Pricing Counterparty Credit Risk at the Trade Level

Michael PykhtinCredit Analytics & MethodologyBank of America

Risk Quant CongressNew York; July 8-9, 2008

2

Disclaimer

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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

( )iV t0t

( ) max[ ( ),0]i iE V

( )iE 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

( ) ( ) max[ ( ),0]i ii i

E t E t V t

NSNS

( ) ( ) max ( ), 0k

k

ik k i

E t E t V t

NSk

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

– 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

0CVA E (1 ) ( )1 T

BR E

B

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

0

ˆCVA (1 ) ( ) ( )T

R dP t e t ( )P t

0ˆ ( ) E ( )te t B B E t t

0ˆ ( ) ( ) E ( )te t e t B B E t

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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.

TradeCVA CVA(Portfolio Trade) CVA(Portfolio)

Trade Trade Trade( ) CVA ( ) ( 0)V x x x x V x

x

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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!

0

ˆCVA (1 ) ( ) ( )T

R dP t e t

CVA CVAii

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

ˆ ˆ( ) ( )ii

e t e t

ˆ ( )ie t

ˆ ( )e t

NS NSˆ ˆ( ) ( ) ( ) ( )k k

k k

E t E t e t e t

0NS NSˆ ( ) E ( )

k k

t

Be t E t t

B

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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” i to trade i so that:

Exposure of an “adjusted” portfolio is

Therefore, exposure is a homogeneous function of weights:

( ) max ( ), 0ii

E t V t

( , ) ( )i i i iV t V t

( , ) ( , )E c t cE t

( , ) max ( ), 0i ii

E t V t

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

01

ˆ ˆ ˆ( ,1 ) ( ,1) ( , )ˆ ( ) lim i

ii

e t u e t e te t

ˆ ( )ie t

1 ii

u

iu

ˆ ( , )e t

1i

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!

0

( ) 0ˆ ( ) E ( ) 1i i

tV t

Be t V t t

B

0ˆ ( , ) E max ( ),0i iit

Be t V t t

B

i

( ) ( )ii

V t V t

1

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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

( ) if ( ) 0( )

0 otherwise

iiV t V t

E t

( ) if ( ) 0( )

0 otherwisei

i

V t V tE t

0 0

( ) 0ˆ ( ) E ( ) E ( ) 1i i i

t tV t

B Be t E t t V t t

B B

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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 is the margin period of risk

Counterparty-level exposure is given by

To simplify the model, we will set = 0 – For liquid trades, typical value of is 2 weeks, and the error in EE

resulting from setting = 0 is small

( ) max ( ) ( ),0E t V t C t

( ) max ( ) ,0C t V t H

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Scenario Approach with Margin Agreements

After setting = 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

( ) 0 ( ) 0 :V t E t

0 ( ) ( )( ) ( )1 ( ) 1

( )i i iV t H V t H

HE t V t V t

V t

0 ( ) ( )( ) 1 ( ) 1

V t H V t HE t V t H

( ) 0iE t

0 ( ) ( ) ( ) :iiV t H E t V t ( ) ( )i iE t V t

( ) ( ) :V t H E t H ( ) ( ) ( )i iE t V t H V t

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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 H

0

0

0 ( )

( )

ˆ ( ) E ( ) 1

E ( ) 1( )

i it

it

V t H

V t H

Be t V t t

B

B HV t t

B V t

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Calculating EE Contributions

Let us assume that exposure is independent of the counterparty credit quality. Then, conditioning on = 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

( ) ( )iiV t V t

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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 (t)

Then, conditional expectation of quantity X can be calculated as

where is the first derivative of the cumulative PD P(t)

0

1E E ( )exp[ ( ) ]

( )

t

X t t s ds XP t

( )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 (tj) for all simulation times j < k, the simulation algorithm for tk might look like this:

– Simulate market factors and intensity (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

( ) ( )iiV t V t

01 1

1

1( )exp ( )( ) ( )

( )k

k

k j j j i kjk t

Bt t t t V t

P t B

01 1

1

1( )exp ( )( ) ( )

( ) ( )k

k

k j j j i kjk t k

B Ht t t t V t

P t B V t

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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 i

To specify the scale, we set for the portfolio

( ) ( ) ( )i i i iV t t t X

( ) ( ) ( )V t t t X

( ) 1t

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No Margin Agreement: Dependence on i

Parameters: 0.05, 0i i

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

4 2 1 0 1 2

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No Margin Agreement: Dependence on i

Parameters: 0.05, 0i i

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

4 2 1 0

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Margin Agreement: Dependence on i

Parameters: 0.05, 0, 0.5i i

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

H=inf H=1 H=0.50 H=0.25 H=0.10

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Margin Agreement: Dependence on i

Parameters: 0.05, 0, 0.5i i

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

H=inf H=1 H=0.50 H=0.25 H=0.10

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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 16-22.