Michael Pykhtin

Report
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
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
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or its contents.
2
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).
3
Exposure at Contract Level
 Market value of contract i with a counterparty is known only for
current date t  0. For any future date t, this value Vi (t ) is
uncertain and should be assumed random.
 prior to the contract
maturity, maximum economic loss equals the replacement cost
of the contract
 If the counterparty defaults at time
– 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.
Ei ( )  max[Vi ( ),0]
 Quantity Ei (t ) is known as contract-level exposure at time t
4
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
E (t )   Ei (t )   max[Vi (t ),0]
i
i
 If there are netting agreements, derivatives with positive value
at the time of default offset the ones with negative value within
each netting set NSk , so that counterparty-level exposure is


E (t )   ENSk (t )   max   Vi (t ), 0 
k
k
iNSk

– Each non-nettable trade represents a netting set
5
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


B0
CVA  E  1 T  (1  R)
E ( ) 
B


– E(t) is the counterparty-level exposure at time t
6
– 
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
CVA and Expected Exposure
 Assuming constant recovery rate R, we can write
T
CVA  (1  R)  dP(t ) eˆ(t )
0
where P(t ) is the risk neutral cumulative probability of
default (PD) between today (time 0) and time t
eˆ(t )  E   B0 Bt  E (t )  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
eˆ(t )  e(t )  E   B0 Bt  E (t ) 
7
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
 CVA Trade  CVA(Portfolio  Trade)  CVA(Portfolio)
 The fair value x of credit risk premium x is calculated from
VTrade( x  x )   CVA Trade( x  x )  VTrade( x 0)
 See Chapter 6 in Arvanitis and Gregory (2001) for details.
8
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:
CVA   CVAi
i
 Recall that counterparty-level CVA is given by
T
CVA  (1  R)  dP(t ) eˆ(t )
0
 Since both recovery rate R and cumulative PD P(t) are the
same for all trades, CVA allocation reduces to EE allocation!
9
EE Allocation

 For each future time t, we need to find allocations eˆi (t ) such
that they
– reflect trade’s contribution to the counterparty-level discounted EE eˆ(t )
– sum up to the counterparty-level discounted EE: eˆ(t )   eˆi (t )
i
 Allocation across netting sets is trivial because
E (t )   ENSk (t )


eˆ(t )   eˆNS
(t )
k
k
where
k
 B0

eˆNSk (t )  E 
ENSk (t )  t 
 Bt


 We will investigate EE allocation within a netting set
10
Homogeneous Exposure
 For convenience, we will assume that all trades with a
counterparty belong to the same netting set:


E (t )  max  Vi (t ), 0 
 i

 Let us assign a “weight” ai to trade i so that:
Vi (a i , t )  a iVi (t )
 Exposure of an “adjusted” portfolio is


E (a , t )  max  a iVi (t ), 0 
 i

 Therefore, exposure is a homogeneous function of weights:
E(ca , t )  cE(a , t )
11
Definition of EE Contributions

 We define EE contribution eˆi (t ) of trade i at time t as
eˆi(t )  lim
eˆ(t ,1    ui )  eˆ(t ,1)
 0

 eˆ(t ,a )

a i a 1

– eˆ (t , a ) is the counterparty-level EE for portfolio with weights a
– ui describes the portfolio consisting of one unit of trade i
– 1   ui
describes the original portfolio ( ai  1 for all i )
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
12
EE Contributions for Homogeneous Exposure
 Counterparty-level EE is given by
 B0



eˆ (a , t )  E  max a iVi (t ),0    t 
 i

 Bt


 Differentiating with respect to
a i and setting a  1 , we obtain
 B0

eˆi (t )  E  Vi (t ) 1V (t )0   t 
 Bt


where V(t) is the portfolio value given by
V (t )  Vi (t )
i
 These EE contributions sum up to the counterparty-level EE!
13
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!
14
Scenario Approach to EE Contributions
 Let us obtain the EE contributions in an alternative way
 Counterparty-level exposure can be written as

 i Vi (t ) if V (t )  0
E (t )  
otherwise

 0
 It is natural to define stochastic exposure contributions as
Vi (t )
Ei (t )  
 0
if V (t )  0
otherwise
 Applying discounting and conditional expectation, we obtain
 B0

 B0

eˆi (t )  E  Ei (t )   t   E  Vi (t ) 1V (t )0   t 
 Bt

 Bt


15
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 :
C (t )  max V (t  )  H ,0
where  is the margin period of risk
 Counterparty-level exposure is given by
E (t )  max V (t )  C (t ),0
 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
16
Scenario Approach with Margin Agreements
 After setting  = 0 , exposure can be written as
E (t )  10V (t ) H V (t )  1V (t )  H  H
 Let us consider three types of scenarios separately:
– V (t )  0  E (t )  0 : we should set Ei (t )  0
– 0  V (t )  H  E (t )   i Vi (t ) : we should set Ei (t )  Vi (t )
– V (t )  H  E (t )  H : it is reasonable to set Ei (t ) Vi (t ) H V (t )
 Combining all three cases, we obtain exposure contributions
H
Ei (t )  Vi (t )10V (t ) H   Vi (t )
1V (t ) H 
V (t )
17
EE Contributions with Margin Agreements
 Applying discounting and conditional expectation, we obtain
 B0

eˆi (t )  E  Vi (t ) 10V (t ) H    t 
 Bt


 B0

H
 E  Vi (t )
1V (t ) H    t 
V (t )
 Bt

 These EE contributions
– sum up to the counterparty-level EE
– converge to the EE contributions for the non-collateralized case
in the limit H  
18
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 V (t )   i Vi (t )
– 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
19
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
t


1
E  X  t  
E l (t ) exp[  l ( s)ds] X 
P(t ) 
0

where P(t ) is the first derivative of the cumulative PD P(t)
20
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 V (t )   i Vi (t )
– For each trade i, update the conditional EE contribution counter:
 if 0 < V(t) ≤ H, add
 if V(t) > H, add
21
 k
B
1
l (tk )exp   l (t j 1 )(t j  t j 1 )  0 Vi (tk )
P(tk )
 j 1
 Btk
 k
 B0
1
H
l (tk )exp   l (t j 1 )(t j  t j 1 )  Vi (tk )
P(tk )
V (tk )
 j 1
 Btk
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
Vi (t )  i (t )   i (t ) X i
to portfolio, whose value (not including trade i) is given by
V (t )   (t )   (t ) X
 Correlation between Xi and X is given by ri
 To specify the scale, we set
22
 (t )  1 for the portfolio
No Margin Agreement: Dependence on i
 Parameters:
4
 i  0.05, ri  0
2
1
0
1
2
0.5
0.4
0.3
0.2
0.1
0.0
-0.5
-0.4
-0.3
-0.2
-0.1
-0.1
-0.2
-0.3
-0.4
-0.5
23
0.0
0.1
0.2
0.3
0.4
0.5
No Margin Agreement: Dependence on ri
 Parameters:
 i  0.05, i  0
4
2
1
0
0.025
0.020
0.015
0.010
0.005
-1.0
-0.8
-0.6
-0.4
0.000
-0.2
0.0
-0.005
-0.010
-0.015
-0.020
-0.025
24
0.2
0.4
0.6
0.8
1.0
Margin Agreement: Dependence on i
 Parameters:
 i  0.05, ri  0,   0.5
H=inf
H=1
H=0.50
H=0.25
H=0.10
0.5
0.4
0.3
0.2
0.1
0.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
-0.1
-0.2
-0.3
25
0.1
0.2
0.3
0.4
0.5
Margin Agreement: Dependence on ri
 Parameters:
 i  0.05, i  0,   0.5
H=inf
H=1
H=0.50
H=0.25
H=0.10
0.020
0.015
0.010
0.005
-1.0
-0.8
-0.6
-0.4
0.000
-0.2
0.0
-0.005
-0.010
-0.015
-0.020
26
0.2
0.4
0.6
0.8
1.0
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
27
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.
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