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is an indicator function for and is the filtration up to time t. The value of the cash flow prior to default is split into two parts, conditioned on whether or not default occurs prior to the maturity T. The recovery value at the time of default is given by

(3.6)

where (V_τ)⁺ and (V_τ)⁻ are the positive and negative parts of the portfolio value respectively and R is the recovery rate or fraction. Note that the portfolio value at close-out, V_τ, is the risk-free close-out discussed above. Hence, using equations (3.5), (3.4) and (3.6), we obtain an expression for the portfolio value at time t when the counterparty is subject to default risk,

      (3.7)

      Using the linearity of expectations and the tower property this reduces to

      (3.8)

      This expression can be simplified by using the fact that V_t = (V_t)⁺ + (V_t)⁻, giving

      (3.9)

      Using the fact that

, the tower property of expectations and with some rearrangement we obtain

      (3.10)

      Hence 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 < t_i < … < t_n = 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

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