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**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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Disclaimer This document is NOT a research report under U.S. law and is NOT a product of a fixed income research department. Opinions expressed here do not necessarily represent opinions or practices of Bank of America N.A. The analyses and materials contained herein are being provided to you without regard to your particular circumstances, and any decision to purchase or sell a security is made by you independently without reliance on us. This material is provided for information purposes only and is not an offer or a solicitation for the purchase or sale of any financial instrument. Although this information has been obtained from and is based on sources believed to be reliable, we do not guarantee its accuracy. Neither Bank of America N.A., Banc Of America Securities LLC nor any officer or employee of Bank of America Corporation affiliate thereof accepts any liability whatsoever for any direct, indirect or consequential damages or losses arising from any use of this report or its contents.

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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).

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**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

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**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

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**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

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**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

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**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.

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**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!

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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

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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:

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**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

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**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!

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**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!

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**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

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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

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**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

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**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

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**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

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**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)

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**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

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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

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**No Margin Agreement: Dependence on mi**

Parameters:

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**No Margin Agreement: Dependence on ri**

Parameters:

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**Margin Agreement: Dependence on mi**

Parameters:

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**Margin Agreement: Dependence on ri**

Parameters:

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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

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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

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