (2015).
In a low number of space dimensions (one to three, say) the PDE can be
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