# A Practical Approach to XVA

## The Evolution of Derivatives Valuation after the Financial Crisis

## Osamu Tsuchiya

- 340 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

# A Practical Approach to XVA

## The Evolution of Derivatives Valuation after the Financial Crisis

## Osamu Tsuchiya

## About This Book

The 2008 financial crisis shook the financial derivatives market to its core, revealing a failure to fully price the cost of doing business then. As a response to this, and to cope with regulatory demands for massively increased capital and other measures with funding cost, the pre-2008 concept of Credit Valuation Adjustment (CVA) has evolved into the far more complex hybrid Cross Valuation Adjustment (XVA).

This book presents a clear and concise framework and provides key considerations for the computation of myriad adjustments to the price of financial derivatives, to fully reflect costs. XVA has been of great interest recently due to heavy funding costs (FVA), initial margin (MVA) and capital requirements (KVA) required to sustain a derivatives business since 2008, in addition to the traditional concepts of cost from counterparty default or credit deterioration (CVA), and its mirror image — the cost of one own's default (DVA).

The book takes a practitioner's perspective on the above concepts, and then provides a framework to implement such adjustments in practice. Models are presented too, taking note of what is computationally feasible in light of portfolios typical of investment banks, and the different instruments associated with these portfolios.

Contents:

- Foreword
- Preface
- About the Author
- List of Figures
- List of Tables
- Introduction
- Fundamentals:
- Underpinnings of Traditional Derivatives Pricing and Implications of Current Environment

- Pricing Adjustments:
- CVA and its Relation to Traditional Bond Pricing
- DVA and FVA — Price and Value for Accountants, Regulators and Others
- Theoretical Framework behind FVA and its Computation
- Ingredients of the Modern Yield Curve and Overlaps with XVA
- Margin Valuation Adjustment (MVA)
- KVA, and Other Adjustments and Costs

- Computing XVA in Practice:
- Typical Balance Sheet and Trade Relations of Banks and Implication for XVA
- Framework for Computing XVA
- Calculation of KVA and MVA

- Managing XVA:
- CVA Hedging, Default Arrangements and Implications for XVA Modeling
- Managing XVA in Practice

- Appendices
- Sample Appendix
- A Brief Outline of Regulatory Capital Charges for Financial Institutions

- Conclusion
- References
- Index

Readership: Professionals in the financial derivatives industry, as well as graduate students of quantitative finance.XVA;CVA;Valuation Adjustments;Counterparty Credit Risk;CCR;KVA;Regulatory Capital0 Key Features:

- The book presents overview of the fundamental principles of XVA
- The book presents the calculation methodology of XVA as concisely as possible
- The book analyzes the features of XVA in the practice of the derivative trading business

## Frequently asked questions

## Information

## Part I

## Fundamentals

## Chapter 1

## Underpinnings of Traditional Derivatives Pricing and Implications of Current Environment

### 1.1.Fundamentals of Derivatives Pricing

**Static replication:**Build a portfolio today whose payout at a future date is (at least approximately) equal to the payout of the derivative.

**Dynamic replication:**Implement a self-financing trading strategy to buy and sell simple instruments, so that the final payout of this portfolio equals the payout of the derivative. Here, the cost incurred in the trading strategy gives the value of the derivative. In general, a model is necessary to inform the trading strategy (i.e. how much of the underlying to hold at a given time subject to market moves).

^{1}but unfortunately its applicability is limited to a far smaller set of derivatives.

#### 1.1.1.*Assumptions of derivatives pricing*

*r*(

*t*).

*S*(

*t*) be

*V*(

*t*,

*S*(

*t*)) and the risk-free short rate be

*r*(

*t*). In the Black–Scholes model, the asset price follows the Geometrical Brownian motion

*t*). The hedging portfolio consists of

*δ*(

*t*) amount of asset,

*β*

_{V}(

*t*) amount of cash to finance the derivative trade and

*β*

_{S}(

*t*) amount of cash to finance the asset, that is

*V*(

*t*,

*S*(

*t*)) + Π(

*t*) = 0.

*V*(

*t*,

*S*(

*t*)) + Π(

*t*) to be hedged against asset movements, it is necessary for

*β*

_{V}(

*t*) is the cash to fund the derivative, we need

*β*

_{S}(

*t*) is given by

*r*(

*t*), and the following is satisfied:

*γ*(

*t*) on top of the risk-free funding.

^{2}

*d*(

*V*(

*t*,

*S*(

*t*)) + Π(

*t*)) = 0, we have the PDE which is called the Black–Scholes equation

*E*

^{Q}[ ] is calculated in a probability measure where the asset price process is given by

*S*to

*S*+

*dS*(for an infinitesimal

*dS*), since how would you otherwise know the price is going up? The result of the above is that the hedging strategy costs more when volatility is higher.

**Continuity of underlying:**Given that we are hedging by getting rid of sensitivity to delta (which only exists if the underlying is continuous), we are clearly assuming a continuous distribution for the underlying. In practice, if the underlying moves sharply, a delta hedge will fail to capture such a move. In practice, a limited set of derivatives have risks that very strongly depend on jumps (e.g. gap risk for CPPI or short-dated options on Credit Default Swaps). The pragmatic approach is often to just add wider margins when in doubt, for cases where dependence is not critical.

**Volatility:**It should be clear from the above that hedging a long position in optionality is more expensive the higher the volatility. Volatility is not an observable, and hedging cost will depend on realized volatility (from date of trade to expiry of option), whereas an option has to be priced before this cost is known. Thus, whether implied volatility (used to price the option and typically what gets charged to the client) is higher than realized volatility will determine whether selling the derivative is profitable in general.

**Funding:**What is however less emphasized in academic texts is that there is a funding rate. Typically, this was conveniently thought of as the “risk-free” rate. But financial institutions could never borrow risk-free (even prior to 2008), so this was conveniently thought of as the Libor rate (for AA borrowing) back then. Post 2008 when funding became a far bigger issue, the exact rate to use has become far more important. This will form the heart of our discussion on FVA subsequently.