Dynamics of basket hedging
(CreditMetrics for baskets – the “Black-Scholes” of the Credit
Derivatives market)
Galin Georgiev
January, 2000
Disclaimer
This report represents only the personal opinions of the author and not those of
J.P.Morgan, its subsidiaries or affiliates
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Summary

Definition of a protection contract on an individual name and a
first-to-default (FTD) protection contract on two names

The CreditMetrics model for baskets: basic definitions and
variations. The basket as a rainbow digital option.

Greeks and dynamic hedging of baskets. Implied vs. realized
correlation.
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Individual protection contract
Suppose risk-free interest rates are zero and denote by t the present time.
A
A protection contract f (t , T ) maturing at time T on company A entitles
the holder to receive $1 at T if A defaults prior to T (and $0 otherwise).
(In the market place, this is called a zero-coupon credit swap, settled at
maturity, with zero recovery).
Note that f (t , T )  P (t , T ) where the latter is the risk-free probability of
A
default of A up to time T, i.e., f (t , T ) is proportional to the credit spread
of A.
A
A
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FTD protection contract
A first-to-default (FTD) protection contract maturing at time T, on
companies A and B, entitles the holder to receive $1 at T if at least one of
the companies defaults prior to T (and $0 otherwise).
A B
(t , T ) equals the probability
The price f
both defaulting before T.
P A B (t , T )
of A or B or
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The market perspective
While the price of protection for individual credits is (more or less) given by
the credit swaps market, there is no liquid market yet for FTD protection
(or, equivalently, FTD probability).
A B
as a rainbow derivative on f
A
One needs a model to price
and f
It depends on the correlation between the underlying spreads in the nodefault state and the correlation between the corresponding default
events.
f
B
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The CreditMetrics formalism
Assumption:
P A (t , T )  Probab( Z A  Kt ,T )  N ( Kt ,T )
A
A
A
A
where Z is a univariate random normal variable and K t ,T is the socalled threshold (defined above; N (.) is the cumulative normal
distribution).
One can be more specific and define
P A (t , T )  Probab( AT  LT )
A
A
A
A
where AT is the normally distributed firm’s asset level and LT is the
(fixed) firm’s liability level (at time T).
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Asset Distribution at Maturity
Default Probability
Liability
Level
Initial
Asset
Level
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Inconsistencies of the CreditMetrics model
Asset Level
Assets <Liabilities
=> Default
Liability
Level 1
Liability
Level 2
Assets >Liabilities =>
No Default
Time
0
T1
T2
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Assuming for simplicity constant volatility (of the assets), one can rephrase
the price of protection in terms of familiar option theory:
A


X
t
,
T
A

P (t , T )  N 
 T t 


A
where X t ,T is a standard Brownian motion which we call normalized
A
threshold (with initial point X 0,T , depending unfortunately on T). This is
A
nothing else but the price of a barrier option (digital) on the underlying X t ,T
struck at 0 and expiring at T.
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The protection contract as a barrier option
A
f
(t , T ) as a contingent
We can therefore think of the protection contract
claim (barrier option) on the underlying X t ,T
A
A


X
A
t ,T
A
A


f (t , X t ,T )  P (t , T )  N
 T t 


Unsurprisingly, it satisfies the (normal version of) the Black-Scholes
equation:
f A 1  2 f A

0
A 2
t 2 ( X t ,T )
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The FTD protection contract as a rainbow barrier
option
The FTD protection price in this context is
f
A B
(t , T )  P A B (t , T )  Probab( AT  LT orAT  LT )
A
A
B
B
which in terms of normalized thresholds means
f
A B
(where
between
For
(t , X t ,T , X t ,T )  1  N 2 (
A
N 2 (.)
X t ,T
 0
A
B
X t ,T
A
T t
,
is the bivariate normal cumulative and
B
and X t ,T ).
, one has
f
A B
X t ,T
B
T t

, )
is the correlation
 f A f B f Af B
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Black-Scholes for the FTD protection
One can easily compute the Greeks and check that our rainbow contingent
claim satisfies the two-dimensional version of the (normal) Black-Scholes
equation:
f AB 1  2 f AB 1  2 f AB
 2 f AB



0
A 2
B 2
A
B
t
2 ( X ) 2 ( X )
X X
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Hedge ratios
Since the normalized thresholds are not traded, we obviously hedge the
FTD protection (the rainbow barrier option) f A B (t , X A , X B )
with
the two individual protection contracts (1-dim barrier options) f A (t , X A )
B
B
and f (t , X ). The corresponding hedge ratios A , B are easily
computable:
A B
A B
A
B
A


f

f

f

X


X

A 

/

N
 T  t 1  2
f A
X A
X A





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Convexity of the hedged portfolio
The hedged FTD portfolio
  f AB  A f A  B f B
is easily seen to have a negative “off-diagonal” convexity:
 2
 2 f A B

0
A
B
A
B
X X
X X
and positive “diagonal” convexity ( 
 0 ):
2 A
 2
 2 f AB

f
A


0
A 2
A 2
A 2
( X )
(X )
(X )
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FTD basket protection
4.000%
3.500%
3.000%
2.500%
2.000%
1.500%
1.000%
0.500%
2.60%
name B
prote ction
0.20%
3.00%
2.60%
0.80%
2.20%
name A prote ction
1.80%
1.40%
1.40%
1.00%
0.60%
2.00%
0.20%
0.000%
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Convexity seen through the effect of individual
tweaks or parallel tweaks on the hedge ratios of 5
name basket
Effect of Tweaking 1 Name
90%
85%
Hedge Ratio
80%
Tweaked name
75%
Other names
70%
65%
60%
55%
50%
0
50
100
150
200
250
Tweaked Spd (bp)
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Effect of Tweaking 5 Names
90%
Tweaked names
85%
Other names
Hedge Ratio
80%
75%
70%
65%
60%
55%
50%
0
50
100
150
200
250
Tweaked Spd (bp)
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Implied vs. realized correlation
If one buys FTD protection and continuously rehedges, the resulting P&L
is
 XA

XB
P & L   n2 
,
,  (    )d
 T  T 

t
T
where  is the realized correlation and n2 (.,.,.) is the bivariate normal
density. If    , one is long convexity and makes money due to
rehedging (but one pays for it upfront because the money earned by
selling the original hedges is less).
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A correlation contract ?
The P&L due to continuous rehedging of the basket is clearly pathdependent. Similarly to the development of the vol contract in standard
option theory, the time will come to develop a “correlation contract”
whose payoff is path-independent and proportional to realized
correlation.
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