Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.1
Chapter 27
Credit Derivatives
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.2Credit Default Swap
? Company A buys default protection from B to
protect against default on a reference bond
issued by the reference entity,C.
? A makes periodic payments to B
? In the event of a default by C
– A has the right to sell the reference bond to B for
its face value,or
– B pays A the difference between the market value
and the face value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.3
CDS Structure
Default
Protection
Buyer,A
Default
Protection
Seller,B
90 bps per year
Payment if default by
reference entity,C
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.4CDS Final Payments
Notation:
L,Face value of bond,notional value of CDS
A(t),Accrued interest on bond per $ of principal at time
t
R,Recovery rate,market price as a percent of face
value plus accrued interest
s,CDS payment rate per year,Annual payment = sL
?,Time since last CDS payment
A pays ?sL and B pays L - RL[1 + A(t)]
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.5
Sample Quotes (Jan 2001)
Company Rating 3yr 5yr 7yr 10yr
Toyota Aa1/AAA 16/24 20/30 26/37 32/53
Merrill Lynch Aa3/AA- 21/41 40/55 41/83 56/96
Ford A+/A 59/80 85/100 95/136 118/159
Enron Baa1/BBB+ 105/125 115/135 117/158 182/233
Nissan Ba1/BB+ 115/145 125/155 200/230 244/274
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.6CDS Valuation
T, Life of cr edit defa ult sw ap
p
i
,Risk - ne utral default prob a bilit y densit y at t i m e t
i
u ( t ),Prese nt value o f $ 1 p e r year on p ayment date s
between ti m e z ero and ti m e t
e ( t ), Prese nt value o f a n ac crual pay m ent at ti me t
v ( t ),Prese nt value o f $ 1 r ece ived at t i m e t
w, To tal pa yments per year m ade by CDS buy er
s, Value of w for wh ic h C DS value is ze ro
?, Risk - ne utral pro ba bilit y o f n o credit even t
du ri ng t he lif e of t he s wap
A ( t ),Accrued int ere st on the r efe rence obligatio n at
ti m e t as a p erc ent of fac e value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.7PV of CDS Payments per $1 of
Notional
? If default event occurs at t < T,PV of
payments is
? If no default event,PV of payments is
? Expected PV is
?
?
???
n
i
iii Tuwtetupw
1
)()]()([
)]()([ tetuw ?
)(Twu
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.8PV of CDS Costs per $1 of
Notional Principal
? If default event occurs at t < T cost is
? Expected cost is
RtARRtA ?)(?1]?)(1[1 ?????
)(]?)(?1[
1
ii
n
i
i tvpRtAR?
?
??
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.9
Value of CDS to Buyer
? Value is expected PV of payments less
expected PV of costs
)(])()([
)(]?)(?1[
1
1
Tutetupw
tvpRtAR
n
i
iii
n
i
iii
????
??
?
?
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.10
CDS Rate
CDS rate sets value to zero
?
?
?
?
???
??
?
n
i
iii
n
i
iii
Tutetup
tvpRtAR
s
1
1
)()]()([
)(]?)(?1[
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.11
CDS Rate continued
When default can happen at any time this
becomes
? ? ? ? ? ?
? ? ? ? ? ? ? ?
0
0
垐1T
T
q t v t R A t R d t
s
q t u t e t d t u T?
????
???
??????
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.12
Approximate CDS Spread
? Let
– y be the yield on bond issued by reference entity
with maturity T
– x be the yield on risk-free bond with maturity T
– a be average value of A(t)
– a* be average value for A(t) if reference bond is a
par-yield bond with maturity T
? ?
? ? ? ?
垐1
?1 1 *
y x R aR
s
Ra
??? ? ?
???
??
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.13
Alternative Uses of the Formula
? To calculate CDS spreads from the
probabilities of default and expected
recovery rate
? To bootstrap the probabilities of default
from CDS spreads and expected
recovery rates
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.14
Sensitivity to Recovery Rate
? Vanilla CDS is not very sensitive to the
recovery rate providing the same
recovery rate is used to estimate default
probabilities and calculate payoffs
? Binary swaps,which provide a fixed
payoff in the event of a default,are
much more sensitive to recovery rates
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.15
First-to-default swaps
? Similar to a regular CDS
? Several reference entities and reference
bonds
? First entity to default triggers a payoff
? Settlement is same as ordinary CDS
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.16
Valuation
? Must use Monte Carlo simulation
? Each reference entity is simulated to
determine when if ever it defaults
? Valuation is sensitive to default correlation
? A conservative (and easy) assumption for the
seller is that all correlations are zero
? The easiest way to build in non-zero
correlations is with the Gaussian copula
model
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.17
Seller Default Risk
? The impact of seller default risk on a
CDS swap can be calculated by jointly
simulating the reference entity and the
seller
? Suppose Y=PV of payoff and C is PV of
payments
? What rules should the simulation have
for calculating Y and C?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.18Total Return Swap
? Company A agrees to pay B the total return
earned on a reference bond issued by the
reference entity,C,over some period of time.
? Total return includes all coupon payments
and any change in the price of the reference
bond,(Usually the latter is made at the end)
? B pays A LIBOR plus a spread on a notional
equal to the initial value of the reference
bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.19
The Structure
Total Return
Payer
Total Return
Receiver
Total Return on Bond
LIBOR plus 25bps
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.20
Uses of a TRS
? Total Return Swaps are usually used a
financing vehicles
? Receiver wants to invest in bond
? Payer (a financial institution) buys the
bond and agrees to the swap
? Payer has less credit exposure than if it
had lent Receiver money to buy bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.21
Valuation of TRS
? If there were no risk of default by
receiver,the value of a TRS would be
difference between value of reference
bond and value of LIBOR bond
? The spread above LIBOR would be zero
? In practice the payer loses money if the
receiver defaults at a time when the
bond value has declined
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.22
Credit Spread Options
? These provide a payoff dependent on
movements in a particular credit spread.
? There is usually no payoff in the event of a
default on the reference asset
? Payoff may be defined in terms of difference
between actual spread and a strike spread or
in terms of the difference between the price of
an FRN and a strike price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.23
Valuation
? European options can be valued using
Black’s model
? This assumes that,conditional on no default,
spread or FRN price is lognormal
? Need a volatility for forward credit spread or
forward FRN price
? Must multiply Black’s formula by risk-neutral
probability of no default
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.24
Collateralized Debt Obligation
? A pool of debt issues are put into a special
purpose trust
? Trust issues claims against the debt in a
number of tranches
– First tranche covers x% of notional and absorbs first
x% of default losses
– Second tranche covers y% of notional and absorbs
next y% of default losses
– …
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.25
Bond 1
Bond 2
Bond 3
?
Bond n
Average Yield
8.5%
Trust
Tranche 1
1st 5% of loss
Yield = 35%
Tranche 2
2nd 10% of loss
Yield = 15%
Tranche 3
3rd 10% of loss
Yield = 7.5%
Tranche 4
Residual loss
Yield = 6%
Collateralized Debt Obligation
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.26
CDOs continued
? Note that average yield on tranches
equals average yield on bonds less fee
taken by trust manager
? Often trust manager holds first tranche
0, 0 5 3 5 % 0, 1 0 1 5 % 0, 1 0 7, 5 % 0, 7 5 6 %
b o n d s i i
i
y w y?
? ? ? ? ? ? ? ?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.27
CDO Applications
? Can provide a range of credit quality
debt objects
? Can create high quality debt from low
quality debt
? Can create high yield debt from average
risk debt
? Can create artificial short by selling
tranches before buying bonds
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.28
Valuing CDO Tranches
? Depends on default correlation of bonds
in portfolio
? Must use Monte Carlo simulation
? It is easiest to handle the default
correlation with the Gaussian copula
model
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.29Quantifying the Cost of Default
on a stand-alone derivatives
contract
Two Categories of Derivatives:
? Those that are always assets to one party
and liabilities to the other (e.g.,options)
? Those that can become assets or liabilities
(e.g.,swaps,forward contracts)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.30
Independence Assumption
? The independence assumption states
that the variables affecting the price of a
derivative are independent of the
variables determining defaults
? This assumption (although not perfect)
makes pricing for default risk possible
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.31
Notation
der i v at i v e of v al u e ac t u al
i m po s s i bl e ar e def au l t s
as su m i n g t i m e at der i v at i v e of v al u e
:)(
:)(
*
tf
ttf
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.32
Contracts that are Assets
TTyTyeff )]()([* *)0()0( ???
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.33
A Simple Interpretation
? Use the,risky” discount rate rather than
the risk-free discount rate when
discounting cash flows in a risk-neutral
world
? Note that this does not mean we simply
increase the interest rate in option
pricing formulas
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.34Credit Exposure for Contracts
That Can be Assets or Liabilities
Exposure
Contract value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.35
The Impact of Defaults
Rt
t
ut
vnit
vuff
i
i
ii
ii
n
i
ii
?
),1
)0()0(
1
1
*
1-b y m u l t i p l i e d
a n d t i m e s b e t w e e n d e f a u l t of p r o b a i l i t y
t h e is a n d t i m e at h a p p e n s d e f a u l t
a if cl a i m e x p e ct e d t h e is (
t i m e s at p l a ce t a ke can d e f a u l t s w h e r e
?
?
??
?? ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.36Example
? 5 year fixed-for-fixed annual-pay currency swap where
interest at 10% in ￡ is exchanged for interest at 5% in $
? Principals are exchanged at the end of the life of the swap
Initial exchange rate = 2.000
Volatility of exchange rate = 15%
￡ principal = 50 $ principal = 100
￡ yield curve flat at 10% pa (ann comp) $ yield curve
flat at 5% pa (ann comp)
? 1-,2-,3-,4-,& 5-year zero-coupon bonds issued by the
counterparty would have yields that are spreads of 25,50,
70,85,& 95 basis points above the risk-free rate
? Defaults can occur only at the end of years 1,2,3,4,& 5
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.37Evaluating the Cost of
Defaults
Maturity when we receive $s when we pay $s
ti ui vi vi uivi
1 0.00250 5.9785 0.0149 5.9785 0.0149
2 0.00745 10.2140 0.0761 5.8850 0.0439
3 0.01083 13.5522 0.1468 5.4939 0.0595
4 0.01265 16.2692 0.2058 5.0169 0.0634
5 0.01296 18.4967 0.2398 4.5278 0.0587
Total 0.6834 0.2404
uivi
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.38Example continued
? The total cost of defaults on a matched pair of
swaps with similar counterparties is
0.6834+0.2404=0.9236% of principal.
? This means that a bid-offer spread of 20 to 21 basis
points is required to compensate for credit risk
? Why do we have more credit risk when we are
receiving dollars in this example?
? From a credit perspective,is it better to receive fixed
or floating in an interest rate swap when yield curve
is upward sloping?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.39Why Cost of Defaults for
Currency Swaps > Interest Rate
Swaps
Currencyswap
Interest rateswap
Time
Expected exposure
on matched pair
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.40
Convertibles
A convertible bond is a corporate bond
that can be exchanged for equity at
certain times in the future at a
predetermined exchange ratio (shares
per bond)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.41
Convertibles continued
? One of the problems in valuing
convertibles is that,in order to value the
corporate bond correctly,it is necessary
to take account of the chance of default
in some way
? Otherwise we are implicitly assuming it
is a no-default Treasury bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.42
Valuing Convertible Bonds
The value at a node is
MAX[ MIN(Q 1,Q 2),Q 3 ]
where
Q 1 is the value given by the rollback
Q 2 is the call price,&
Q 3 is the value if conversion takes
place.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.43
Valuing Convertible Bonds
(continued)
? We divide the value of the bond at each
node into two components
– a component that arises from situations
where the bond ultimately ends up as
equity
– a component that arises from situations
where the bond ultimately ends up as debt
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.44
Example 27.6
? 9-month zero-coupon bond with face value of $100
? Convertible into 2 shares
? Callable for $115 at any time
? Initial share price = $50,volatility = 30%,no
dividends
? Risk-free rates all 10%
? Yields on issuer’s non-convertible bonds = 15%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.45The Tree
(Numbers at each node in descending order are the stock
price,equity component,debt component & total value)
A
B
C
D
E
F
50.00
76.55
28.40
104.95
67.49
134.98
0
134.98
58.09
116.18
0
116.18
50.00
61.95
37.04
0
96.32
96.32
43.04
33.03
65.05
98.08
43.66
105.61
78.42
156.84
0
156.84
58.09
116.18
0
116.18
43.04
0
100.00
100.00
31.88
0
100.00
100.00
27.1
Chapter 27
Credit Derivatives
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.2Credit Default Swap
? Company A buys default protection from B to
protect against default on a reference bond
issued by the reference entity,C.
? A makes periodic payments to B
? In the event of a default by C
– A has the right to sell the reference bond to B for
its face value,or
– B pays A the difference between the market value
and the face value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.3
CDS Structure
Default
Protection
Buyer,A
Default
Protection
Seller,B
90 bps per year
Payment if default by
reference entity,C
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.4CDS Final Payments
Notation:
L,Face value of bond,notional value of CDS
A(t),Accrued interest on bond per $ of principal at time
t
R,Recovery rate,market price as a percent of face
value plus accrued interest
s,CDS payment rate per year,Annual payment = sL
?,Time since last CDS payment
A pays ?sL and B pays L - RL[1 + A(t)]
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.5
Sample Quotes (Jan 2001)
Company Rating 3yr 5yr 7yr 10yr
Toyota Aa1/AAA 16/24 20/30 26/37 32/53
Merrill Lynch Aa3/AA- 21/41 40/55 41/83 56/96
Ford A+/A 59/80 85/100 95/136 118/159
Enron Baa1/BBB+ 105/125 115/135 117/158 182/233
Nissan Ba1/BB+ 115/145 125/155 200/230 244/274
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.6CDS Valuation
T, Life of cr edit defa ult sw ap
p
i
,Risk - ne utral default prob a bilit y densit y at t i m e t
i
u ( t ),Prese nt value o f $ 1 p e r year on p ayment date s
between ti m e z ero and ti m e t
e ( t ), Prese nt value o f a n ac crual pay m ent at ti me t
v ( t ),Prese nt value o f $ 1 r ece ived at t i m e t
w, To tal pa yments per year m ade by CDS buy er
s, Value of w for wh ic h C DS value is ze ro
?, Risk - ne utral pro ba bilit y o f n o credit even t
du ri ng t he lif e of t he s wap
A ( t ),Accrued int ere st on the r efe rence obligatio n at
ti m e t as a p erc ent of fac e value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.7PV of CDS Payments per $1 of
Notional
? If default event occurs at t < T,PV of
payments is
? If no default event,PV of payments is
? Expected PV is
?
?
???
n
i
iii Tuwtetupw
1
)()]()([
)]()([ tetuw ?
)(Twu
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.8PV of CDS Costs per $1 of
Notional Principal
? If default event occurs at t < T cost is
? Expected cost is
RtARRtA ?)(?1]?)(1[1 ?????
)(]?)(?1[
1
ii
n
i
i tvpRtAR?
?
??
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.9
Value of CDS to Buyer
? Value is expected PV of payments less
expected PV of costs
)(])()([
)(]?)(?1[
1
1
Tutetupw
tvpRtAR
n
i
iii
n
i
iii
????
??
?
?
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.10
CDS Rate
CDS rate sets value to zero
?
?
?
?
???
??
?
n
i
iii
n
i
iii
Tutetup
tvpRtAR
s
1
1
)()]()([
)(]?)(?1[
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.11
CDS Rate continued
When default can happen at any time this
becomes
? ? ? ? ? ?
? ? ? ? ? ? ? ?
0
0
垐1T
T
q t v t R A t R d t
s
q t u t e t d t u T?
????
???
??????
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.12
Approximate CDS Spread
? Let
– y be the yield on bond issued by reference entity
with maturity T
– x be the yield on risk-free bond with maturity T
– a be average value of A(t)
– a* be average value for A(t) if reference bond is a
par-yield bond with maturity T
? ?
? ? ? ?
垐1
?1 1 *
y x R aR
s
Ra
??? ? ?
???
??
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.13
Alternative Uses of the Formula
? To calculate CDS spreads from the
probabilities of default and expected
recovery rate
? To bootstrap the probabilities of default
from CDS spreads and expected
recovery rates
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.14
Sensitivity to Recovery Rate
? Vanilla CDS is not very sensitive to the
recovery rate providing the same
recovery rate is used to estimate default
probabilities and calculate payoffs
? Binary swaps,which provide a fixed
payoff in the event of a default,are
much more sensitive to recovery rates
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.15
First-to-default swaps
? Similar to a regular CDS
? Several reference entities and reference
bonds
? First entity to default triggers a payoff
? Settlement is same as ordinary CDS
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.16
Valuation
? Must use Monte Carlo simulation
? Each reference entity is simulated to
determine when if ever it defaults
? Valuation is sensitive to default correlation
? A conservative (and easy) assumption for the
seller is that all correlations are zero
? The easiest way to build in non-zero
correlations is with the Gaussian copula
model
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.17
Seller Default Risk
? The impact of seller default risk on a
CDS swap can be calculated by jointly
simulating the reference entity and the
seller
? Suppose Y=PV of payoff and C is PV of
payments
? What rules should the simulation have
for calculating Y and C?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.18Total Return Swap
? Company A agrees to pay B the total return
earned on a reference bond issued by the
reference entity,C,over some period of time.
? Total return includes all coupon payments
and any change in the price of the reference
bond,(Usually the latter is made at the end)
? B pays A LIBOR plus a spread on a notional
equal to the initial value of the reference
bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.19
The Structure
Total Return
Payer
Total Return
Receiver
Total Return on Bond
LIBOR plus 25bps
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.20
Uses of a TRS
? Total Return Swaps are usually used a
financing vehicles
? Receiver wants to invest in bond
? Payer (a financial institution) buys the
bond and agrees to the swap
? Payer has less credit exposure than if it
had lent Receiver money to buy bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.21
Valuation of TRS
? If there were no risk of default by
receiver,the value of a TRS would be
difference between value of reference
bond and value of LIBOR bond
? The spread above LIBOR would be zero
? In practice the payer loses money if the
receiver defaults at a time when the
bond value has declined
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.22
Credit Spread Options
? These provide a payoff dependent on
movements in a particular credit spread.
? There is usually no payoff in the event of a
default on the reference asset
? Payoff may be defined in terms of difference
between actual spread and a strike spread or
in terms of the difference between the price of
an FRN and a strike price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.23
Valuation
? European options can be valued using
Black’s model
? This assumes that,conditional on no default,
spread or FRN price is lognormal
? Need a volatility for forward credit spread or
forward FRN price
? Must multiply Black’s formula by risk-neutral
probability of no default
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.24
Collateralized Debt Obligation
? A pool of debt issues are put into a special
purpose trust
? Trust issues claims against the debt in a
number of tranches
– First tranche covers x% of notional and absorbs first
x% of default losses
– Second tranche covers y% of notional and absorbs
next y% of default losses
– …
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.25
Bond 1
Bond 2
Bond 3
?
Bond n
Average Yield
8.5%
Trust
Tranche 1
1st 5% of loss
Yield = 35%
Tranche 2
2nd 10% of loss
Yield = 15%
Tranche 3
3rd 10% of loss
Yield = 7.5%
Tranche 4
Residual loss
Yield = 6%
Collateralized Debt Obligation
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.26
CDOs continued
? Note that average yield on tranches
equals average yield on bonds less fee
taken by trust manager
? Often trust manager holds first tranche
0, 0 5 3 5 % 0, 1 0 1 5 % 0, 1 0 7, 5 % 0, 7 5 6 %
b o n d s i i
i
y w y?
? ? ? ? ? ? ? ?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.27
CDO Applications
? Can provide a range of credit quality
debt objects
? Can create high quality debt from low
quality debt
? Can create high yield debt from average
risk debt
? Can create artificial short by selling
tranches before buying bonds
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.28
Valuing CDO Tranches
? Depends on default correlation of bonds
in portfolio
? Must use Monte Carlo simulation
? It is easiest to handle the default
correlation with the Gaussian copula
model
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.29Quantifying the Cost of Default
on a stand-alone derivatives
contract
Two Categories of Derivatives:
? Those that are always assets to one party
and liabilities to the other (e.g.,options)
? Those that can become assets or liabilities
(e.g.,swaps,forward contracts)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.30
Independence Assumption
? The independence assumption states
that the variables affecting the price of a
derivative are independent of the
variables determining defaults
? This assumption (although not perfect)
makes pricing for default risk possible
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.31
Notation
der i v at i v e of v al u e ac t u al
i m po s s i bl e ar e def au l t s
as su m i n g t i m e at der i v at i v e of v al u e
:)(
:)(
*
tf
ttf
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.32
Contracts that are Assets
TTyTyeff )]()([* *)0()0( ???
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.33
A Simple Interpretation
? Use the,risky” discount rate rather than
the risk-free discount rate when
discounting cash flows in a risk-neutral
world
? Note that this does not mean we simply
increase the interest rate in option
pricing formulas
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.34Credit Exposure for Contracts
That Can be Assets or Liabilities
Exposure
Contract value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.35
The Impact of Defaults
Rt
t
ut
vnit
vuff
i
i
ii
ii
n
i
ii
?
),1
)0()0(
1
1
*
1-b y m u l t i p l i e d
a n d t i m e s b e t w e e n d e f a u l t of p r o b a i l i t y
t h e is a n d t i m e at h a p p e n s d e f a u l t
a if cl a i m e x p e ct e d t h e is (
t i m e s at p l a ce t a ke can d e f a u l t s w h e r e
?
?
??
?? ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.36Example
? 5 year fixed-for-fixed annual-pay currency swap where
interest at 10% in ￡ is exchanged for interest at 5% in $
? Principals are exchanged at the end of the life of the swap
Initial exchange rate = 2.000
Volatility of exchange rate = 15%
￡ principal = 50 $ principal = 100
￡ yield curve flat at 10% pa (ann comp) $ yield curve
flat at 5% pa (ann comp)
? 1-,2-,3-,4-,& 5-year zero-coupon bonds issued by the
counterparty would have yields that are spreads of 25,50,
70,85,& 95 basis points above the risk-free rate
? Defaults can occur only at the end of years 1,2,3,4,& 5
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.37Evaluating the Cost of
Defaults
Maturity when we receive $s when we pay $s
ti ui vi vi uivi
1 0.00250 5.9785 0.0149 5.9785 0.0149
2 0.00745 10.2140 0.0761 5.8850 0.0439
3 0.01083 13.5522 0.1468 5.4939 0.0595
4 0.01265 16.2692 0.2058 5.0169 0.0634
5 0.01296 18.4967 0.2398 4.5278 0.0587
Total 0.6834 0.2404
uivi
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.38Example continued
? The total cost of defaults on a matched pair of
swaps with similar counterparties is
0.6834+0.2404=0.9236% of principal.
? This means that a bid-offer spread of 20 to 21 basis
points is required to compensate for credit risk
? Why do we have more credit risk when we are
receiving dollars in this example?
? From a credit perspective,is it better to receive fixed
or floating in an interest rate swap when yield curve
is upward sloping?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.39Why Cost of Defaults for
Currency Swaps > Interest Rate
Swaps
Currencyswap
Interest rateswap
Time
Expected exposure
on matched pair
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.40
Convertibles
A convertible bond is a corporate bond
that can be exchanged for equity at
certain times in the future at a
predetermined exchange ratio (shares
per bond)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.41
Convertibles continued
? One of the problems in valuing
convertibles is that,in order to value the
corporate bond correctly,it is necessary
to take account of the chance of default
in some way
? Otherwise we are implicitly assuming it
is a no-default Treasury bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.42
Valuing Convertible Bonds
The value at a node is
MAX[ MIN(Q 1,Q 2),Q 3 ]
where
Q 1 is the value given by the rollback
Q 2 is the call price,&
Q 3 is the value if conversion takes
place.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.43
Valuing Convertible Bonds
(continued)
? We divide the value of the bond at each
node into two components
– a component that arises from situations
where the bond ultimately ends up as
equity
– a component that arises from situations
where the bond ultimately ends up as debt
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.44
Example 27.6
? 9-month zero-coupon bond with face value of $100
? Convertible into 2 shares
? Callable for $115 at any time
? Initial share price = $50,volatility = 30%,no
dividends
? Risk-free rates all 10%
? Yields on issuer’s non-convertible bonds = 15%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
27.45The Tree
(Numbers at each node in descending order are the stock
price,equity component,debt component & total value)
A
B
C
D
E
F
50.00
76.55
28.40
104.95
67.49
134.98
0
134.98
58.09
116.18
0
116.18
50.00
61.95
37.04
0
96.32
96.32
43.04
33.03
65.05
98.08
43.66
105.61
78.42
156.84
0
156.84
58.09
116.18
0
116.18
43.04
0
100.00
100.00
31.88
0
100.00
100.00
