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XVA. Green Andrew

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XVA - Green Andrew

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by the definition in equation (3.1), the unilateral CVA is given by

(3.11)

Up to this point I have made no assumptions about the behaviour of the underlying state variables. To proceed further and obtain the standard CVA formula we need to assume that the credit risk is independent of any market risk factors.21 Once this is done the expectation in equation (3.11) can be represented as an integral over time,

(3.12)

      where λC(s) is the counterparty hazard rate, r is the short rate,

is the filtration to time t containing only market state variable information and the inner expectation is made in the risk-neutral measure with the money market account as numeraire. Introducing a time partition t = t0 < ti < … < tn = T allows the integral to be approximated using a summation,

(3.13)

where Φ(τ > t) is the survival probability up to time t of the counterparty and I have assumed that the recovery rate is deterministic. Equation (3.13) is the standard expression for CVA and is very similar to the form used by the Basel Committee in Basel III (2011).22 An example unilateral CVA calculation can be found in Table 3.1.

Table 3.1 An example unilateral CVA calculation for a five-year GBP interest rate swap. The counterparty CDS spread is set at 300 basis points for all maturities and the counterparty recovery is 40 %.

      3.2.2 Unilateral CVA by Replication

The same CVA formula can also be derived using replication in the same way as the Black-Scholes model. To obtain an expression for unilateral CVA, the risk of counterparty default must be considered alongside the Brownian motion driving the stock price. A summary of the notation used in this section and in the equivalent bilateral CVA replication model can be found in Table 3.2.

Table 3.2 Notation used in unilateral and bilateral CVA replication models. The same notation, with some additions, will also be used in the FVA model described in Chapter 9 and also in the KVA model discussed in Chapter 13.

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      1

      Of course in reality there may be reasons why it is not possible to actually execute the trade, such as lack of access to both markets, regulatory prohibitions and so on.

      2

      See section 3.3.3.

      3

      In my view it is precisely this break with the past that has led to such a vigorous debate around funding valuation adjustment, a subject that is discussed in section 1.4.1.

      4

1

Of course in reality there may be reasons why it is not possible to actually execute the trade, such as lack of access to both markets, regulatory prohibitions and so on.

2

See section 3.3.3.

3

In my view it is precisely this break with the past that has led to such a vigorous debate around funding valuation adjustment, a subject that is discussed in section 1.4.1.

4

See Chapter 22 for a broader discussion of organisational design.

5

See Chapter 8.

6

Chapter 8 discusses the construction of OIS discounting curves in more detail. It should be noted that collateral substitution where one piece of collateral is exchanged for another depends on the local legal framework and that collateral can be viewed as “sticky” in some jurisdictions. The pricing of the embedded cheapest-to-deliver option is not straightforward, therefore.

7

Except for repo transactions where the margin period of risk is set at five days.

8

Of course with sufficiently large market moves, almost any initial margin can be exceeded as is clear from the removal of the CHF-EUR peg by the Swiss National Bank on 15 January 2015.

9

Note that sometimes this spot capital requirement is referred to in terms of RWA or Risk-Weighted Assets as this is the regulatory asset value against which the capital must be held. In reality the Basel framework is written partially in terms of RWA and partly in terms of direct capital requirements but RWA is frequently used in finance as a shorthand way of describing all capital requirements.

10

OIS discounting models are discussed in Chapter 9.

11

The regulatory EAD is a separate but related quantity that I will discuss later in Chapter 12.

12

Or sometimes simply expected exposure (EE), although this can be confused with the forward value Vt which is sometimes also known as expected exposure.

13

is the filtration to time t. This is a technical element of stochastic calculus and the filtration simply represents all information such as asset prices up to time t. Where necessary I will include technical elements like this but they will be left out where appropriate and when it simplifies the presentation.

14

The specification of the recovery as a percentage of price is the most common approach but other specifications are possible, see Schönbucher (2003).

15

International Swaps and Derivatives Association.

16

A reverse repo is the opposite transaction where one counterparty buys bonds and agrees to sell them again at a fixed price at an agreed date in the future (see Figure 2.3). Hence each repo transaction is paired with a reverse repo for the counterparty.

17

This

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