Abstract: The value of an option on a credit default index swap consists of two parts. The first one is the protection value due to potential default of the reference names before option expiry date. The second one depends on the value of the underlying credit default index swap where the index consists of the remaining survived names at the option expiry date. This article presents a valuation framework driven by the distribution of the credit default index swap rate conditioned on the state of the reference pool at the option expiry date. As an example, the one-factor Gaussian Copula is used to model the state of the reference pool of the index at the option expiry date. Conditional on the state of the reference pool at the expiry date, the option is valued under the assumption that the conditional forward credit default index swap rate follows a displaced diffusion (shifted lognormal) law.
Default Index Swaps
*
*The authors thank Constantinos Katsileros, Sam Phillips, Ebbe Rogge and Martijn van der Voort for their valuable comments and suggestions.
receive an up-front amount to enter a contract. For instance, if an index has a fixed coupon of 100 basis points (bps) and it is traded at 110bps then the protection seller will receive an up-front cash payment equivalent to 10bps for the duration of the contract. This up-front payment is usually calculated using Bloomberg’s CDS pricer.
Options on CD index swaps give investors the right to buy or sell risks at the strike spread. A payer swaption is an option that gives the holder the right to be the premium payer (protection buyer), while a receiver swaption holder has the right to be the premium receiver (protection seller). A payer swaption is often referred to as a put (right to sell risk), a receiver swaption as a call (right to buy risk). Also traded on the market is a straddle which entitles the holder to choose whether to buy or to sell risk at a specified strike spread. Options on CD index swaps are traded as knock-in. In the case of a payer swaption, should the option holder exer-cise it at the expiry date the holder will be compensated for losses incurred from any default of the reference names between the trade date and the expiry date. In the case of a receiver swaption, the option holder
Yunkang Liu and Peter Jäckel
Credit, Hybrid, Commodity and Inflation Derivative Analytics
Structured Derivatives
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1 Introduction
A credit default swap on an index (CD index swap) is to some extent sim-ilar to a single name credit default swap (CDS). The index is a portfolio of defaultable reference names with equal weights. Like a single name CDS, a CD index swap consists of a protection leg and a premium leg. The cash flows on the protection leg are contingent on losses incurred from cred-it events of the reference names. The premium leg consists of scheduled coupon payments with a fixed coupon rate. The sizes of these cash flows depend on the recovery rate and the notional amount outstanding. For example, consider an index of 100 reference names with a total notional of 100 millions, if one name defaults then the notional amount for premium calculation will be reduced by one million, the protection buyer will deliv-er one million principal amount of the bonds of the refdeliv-erence name suf-fering the credit event to the protection seller in return for one million in cash. A CD index swap usually carries a fixed coupon rate. When it is trad-ed on the market at a different level, there is a requirement to pay or
will have to pay for the losses should he exercise the option. By contrast, a swaption on a single name CDS is usually traded as knock-out, i.e. if the reference name defaults before the option expiry date the trade is termi-nated and there is no exchange of cash flows. A knock-in payer swaption is always more valuable than its knock-out version. For a CD index swap, a knock-in receiver swaption is less valuable than its knock-out version.
This paper deals with the pricing of options on CD index swaps. Section 2 presents a valuation framework driven by the distribution of the credit default index swap rate conditioned on the state of the refer-ence pool at the option expiry date. By conditioning the option price on the state of the reference pool we have effectively followed the survival measure approach used by [Schönbucher 2003] for pricing options on sin-gle name CD swaps. Section 3 presents an alternative pricing model that is currently used by some market practitioners.
2 A Survival Measure Based Option
Pricing Model
Consider a forward-starting CD index swap with a forward start date t0,
coupon payment dates t1, . . . ,tn, and a fixed coupon rate of C. The index
is treated as if it were a single name for the purpose of making assump-tions on the recovery rate and for bootstrapping the default probability from the market prices of traded index swaps. This is equivalent to approximating the underlying reference pool with a homogeneous one. We will comment on the non-homogeneous case at the end of this sec-tion. Furthermore, the interest rate and default intensity are assumed to be independent from each other.
Denote
δi the year fraction between ti−1and ti,
D(t,u) the risk free discount factor, i.e., the price at t of a zero bond that pays 1 at u,
R the recovery rate of a reference name,
p(t,u) the default probability at time u≥tof a reference name
conditional on survival att and for t≤t0let
¯ A(t)= n i=1 δi D(t,ti)(1−p(t,ti))+ ti ti−1 u−ti−1 ti−ti−1 D(t,u)dp(t,u) (1)
be the risky annuity1, and ¯
B(t)=(1−R)
tn t0
D(t,u)dp(t,u) (2)
the expected value of the protection per unit notional. Because of the assumption that the interest rate is independent of the default intensity, the discount factor D(t,u)is assumed to be deterministic.
Let us consider an option to enter the aforementioned CD index swap at the expiry date T=t0. If there are defaults amongst the reference names
before the option expiry date, the reference pool contains fewer reference names at the option expiry date Tthan at the valuation date. For this rea-son, the valuation of the option should be conditional on the state of the reference pool at the expiry date. Let Nbe the notional amount of each name and Mthe number of reference names remaining at the valuation date. Denote by m the set of exactly mdefaults in the reference pool
between the valuation date and the option expiry date T. Noting that the
reference pool is assumed to be (or approximated by) a homogenous one, it suffices to consider the state of the portfolio mfor 0≤m≤M. Conditional
on the reference pool being in a state in mat the expiry date T, the value at
time t≤Tof the underlying index swap to the protection buyer is
(M−m)N[B(t)−CA(t)], (3) where A(t)= 1 1−p(t,T)A¯(t), B(t)= 1 1−p(t,T)B¯(t) (4)
are the annuity and protection value per unit notional conditional on the survival of the rest of the reference names at the expiry date T. It is the division by 1−p(t,T), the survival probability at T, in (4) that turns the risky annuity and protection value into annuity and protection value conditional on survival at T.
In the case of a payer swaption with a strike coupon rate of K, the m
-conditional pay-off at Tis
max((M−m)N[B(T)−CA(T)+(C−K)A˜]+Lm,0), (5)
where
Lm= ¯L+(1−R)mN (6)
is the aggregated loss due to default with L¯being the accumulated loss
due to default between the trade date and the valuation date t, and
˜ A= n i=1 δi D(t0,ti)(1− ˜p(t0,ti))+ ti ti−1 u−ti−1 ti−ti−1 D(t0,u)dp˜(t0,u) (7)
is a risky annuity with p˜(t0,t)being the default probability bootstrapped
from a flat CDS curve with CDS rates fixed as K. The quantity (C−K)A˜ represents the upfront cash settlement value per unit notional to the
payer for entering a CD index swap with a fixed coupon C, even though K
is the agreed coupon rate by the payer and the receiver of the swap. We
note that had it been settled according to the index CDS curve at T the
m-conditional pay-off would simply be
max((M−m)N[B(T)−KA(T)]+Lm,0). (8)
On the other hand, when the strike Kis close to either the fixed coupon
rate C or the forward rate S(t), (8) is a good approximation of (5). For 0≤m<M, by using A(t)as the numéraire, we see that the value at time tof the m-conditional pay-off is
Vpayer m (t)=(M−m)NA(t)EM (A)max (S(T)−cm,0)], (9)
^
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less at time t, and
cm=C−
(C−K)(M−m)NA˜+Lm
(M−m)NA(T) . (11)
To be able to calculate the value of the pay-off, we need to express 1/A(T)
in cmin terms of S(T). Motivated by the linear swap rate model [Hunt and
Kennedy 2000], we propose the following approximation D(u,T) A(u) ≈a(t)+b(t)S(u), t≤u≤T, (12) where a(t)= nD(t,T) i=1 δiD(t,ti) , b(t)= 1 S(t) D(t,T) A(t) −a(t) . (13)
Roughly speaking, a(t)is chosen so that both sides of (12) match when p(u,v)=0and thus S(u)=0for v≥u≥t, and b(t)is chosen so that
D(t,T) A(t) =E M(A) D (T,T) A(T) =EM(A)[a(t)+b(t)S(T)]=a(t)+b(t)S(t), (14)
noting that D(u,T)/A(u)and S(u)are martingales under the probability measure M(A). Hence, EM(A) [max(S(T)−cm,0)]=(1+bm)EM(A)[max(S(T)−Km,0)], (15) where bm= (C−K)(M−m)NA˜+Lm (M−m)N b(t) (16) and Km= 1 1+bm C−(C−K)(M−m)NA˜+Lm (M−m)N a(t) . (17)
There is no swap left when all the reference names default before T. Thus, the value at time tof the M-conditional pay-off is
VMpayer(t)=LMD(t,T). (18)
To summarize, the value at time tof the payer swaption is
Vpayer(t)= M−1 m=0 Pm(M−m)NA(t)(1+bm)EM(A)[max(S(T)−Km,0)] +PMLMD(t,T), (19)
where Pmdenotes the probability of m.
M m=0 Pm=1, M m=0 mPm=Mp(t,T), EM(A)[S(T)]=S(t) (21)
that the values of payer and receiver swaptions obey a call-put parity of the form Vpayer(t)−Vreceiver(t)=MNA¯(t)(S¯(t)−K), (22) where ¯ S(t)=S(t)+ (L¯/MN+(1−R)p(t,T))D(t,T) +(C−K)(AD˜ (t,T)−A(t))(1−p(t,T)) (1−p(t,T))A(t) (23)
is the default-and-settlement-adjusted forward rate, recalling that A¯, A and Sare defined in (1), (4) and (10), respectively. Because of this call-put parity we only consider payer swaptions in the rest of this paper. For ease of comparison, we shall call a CD index swaption at-the-money if the default-and-settlement-adjusted forward rate S¯(t)equals the strike K.
Empirical research such as [McGinty and Watts 2003] suggests that the dynamical behaviour of credit default swap rates tends to be complex and neither completely lognormal nor normal. In general, it is close to a normal distribution for high credit quality reference names and to a log-normal distribution for low quality reference names. Therefore, a reason-able assumption might be that given tas the valuation date, S(u)follows the law generated by a displaced diffusion process
dS(u)=σ[βS(u)+(1−β)S(t)]dWu, t≤u≤T (24)
where β≥0is a displacement parameter and σ is a volatility parameter.
Note that there is no drift term in (24) because the swap rate we are con-cerned with is a martingale with respect to the probability measure M(A).
When β=1, the process (24) is lognormal. When β=0, it becomes a
nor-mal process. The displaced diffusion process is frequently used as a con-venient method to model implied volatilities both in the equity option markets as a result of leverage [Rubinstein 1983], and for interest rate deriv-atives [Khuong-Huu 1999]. A discussion of the displaced diffusion process and its properties can be found in [Jäckel 2002], [Rebonato 2002] and many other reference books. However, one needs to take extra care in the case of
β <1, since the swap rate S(u)modelled by (24) can be negative with a small but positive probability. The beauty of using the displaced diffusion process is that it is no harder to get analytic option price formulae than it is in the case of a lognormal process. Knowing that (βS(u)+(1−β)S(t))is lognormal when β >0and that S(u)is normal when β=0, we have
EM(A)
^
where we define Call(S,K,T, σ, β)= σ√√TS 2πe −1 2d 2 3−(K−S)(d3) , β=0 1 β S(d1)−Kβ(d2) , β >0, Kβ >0 S−K, β >0, Kβ≤0 (26) with Kβ =βK+(1−β)S, and d1= log(S/Kβ) βσ√T + 1 2βσ √ T, d2=d1−βσ √ T, d3= S−K σ√TS. (27) In order to calculate the distribution of the number of defaults at the expiry T in a way that is consistent with the market practice for CDO tranche pricing, we choose the homogeneous one-factor Gaussian Copula model (see, e.g., [Schönbucher 2003]), noting that the reference pool istreated as a homogeneous one. Let ρ≥0be a correlation parameter,
the cumulative density function of the standard normal distribution, and q(x)= −1(p(0,T))− √ρx √ 1−ρ (28)
the default probability at time Tconditional on a given sample xof the
standard normal distribution. Then Pm= M m ∞ −∞ q(x)m(1−q(x))M−mϕ(x)dx, m=0,1,· · ·,M, (29)
where ϕis the density function of the standard normal distribution. We
note that the integral in (29) can be solved numerically with Gauss-Hermite quadrature (see, e.g., [Press et al 1992]). For a short-dated option, it is reasonable to assume that Pm is very small for large m. Hence, one
may choose ρto be the implied correlation of the
trad-ed equity tranche of the corresponding index.
By combining formulae (19), (25), and (29), we are able to price payer swaptions on credit default indices. The resulting formula resembles the equity call option formula in [Merton 1976, 1990] where the underlying process is assumed to be a jump diffusion of the form
dS(t)
S(t) =µdt+σdWt+(J−1)dN(t) (30)
where Jis the jump size, N(t)is a Poisson process. Of course, in our case there are only a finite number of jumps, each jump corresponds to a state of the refer-ence pool at the option expiry, the sizes of these jumps are not identical, and the arrival of the jumps are not Poisson. It is worth noting that (19) is a gener-ic prgener-icing formula and one can choose a different model for calculating Pm and a different distribution
law for S(T).
Example 1.To illustrate the impact of the correlation parameter ρon the option price, we consider payer swaptions that give the option hold-ers the right to buy protection in three months time on a 4 Year 9 Month index CD swap. Suppose the index consists of 30 reference names with a total notional of 10 million. It is traded at 100bps flat and is assumed to have 40% recovery rate. The interest swap rate is 4%. The model parame-ters are σ =50% and β=1. Figure 1 shows that correlation has little effect on the option value unless it is deeply out-of-money.
Example 2.When β=1, the option pricing model presented here can generate volatility skews, i.e., the implied Black volatility (the equivalent volatility for β=1) can display certain levels of skewness. Consider the
same put option as in Example 1 but priced with ρ=20%and different
values of β. Figure 2 shows that the two different shapes of skewness, one for β <1and the other for β >1.
To model volatility smiles, one may choose to replace the displaced diffusion law generated by process (24) with stochastic volatility models such as Heston or SABR. Due to the simple and modular setup of our
approach, the Heston or SABR densities can be used as a drop-in
replace-ment for the distribution of the forward credit index default swap rate. One may also choose to model the state of the portfolio at the option expiry date using a different type of copula. After all, the copula model is merely used to generate the distribution of the number of defaults at the option expiry.
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-10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000 0% 10% 20% 30% 40% 50% 60% 70% 80% 90%Correlation
Option Value
Strike=90bps Strike=100bps Strike=110bps Strike=120bps Strike=130bps Strike=140bps Strike=150bpsThe ideas presented in this section can be extended to cover the case when the reference pool is not homogeneous. If the reference pool is relatively homogeneous, we can approximate the pool by an homogeneous one and then price the option accordingly. Otherwise, we need to condition the option pricing on all the distinguishable states of the reference pool instead of merely on the number of
defaults. The total number of distinguishable states is 2M, which can
be very large even for a relatively small Msuch as 30 (the number of names for the DJ iTraxx Europe HiVol and the Crossover indices). To be computationally practical, one may have to approximate the original reference pool by a homogeneous pool for cases where there are more than a couple of defaults. In either case, we may have to make adjust-ment to the CDS curves of the underlying reference names. This is because, while we can infer the fair credit default swap rate of an index CD swap directly from the CDS curves of the underlying refer-ence names, the market may trade the index swap at a different level due to liquidity issue and different definitions of credit event for indices and for single names.
We emphasize that we have not made any attempt to model the effect of the default of any single name on the remaining survival names, a sub-ject that is both difficult and interesting. However, recent events such as the default of Pamalat show that it is questionable whether the default of one name needs to be modelled as causing significant spread jumps to other names within a portfolio of names.
recall that A¯ and ¯Sare defined in (1) and (23), respectively. Assuming the default_and_settlement_adjusted forward rate
¯
S(t)follows the displaced diffusion process (24), by using A¯(t)as a “numéraire” we find that the present value of the pay-off is
MNA¯(t)Call(S¯(t),K,T−t, σ, β). (32) The payer swaption is effectively valued by conditioning on the “expected state’’ of the reference pool at the expiry date. Note that the adjusted forward S¯(t)is like a big carpet where the default and settlement adjustments are put underneath it.
The lognormal case of (32) or its variations seem to be popular with some market practitioners. Because of its sim-plicity, (32) is a suitable candidate for quoting purposes. However, it does have problems, in theory as well as in prac-tice, for CD index swaption valuation.
First of all, there is a technical flaw in the derivation of (32) in that the risky annuity A¯(t)is not a valid numéraire since it can become zero upon the default of all underlying assets (read-ers are referred to [Schönbucher 2003] for an excellent discussion on the survival measure approach for pricing of single name CD swaption.). In the special case of an index with a single reference name (M=1 and
K =C) the pricing formula (32) is inconsistent with the obvious choice
NA¯(t)Call(S(t),K,T−t, σ, β)+(1−R)Np(t,T)D(t,T), (33) which is simply the option value in the knock-out case (see, e.g., [Hull and White 2002]) and [Schönbucher 2003] for the lognormal case) plus the present value of the expected loss due to default by the option expiry date.
Even though the model parameters σ and βin (32) have slightly different
meanings from those in (33), it does show that (32) is less than ideal. Secondly, (32) may not be adequate in capturing the distributional nature of the number of default at the option expiry, especially when the option is deeply out-of-money. To be more precise, (32) may underesti-mate the values of out-of-money options as is shown by the following example.
Example 3. Consider a payer swaption that gives the option holder the right to buy protection at 200bps in three months time on a 4 Year 9 Month CD index swap of two reference names with a total notional of 10 million and a fixed coupon rate of 100bps. Assume that the interest rate is 4% and both reference names’ CDS curves are traded at 100bps flat and are assumed to have 40% recovery rate. This means that the probability of a sin-gle name default within 3 months is approximately 1−exp(−0.01/ (1−0.4)∗0.25)≈0.42%, the forward swap rate for the index is 100bps, the default-adjusted forward swap rates is 106bps, and the annuity is 4.15.
40% 42% 44% 46% 48% 50% 52% 54% 70 80 90 100 110 120 130 140 Strike (bps)
Implied Black Volatility
Beta=1.2 Beta=1.6
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W
Let us estimate the option value from first principles. If there is one default before the option expiry and the CDS spread on the remaining survived name has not changed dramatically, the option holder would exercise the option to receive a protection payment of (1−0.4)∗5,000,000= 3,000,000at the option expiry date. The present value of the payment is approximately 3,000,000∗exp(−4%∗0.25)≈2,970,000. At the same time the option holder makes a mark-to-market loss of approximately
(200−100)/10000∗4.15∗5,000,000=207,500 on the remaining CD
index swap. The total value to the option holder would be 2,762,500in
this case. If both names default then the value to the option holder would be even greater. Suppose defaults between the two reference names are independent of each other. The probability of having at least one default is 1−(1−0.42%)2≈0.838%. Therefore, the expected value to the option
holder would be slightly greater than 2,762,500∗ 0.838%≈23,150. However, the swaption price given by (32) is approximately 260 if the log-normal volatility of the default-adjusted forward CD index swap rate is assumed to be σ =50%. To have the option price given by (32) exceed 23,150, a lognormal volatility of over 120% is needed. By contrast, we are able to generate an option price of 23,500by using (19), (25) and (29) with a lognormal volatility of 50% and zero correlation. A higher correlation generates a slightly higher option value. For example, the option value obtained from (19), (25) and (29) is 24,300when the correlation is 90%. Because the option value in this case is dominated by the value of the expected loss which is approximately 25,000, it is hardly surprising to see that the option value is almost insensitive to correlation.
1.If the coupon is paid on the notional remaining at the coupon date, i.e., contingent accrued coupon is not paid, then the annuity reduces to
¯
A(t)=ni=1δiD(t,ti)(1−p(t,ti)).
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