Pricing Counterparty Credit Risk at the Trade Level

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

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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 exposureat 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-levelexposure equals the sum of contract-levelexposures • 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 adjustmentis 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 Twhere • 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 writewhere is the risk neutral cumulative probability of default (PD) between today (time 0) and time tis 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 becausewhere • 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 obtainwhere 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 collateralC(t) whenever portfolio value exceeds the thresholdH :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 • 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 aswhere 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 intensityl(tk) for simulation time tkjointly • 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 RiskGARP Risk Review, July/August, pages 16-22.