Presentation on theme: "Pricing Counterparty Credit Risk at the Trade Level"— Presentation transcript:
1 Pricing Counterparty Credit Risk at the Trade Level Michael PykhtinCredit Analytics & MethodologyBank of AmericaRisk Quant CongressNew York; July 8-9, 2008
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3 IntroductionCounterparty 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 contractIf 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 recoveryIf counterparty risk is not mitigated in any way, counterparty-level exposure equals the sum of contract-level exposuresIf 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 isEach 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 whereE(t) is the counterparty-level exposure at time tt is the counterparty’s default timeR is the counterparty-level recovery rateBt 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 CVAThe fair value of credit risk premium x is calculated fromSee 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 theyreflect trades’ contributions to the counterparty-level CVAsum up to the counterparty-level CVA:Recall that counterparty-level CVA is given bySince both recovery rate R and cumulative PD P(t) are the same for all trades, CVA allocation reduces to EE allocation!
10 EE AllocationFor each future time t, we need to find allocations such that theyreflect trade’s contribution to the counterparty-level discounted EEsum up to the counterparty-level discounted EE:Allocation across netting sets is trivial because whereWe will investigate EE allocation within a netting set
11 Homogeneous ExposureFor 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 isTherefore, exposure is a homogeneous function of weights:
12 Definition of EE Contributions We define EE contribution of trade i at time t asis the counterparty-level EE for portfolio with weightsdescribes the portfolio consisting of one unit of trade idescribes the original portfolio ( for all i )EE contributions sum up to the counterparty-level EE by Euler’s theoremMotivation for this definition comes from allocation of economic capital for loan portfoliossee Chapter 4 in Arvanitis and Gregory (2001) for details
13 EE Contributions for Homogeneous Exposure Counterparty-level EE is given byDifferentiating with respect to and setting , we obtain where V(t) is the portfolio value given byThese 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 anymoreThe 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 EELet us consider a margin agreement and assume that the portfolio value is above the threshold. ThenCounterparty-level exposure equals thresholdInfinitesimal change of the weight of any trade does not change the counterparty-level exposureTherefore, 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 wayCounterparty-level exposure can be written asIt is natural to define stochastic exposure contributions asApplying discounting and conditional expectation, we obtain
16 Margin AgreementsLet us consider a counterparty with a netting agreement supported by a margin agreementUnder a margin agreement, the counterparty must post collateral C(t) whenever portfolio value exceeds the threshold H : where D is the margin period of riskCounterparty-level exposure is given byTo simplify the model, we will set D = 0For 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 asLet us consider three types of scenarios separately:we should setit is reasonable to setCombining all three cases, we obtain exposure contributions
18 EE Contributions with Margin Agreements Applying discounting and conditional expectation, we obtainThese EE contributionssum up to the counterparty-level EEconverge 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 tFor each trade i, calculate trade value Vi (t)Calculate portfolio valueFor each trade i, update its EE contribution counter:if 0 < V(t) ≤ H, add Vi (t) B0/Btif V(t) > H, add Vi (t) H /V(t) B0/BtAfter 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 qualityIf 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 jointlyAssuming 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 jointlyFor each trade i, calculate trade value Vi (tk)Calculate portfolio valueFor each trade i, update the conditional EE contribution counter:if 0 < V(t) ≤ H, addif V(t) > H, add
22 Set-Up for ExamplesIf we assume that all trades’ values are normally distributed, then EE contributions can be evaluated in closed formWe will look at the EE contribution of trade i of value to portfolio, whose value (not including trade i) is given byCorrelation between Xi and X is given by riTo specify the scale, we set for the portfolio
23 No Margin Agreement: Dependence on mi Parameters:
24 No Margin Agreement: Dependence on ri Parameters:
27 SummaryDiscrete marginal approach should be used for pricing counterparty risk in new tradesCVA contributions of existing trades to the counterparty-level CVA can be calculated from the EE contributionsContinuous marginal approach works when counterparty-level exposure is homogeneous function of trades’ weightsScenario-based approach is needed to handle non-homogeneous cases (such as margin agreements)EE contributions can be easily included in the exposure simulating processNormal approximation gives closed-form results
28 ReferencesA. Arvanitis and J. Gregory, 2001, “Credit: The Complete Guide to Pricing, Hedging and Risk Management”, Risk BooksD. Brigo and M. Masetti, 2005, Risk Neutral Pricing of Counterparty Risk in “Counterparty Credit Risk Modelling” (M. Pykhtin, ed.), Risk BooksE. Canabarro and D. Duffie, 2003, Measuring and Marking Counterparty Risk in “Asset/Liability Management for Financial Institutions” (L. Tilman, ed.), Institutional Investor BooksB. De Prisco and D. Rosen, 2005, Modelling Stochastic Counterparty Credit Exposures for Derivatives Portfolios in “Counterparty Credit Risk Modelling” (M. Pykhtin, ed.), Risk BooksE. 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 BooksM. Pykhtin and S. Zhu, 2007, A Guide to Modelling Counterparty Credit Risk GARP Risk Review, July/August, pages