1. Pricing Counterparty Credit Risk at the Trade Level
Michael Pykhtin
Credit Analytics & Methodology
Bank of America
Risk Quant Congress
New York; July 8-9, 2008

2. Disclaimer
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3. Introduction
Counterparty credit risk is the risk that a counterparty in an OTC derivative transaction will default prior to the expiration of the contract and will be unable to make all contractual payments.
Exchange-traded derivatives bear no counterparty risk.
The primary feature that distinguishes counterparty risk from lending risk is the uncertainty of the exposure at any future date.
Loan: exposure at any future date is the outstanding balance, which is certain (not taking into account prepayments).
Derivative: exposure at any future date is the replacement cost, which is determined by the market value at that date and is, therefore, uncertain.
For the derivatives whose value can be both positive and negative (e.g., swaps, forwards), counterparty risk is bilateral.
See Canabarro & Duffie (2003), De Prisco & Rosen (2005) or Pykhtin & Zhu (2007).

4. Exposure at Contract Level
Market value of contract i with a counterparty is known only for current date For any future date t, this value is uncertain and should be assumed random.
If the counterparty defaults at time prior to the contract maturity, maximum economic loss equals the replacement cost of the contract
If the contract value is positive for us, we do not receive anything from defaulted counterparty, but have to pay this amount to another counterparty to replace the contract.
If the contract value is negative, we receive this amount from another counterparty, but have to forward it to the defaulted counterparty.
Quantity is known as contract-level exposure at time t

5. Exposure at Counterparty Level
Counterparty-level exposure at future time t can be defined as the loss experienced by the bank if the counterparty defaults at time t under the assumption of no recovery
If counterparty risk is not mitigated in any way, counterparty-level exposure equals the sum of contract-level exposures
If there are netting agreements, derivatives with positive value at the time of default offset the ones with negative value within each netting set , so that counterparty-level exposure is
Each non-nettable trade represents a netting set

6. Credit Value Adjustment (CVA)
Credit value adjustment is the price of counterparty credit risk.
See Arvanitis & Gregory (2001), Brigo & Masetti (2005) or Picoult (2005).
CVA can be calculated as the risk neutral expectation of the discounted loss over the life of the longest transaction T where
E(t) is the counterparty-level exposure at time t
t is the counterparty’s default time
R is the counterparty-level recovery rate
Bt is the value of the money market account at time t

7. CVA and Expected Exposure
Assuming constant recovery rate R, we can write where is the risk neutral cumulative probability of default (PD) between today (time 0) and time t is risk-neutral discounted expected exposure (EE) at time t conditional on counterparty defaulting at time t.
If both exposure and money market account are independent of counterparty credit state (there is no wrong-way risk), then

8. Portfolio Pricing for New Trades
Suppose, we have a portfolio of derivatives with a counterparty and we want to add a new trade. How should we price the counterparty risk for this trade?
The price of counterparty risk of the new trade is calculated as the marginal contribution to the portfolio CVA
The fair value of credit risk premium x is calculated from
See Chapter 6 in Arvanitis and Gregory (2001) for details.

9. Allocating CVA to Existing Trades
CVA is defined and calculated for the entire portfolio. Can we allocate the counterparty-level CVA to individual trades?
We need to find allocations CVAi such that they
reflect trades’ contributions to the counterparty-level CVA
sum up to the counterparty-level CVA:
Recall that counterparty-level CVA is given by
Since both recovery rate R and cumulative PD P(t) are the same for all trades, CVA allocation reduces to EE allocation!

10. EE Allocation
For each future time t, we need to find allocations such that they
reflect trade’s contribution to the counterparty-level discounted EE
sum up to the counterparty-level discounted EE:
Allocation across netting sets is trivial because where
We will investigate EE allocation within a netting set

11. Homogeneous Exposure
For convenience, we will assume that all trades with a counterparty belong to the same netting set:
Let us assign a “weight” ai to trade i so that:
Exposure of an “adjusted” portfolio is
Therefore, exposure is a homogeneous function of weights:

12. Definition of EE Contributions
We define EE contribution of trade i at time t as
is the counterparty-level EE for portfolio with weights
describes the portfolio consisting of one unit of trade i
describes the original portfolio ( for all i )
EE contributions sum up to the counterparty-level EE by Euler’s theorem
Motivation for this definition comes from allocation of economic capital for loan portfolios
see Chapter 4 in Arvanitis and Gregory (2001) for details

13. EE Contributions for Homogeneous Exposure
Counterparty-level EE is given by
Differentiating with respect to and setting , we obtain where V(t) is the portfolio value given by
These EE contributions sum up to the counterparty-level EE!

14. Non-Homogeneous Exposure
If there is an exposure-limiting agreement between the bank and the counterparty (e.g., a margin agreement), exposure is not a homogeneous function of trades’ weights anymore
The incremental definition of EE contributions is bound to fail!
Conditions of Euler’s theorem are not satisfied, and the incremental EE contributions will not sum up to the counterparty-level EE
Let us consider a margin agreement and assume that the portfolio value is above the threshold. Then
Counterparty-level exposure equals threshold
Infinitesimal change of the weight of any trade does not change the counterparty-level exposure
Therefore, according to the incremental definition, exposure contribution of any trade is zero!

15. Scenario Approach to EE Contributions
Let us obtain the EE contributions in an alternative way
Counterparty-level exposure can be written as
It is natural to define stochastic exposure contributions as
Applying discounting and conditional expectation, we obtain

16. Margin Agreements
Let us consider a counterparty with a netting agreement supported by a margin agreement
Under a margin agreement, the counterparty must post collateral C(t) whenever portfolio value exceeds the threshold H : where D is the margin period of risk
Counterparty-level exposure is given by
To simplify the model, we will set D = 0
For liquid trades, typical value of D is 2 weeks, and the error in EE resulting from setting D = 0 is small

17. Scenario Approach with Margin Agreements
After setting D = 0 , exposure can be written as
Let us consider three types of scenarios separately:
we should set
it is reasonable to set
Combining all three cases, we obtain exposure contributions

18. EE Contributions with Margin Agreements
Applying discounting and conditional expectation, we obtain
These EE contributions
sum up to the counterparty-level EE
converge to the EE contributions for the non-collateralized case in the limit

19. Calculating EE Contributions
Let us assume that exposure is independent of the counterparty credit quality. Then, conditioning on t = t is immaterial.
The simulation algorithm might look like this:
Simulate market scenario for simulation time t
For each trade i, calculate trade value Vi (t)
Calculate portfolio value
For each trade i, update its EE contribution counter:
if 0 < V(t) ≤ H, add Vi (t) B0/Bt
if V(t) > H, add Vi (t) H /V(t) B0/Bt
After running large enough number of market scenarios, divide each EE contribution counter by the number of scenarios

20. Accounting for Wrong/Right-Way Risk
Let us assume that trade values are dependent on the counterparty credit quality
If exposure tends to increase (decrease) when the counterparty credit quality worsens, the risk is said to be wrong-way (right-way).
Let us characterize counterparty credit quality by intensity l(t)
Then, conditional expectation of quantity X can be calculated as where is the first derivative of the cumulative PD P(t)

21. Calculating Conditional EE Contributions
Paths of trade values and of intensity process are simulated jointly
Assuming that we have already simulated l(tj) for all simulation times j < k, the simulation algorithm for tk might look like this:
Simulate market factors and intensity l(tk) for simulation time tk jointly
For each trade i, calculate trade value Vi (tk)
Calculate portfolio value
For each trade i, update the conditional EE contribution counter:
if 0 < V(t) ≤ H, add
if V(t) > H, add

22. Set-Up for Examples
If we assume that all trades’ values are normally distributed, then EE contributions can be evaluated in closed form
We will look at the EE contribution of trade i of value to portfolio, whose value (not including trade i) is given by
Correlation between Xi and X is given by ri
To specify the scale, we set for the portfolio

23. No Margin Agreement: Dependence on mi
Parameters:

24. No Margin Agreement: Dependence on ri
Parameters:

25. Margin Agreement: Dependence on mi
Parameters:

26. Margin Agreement: Dependence on ri
Parameters:

27. Summary
Discrete marginal approach should be used for pricing counterparty risk in new trades
CVA contributions of existing trades to the counterparty-level CVA can be calculated from the EE contributions
Continuous marginal approach works when counterparty-level exposure is homogeneous function of trades’ weights
Scenario-based approach is needed to handle non-homogeneous cases (such as margin agreements)
EE contributions can be easily included in the exposure simulating process
Normal approximation gives closed-form results

28. References
A. Arvanitis and J. Gregory, 2001, “Credit: The Complete Guide to Pricing, Hedging and Risk Management”, Risk Books
D. Brigo and M. Masetti, 2005, Risk Neutral Pricing of Counterparty Risk in “Counterparty Credit Risk Modelling” (M. Pykhtin, ed.), Risk Books
E. Canabarro and D. Duffie, 2003, Measuring and Marking Counterparty Risk in “Asset/Liability Management for Financial Institutions” (L. Tilman, ed.), Institutional Investor Books
B. De Prisco and D. Rosen, 2005, Modelling Stochastic Counterparty Credit Exposures for Derivatives Portfolios in “Counterparty Credit Risk Modelling” (M. Pykhtin, ed.), Risk Books
E. Picoult, 2005, Calculating and Hedging Exposure, Credit Value Adjustment and Economic Capital for Counterparty Credit Risk in “Counterparty Credit Risk Modelling” (M. Pykhtin, ed.), Risk Books
M. Pykhtin and S. Zhu, 2007, A Guide to Modelling Counterparty Credit Risk GARP Risk Review, July/August, pages