of credit derivatives: credit default swaps on single names, credit default swaps ... long a credit default swap is equivalent to owning a risky bond: you get extra ...
CREDIT DERIVATIVES PATRICK S. HAGAN Abstract. Here we sketch the market conventions, schedule and payment generation, and pricing for the standard types of credit derivatives: credit default swaps on single names, credit default swaps on indices, European options on single names, European options on indices, and synthetic CDOs. In the Appendices we present pricing algorithms in detail.
1. Single name credit default swaps. The backbone of the credit markets in the single name credit default swap. A credit default swap exchanges a coupon leg (…xed leg) for a protection leg. It is de…ned by its start date, its end date, its coupon (also called the spread) C, the notional (face value) N , and the name (reference entity) E. In addition, contracts stipulate exactly what constitutes a default, usually the failure of the name to make a legitimate payment of over 1 million USD, or the equivalent. Some explicitly specify the reference paper (a loan or bond) B as the deliverable; others specify the seniority class of the paper. 1.1. Deal mechanics. The protection seller (the receiver ) receives the coupon leg (…xed leg) and pays the protection leg (also called the insurance or ‡oating leg). The coupon leg consists of …xed payments with coupon rate (…xed rate) C on the notional N: These coupons are quoted as an annual rate, but are invariably paid quarterly with an Act360 basis. These payments are made until either the end date is reached, or the reference entity E defaults. Upon default, the …xed leg owes an amount representing the coupon that has accrued between the previous payment and the default date. This amount is not paid seperately. Instead it is deducted from the amount the protection seller pays to settle the default. See below. The protection (‡oating) leg makes no payments unless a qualifying default event occurs before the end date of the swap. Upon receiving notice of default (the payer normally has a 14 day window for providing notice), the protection leg makes a payment equivalent to N (1¡R), the notional less the recovery. Depending on the contract, the mechanics of this payment can happen in a variety of ways: ¡ If the swap is physically settled, the protection leg pays the notional N and receives the reference paper B with face value N . So the protection leg makes good the loss upon default of the bond or loan B. (The person receiving the paper usually turns around and sells it to a vulture fund for whatever he can get.) Contracts may specify only a seniority class of paper instead a particular bond or loan B. This gives the protection buyer a choice (the “delivery option”) of which paper is delivered to the ‡oating leg; clearly he will deliver the cheapest paper he can …nd. ¡ If the swap is cash-settled, the ‡oating leg pays N less the “recovery.” The recovery is the post-failure market value of the reference paper with face amount N . The contract states how this recovery value is to be determined. For example, it may be determined by polling …ve dealers a certain number of business days (often …ve or ten) after the credit event. ¡ It can be unclear which legal entity (parent, subsidiary, ...) is the reference entity for a credit derivative, and which bond or loan acts as the reference paper. Mark-It partners (and soon Bloomberg) will have RED, the reference entity database of o¢cial legal entities and reference paper for credit derivatives. The payment-upon-default may occur anywhere from 3 to 90 days after the notice of default is received, depending on the contract. From this default settlement N (1 ¡ R), the protection leg also calculates and deducts the coupon that accrued between the last coupon date and the default date. Being long a credit default swap means you receive the coupon leg and pay the protection leg. Being long a credit default swap is equivalent to owning a risky bond: you get extra coupon until the company goes bust; then you lose the di¤erence between the notional and the recovery. 1.2. CD swap schedule. A credit default swap speci…es a start date (e¤ective date) and an end date, t0 and tn : Protection starts at midnight GMT (before the market opens) on the start date t0 , and runs until midnight GMT of the end date tn . If the start date is not speci…ed, protection starts at midnight GMT of 1
the trade date. That is, the e¤ective date is one calendar day (not business day!) after the trade date. The end date is calculated slightly di¤erently depending on whether IMM dates are used. Standard credit default swaps end on an IMM date: Mar. 20, Jun. 20, Sep. 20, and Dec. 20. If a trader speci…es, say a 24 month CD swap, one would add 24 months to the trade date, and the swap’s end date tn would be the …rst IMM date on or after that date. So a 12 month CD swap traded on June 20 would have an e¤ective date of June 21 and end on the following June 20, but if it traded on June 21, it would have an e¤ective date of June 22 and end on Sept 20 of the following year. CD swaps with non-IMM dates are traded less commonly now, but they are still quite liquid. As above, the protection becomes e¤ective on midnight GMT of the trade date, and the start date (e¤ective date) is trade plus one. The end date is obtained by adding the tenor (length) of the CD swap to the e¤ective date. So a 12 month CD swap traded on June 20 would have an e¤ective date of June 21, and would end on June 21 of the following year, not on June 20th. The coupon leg is constructed backward from the end date. The …rst period is odd if an odd period is needed. If the …rst period is shorter than one month, it is combined with the next period to make a long …rst coupon. Otherwise a short …rst period (stub) is used. Let the theoretical coupon days be th th th tth 0 < t 1 < ¢ ¢ ¢ < tn¡1 < t n :
(1.1)
th The start date tth 0 and end date t n require special handling. For all other dates, the actual dates tk are obtained from the theoretical dates tth k using the modi…ed following business day rule and the standard …xed leg holiday centers. E.g., for USD denominated CD swaps, NY and LDN banking holidays are used, and for EUR denominated swaps, TGT holidays are used. Defaults can occur on non-business days, so neither the start date nor the end date of the deal is adjusted: th t0 = tth 0 and t n = tn . Although the deal ends on the end date, if this is a bad date the …nal payment is delayed to the next good business day. No coupon accrues and no protection is given for the period between the end date and the delayed pay date. Single name CD swaps are normally traded at par. If not, settlement is usually trade plus three business days in the center transacting the deal. Unless explicitly stated otherwise, credit default coupons are adjusted in that the coupon accrues from actual date to actual date. Let
(1.2)
®k = cvg(tk¡1 :tk ):
be the year fraction of the k th interval tk¡ 1 to tk . (The basis is invariably Act360). If no default occurs, the coupon leg pays the C® k N at each pay date tk for k = 1; 2; :::; n. If a default occurs, the coupon leg owes the interest that has accrued from the last coupon date to the default date tdef , and then pays nothing more. So the kth coupon leg payment is (1.3a) (1.3b) (1.3c)
paid at tk
N C® k 0
if no default before tk ;
paid at tk
N C cvg(tk¡1 ; tdef )
if tdef < tk¡1 ; paid at tdef + ± def
if tk¡ 1 < tdef < tk :
Here ± def is the assumed delay between the default date and the settlement upon default. If the entity defaults at date tdef , the protection leg (‡oating leg) e¤ectively pays the swap’s notional N less the recovery RN . So the protection leg payo¤ is (1.4)
N (1 ¡ R)
paid at tdef + ±def
if t0 · tdef < tn :
For valuation, the assumed recovery rate R is a trader-enterred number, usually 40%. 2
1.3. CD swap valuation. Let D(t) be the usual discount curve for the swap market. De…ne the survival curve (1.5a)
Q(t) = probability the entity does not default by time t.
Then, (1.5b)
Q(t) ¡ Q(t + ¢t) = ¡Q0 (t)¢t = prob of default between t and t + ¢t:
In particular, Q(tk¡1 ) ¡ Q(tk ) is the probability that entity defaults between tk¡1 and tk . The value of the protection leg is n Z tk X (1.6) Vprot = fpayment if default occurs at tdef g k= 1
tk¡1
¤D(tdef + ±def ) ¤ fprob of default at tdef g dtdef
Since ¡Q 0(tdef )¢tdef is the probability of a default between tdef and tdef + ¢tdef , clearly n Z tk X (1.7) Vprot = N (1 ¡ R)D(tdef + ± def ) f¡Q0 (tdef )g dtdef k= 1
tk¡1
= N (1 ¡ R)
n Z X k=1
tk
tk¡1
D(tdef + ±def ) f¡Q0 (tdef )g dtdef :
Let us now value the coupon leg. Q(tk ) is the probability that the entity do es not default before tk . In this case we receive the full coupon N C ®k at tk . The quantity ¡Q0 (tdef )¢tdef is the probability that default occurs between tdef and tdef + ¢tdef . In this case we only get the coupon that accrues up to tdef . We assume that accrual is linear in t (this is certainly true for Act360, but only approximately true for, say, 30/360). Since ®k is the amount that accrues over the full interval, the amount accrued at tdef is ® k (tdef ¡ tk¡ 1 )=(tk ¡ tk¡1 ). The value of the coupon leg is thus ( ) Z tk n X tdef ¡ tk¡ 1 0 (1.8) V cpn = N C ® k Q(tk )D(tk ) + D(tdef + ±de f ) f¡Q (tdef )g dtdef tk ¡ tk¡1 tk¡1 k=1
The value of the CD receiver swap is (eliminating the tdef subscript for simplicity.) ( ) Z tk n X t ¡ tk¡ 1 0 (1.9) Vre c = N C ® k Q(tk )D(tk ) + D(t + ±def ) f¡Q (t)g dt tk ¡ tk¡1 tk¡1 k=1 n Z tk X ¡N (1 ¡ R) D(t + ±def ) f¡Q0 (t)g dt: k=1
tk¡1
1.3.1. Standard approximation. We can simplify this formula for CD swaps by making a standard set of approximation. These are: ¢ in the protection leg, D(t) is approximated by its average value in each interval k, ³ ´ 1 ~ k¡1 + D ~k (1.10a) [D(tk¡1 + ± def ) + D(tk + ±def )] = 1 D 2
2
¢ in the coupon leg one assumes that defaults occur, on average, halfway through the interval, so (1.10b)
t ¡ tk¡1 tk ¡ tk¡1 3
is replaced by 12 . One also approximates D(t + ±de f ) in interval k by its …nal value D(tk + ± def ). Together these approximations yield the following value for the CD swap: (1.11)
V rec = N C
n X
k=1
n n o X ~ k (Q k¡1 ¡ Q k ) ¡ N (1 ¡ R) ®k Dk Qk + 12 D
1 2
k=1
³
´ ~ k¡1 + D ~ k (Qk¡1 ¡ Q k ) : D
~ k = D(tk + ±def ). This value can be written more suggestively as Here Q k = Q(tk ); Dk = D(tk ); and D (1.12a)
V rec = N L(C ¡ C be );
(1.12b)
V pay = ¡V rec = N L(Cbe ¡ C):
where the risky level (also known as the risky PV01 ) is (1.12c)
L=
n X
k= 1
n o ~ k (Q k¡1 ¡ Q k ) ; ® k Dk Qk + 12 D
and the break-even coupon (market CDS coupon ) is ³ P (1.12d)
Cbe = (1 ¡ R)
n 1 k=1 2
´ ~ k¡ 1 + D ~ k (Qk¡1 ¡ Qk ) D L
;
1.3.2. More accurate valuation formulas. The approximation used in the “standard approximation,” are pretty much ad hoc. Systematic approximations, of ever increasing re…nement, can be used instead. This leads to increasingly accurate formulas, which are also increasingly complex. These formulas are derived in Appendix A. Here we note that these approximations all yield CD swap prices of the form, (1.13a)
V rec = N L(C ¡ C be );
(1.13b)
V pay = ¡V rec = N L(Cbe ¡ C);
with slightly di¤erent formulas for the risky level L and the break-even coupon C be . The risky level and break even spread are always of the form
(1.13c)
L=
I X
b iQ(¿ i )
C be =
i=0
I (1 ¡ R) X h iQ(¿ i ) L i=0
where the coe¢cients b i and h i are derived in Appendix A. Here, the set of dates (1.13d)
t0 = ¿ 0 ; ¿ 1 ; :::; ¿ I = tn
are the coupon dates tk , except for approximation methods which split the coupon intervals into …ner subdivisions. Then the sub-interval dates are also included in the ¿ i . 1.4. Stripping the risky curve. We write the survival probability Q(t) as (1.14)
Q(t) = e¡
Rt 0
¸(t0 )dt0
;
where ¸(t) is the hazard rate, the rate at which defaults occur at time t. For each entity one has market quotes for CD swaps of di¤ering lengths (tenors). For single name CD swaps, typically the break even spreads C are quoted. See below. 4
tenor 12m 24m 36m 48m 60m
C in bps 65 71 72 72 68
V in bps 0 0 0 0 0
One strives to …nd a “nice” curve for ¸(t) which values all the CD swaps to zero. The standard market practice is to use piecewise constant hazard rates ¸(t); where the breaks occur on the end dates of the CD swaps. So suppose one has m CD swaps with coupons Cj and end dates T j , for j = 1; 2; :::; J. Then,
(1.15)
¸(t) = ¸ 1
for t < T 1 ;
¸(t) = ¸ j
for Tj¡ 1 < t < T j for T m¡ 1 < t
¸(t) = ¸ m
j = 2; 3; :::; m ¡ 1
This means that the survival rates are: Q(t) = e¡¸ 1t
(1.16a)
for t < T 1 ;
Q(t) = e¡¸ 2(t¡T1 )e ¡¸1T 1
(1.16b)
= e¡¸ 2(t¡T1 )Q(T 1 )
for T 1 < t < T 2
Q(t) = e¡¸ 3(t¡T 2) e¡¸2 (T2¡ T1) e¡¸ 1T1
(1.16c)
= e¡¸ 3(t¡T 2) Q(T 2 )
for T 2 < t < T 3 ;
and so on. Clearly, (1.17a) (1.17b) (1.17c)
Q(t) = e¡¸ 1t
for t < T 1 ;
¡¸ j (t¡Tj ¡1)
Q(t) = e
¡¸ m( t¡T2)
Q(t) = e
for T j¡1 < t < Tj
Q(T j¡1 )
Q(T m )
for j = 2; 3; :::; m
for Tm < t
1.4.1. Stripping algorithm. To strip the curve, we assume that the CD swaps all share a common set of coupon dates, so that the J th credit default swap has the actual dates (1.18)
t0 ; t1 ; :::; tn j
for J = 1; 2; :::; m:
(In rare situations we may need to cheat a little with the dates sequence to get the dates lined up). Regardless of the approximation method used, the value of the J th swap can be written as (1.19a)
VJ = C J
IJ X i=0
=
IJ X i=0
bi Q(¿ i) ¡ (1 ¡ RJ )
IJ X i= 0
[C J b i ¡ (1 ¡ RJ ) hi ] Q(¿ i ) 5
h iQ(¿ i)
where the dates (1.19b)
t0 = ¿ 0 ; ¿ 1 ; :::; ¿ I J = T J :
are the coupon interval (or subinterval dates if re…nement is used) for the J th credit default swap. In particular, with coupon dates aligned, all the valuation methods proposed in Appendix A use a common set of subinterval dates ¿ i . for j = 1; 2; :::; m; so we can write the stripping problem as Ij J X X
(1.20)
j=1 i=I j¡1
for J = 1; 2; :::; M
[CJ bi ¡ (1 ¡ RJ )hi ] Q(¿ i) = V J
We solve for the hazard rates ¸j by bootstrapping. Suppose that we have found ¸ 1 ; ¸2 , ...,¸ k¡1 by solving ?? for J = 1; 2; :::; k ¡ 1. We then know Q(t) for all t · ¿ Ik¡1 = T k¡1 . We also know that Q(t) = Q(Tk¡ 1 )e¡¸ k( t¡Tk¡1 )
(1.21)
for T k¡1 < t · T i :
We can re-write the kth problem as (1.22)
Ik X
i=I k¡1+ 1
I k¡1 ¡¸ k( ¿ i¡Tk¡1)
[Ck bi ¡ (1 ¡ Rk ) h i] Q(T k¡ 1 )e
= Vk ¡
X i=1
[C k b i ¡ (1 ¡ R k ) h i ] Q(¿ i ):
Observe that the right hand side only involves Q(t) for t · Tk¡ 1 , and these are all known. The left hand side involves Q(T k¡1 ), which is known, and the new hazard rate ¸ k . This equation is of the form (1.23a)
F (¸ k ) = target;
where (1.23b)
F (¸k ) =
Ik X
i=Ik¡1+1
(1.23c)
F 0(¸k ) = ¡
Ik X
[C k b i ¡ (1 ¡ Rk ) hi ] Q(T k¡1 )e ¡¸k (¿ i¡ Tk¡1) ; (¿ i ¡ T k¡1 ) [Ck b i ¡ (1 ¡ R k ) h i ] Q(T k¡1 )e¡ ¸k (¿ i¡T k¡1) :
i=I k¡1+ 1
Using a (global) Newton scheme to …nd ¸k is now straightforward, and then we can move on to interval k + 1; : : : : 1.4.2. Representation of the curve. We don’t have to store both Q(T j ) and ¸ j for each interval, since (1.24)
log Q(t) = log Q(Tj¡ 1 ) +
t ¡ T j¡1 [log Q(Tj ) ¡ log Q(T j¡1 )] : Tj ¡ T j¡1
A more e¤ective representation is to store the vector of Aj ’s, where (1.25)
Aj = log Q(T j )
for j = 0; 1; 2; :::; m:
Noting that A(0) = 0, one has (1.26a)
Q(t) = e¡ A(t) ; 6
where t ¡ T0 (A1 ¡ A0 ) for t · T 1 ; T1 ¡ T 0 t ¡ T j¡1 A(t) = Aj¡1 + (Aj ¡ Aj¡1 ) for T j¡1 · t · Tj ; T j ¡ T j¡1 t ¡ T m¡1 A(t) = Am¡1 + (Am ¡ Am¡ 1 ) for T m¡1 · t T m ¡ T m¡1
(1.26b)
A(t) = A0 +
(1.26c) (1.26d)
j = 2; 3; :::; m ¡ 1
1.4.3. Stripping (risk free) discount factors. As long as we are willing to use piecewise constant forward rates, the same stripping software will also strip yield curves. Recall that today’s discount factor D(t) is D(t) = e ¡
(1.27)
Rt 0
f ( t0)dt0
;
where f (t) is today’s (instantaneous) forward rate curve. We obtain the forward rate by requiring the stripping instruments to match their market value. For two period stripping instruments (deposits/loans and EDF’s), this yields (j)
(j)
( j)
D(t0 ) = (1 + ® (j)R f wd )D(t1 );
(1.28) (j)
where Rf wd is the forward rate. Here the forward rate is the quoted deposit/loan rate, or the future’s rate corrected for the convexity adjustment. For swaps, we have (j) D(t0 )
(1.29)
=
Rjsw
nj X
(j)
® (j) D(tk ) + D(t(j) nj ):
k=1
We re-order the stripping instruments in increasing maturities. Let these maturities be T 1 ; T 2 ; : : : ; T m . We take the forward rate f (t) to be
(1.30)
f (t) = f 1
for t < T 1 ;
f (t) = f j f (t) = f m
for T j¡1 < t < T j for Tm¡1 < t;
j = 2; 3; :::; m ¡ 1;
so the discount factors are (1.31a) (1.31b) (1.31c)
D(t) = e ¡f1t
for t < T 1 ;
D(t) = e
¡fj (t¡T j¡1)
D(t) = e
¡fm (t¡T 2)
D(T j¡ 1 )
D(T m )
for T j¡1 < t < T j
for j = 2; 3; :::; m;
for T m < t:
We can now bootstrap to …nd each of the f k ’s in turn. 1.5. Risks and hedges. Let us select a set of M CD swaps as our hedging instruments for the entity in question. We require these CD swaps to have a common set coupon dates. Let the interval or (or subinterval dates if re…nement is used) for the J th credit default swap be (1.32a)
t0 = ¿ 0 ; ¿ 1; :::; ¿ iJ = T J 7
for J = 1; 2; :::; M . Normally one chooses the same CD swaps that are used for stripping the survival curve, but this need not be the case. The valuation equation for CD swap J is of the form
(1.32b)
CJ
8 ij J < X X j=1
where
:
bi Qi
i=i j¡1
9 = ;
¡ (1 ¡ RJ )
(1.32c)
8 ij J < X X j=1
:
h iQ i
i=ij ¡1
9 = ;
= VJ
for J = 1; 2; :::; M
Qi = Q(¿ i ):
As indicated notationally, the coe¢cients b i and h i are constants that do not depend on the hazard rates ¸. De…ne ij X
L( j) =
(1.33a)
bi Qi ;
i=i j¡1
(1.33b)
H
( j)
=
ij X
hi Qi ;
i=i j¡1
Then CD swap J becomes (1.33c)
VJ = CJ
J X j=1
L(j) ¡ (1 ¡ RJ )
J X
H (j)
for J = 1; 2; :::; M
j=1
Here Lj and Hj are the terms in the level (PV01) and protection which involve dates ¿ i in the interval T j¡1 · ¿ i · T j :
1.5.1. Basic risk scenarios. We generate our fundamental risks by piecewise constant shifts of the hazard rate.The k th scenario is ½ ¸k 0 for k 0 6= k (1.34a) ¸ k 0 ¡! ; ¸k + ± for k 0 = k for k = 1; 2; :::M:The change in the survival rate for scenario 8 0 < (1.34b) ¢Q(t) = ¡Q(t) (t ¡ T k¡1 ) : (T k ¡ T k¡1 ) for k = 1; 2; ::; M ¡ 1 and (1.34c)
¢Q(t) = ¡Q(t)
½
0 (t ¡ T k¡1 )
for k = M . At the coupon (or subinterval) dates, this is 8 0 < (1.34d) ¢Q i = ¡Q i (¿ i ¡ T k¡1 ) : (T k ¡ Tk¡ 1 ) 8
k is ± ¢ ¢Q(t), where for t · Tk¡ 1 for Tk¡ 1 · t < T k for T k < t
for t · T k¡1 for T k¡1 · t for i · i k¡1 for ik¡1 · i < i k for ik < i
1.5.2. Jacobian. We now apply the risk scenarios to the CD swaps used in stripping. The change in the partial levels L(j) and partial protection terms H (j) for scenario k are L(j) ¡! L(j) + ± ¢ ¢k L( j) ;
(1.35a)
H (j) ¡! H (j) + ± ¢ ¢k H (j);
where (1.35b)
¢k L(j) = 0;
for j < k
¢Hj = 0 ij
(1.35c) (1.35d)
¢k L
(j)
=¡
X
i=i j¡1
b iQ i(¿ i ¡ T k¡1 );
¢k L(j) = ¡(T k ¡ T k¡ 1 )L(j) ;
¢k H
( j)
=¡
ij X
i=i j¡1
h iQ i(¿ i ¡ T k¡1 )
¢k H (j) = ¡(T k ¡ Tk¡ 1 )H (j)
for j = k
for j > k
The change in the value of CD swap J under scenario k is (1.36a)
VJ ¡! VJ + ± ¢ SkJ
where (1.36b)
SkJ = CJ
J X j=k
(1.36c)
¢k L(j) ¡ (1 ¡ RJ )
SkJ = ¢k V J = 0
for k > J
J X
¢k H (j)
j=k
for k · J
Observe that the CD swap sensitivity matrix SkJ is upper triangular. The Jacobian is de…ned to be the inverse of this matrix: (1.37a)
JAC J k = [SkJ ]¡1
The Jacobian is also upper triangular: for J > k
JAC J k = 0
1.5.3. Hedging with the Jacobian. Suppose we have a book of CD swaps and other credit-sensitive instruments based on this entity. After stripping the CD curve, suppose we …nd that the value of these deals is V bo ok . To hedge the credit risk, suppose we apply the risk scenarios one-by-one, discovering that the book had the following vector of risks: 0 1 0 1 ¢1 V book Risk 1 B C B C .. .. (1.38) R=±@ A = ±@ A . . ¢m V book Riskm We hedge this risk by buying/selling a linear combination of the CD swaps VJ that were used in stripping the instruments. Our portfolio is (1.39a)
¼ = V book ¡
M X
aJ VJ ;
J =1
where aJ is the notional of CD swap J . The risk of the portfolio to scenario k is ( ) M M X X (1.39b) ¢k ¼ = ± ¢ Riskk ¡± aJ ¢k VJ = ± Riskk ¡ SkJ aJ : J =1
J =1
9
Clearly the hedges are determined by inverting the sensitivity matrix: (1.39c)
aJ =
M X
k=1
=
M X
k=J
JACJ k ¢ Riskk for J = 1; 2; :::; M
JAC J k ¢ Riskk
1.5.4. Alternative risk scenarios. The hedging instruments are normally chosen to be the same CD swaps used in stripping the survival curve. The most common set of risk scenrios are obtained by bumping the break even coupon CJ for each CD swap, and re-stripping the curve to get the bumped survival curve. We can …nd s linear combination of our piecewise constant shift scenarios that reproduces the risks of these bumped scenarios. Recall that our CD swap valuation can be written as
(1.40)
VJ = C J
J X
j=1
L(j) ¡ (1 ¡ RJ )
J X
H (j)
for J = 1; 2; :::; M :
j=1
Normally par CD swaps are used, so the values V J are zero. Suppose we bump the coupons of the CD swaps, (1.41a)
for J = 1; 2; :::; M ;
CJ ¡! C J + ±CJ
and perturb the hazard rates, (1.41b)
for k = 1; 2; :::; M :
¸ k ¡! ¸ k + ±¸ k
The changes in the CD swap values are (1.42a)
±VJ = ±CJ ¢ LJto t +
J X
SkJ ±¸ k
for J = 1; 2; :::; M;
k= 1
where L Jtot =
(1.42b)
J X
L(j)
j=1
are the risky PV01’s of each CD swap. Restripping the curve to match the original CD swap prices is equivalent to setting ±V J = 0 for all J . Clearly the stripping would yield the scenario (1.43a)
for k = 1; 2; :::; M
¸k ¡! ¸ k + ±¸k
where (1.43b)
±¸k = ¡
M X
J =k
±CJ ¢ LJtot JACJ k
The change in the survival rate is ( i¡1 ) X (1.44) ¢Q(t) = ¡Q(t) (Tk ¡ T k¡1 ) ±¸ k + (t ¡ Ti¡1 ) ±¸ i k=1
10
for T i¡1 < t · T i
for i = 1; 2; :::; M . (The case i = M also includes t > T M ). Consider the scenario in which we bump just the J th coupon by ±, and re-strip the curve. The change in the survival probabilities is ( i¡1 ) X (1.45a) ¢Q(t) = ±Q(t)LJtot (Tk ¡ T k¡1 ) JACJ k + (t ¡ Ti¡1 ) JAC J i for T i¡ 1 < t · T i k=1
Let ± RiskJ be the change in the value of our book using the new probabilities, so 0 1 0 1 ¢ 1 Vbook Risk1 C B C .. .. ^ = ±B (1.45b) R @ A = ±@ A . . ¢M V book RiskM is the complete set of risks.Clearly, the hedged portfolio is (1.46a)
¼ = V book ¡
M X
aJ VJ ;
J =1
where the notional aJ of CD swap J is (1.46b)
aJ = ¡ RiskJ =LJtot ;
for J = 1; 2; :::; M
1.6. Forward strarting CD swaps. When an entity defaults, qualifying trigger events continue to happen for months or years. This means that a poorly executed forward starting swap will e¤ectively provide protection before it’s start date, since the payer can provide notice of a trigger events that occur after the forward start date. The simplest way to buy/sell protection from a forward date tst to tend is to take opposite positions in two CD swaps. To trade a forward starting receiver at spread C, one enters a receiver at C from today to tend , and enters a payer at C from today to tst. The opposite positions would create a forward starting payer. Alternatively, one can include the phrase “either party can cancel should a qualifying default event occur before tst.” 2. CD swaptions. A CD swaption is a European option on a credit derivative swap. A receiver is the option to receive the coupon leg and pay the ‡oating leg. I.e., an option to go long the credit derivative swap, or, in other words, an option to provide protection at a pre-set coupon (spread). A payer is the option to pay the coupon leg and receive the ‡oating leg. I.e., it is the option to get protection against the entity’s default at a pre-determined coupon. 2.1. Mechanics of CD swaptions. A European option is de…ned by the exercise date tex, the strike (coupon) C, and the tenor (length) ` of the underlying credit default swap. Unless otherwise speci…ed during the trade, the CD swap’s schedule is generated exactly as above: the start date t0 is one (calendar) day after the exercise date, the end date is the …rst IMM date on or after the date found by adding the tenor ` to the exercise date te x, the schedule is quarterly, the day count basis is Act360, and so on. The only wrinkle arises if the underlying entity defaults on or before the exercise date (i.e., before the CD swap starts). Contracts can specify one of four possible outcomes should an early default occur. The outcome speci…ed by the overwhelming majority of credit default swaptions is the knock out : if the entity defaults on or before the exercise date, the option knocks out and cannot be exercised. This is implemented via the phrase “either party can cancel the swap should a qualifying default event occur before the exercise date” in the contract. 11
This outcome e¤ectively gives payer CD swaptions a digital feature: as an entity’s credit deteriorates, the payer option becomes increasingly valuable, unless it actually defaults before the exercise date. If the entity defaults one day before the exercise date, the option is worthless, and if it defaults one day after the exercise date, the option is at it’s most valuable. Some options are written which are default-protected, with delayed recovery to avoid this awkwardness. If the entity defaults before the exercise date tex , these options allow the holder to wait until the CD swap’s start date to deliver notice of the credit event. If a pre-exercise default occurred, a payer option would always be exercised, and the default payment N (1 ¡ R) would be paid on tex + ± def , just as if the entity had defaulted immediately after exercise. No coupon payment would have accrued, since it only accrues after tex and before tdef . If a pre-exercise default occurred, a receiver option would be worthless, since exercising it would make one liable for loss N (1 ¡ R) without getting any compensating coupon payments. Some payer options are written which are default-protected with immediate recovery. If the entity defaults before the exercise date tex , these options allow the payer option to immediately present notice of default, so N (1 ¡ R) would be paid on tdef + ± def , exactly as if he had already enterred the CD swap. This allows both parties to clean up there books as soon as possible after the default occurs. The receiver option is, of course, worthless. On rare occasions, options on CD swaps are written to be knock-out with fee. If the entity defaults before the credit default swap begins, these options pay a pre-speci…ed fee F to the option holder, and then knock-out. The contract would specify whether the fee would be paid on tdef + 1 or on tex + 1. This fee is often the orginal premium paid to purchase the option, so the option purchaser gets reimbursed for his “lost”.loption 2.2. Pricing CD swaptions via Black’s model. We …rst price a CD swaption, assuming that the option knocks out upon early default, since this is by far the most common deal type. Recall that the CD swap valuation formulas are of the form (2.1)
V rec (t) = N L(t) [C ¡ C be (t)] ;
where L is the risky level. We use the risky level L as the numeraire. (The risky level is just a linear combination of zero coupon bonds issued by the entity, which, in principle, are tradeable instruments). According to the fundamental theorem of arbtirage free pricing, there exists a probability measure such that the value of any tradeable instrument divided by the level L is a Martingale. In particular, the ratio of the CD swaption and L is a Martingale, so ¯ ½ o pt ¾ Vr ec (T ) ¯¯ opt (2.2) V rec (t) = L(t) E all info available at t for all T > t L(T ) ¯
Since this holds for all T > t, let us evaluate it at the exercise date tex. At tex ; the value of the CD swaption is clearly +
pt Vroec (tex ) = N L(tex) [C ¡ C be (tex )]
So (2.3a)
pt Vroec (t) = N L(t) E
n
¯ o +¯ [C ¡ C be (tex )] ¯ all info available at t :
Now, the CD swap valuation formulas show that the break-even coupon Cbe is a linear combination of risky zero coupon bonds divided by the risky level L. Therefore Cbe is a Martingale. By the Martingale representation theorem, we know that (2.3b)
dCbe = A(¤)dW 12
for some random function A(¤). The standard pricing model CDS assumes the break even coupon is log normal: (2.4)
dCbe = ¾ c Cbe dW:
This then yields Black’s formula (2.5a) with (2.5b)
© ª opt 0 V rec (t) = N L(t) C N (d1 ) ¡ Cbe N (d2 ) d1;2 =
(knock-out)
0 ln C=C be § 12 ¾ 2c (tex ¡ t)
¾ c (tex ¡ t)1=2
0 where C be is today’s (forward) CD coupon for the CD swap beginning at tst and ending at tend . The value of the payer option is obtained by interchanging C and Cbe everywhere: © 0 ª o pt (2.6a) V pay (t) = N L(t) Cbe N (d¤1 ) ¡ C N (d¤2 ) (knock-out); ¡ ¢ o pt 0 (2.6b) = Vr ec (t) ¡ N L(t) C ¡ C be ;
with
(2.6c)
0 ln Cbe =C § 12 ¾ 2c (tex ¡ t)
d¤1;2 =
1=2
¾ c (tex ¡ t)
:
Note that pricing CD swaps at two di¤erent coupons will determine both the break-even coupon Cbe (t) and the risky level, L(t). The only other quantity needed is the “spread volatility” ¾ c . These can be implied from the market prices for CD swaps. Although Black’s model forms the basis for quoting CD swaption prices, the model is a poor description of the dynamics of C, since credit spreads can, with nonzero probability, jump to very high levels. Consequently one expects the implied volatilities of CD swaptions to have pronounced skews and smiles. Shortly we will take up the question of more realistic models. 2.2.1. Early default protected swaptions. Consider now a default protected option. A receiver option would never be exercised if the default occurs before t0 , since the counterparty would deliver notice of default, resulting in an immediate loss of N (1 ¡ R). Thus, for receiver CD swaptions there is no di¤erence between default-protected and knockout options. If early default occurs for a payer swaption, one would exercise the option on t0 , and immediately claim protection. Since [1 ¡ Q(t0 )] is the probability of early default, default protection adds [1 ¡ Q(t0 )] D(t0) to the value of a payer CD swaption. So for early default protection: © ª pt 0 (2.7a) Vroec (t) = N L(t) CN (d1 ) ¡ Cbe N (d2 ) (early default protected) with
(2.7b)
d1;2 =
and (2.8a) with (2.8b)
0 ln C=C be § 12 ¾ 2 (tex ¡ t) 1=2
¾ (tex ¡ t)
opt V pay (t) = N (1 ¡ R) [1 ¡ Q(t0 )] D(t0 ) © 0 ª +N L(t) Cbe N (d¤1 ) ¡ C N (d¤2 )
d¤1;2 =
(early default protected);
0 ln C be =C= § 12 ¾ 2 (tex ¡ t) 1=2
¾ (te x ¡ t) 13
:
2.2.2. CD swaptions with knock-out fees. Consider the holder of the CD payer swaption. He purchased the option of protecting himself from the default of the entity E. If this default happens on t0 , the option holder can exercise and receive N (1 ¡ R); if it defaults one day earlier, he receives nothing. To make up for this percieved unfairness, some deals have knock-out fees paid to the option holder if default occurs before t0 . Such fees are fairly rare and are usually only found in CD payer swaptions. Suppose we have a fee f paid upon early default. Depending on the deal, this fee may be paid on t0 , the earliest start date of the deal, or immediately after default. If it is paid at t0 , then clearly the value of the CD payer swaption is: (2.9a)
VP ayerO pt (t) = f [1 ¡ Q(t0 )] D(t0 ) © 0 ª +N L(t) C be N (d¤1 ) ¡ CN (d¤2 )
(with ko fee f at t0 ),
Alternatively, if the value is paid immediately upon default, then (2.9b)
V P ayer Opt (t) = V CDswap © 0 ª +N L(t) Cbe N (d¤1 ) ¡ CN (d¤2 )
(with ko fee f at tdef ),
where V CDswap is the value of a credit default payer swap which has zero coupon, which starts today and runs until t0 . I.e., the payer option is equivalent to a straight knockout payer swaption, plus a credit default swap with the value of the protection built into the up-front option price instead of being paid as an annual coupon. 2.3. Other models for CD swaptions. not done yet ... 3. CD swaps on indices. 3.1. Mechanics of CD swaps. For single name credit default swaps, the coupon is normally set at the current break-even rate, so the deal can be executed with no initial payment. For credit default swaps on indices, however, the coupon Cind and schedule (and all other terms of the deal) are pre-determined at the indices’ “issue date,” which may be several months (or years) in the past. A CD swap on an index generally has a non-zero value. If the value of the CD swap is positive (negative), one pays (receives) the value of the CD swap as an “up front payment.” Market practice is to calculate the up front payment by using model J of Bloomberg’s CDSW screen to value the CD swap. For example, consider a CD swap on, say, the ITRAXX Eur index, series 2EU1. This index has a set schedule (end date of 9/20/10, quarterly coupon intervals, ...) and a set annual coupon of 35 bps. Suppose one agrees with the counterparty to be the receiver on a CD swap at a coupon of 30bps. Since the counterparty will be paying you a coupon of 35bps per year (and since the …rst coupon is generally not a full period away), one must make an up front payment equal to the value of the CD swap. This value is found by using Bloomberg’s CDSW screen to value the CD swap: one brings up the index, goes to the CDSW screen, and enters a ‡at CDS spread of 30bps with IMM set to Y, sets the model to J and pays the counter-party the total value. Let the index be composed of J names. Let (3.1a) (3.1b)
Nj ´ notional of each name, N tot =
J X
N j = total initial notional of basket,
j=1
and de…ne (3.1c)
Ij (t) =
½
1 0
if entity j defaults on or before t if entity j has not defaulted by t 14
¾
:
The fraction of the basket’s notional that survives to time t is N (t) =
J X
j= 1
[1 ¡ Ij (t)]
Nj = surviving fraction of notional. N tot
and the defaultable part of this surviving notionalis (3.2a)
M (t) =
J X
(1 ¡ Rj ) [1 ¡ Ij (t)]
j=1
Nj ; N tot
where (3.2b)
Rj = recovery rate of name j:
On each coupon date, the receiver gets the index’s coupon paid on the fraction of the notional surviving to the coupon date: (3.3)
N (tk )®(tk¡1 ; tk )C ind =
J X [1 ¡ Ij (tk )] Nj ®(tk¡1 ; tk )Cind Nto t j= 1
rec’d on tk
for k = k1 ; k1 + 1; : : : ; n. Here tk1 is the …rst coupon date strictly after the trade date, and ®(tk¡1 ; tk ) is the coverage of period k. (Notice that the receiver gets the full …rst coupon, even if the trade date is only a few days before the next coupon date tk 1. Of course the value of this payment is accounted for in the up front payment). Whenever a default occurs, the receiver also get the coupon that has accrued from the last coupon date on the defaulting portion of the notional: h i + (3.4) N (t¡ def ) ¡ N (t def ) ®(tk¡ 1 ; t def )Cind =
J i X Nj h ¡ Ij (t+ def ) ¡ Ij (t def ) ®(t k¡ 1 ; tdef )C ind Nto t
rec’d on tdef :
j=1
where tk¡1 is the last coupon date before the default. Upon each default, the coupon leg must make good the loss: (3.5)
+ M (t¡ def ) ¡ M (t def ) =
J h i X Nj ¡ (1 ¡ Rj ) Ij (t+ ) ¡ I (t ) j de f def N tot j=1
paid on tdef :
3.2. Pricing CD swaps on indices. CD index swaps always have completely standard schedules. In particular, the coupon legs are quarterly and end on IMM dates. (See the discussion above about schedule generation for single name credit default swaps.) Let ti0 be the last coupon date before the trade (execution date), and let tk for k = k0 + 1; k0 + 2; : : : ; n be the rest of the coupon dates. A CD index swap’s payment are just the total of the payments of the single name CD swaps for the individual names in the basket. Therefore the value of a index CD swap is exactly the sum of the values of its component single name CD swaps: (3.6a)
CDS V ind =
J X
j=1
15
VjC DS ;
where (3.6b)
V jCDS
n X Nj = C ind ®k N tot k= k 1
Nj ¡(1 ¡ Rj ) N tot
(
Z
tk
ª t ¡ tk¡1 © Qj (tk )D(tk ) + D(t) ¡Q0j (t) dt t ¡ t k k¡1 tk¡1 Z n t k X © ª D(t) ¡Q0j (t) dt: k=1
)
tk¡1
Recall that the value of a single name CD swap could be written as ¡ ¢ Nj VjC DS = Cind ¡ C jbe Lj ; N tot
(3.7a) where risky level (PV01) is (3.7b)
Lj =
n X
k=k 1
®k
(
) ª t ¡ tk¡1 © 0 Q j (tk )D(tk ) + D(t) ¡Qj (t) dt ; tk ¡ tk¡ 1 tk¡1 Z
tk
and the break even coupon is
C jbe =
(3.7c)
(1 ¡ R j )
n Z X k= 1
tk tk¡1
© ª D(t) ¡Q0j (t) dt
Lj
:
Thus the value of the index CDS is ¡ ¢ CDS be V ind = Cind ¡ C ind Lind ;
(3.8a) where
(3.8b)
Lind =
J X j=1
Nj Lj ; N tot
be C ind =
J X
j=1
Cjbe
Nj Lj N tot
Lind
:
So the risky level for the index is just the total of the individual risky levels, and the break-even coupon is the weighted average of the individual coupons. 3.2.1. Pricing Index CDS’s as single name CDS’s. The value of an index CD swap is exactly the sum of the value of its constiuent single name CD swaps. However, for simplicity, one usually treats the basket as a single entity and prices an index CD swap as if it were a single name CD swap. To do this, one de…nes (3.9a)
N¹ (t) =
J X Nj Qj (t) = expected basket notional at t; N tot j=1
(3.9b)
¹ (t) = M
J X
j=1
(1 ¡ Rj )
Nj Q j (t) = expected defaultalbe basket notional at t: N tot 16
The value of the index CDS is (3.9c)
CDS V ind (t)
= C ind
n X
®k
k=k1
¡
n Z X k=1
tk
tk¡1
(
N¹ (t)D(tk ) +
Z
tk
t ¡ tk¡1 © ¹ 0 ª D(t) ¡N (t) dt tk ¡ tk¡ 1 tk¡1
)
© ª ¹ 0 (t) dt: D(t) ¡M
Aside:CDSW and the upfront fee. When we use CDSW to price index swaps, we use the same recovery rate for each name in the basket. In this case, ¹ (t) = (1 ¡ Rind )N¹ (t) M
(3.10)
for some basket averaged recovery rate Rind . This formula is identical to the single name CD swap valuation formula. We can use any of the single name approximations for the index. Indeed, if the break-even coupons were quoted for index CDS’s of various maturities, then we could strip an overall “survival curve” for the basket. This is how CDSW determines the value of the CD index swap. The upfront fee is calculated using ‡at spread and a …xed 40% recovery rate. SWPM uses ...what does SWPM do exactly for inter-period approximations? 3.2.2. Standard approximations. Here we shall not use the approximation of a basket-averaged ¹ (t) and N¹ (t) are distinct curves. Even so, we can make the same approximations within recovery rate, so M each interval as we did in the single name case. The standard approximations are: ¢ in the protection leg, approximate D(t) in interval k by its average value 12 [D(tk¡1 ) + D(tk )]. ¢ in the coupon leg, one replaces (t ¡ tk¡ 1 )=(tk ¡ tk¡1 ) with 12 . One also approximates D(t) by D(tk ) in interval k. These approximations yield (3.11)
CDS V ind = C ind
n X
k=1
That is,
1 2
¡
n X ¢ N¹ k¡1 + N¹ k ®k Dk ¡
1 2
k=1
¡
¢ ¹ k¡1 ¡ M ¹ k (Dk¡1 + Dk ) : M
CDS be Vind = (C ind ¡ C ind )L
(3.12a)
where the risky level (PV01) and break even coupon are (3.12b)
Lind =
n X
k=1
(3.12c)
be Cind =
Xn
1 k=1 2
1 2
¡
¢ N¹ k¡1 + N¹ k ®k Dk
¡ ¢ ¹ k¡ 1 ¡ M ¹k (Dk¡1 + Dk ) M Lind
;
3.2.3. Linear approximation. Instead of the ad hoc approximations used in the “standard approxi¹ (t), and the discount factor mation,” one can approximate the expected notional N¹ (t), the anticipated loss M D(t) as linear in each interval, (3.13a) (3.13b) (3.13c)
N¹k ¡ N¹ k¡1 N¹ (t) = N¹ k¡1 + (t ¡ tk¡1 ) tk ¡ tk¡1 ¹ ¹ ¹ (t) = M ¹ k¡1 + Mk ¡ Mk¡ 1 (t ¡ tk¡ 1) M tk ¡ tk¡ 1 Dk ¡ Dk¡1 D(t) = Dk¡1 + (t ¡ tk¡1 ) tk ¡ tk¡ 1 17
for tk¡1 < t · tk ; for tk¡ 1 < t · tk ; for tk¡1 < t · tk :
This yields the same formulas, except that the risky level is now (3.14)
L=
n X
®k
k=1
©1 ¡ 2
¢ N¹k¡ 1 + N¹k Dk +
1 6
¡
¢ ª N¹ k¡ 1 ¡ N¹ k (Dk¡ 1 ¡ Dk )
3.2.4. Exponential approximation. Suppose we approximate Q(t) and D(t) as piecewise exponential:
(3.15b)
N¹ (t) = N¹ k¡ 1 e¡¸k (t¡ tk¡1) ¹ (t) = M ¹ k¡1 e¡º k (t¡t k¡1) M
for tk¡1 < t · tk ;
(3.15c)
D(t) = Dk¡1 e¡f k( t¡tk¡1)
for tk¡1 < t · tk :
(3.15a)
for tk¡1 < t · tk ;
where the forward hazard rates and interest rate are given by: (3.15d)
¸k =
¹k log N¹ k¡1 = N ; tk ¡ tk¡1
ºk =
¹ k¡1 = M ¹k log M ; tk ¡ tk¡ 1
fk =
log Dk¡1 =Dk : tk ¡ tk¡ 1
This yields (3.16) V rec = Cind
n X
® k Dk N¹ k
k=1
(
1+
1 ¸ 2 k
(tk ¡ tk¡ 1 )
e( ¸k +fk )(tk ¡tk¡1) ¡1 ¡ (¸ k + fk ) (tk ¡ tk¡1 )
n ´ X ¹ k ³ (º +f )(t ¡t ) º k Dk M e k k k k¡1 ¡1 : ¡ º k + fk
1 2
2
(¸ k + f k ) (tk ¡ tk¡1 )
2
)
k=1
Using these approximations, we have be Vre c = (Cind ¡ C ind )Lind ;
(3.17a) with
(3.17b)
Cbe
¡ ¢ ºk ¹ k¡1 Dk¡ 1 ¡ M ¹ k Dk M k=1 º k + f k = ; Lind n P
and (3.17c)
Lind =
n X
k=1
® k Dk N¹ k
(
1 + 12 ¸ k (tk ¡ tk¡ 1 )
e( ¸k +fk )(tk ¡tk¡1) ¡1 ¡ (¸ k + fk ) (tk ¡ tk¡1 ) 1 2
(¸ k + f k ) 2 (tk ¡ tk¡1 ) 2
)
3.2.5. Interval re…nement. We can obtain increased accuracy by re…ning the pricing to monthly intervals. We don’t write these formulas here since they are a repeat of the single name results. 4. Options on CD Index Swaps. Recall that standard options on single name CD swaps are nearly always knock-out: if the entity defaults on or before the start-upon-exercise date, the option is torn up and the option purchaser is out his premium with nothing to show for it. In contrast, European options on CD index swaps are default protected : if the option is exercised, the basket losses that occur between the trade and exercise date are included in the “upfront” fee as bene…ting the protection leg. TO BE COMPLETED LATER The value of the option is ... 18
5. Synthetic CDOs. Consider a (synthetic) CDO written on J names. De…ne (5.1a) (5.1b) (5.1c)
N j ´ notional of each name j,
Rj ´ recovery rate of each name j,
Qj (t) = probability that name j does not default on or before t;
and de…ne the random variable Ij (t) to be the indicator of default: ½ ¾ 1 if entity j defaults on or before t (5.2) Ij (t) = :: 0 if entity j has not defaulted by t We let (5.3a)
N tot =
J X
Nj
j=1
be the total notional in the basket. Then (5.3b)
a j = (1 ¡ Rj )
Nj N tot
is the basket’s loss if entity j defaults, and the basket’s total (per cetntage) loss at time t is (5.3c)
X(t) =
J X
Ij (t)aj = basket’s per centage loss.
j=1
Also de…ne P (t; x), (5.3d)
P (t; x) = prob fx · X(t) < x + dxg
as the probability density for the basket’s per centage loss being x at date t. For reasons that will later become apparent, P (t; x) is called the basket’s unconditional default density 5.1. The mechanics of (synthetic) CDO swaps. The underlying for a synthetic CDO is a basket, and an “attachment” point A and “exhaustion point” B. The notional for this traunche, N tr (t), is de…ned by N tr (t) = µ AB (X(t));
(5.4) where (5.5)
µAB (X) =
8 > < > :
1 B¡X B¡A 0
if X(t) < A if A < X < B if B < X
This traunche covers the losses in the basket which exceed fraction A of the original notional, and which are less than fraction B. More junior traunches absorb any losses up to A, while this traunche protects more senior pieces up to a total basket loss of B: In a CDO swap the …xed leg (coupon leg) receives …xed rate payments at an agreed upon rate C on the traunche notional. On each coupon date tk , the …xed leg gets (5.6)
® k C N tr (tk )
rec’d on tk ; 19
for k = 1; 2; :::; n
Whenever a default occurs which changes the traunche notional, two things happern. First, the coupon accrual is caught up on the lost part of the notional. So if a default occurs, say, at date ¿ in interval k; the …xed leg receives, £ ¤ (5.7) cvg(tk¡1 ; ¿ )C N tr(¿ ¡ ) ¡ N tr (¿ + ) rec’d on ¿ . Second is that the …xed leg must pay its counter-party the amount of the traunche’s loss: £ tr ¡ ¤ N (¿ ) ¡ N tr (¿ +) paid on ¿ caused by the default. (These payments are the “‡oating”, or “protection” leg).
5.1.1. CDO schedules. CDO schedules are set up exactly like the single name credit default swap: ¢ by default, the start date is the trade date, with the e¤ective date (when protection starts) being trade+1 (actually, it is usually midnight of the trade date); ¢ if the tenor of the CDO is listed as tenor months, the end date is the …rst IMM date (the 20t h of March, June, Sept., or Dec.) which is on or after tenor months from the trade date; ¢ the dates are built backwards from the end date, using quarterly intervals and an Act360 day count basis; ¢ the odd period, if any is the …rst period. This period is a stub unless the stub would be less than 1 month. Then “long …rst” is used. ¢ for all days but the start day and the end date, the modi…ed business day convention is used. For USD denominated CDOs, both the USD and GBP holidays to disqualify good business days; for EUR denominated swaps, just the TGT holidays are used; ¢ the coupons are adjusted in that they accrue interest from actual day to actual day, not from theoretical date to theoretical date. 5.2. CDO valuation structure. To value CDOs, de…ne M (t) to be the expected value of the traunche notional at t. Then Z 1 M (t) = E fN ¿ r (t)g = E fµ AB (X(t))g = µ AB (x)P (t; x)dx; 0
where P (t; x) is the “unconditional loss density” of the basket. The value of the (receiver) CDO is ( ) Z tk n X t ¡ tk¡1 0 (5.8) VC DO = C ®k M (tk )D(tk ) + D(t) f¡M (tk ) (t)g dt tk ¡ tk¡1 tk¡1 k=1 n Z tk X ¡ D(t) f¡M 0 (t)g dt: tk¡1
k=1
Clearly, the CDO value depends on the loss distribution only through the expected traunche notional M (t). We can now use the same suite of approximations for M (t) within each interval as used for the the single name CD swap. At the risk of being highly redundant, we re-write the resulting formulas here. 5.2.1. Standard approximation. Using the standard approximations, we write the value of the CDO as (5.9)
rec V CDO =C
n X
k=1
1 2
(Mk¡1 + M k ) ® k Dk ¡
n X
k=1
1 2
(M k¡ 1 ¡ M k ) (Dk¡1 + Dk ) :
We re-write this as (5.10)
rec V CDO = (C ¡ Cbe )L;
20
where the break-even coupon is (5.11)
Pn
1 k=1 2
C be =
(M k¡1 ¡ Mk ) (Dk¡ 1 + Dk ) ; L
and the risky level (risky PV01 ) is (5.12)
L=
n X
1 2
(M k¡1 + M k ) ®k Dk :
k= 1
With these approximations, the traunche value only depends on the (unconditional) expected traunche notionals at the coupon dates. That is, only on M k for k = 1; 2; :::; n. 5.2.2. Linear approximation. Suppose we approximate both the expected traunche notional and discount factors as being linear in each interval. Then (5.13) (5.14)
Mk ¡ Mk¡ 1 (t ¡ tk¡ 1) tk ¡ tk¡ 1 Dk ¡ Dk¡1 D(t) = Dk¡1 + (t ¡ tk¡1 ) tk ¡ tk¡ 1
for tk¡ 1 < t · tk ;
M (t) = Mk¡1 +
for tk¡1 < t · tk :
This yields rec V CDO = (C ¡ Cbe )L
(5.15) with (5.16)
C be =
Pn
1 k=1 2
(M k¡1 ¡ Mk ) (Dk¡ 1 + Dk ) ; L
and with the risky level being (5.17)
L=
n X
®k
k=1
©1 2
(Mk¡1 + M k ) Dk +
1 6
ª (M k¡1 ¡ M k ) (Dk¡1 ¡ Dk )
Apart from the extra term in the risky level, these are the same formulas as in the “standard” case. Once again, these formulas only depend on Mk for k = 1; 2; :::; n. 5.2.3. Exponential approximation. Suppose we approximate M (t) and D(t) as piecewise exponential: (5.18) (5.19)
M (t) = Mk¡ 1 e¡¸k (t¡ tk¡1) D(t) = Dk¡1 e¡fk (t¡ tk¡1)
for tk¡1 < t · tk ;
for tk¡1 < t · tk :
where ¸ k and fk are de…ned by: (5.20)
¸k =
log M k¡1 =M k ; tk ¡ tk¡1
fk =
log Dk¡1 =Dk : tk ¡ tk¡ 1
This yields (5.21)
rec V CDO = (C ¡ Cbe )L
21
with (5.22)
Cbe =
n 1 X ¸k (M k¡ 1 Dk¡1 ¡ M k Dk ) ; L ¸ k + fk k=1
and (5.23)
L=
n X
® k Dk M k
k= 1
(
1 + 12 ¸ k (tk ¡ tk¡ 1)
e( ¸k +fk )(tk ¡tk¡1) ¡1 ¡ (¸ k + fk ) (tk ¡ tk¡ 1 ) 1 2
2
(¸ k + f k ) (tk ¡ tk¡1 )
2
)
5.2.4. Interval re…nement - standard approximation. One can sub-divide each coupon interval into …ner divisions. Suppose one divides each interval into three sub-intervals. Tpically these would be monthly. Using the standard approximation, the value of the CD swap is rec V CDO = (C ¡ Cbe )L
(5.24) where the risky level is (5.25)
L=
n X
®k
k=1
and the break-even coupon is n
(5.26)
C be
1 X = f L
1 2
k=1
¡
©1
6 Dk¡2=3 Mk¡ 1
¡ ¢ + ¡ 16 Dk¡2=3 + 12 Dk¡ 1=3 Mk¡ 2=3
¡ ¢ ª + ¡ 12 Dk¡ 1=3 + 56 Dk M k¡1=3 + 16 Dk M k ;
¢ Dk¡1 + Dk¡2=3 M k¡1 ¡
1 2
¡
¢ Dk¡1 ¡ Dk¡1=3 M k¡2=3
¡ ¢ ¡ 12 Dk¡2=3 ¡ Dk M k¡1=3 ¡
1 2
¡
¢ Dk¡1=3 + Dk M k g:
Similarly, using piecewise linear approximations on each sub-interval yields the same formulas, save that the risky level would be (5.27)
L=
n X
®k
k=1
©¡
1 D 18 k¡1
¢ ¡ 1 + 19 Dk¡2=3 M k¡1 + ¡ 18 Dk¡ 1 + 19 Dk¡ 2=3 +
5 D 18 k¡ 1=3
¢
Mk¡ 2=3
¡ ¢ ¡ 7 ¢ ª + ¡ 29 Dk¡2=3 + 19 Dk¡1=3 + 49 Dk M k¡1=3 + ¡ 18 Dk¡1=3 + 59 Dk M k
The break-even coupon formula would not change: n
(5.28)
C be =
1 X f L
k=1
1 2
¡
¢ Dk¡1 + Dk¡2=3 M k¡1 ¡
1 2
¡
¢ Dk¡1 ¡ Dk¡1=3 M k¡2=3
¡ ¢ ¡ 12 Dk¡2=3 ¡ Dk M k¡1=3 ¡
1 2
¡
¢ Dk¡1=3 + Dk M k g:
5.3. Using market models to complete the pricing. We have reduced the pricing of CDOs to …nding the expected traunche notionals, (5.29)
M (t) =
Z
1
µAB (x)P (t; x)dx; 0
22
with (5.30)
µAB (X) =
8 > < > :
1 B¡X B¡A 0
if X(t) < A if A < X < B if B < X
We need to …nd the (unconditional) loss distribution P (t; x), and thus M (t), at each coupon date (if re…nement is not used), and monthly otherwise. Now, stripping single name credit default swaps will enable us to deduce the (market implied) default probabilities p k (t) of each name. But the loss density and expected traunche notionals depend heavily on the correlations of default events. That is, they are very sensitive to the tendency of defaults to “clump” together statistically. Now a copula describes the joint probability distribution of a set of random variables, given that one knows the marginal distributions, the distribuitions of each variable seperately. To describe the joint distribution of defaults, we use a special set of copula models, the market models. These models will determine the basket’s loss denisty P (t; x) from knowledge of the individual loss probability’s p k (t). Consider an ` factor market model. (` is nearly always 1 in practice). Then one has a ` random variables, U1 ; : : : ; U` which encapsulate the state of the economy and various industries. Let (5.31)
f (t; u)du; : : : du ` = f (t; u1 ; u 2 ; :::; u ` ) = prob fu 1 < U 1 (t) < u 1 + du 1 ; u` < U` (t) < u ` + du` g
be the probability density of the market variables. For a market model (i) the default probabilities of each individual name depend on the market variables, (5.32)
p j (t; u) = p j (t; u1 ; : : : ; u` ) = prob fname j defaults on or before t j U = ug :
Of course, integrating these “conditional” default probabilities p j (t; u) over all possible values of the market variables must match the known default probability for each name j : Z 1 Z 1 (5.33) p j (t) = ¢¢¢ p j (t; u)f (t; u)du 1 ¢ ¢ ¢ du` : ¡1
¡1
(ii) given the values of these market variables, the defaults of each name are independent of each other. I.e., the only interdependence of defaults is through their dependence on the market variables. Recall that ½ ¾ 1 if name j defaults on or before date t Ij (t) = 0 if name j does not default by date t Given the values u of the market variables, the probability of any set of defaults is (5.34)
prob fI1 (t) = i 1 ; I2 (t) = i 2 ; : : : ; IJ (t) = i j j U = ug =
J Y
j= 1
i
The unconditional probability of this set of defaults is: (5.35)
prob fI1 (t) = i1 ; I2 (t) = i2 ; : : : ; IJ (t) = ij g Z 1 Z 1 Y J i 1¡ i = ¢¢¢ [p j (t; u)] j (1 ¡ p j (t; u)) j f (t; u)du1 ¢ ¢ ¢ du ` : ¡1
¡1 j=1
23
1¡i j
[pj (t; u)] j (1 ¡ p j (t; u))
5.3.1. One factor Gaussian copula. The most common market model used in practice is the (one factor) Gaussian copula. For this model we say that name j defaults on or before t if a random variable Yj is below some barrier: (5.36)
Yj < ´ j (t):
Here y k is a Gaussian random variable with mean zero and variance one. We must choose the value of ´ j (t) as (5.37)
p j (t) = N (´ j (t))
where N is the cumulative normal distribution. That way breaching this barrier gives the same default probability as obtained from the single name CD swaps. To complete the model we assume that each name is made up of a “market” factor U and a name-speci…c factor Zj : q p (5.38) Y j = ½ j U + 1 ¡ ½ j Zj ;
Here U , and Z1 ; Z2 ; :::ZJ are assumed to be independent Gaussian random variables with mean 0 and variance 1. The constant ½j is the name’s correlation. So given the value u of the market variable U , the defaults of the names are independent of each other, and occur with probability à ! p ´ j (t) ¡ ½j u (5.39) p j (t; u) = N p = prob fname j defaults on or before t given U = ug : 1 ¡ ½j The (unconditional) probability of any set of defaults at t, is thus (5.40)
prob fI1 (t) = i1 ; I2 (t) = i2 ; : : : ; IJ (t) = i j g Z 1 1 1 2 = p e¡ 2 u [pj (t; u)]ij (1 ¡ pj (t; u))1¡ij du: 2¼ ¡1
5.4. Pricing procedure. In the next section we will detail the convolution method, which is a direct method calculating the conditional loss density (5.41)
P (t; x; u)dx = prob fx · X(t) < x + dx j U = ug
for a given value u of the market variables. One then needs to integrate to …nd the unconditional density Z 1 Z 1 (5.42) P (t; x) = ¢¢¢ P (t; x; u)f (t; u)du1 ¢ ¢ ¢ du ` ; ¡1
¡1
and again to …nd the expected traunche notional, Z 1 (5.43) M (t) = µAB (x)P (t; x)dx: 0
Note this has to be done for every coupon date (if the re…ned intervals are not used) or for every month (if re…ned intervals are used). So here are the steps for calculating the CDO price (a) One chooses an interpolation method (standard, linear, exponential, unre…ned or re…ned intervals). This 24
determines the dates tk on which one needs the traunche notional M (tk ); (b) One discretizes the market variables u into ui for i = 1; :::; ni ; (c) For each date tk and for each point u i, one needs to uses the convolution method detailed below to calculate the conditional loss density P (tk ; x; u i); (d) One then integrates (sums) over the market variables to get the unconditional loss density (5.44)
P (tk ; x) =
ni X
P (tk ; x; ui )f (t; ui)¢ui ;
i¡1
(e) From this one calculates the trauche notionals Z 1 (5.45) M (tk ) = µ AB (x)P (t; x)dx; 0
(f) One then uses these notionals to calculate the CDO values. At this point one must often calculate the delta risks: the change in the CDO’s price for a change in each of the underlying credit spreads. For each name j, one changes the single name default probability by a constant increase ±¸ in the underlying hazard rate: (5.46)
pj (t) =) pj (t) + [1 ¡ p j (t)] [t ¡ t0 ] ±¸:
One then dertmiThis generates a perturbed conditional default probabilities (5.47)
p j (tk ; ui ) =) p j (tk ; ui) + ±p j (tk ; ui ):
One then has to …nd out how this perturbation of the j th name changes the conditional loss distributions (5.48)
P (tk ; x; u i) =) P (tk ; x; u i) + ±j P (tk ; u i):
at the coupon dates and the discretized u values. One then needs to integrate to …nd the unconditional density (5.49)
P (tk ; x) + ±j P (tk ) =
ni X i¡ 1
fP (tk ; x; u i) + ± j P (tk ; u i)g f (t; ui )¢u i;
and …nally integrate again to …nd the change in the expected traunche notionals Z 1 (5.50) M (tk ) + ±j M (tk ) = µ AB (x) fP (tk ; x) + ± j P (tk )g dx; 0
due to a small change ±¸ in the hazard rate of the j th underlying name. Substituting this into the valuation formulas then shows the dollar gain or loss to the traunche due to the hazard rate of the j th name increasing by a small amount. To caculate the hedge, one determines the change in value of a single name CD swap on name j caused by the same shift (5.51)
pj (t) =) pj (t) + [1 ¡ p j (t)] [t ¡ t0 ] ±¸:
The notional needed to hedge out the CDO’s risk to name j is then obtained by dividing the CDO’s risk by this single name CD swap’s risk In order to generate these risks in a reasonable amount of time, we need an algorithm that not only calculates the conditional loss density quickly, but also e¢ciently calculates the change in the conditional loss density due to small changes in the default probability of each underlying name. 25
5.5. Using convolution to calculate the conditional loss density. We …rst describe the convolution algorithm in general terms, and then specialize it to the one factor Gaussian copula model. The convolution method works by induction. De…ne P (j) (t; x; u) = conditional loss density from the …rst j names.
(5.52)
Suppose we already know P (j¡1)(t; x; u), the conditional loss density for the …rst j ¡ 1 names. With probability pj (t; u), the name j defaults and basket loss increases by (5.53)
aj = (1 ¡ Rj )
Nj : N tot
With probability 1 ¡ p j (t; u) name j does not default, and the loss stays the same. So ½ [1 ¡ p j (t; u)] P ( j¡1) (t; x; u) + p j (t; u)P ( j¡ 1) (t; x ¡ aj ; u) for x ¸ aj P (j)(t; x; u) = [1 ¡ p j (t; u)] P ( j¡1) (t; x; u) for 0 · x < aj Clearly, with no names, the probability density is all at 0 loss, P (0) (t; x; u) = ±(x):
(5.54)
and we can build up the loss distribution inductively. 5.5.1. Algorithm: conditional loss. We …rst select a small percentage h, certainly small enough to be less than half the smallest loss: h < 12 aj
(5.55)
for all j:
(Some experimentation is needed to …nd the optimal value of h). De…ne the “bins” n o (5.56) f (j) (t; u; 0) = prob X ( j) (t) = 0 j U = u n o (5.57) f (j) (t; u; k) = prob (k ¡ 1)h < X (j) (t) · kh j U = u ; for k = 1; 2; :::
where X (j) (t) is the loss of the …rst j names. We also keep track of the average value of the loss within each bin: (5.58) (5.59)
m (j) (t; u; 0) = 0
n o m( j)(t; u; k) = E X (j) (t) j (k ¡ 1)h < X (j)(t) · kh and U = u ;
for k = 1; 2; ::::
We shall approximate all probability in the bin as occurring at (j + m)h: The initial condition for the loss distribution is (5.60)
f ( 0) (t; u; k) = ± k0 ;
m (0) (t; u; k) = 0;
for k = 0; 1; 2; :::
For each j = 1; 2; :::; J we (a) de…ne (5.61)
i j + °j = aj =h;
with 0 · °j < 1 and ij being an integer;
(b) and set (5.62)
f ( j) (t; u; k) = [1 ¡ p j (t; u)] f (j¡1)(t; u; k) 26
for k = 0; 1; 2; ::::
(c) For each k = 0; 1; 2; ::: in turn, we de…ne ½ 1 (5.63) k new = k + i j + 0 (5.64) (5.65)
if m(t; u; :k) + °j ¸ 1; if m(t; u; :k) + °j < 1;
(j)
¾
;
µ k = k + aj =h ¡ k new ½ ¾ m(t; u; k) + °j ¡ 1 if m(t; u; :k) + ° j ¸ 1; = m(t; u; k) + °j if m(t; u; :k) + ° j < 1;
and set (j)
(5.66)
(5.67)
m(t; u; knew ) !
f (j) (t; u; knew )m(t; u; k new ) + p j (t; u)f (j¡ 1) (t; u; k)µ k ; f (j) (t; u; k new ) + p j (t; u)f (j¡1) (t; u; k)
f (t; u; knew ) ! f ( j) (t; u; knew ) + p j (t; u)f (j¡ 1) (t; u; k):
5.5.2. Algorithm: Calculating the perturbed conditional losses for risk calculations. To …nd the risk for name j, we have to remove the e¤ects of name j from the calculation, and then reintroduce it with slightly di¤erent probability, (5.68)
p~j (tk ; ui) = p j (tk ; ui ) + ±pj (tk ; u i):
Suppose that P ( J) (t; x; u) is the probability density after accounting for all names J . To remove name j, we need to solve P ¡j (t; x; u) = P ¡j (t; x + aj ; u) =
P (J ) (t; x; u) 1 ¡ p j (t; u)
for 0 · x < aj
P (J ) (t; x + aj ; u) pj (t; u) ¡ P ¡j (t; x; u) 1 ¡ p j (t; u) 1 ¡ p j (t; u)
for x ¸ 0
Then P ¡j (t; x + aj ; u) is the distribution of all names but name j. So let f (t; u; k) and m(t; u; k) be the probability and position within each bin after all J names have been included; this is our numerical approximation to P . Again let us de…ne (5.69)
i j + °j = aj =h;
with 0 · °j < 1 and ij being an integer:
We set g j (t; u; k) =
(5.70)
f (t; u; k) 1 ¡ pj (t; u)
for k = 0; 1; 2; :::;
as before. For each k = 0; 1; 2; ::: in turn, we de…ne ½ ¾ 1 if m(t; u; k) + ° j ¸ 1; new (5.71) k = k + ij + ; 0 if m(t; u; k) + ° j < 1; (5.72) (5.73)
(j)
µ k = k + aj =h ¡ k new ½ ¾ m(t; u; :k) + ° j ¡ 1 if m(t; u; k) + ° j ¸ 1; = m(t; u; :k) + ° j if m(t; u; k) + ° j < 1;
and we set 27
(j)
m(t; u; k new ) !
(5.74)
(1 ¡ p j (t; u))g j (t; u; knew )m(t; u; knew ) ¡ pj (t; u)g j (t; u; k)µk ; (1 ¡ p j (t; u))g j (t; u; knew ) ¡ pj (t; u)gj (t; u; k):
g j (t; u; k new ) ! g j (t; u; knew ) ¡
(5.75)
p j (t; u) g j (t; u; k): 1 ¡ p j (t; u)
The density g j (t; u; k) is the numerical approximation of the loss density for all names in the basket except for name j. We now add it back into the basket with a slightly di¤erent probability p~j (tk ; u i): We …rst set h (j) (t; u; k) = [1 ¡ p~j (t; u)] g j (t; u; k)
(5.76)
for k = 0; 1; 2; :::;
Then for each k = 0; 1; 2; ::: in turn we use exactly the same k new (k) as before, and de…ne ( j)
(j)
µk = µ k = k + aj =h ¡ knew : We then set (j)
m(t; u; k new ) !
(5.77)
h (j) (t; u; k new )m(t; u; knew ) + p~j (t; u)gj (t; u; k)µk ; h (j)(t; u; knew ) + p~j (t; u)g j (t; u; k)
and h( j) (t; u; knew ) ! h (j)(t; u; knew ) + p~j (t; u)g j (t; u; k):
(5.78)
h (j) (t; u; k) then represents the conditonal loss density after the j th credit curve is bumped a little bit. 5.5.3. Identical recovery rates and notionals. The convolution method simpli…es greatly if the loss-given-default is the same for each name aj = (1 ¡ Rj )N j = a
for all j.
In this case we de…ne f (j) (t; u; k) = prob fexactly k defaults by t j U = ug to be the probability of k defaults in the …rst j names. Starting with no defaults, f (0) (t; u; k) = ± k0 ;
(5.79) we itereate (5.80) (5.81)
f (j) (t; u; 0) = [1 ¡ pj (t; u)] f (j¡ 1) (t; u; k)
f (j) (t; u; k) = [1 ¡ pj (t; u)] f (j¡ 1) (t; u; k) + pj (t; u)f ( j¡ 1) (t; u; k ¡ 1)
for k = 1; 2; 3; :::; j
for j = 1; 2; :::; J . Then f (J ) (t; u; k) is the conditional probability of the basket loss being ak. 28
5.5.4. Identical names. Suppose that both the default probabilties of each name are all equal, (5.82)
for all j,
pj (t; u) = p(t; u)
as well as the loss given default being equal: (5.83)
aj = (1 ¡ Rj )N j = a
for all j.
Then there is a closed form expression, f (J )(t; u; k) =
(5.84)
µ ¶ J J ¡k p k (t; u) [1 ¡ p(t; u)] ; k
for the probability that the loss is exactly ak. 5.6. Large basket approximations. Recall that the probability of j of the n names defaulting before t is (5.85a)
µ ¶ n n¡j P j (t j ») = prob fj defaults on or before t j »g = pj (tj») [1 ¡ p(tj» )] ; j
where (5.85b)
p(tj» ) = N
µ
p ¶ º(t) ¡ ½» p : 1¡½
De…ne ª(t; µ j » ) to be the probability that the number of defaults is µn or less at time t : (5.86)
ª(t; µ j ») =
µn X
j= 0
In the limit n ! 1, (5.87)
ª(t; µ j ») =
µ ¶ n n¡j p j (tj» ) [1 ¡ p(tj»)] : j ½
if p(tj») < µ : if p(tj») ¸ µ
1 0
De…ning y by µ = N (y), we can write the unconditional probability ª(t; µ) as ½ p ¾ ½ p ¾ º (t) ¡ ½» º(t) ¡ 1 ¡ ½y (5.88) ª(t; µ) = prob p < y = prob » > : p 1¡½ ½ So (5.89a)
ª(t; µ) = N
µp
1 ¡ ½y ¡ º (t) p ½
¶
where (5.89b)
µ = N (y)
and where the barrier º(t) is de…ned implicitly in terms of the single name default probability by (5.89c)
p(t) = N (º(t)): 29
De…ne F (t; x) as the probability that the basket loss is x or less at date t. Then µp ¶ 1 ¡ ½y(x) ¡ º(t) (5.90a) F (t; x) = N p ½ with y(x) de…ned by (5.90b)
N (y) =
x : 1¡ R
For the CDO traunche K1 to K2 , the expected traunche notional at date t is Expected notional = F (t; K1 ) ¶½ ¾ L µ X 2L ¡ 2` + 1 ` `¡1 + F (t; K2 ¡ (K2 ¡ K1 ) ) ¡ F (t; K 2 ¡ (K2 ¡ K 1 ) ) 2L L L `= 1
Appendix A. Alternative pricing formulas for single name CD swaps. In the main body we showed that the value of CD receiver swap is ( ) Z tk n X t ¡ tk¡ 1 0 (A.1) Vre c = N C ® k Q(tk )D(tk ) + D(t + ±def ) f¡Q (t)g dt tk ¡ tk¡1 tk¡1 k=1 n Z tk X ¡N (1 ¡ R) D(t + ±def ) f¡Q0 (t)g dt: k=1
tk¡1
As noted there, this formula is commonly simpli…ed by using the following standard approximations: ¢ in the protection leg, D(t) is approximated by its average value in each interval k, ³ ´ 1 1 ~ ~ 2 [D(t k¡1 + ± def ) + D(t k + ±def )] = 2 Dk¡1 + Dk ¢ in the coupon leg one assumes that defaults occur, on average, halfway through the interval, so t ¡ tk¡1 tk ¡ tk¡1 is replaced by 12 . One also approximates D(t + ±de f ) in interval k by its …nal value D(tk + ± def ). Together these approximations yield the following value for the CDS: (A.2a)
V rec = N C
n X
k=1
n n o X ~ k (Qk¡ 1 ¡ Qk ) ¡ N (1 ¡ R) ®k Dk Q k + 12 D
1 2
k=1
³
´ ~ k¡1 + D ~ k (Q k¡1 ¡ Q k ) : D
~ k = D(tk + ±def ). This value can be written Here Q k = Q(tk ); Dk = D(tk ); and D (A.3a)
V rec = N L(C ¡ C be );
(A.3b)
V pay = ¡V rec = N L(Cbe ¡ C):
where the risky level (also known as the risky PV01 ) is (A.3c)
L=
n X
k= 1
n o ~ k (Q k¡1 ¡ Q k ) ; ® k Dk Qk + 12 D 30
and the break-even coupon (market CDS coupon ) is ³ P (A.3d)
Cbe = (1 ¡ R)
n 1 k=1 2
´ ~ k¡ 1 + D ~ k (Qk¡1 ¡ Qk ) D L
;
The approximation used in the “standard approximation,” are pretty much ad hoc. Systematic approximations, of ever increasing re…nement, can be used instead. This leads to increasingly accurate formulas which are also increasingly complex. Below we derive these pricing formulas for several di¤erent sets of re…nements. Here we note that these approximations all yield CD swap prices of the form, (A.4a)
V rec = N L(C ¡ C be );
(A.4b)
V pay = ¡V rec = N L(Cbe ¡ C);
with slightly di¤erent formulas for the risky level L and the break-even coupon C be . Moreover, the risky level and break even spread are always of the form
(A.4c)
L=
N X
b iQ(¿ i )
C be =
i=0
N (1 ¡ R) X h iQ(¿ i ) L i=0
for some set of dates (A.4d)
t0 = ¿ 0 ; ¿ 1 ; :::; ¿ N = tn :
These dates ¿ 1 are the coupon dates, except for the approximation methods which split the coupon intervals into …ner subdivisions. Then the sub-interval dates are included in the ¿ i, as well as the coupon dates A.0.1. Linear approximation. Instead of the ad hoc approximations used in the “standard approximation,” one can approximate the survival curve Q(t) and discount curve D(t) as linear in each interval, (A.5a) (A.5b)
Qk ¡ Qk¡ 1 (t ¡ tk¡ 1) tk ¡ tk¡1 ~ ~ ~ k¡1 + Dk ¡ Dk¡1 (t ¡ tk¡1 ) D(t + ± def ) = D tk ¡ tk¡1 Q(t) = Q k¡1 +
for tk¡ 1 < t · tk ; for tk¡1 < t · tk :
This yields the level (A.6a)
L=
n X
k= 1
n ³ ´ o ~ k¡ 1 + 1 D ~ ® k Dk Qk + 16 D 3 k (Q k¡1 ¡ Q k )
and the break-even spread (A.6b)
Cbe = (1 ¡ R)
Pn
1 k=1 2
³
´ ~ k¡ 1 + D ~ k (Qk¡1 ¡ Qk ) D L
;
A.0.2. Exponential approximation. Suppose we approximate Q(t) and D(t) as piecewise exponential: (A.7a) (A.7b)
Q(t) = Q k¡1 e¡¸ k( t¡tk¡1) for tk¡1 < t · tk ; ¡fk (t¡tk¡1¡± def ) ~ D(t) = Dk¡1 e for tk¡ 1 + ±def < t · tk + ± def ; 31
where the forward hazard rate and interest rate are given by: (A.7c)
¸k =
log Q k¡1 =Q k ; tk ¡ tk¡1
fk =
~ k¡1 = D ~k log D : tk ¡ tk¡1
This yields the level and break even coupon, ( ) n X e(¸k +fk )(tk ¡ tk¡1) ¡1 ¡ (¸k + f k ) (tk ¡ tk¡1 ) ~ (A.8) L= ® k Dk Qk + ¸ k Dk Q k 2 (¸ k + f k ) (tk ¡ tk¡1 ) k= 1 (A.9)
C be =
n ³ ´ (1 ¡ R) X ¸k ~ k¡1 Q k¡1 ¡ D ~ k Qk ; D L ¸k + f k k=1
A.0.3. Interval re…nement. One can sub-divide each coupon interval into …ner divisions. Suppose one divides each interval into three sub-intervals. Typically these subintervals would be monthly. Using the standard approximation, on these subintervals yields the risky level (A.8a)
L=
n X
®k
k=1
n
1 ~ D Q 6 k¡ 2=3 k¡1
³ ´ o ~ k¡1=3 + 1 D ~ k Qk¡1=3 + (¡ 1 D ~ k + Dk )Qk ; + ¡ 16 D 6 6
and break-even coupon (A.8b)
C be
³ ´ ~ k¡2=3 + 1 D ~ k¡ 1=3 Qk¡2=3 + ¡ 16 D 6
n ´ ³ ´ 1¡R X 1 ³~ 1 ~ ~ ~ = f 2 Dk¡1 + Dk¡2=3 Q k¡1 ¡ 2 Dk¡1 ¡ Dk¡1=3 Q k¡2=3 L k=1 ³ ´ ³ ´ ~ k¡2=3 ¡ D ~ k Qk¡1=3 ¡ 1 D ~ k¡1=3 + D ~ k Qk g: ¡ 12 D 2
Using piecewise linear approximations on each sub-interval yields (A.9a) L =
n X
k=1
®k
n³
1 ~ 18 Dk¡ 1
´ ³ 1 ~ ~ k¡ 2=3 Q k¡1 + ¡ 1 D ~ + 19 D 18 k¡ 1 + 9 Dk¡2=3 +
5 ~ 18 Dk¡1=3
´
Q k¡ 2=3
³ ´ ³ ´ o ~ k¡2=3 + 1 D ~ k¡1=3 + 4 D ~ k Q k¡ 1=3 + ¡ 7 D ~ k¡1=3 + 5 D ~ k + Dk Qk + ¡ 29 D 9 9 18 9
and the same formula for the break-even coupon: n
(A.9b)
C be =
1¡R X f L
k=1
1 2
³
´ ~ k¡1 + D ~ k¡2=3 Q k¡1 ¡ D
1 2
³
´ ~ k¡1 ¡ D ~ k¡1=3 Q k¡2=3 D
³ ´ ~ k¡2=3 ¡ D ~ k Qk¡1=3 ¡ ¡ 12 D
1 2
³
´ ~ k¡1=3 + D ~ k Qk g: D
Clearly we could use piecewise exponential approximations on subintervals to obtain yet greater accuracy in our pricing formulas. As a practical matter, however, this gives us a very slight gain in the price at the cost of greatly increasing the complexity of the valuation function. Regardless of which approximation is used, to avoid a pricing bias, it is essential to use the same approximation in stripping the credit survival curve Q(t). Appendix B. CDSW pricing. Not ready yet 32
