Transcript Document

Counterparty Risk
Advanced Methods of Risk Management
Umberto Cherubini
Learning Objectives
• In this lecture you will learn
1. The difference between futures-style and
Over-the-Counter markets.
2. The Credit Valuation Adjustment (CVA) of
derivative transactions (linear/non linear)
3. The impact of dependence between risk of
the underlying asset and default of the
counterparty.
OTC vs Futures-style
• Over-the-Counter
• Bilateral relationship
• Customized
products
• Low basis risk
• Low liquidity
• Relevant risk
– Market
– Counterparty
• Futures-style
• Organized market
• Standardized
products
• High basis risk
• High liquidity
• Relevant risks
– Market
– Basis risk
Derivatives on OTC markets
• Most of financial derivative contracts, and
particularly those with retail counterparties
are traded on what is called Over-the-Counter
(OTC) market
• The OTC market allows the construction of
customized positions for hedging or
investment purposes
• The cost is illiquidity and credit risk
(counterparty risk)
The simplest example
• Consider a lineare OTC contract, i.e. forward,
determined at time 0.
• Remember that if we only focus on the risk of
price changes in the underlying asset we
have
CF(t) = v(t,T)EQ[S(T) –F(0)]
= S(t) – P(t,T)F(0)
where F(0) is the forward price at time 0.
• Notice that the product is linear, meaning
delta = 1 and the replicating portfolio is static.
Counterparty risk
• We make the assumption that: default occurs at time
T, which is the maturity of the contract: this is a
simplifying assumption that will be relaxed later one.
• We assume that if at maturity the marked-to- market
value of the derivative contract is positive for the
counterparty which goes in default, the other party is
compelled to abide by the contract and pay its
obligation. On the other side, if the value of the
contract is negative for the counterparty in default the
other party has an exposure equal to that value, with
the same degree of seniority of the other liabilities.
This assumption corresponds to the reality of legal
provisions of counter party risk.
Long and short positions
• The value of the impact of counter party risk requires
to distinguish the sign, long or short of the position.
This is because counterparty risk is triggered by two
events:
– Default of the counterparty
– The contract is “out-of-the-money” for the party in
dafault, that it the contract has negative value for
the counterparty in default.
• So, in case on the delivery date we have S(T) > F(0)
the contract is in-the-money for the party long in the
contract. If instead it is S(T) < F(0) the contract is inthe-money for the short party. In the former case the
long party in the contract will be exposed to default
risk, in the latter the short one will.
Default and loss
• Denote A the long party of the contract and B
the short one.
• Let us introduce characteristic functions 1A
and 1B assuming value 1 if the party A or B is
in a default state and zero otherwise.
• Definiamo RRA e RRB i tassi di recupero delle
due controparti. Nello stesso modo definiamo
le loss-given-default LgdA = 1 – RRA e LgdB =
1 – RRB.
Risk of the long party
• The pay-off value of the forward contract
must take into account both its sign and its
value in caso of default of the relevant
counterparty.
• From the viewpoint of the long end of the
contract we have
CFA(T) = max[S(T) – F(0),0](1 –1B) +
max[S(T) – F(0),0]RRB1B –
– max[F(0) –S(T),0] =
CF(T) – LgdB1Bmax[S(T) – F(0),0]
Risk of the short party
• For the short end of the contract, the default
event is relevant only in the hypothesis that
the contract ends in-the-money.
• From the viewpoint of the counterparty
CFB(T) = max[F(0) – S(T),0](1 –1A) +
max[F(0) – S(T),0]RRA1A –
– max[S(T) – F(0),0] =
– CF(T) – LgdA1Amax[F(0) – S(T),0]
Counterparty risk
• Counterparty risk corresponds to a short
position is options.
• The option is of the call type for the long
endo of the contract and of the put type for
the other end of the contract.
• Exercise of the option is contingent on two
events
– The value of the underlying asset at time T
– Default event of the relevant counterparty
Contract evaluation
• The value of the product from the point of
view of the long end of the contract will be
given by
CFA(t) = S(t) – v(t,T)F(0) –
EQ[v(t,T)LgdB1Bmax[S(T) – F(0),0]]
• From the viewpoint of the short end of the
contract we will then have
CFB(T) = – S(t) + v(t,T)F(0) –
EQ[v(t,T)LgdA1Amax[F(0) – S(T),0]]
Risk factors
• Counterparty risk is represented by
EQ[v(t,T)Lgdi1imax[(S(T) – F(0)),0]]
with i = A, B and  = 1(–1) for call (put)
options
• Counterparty risk is made by
–
–
–
–
Interest rate risk
Market risk of the underlying
Credit risk of the counterparty
Recovery risk
• All these factors may be correlated.
A simple model
• In what follows we will assume that
– Interest rate is independent of the other
risk factors
– Default risk of the counterparty is not
dependent on other risk factors.
Evaluation
• The value of counterparty risk is then
v(t,T)EQ[Lgdi1i] EQ[max[(S(T) – F(0)),0]]
• Notice tha in case of a zero-coupon-bond issued by
the party i we have
Di = v(t,T) – v(t,T)EQ[Lgdi1i],
or
Di = v(t,T) – v(t,T)ELi,
with ELi = EQ[Lgdi1i] the expected loss.
• In case of independence, then
ELi v(t,T) EQ[max[(S(T) – F(0)),0]
Effects of counterparty risk
• Effect 1: ruling out counterparty risk leads to
undervaluation of the overall exposure to credit risk
• Effect 2: if one does not consider counterparty risk,
he comes out with the wrong price, and the wrong
hedge.
• Effect 3: counterparty risk makes linear product non
linear, so that changes in volatility may affect the
value of the contract even though it is linear and one
would not expect any effect.
Greek letters
• The sensitivity of the contract to small
changes in the underlying is no more that of a
linear contract. We get in fact
A = 1 – ELBN()
B = – 1 + ELAN(– )
with
 =(ln(S(t)/F(0))+(r + ½ 2)(T – t))/[(T – t)1/2 ]
Gamma and Vega
• The second order effect of finite changes in the
underlying is now given by
– n()/ [S(t)(T – t)1/2]
• Changes in volatility affect the value of the
position through a vega effect
– S(t)n()/ [(T – t)1/2 ]
 =(ln(S(t)/F(0))+(r – ½ 2)(T – t))/[(T – t)1/2 ]
An example
• Forward contract
– Notional 1 million
– Volatility 20%
– Maturity 1 year
• Counterparty
– Loss given default (Lgd): 100%
– 1 year default probability: 5%
• Counterparty risk at the origin of the contract,
for both the long and the short end of the
contract: 3983
Long position
500000
400000
300000
200000
100000
0
0,5
-100000
-200000
-300000
-400000
-500000
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
Long position delta
1,01
1
0,99
0,98
0,97
0,96
0,95
0,94
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
Counterparty risk (long)
30000
25000
20000
15000
10000
5000
0
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
Short position
500000
400000
300000
200000
100000
0
0,5
-100000
-200000
-300000
-400000
-500000
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
Short position delta
-0,94
0,5
-0,95
-0,96
-0,97
-0,98
-0,99
-1
-1,01
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
Counterparty risk (short)
30000
25000
20000
15000
10000
5000
0
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
Counterparty risk and vol. (long)
30000
25000
20000
0,1
0,2
0,3
0,4
15000
10000
5000
0
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
Counterparty risk and vol. (short)
30000
25000
20000
0,1
0,2
0,3
0,4
15000
10000
5000
0
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
Default before maturity
• Assume now that default may occur before maturity,
for example by a time .
• The value of exposure for the long position is now
max[S() – P( ,T)F(0), 0 ]
and for the short position
max[P( ,T)F(0) – S(), 0 ]
• The value of exposure is given by a sequence of
options that will be multiplied times the value of the
default probability of the counterparty in the subperiods.
Counterparty risk
• Partition the lifetime of the contract in a grid of
dates {t1,t2,…tn}
• Denote Gj(ti) the survival probability of
counterparty j = A, B beyond time ti.
• Compute
[GB(ti -1) – GB(ti) ]Call(S(ti), ti; P(ti ,T)F(0), ti )
[GA(ti -1) – GA(ti) ] Put(S(ti), ti; P(ti ,T)F(0), ti )
respectively for long and short positions
• Aggregate the values obtained in this way
from 1 through n.
Counterparty risk in swap contracts
• In a swap cotnract both the legs are exposed to
counterparty risk.
• In the event of default of one of the two parties the
other takes a loss equal to the marked to market
value of the contract, equal to the net value of the
cash-flows.
• Remember that the net value of the swap contract is
positive for the long end of the contract if the swap
rate on the day of default of the contract is greater
than the rate on the origin of the contract.
Swap counterparty risk exposure
• Assume the set of dates at which swap payments
are made be {t1, t2,…, tn} and default of the
counterparty that receives fixed payments (B) took
place between time tj-1 and tj. In this case, the loss
for the party paying fixed is given by
Lgd B  Pt , ti 1  max sr t j , tn   k ,0
n -1
i j
where sr is the swap rate at time tj and k is the swap
rate at the origin of the contract. Notice that this is
the payoff of a payer swaption (a call option on a
swap).
Swap counterparty risk exposure
• Assume the set of dates at which swap payments
are made be {t1, t2,…, tn} and default of the
counterparty that pays fixed payments (A) took place
between time tj-1 and tj. In this case, the loss for the
party receiving fixed is given by
Lgd A  Pt , ti 1  max k  sr t j , tn ,0
n -1
i j
where sr is the swap rate at time tj and k is the swap
rate at the origin of the contract. Notice that this is
the payoff of a receiver swaption (a put option on a
swap).
Credit risk: long party
Vulnerable Call Swaptions: Financial Institution Paying Fixed
0,012
0,01
0,008
Independence
Perfect positive dependence
0,006
0,004
0,002
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Swap credit risk: Baa
Correl.
Rho
0
0,25
0,5
0,75
1
Correl.
Rho
0
0,25
0,5
0,75
1
5years
0,0259%
0,0536%
0,0813%
0,1090%
0,1367%
5years
0,0040%
0,0030%
0,0020%
0,0010%
0,0000%
10years
0,0686%
0,1448%
0,2210%
0,2971%
0,3733%
10years
0,0150%
0,0113%
0,0075%
0,0038%
0,0000%
Fixed-Payer
15years
20years
0,0935%
0,2178%
0,3420%
0,4663%
0,5905%
0,1141%
0,2574%
0,4007%
0,5440%
0,6873%
Fixed-Receiver
15years
20years
0,0355%
0,0266%
0,0177%
0,0089%
0,0000%
0,0429%
0,0322%
0,0215%
0,0107%
0,0000%
25years
0,1385%
0,3056%
0,4726%
0,6397%
0,8068%
25years
0,0491%
0,0368%
0,0245%
0,0123%
0,0000%
30years
0,1678%
0,3658%
0,5637%
0,7617%
0,9597%
30years
0,0556%
0,0417%
0,0278%
0,0139%
0,0000%
Dependence structure
• A more general approach is to account
for dependence between the two main
events under consideration
– Exercise of the option
– Default of the counterparty
• Copula functions can be used to
describe the dependence structure
between the two events above.
Vulnerable digital call option
• Consider a vulnerable digital call (VDC)
option paying 1 euro if S(T) > K (event A). In
this case, if the counterparty defaults (event
B), the option pays the recovery rate RR.
• The payoff of this option is
VDC = v(t,T)[H(A,Bc)+RR H(A,B)]
= v(t,T) [Ha – H(A,B)+RR H(A,B)]
= v(t,T)Ha – (1 – RR)H(A,B)
= DC – v(t,T) Lgd C(Ha, Ha)
Vulnerable digital put option
• Consider a vulnerable digital put (VDP) option paying 1
euro if S(T) ≤ K (event Ac). In this case, if the counterparty
defaults (event B), the option pays the recovery rate RR.
• The payoff of this option is
VDP = DP – v(t,T)(1 – RR)H(Ac,B)
= P(t,T)Ha – v(t,T)(1 – RR)H(Ac,B)
= P(t,T)Ha – v(t,T)(1 – RR)[Hb – C(Ha, Hb)]
= v(t,T)(1 – Ha) – v(t,T) Lgd [Hb – C(Ha, Hb)]
= v(t,T) – VDC – v(t,T) Lgd Hb
Vulnerable digital put call parity
• Define the expected loss EL = Lgd Hb.
• If D(t,T) is a defaultable ZCB issued by the
counterparty we have
D(t,T) = v(t,T)(1 – EL)
• Notice that copula duality implies a clear noarbitrage relationship
VDC + VDP = v(t,T) – v(t,T) EL = D(t,T)
• Buying a vulnerable digital call and put option
from the same counterparty is the same as
buying a defaultable zero-coupon bond
Vulnerable call and put options

VC S , t : K , T    VDC  d 
K


K
K
  DC  d  vt , T Lgd  C 1  Q , H b d 

 C S , t : K , T   vt , T Lgd  C 1  Q , H b d
K

VPS , t : K , T   PS , t : K , T   vt , T Lgd  C Q , H b d
K

C  H bC 1  Q , H b 
0
Vulnerable put-call parity

VPS , t : K , T   S t   PS , t : K , T   S t   vt , T Lgd  C Q , H b d 
K
0

 C S , t : K , T   Kvt , T   Pt , T Lgd  C Q , H b d
K
0
K
 C S , t : K , T   Kvt , T 1  EL   vt , T Lgd  C 1  Q , H b d
0

 VC S , t : K , T   KDt , T   vt , T Lgd  C 1  Q , H b d
0
Credit risk mitigation
• Several techniques are used on the market to
mitigate countrerparty risk. The ispiration of these
techniques is the structure of futures-style, markets,
based on three key principles
– Margins
– Evaluation (marking-to-market) and settlemen t of
profits and losses before maturity of the contract.
– Compensation of profits and losses on different
positions
• Risk mitigation clauses make more the computation
of CVA more involved. Unfortunately, there is not
much literature on the subject.
CRM: theory
•
In principle one can think of different
techniques to mitigate counterparty risk
1. Margin deposit at the origin of the contract
2. Position evaluation at daily on weekly period and
requirement of the payment of a collateral.
3. Netting agreement so that in case of default the
net exposure between the counterparty is
liquidated.
CRM: practice
• According to the so-called ISDA Agreement
the credit mitigating techniques used apply
netting and the Credit Annex requiring
periodic marking-to-market of the
exposures.
• Unfortunately, there is no evidence on the
diffusion of these techniques in the market
practice (for example) it seems that
Goldman Sachs did not use them.
A simple example
• Assume a counterparty A with CFi positions in
forward contracts i = 1, 2,…,p, with delivery
prices Fi and delivery dates Ti with the same
counterparty B.
• The value of each position is
CFi = [Si(t) – v(t,Ti)Fi]
where  = 1 represents long positions and 
= – 1 short positions.
CVA with netting
• Assume that the counterparty B get into default
at time . The value of the exposure at that date
is equal to the pay-off of a basket option
 p

max  Si    A ,0
 i 1

p
A    v , Ti Fi
i 1
Monte Carlo simulation
• As it is well known basket options can only be
evaluated by Monte Carlo simulation.
• The idea is then to select a grid of dates {t1,t2,…tn}
and for each one of these to evaluate a basket
option, with strike A(ti). CVA is now computed, for
each date, as
[G(ti-1) – G(ti)]Basket Option(S1, …Sp, ti; A(ti), ti)
where G(ti) is the survival probability of counterparty
beyond time ti.
• Extension of the use of collateral occur according to
the same lines as in the univariate exposure.