Credit Default Index Swap Options
Overview
A credit default index swap (CDIS) is a credit default
swap (CDS) on a portfolio of defaultable entities, or more specifically, a
portfolio of single-entity CDSs. It can
be seen as an extension of a CDS on a single entity to a portfolio of
entities. Like a single entity CDS a
CDIS has a payoff leg and a premium leg that has a fixed coupon rate. The basic difference is that in a single-entity
CDS the notional is fixed during the life of the CDS and the protection buyer
is compensated at most once, while in a CDIS the premium notional is variable. Whenever a default in the portfolio occurs, the
premium notional is reduced by the loss amount of the defaulted entity and at
the same time the protection buyer gets compensated by the lost amount. The most popular credit default index swaps
are the so-called standardized credit default index swaps. In these standardized contracts the reference
credit pool is homogeneous, that is, all the reference entities have the same
notional and the same recovery rate. Typical
examples of standardized CDISs are the CDX index and the ITRAXX index. For more details on the credit default index
swaps see the Credit Default Index Swaps (CDS Indices)
FINCAD Math Reference document.
A credit default index swap option (CD index swap option,
or CD index swaption, or CDS index option) is an option to buy or sell the
underlying CDIS at a specified date. A
payer swaption gives the holder of the option the right to buy protection (pay
premium) and a receiver swaption gives the holder of the option the right to
sell protection (receive premium).
This basic definition of a CD index swaption is similar to
that of an option on a single-entity CDS.
Unlike a CD index swap, which is a natural extension of a
CDS on a single-entity to a CDS on a portfolio of entities, a CD index swaption
is significantly different from a CDS option, an option on a single-entity CDS
(For options on single-entity CDSs see the Credit
Default Swap Options FINCAD Math Reference document). In the case of an option on a single-entity,
if the reference entity defaults before the option’s expiry, the option will be
knocked out, which means that the option becomes worthless. For an option on a CDIS, when a reference
entity defaults before the option’s expiry, the loss will be paid by the
protection seller to the protection buyer when the option is exercised. Even if there is only one entity in the
portfolio, a CD index swaption is still different from a single-entity CDS
option: if the entity defaults before
expiry, the option’s seller will pay to the protection buyer the lost amount at
expiry. Clearly, a CD index swaption is
always more valuable than a single-entity CDS option.
Another fundamental difference between a CD index swaption
and a CD swaption is that the former has a fixed premium rate and a strike
spread while the later has only a fixed premium rate. The strike spread is used to calculate an
upfront cash settlement amount of the underlying CDIS. Only when the strike spread is the same as the
fixed premium rate, is this upfront cash settlement amount equal to zero.
FINCAD provides tools to value CD index swaptions and
derive implied volatilities given option prices. The basic assumptions are that the underlying
CD index swaps are standardized CD index swaps and the par CDS spread of the
underlying CDIS follows a lognormal model (Black model), or more generally, a displacement
diffusion model.
Formulas & Technical Details
Consider a CD index payer swaption. The swaption has a
time to expiration 
and the underlying
credit default index swap becomes effective immediately after the option is
exercised. Suppose there are 
remaining entities in
the credit reference pool at the valuation time (= 0). Let 
be the notional and 
the recovery rate of
each entity. Assume that all the
reference entities have the same default probability curve. Let 
be the fixed annual
premium rate and 
the option’s strike
spread. Let 
denote the time 
value of the premium
leg of the underlying CDIS with an annual coupon rate of 1 and 
the time value of the payoff
leg. Then given that there are exactly 
defaults between the
valuation time and the option’s expiration the payoff at the option’s expiry
is:
Equation 1
where
is the time value of the
underlying CDIS with an annual premium rate of 1 and valued with a probability
curve bootstrapped from the strike spread 
and
is the accumulated
loss of the reference pool up to time :
with 
the accumulated loss
of the pool up to the valuation date.
In the above Equation 1
the amount 
represents the upfront
cash settlement value for a payer to
enter into a CD index swap with a fixed coupon 
If the option doesn’t have such an upfront
payment, one can simply set 
in the formula. In this case, given that there are exactly defaults between the
valuation time and the option’s expiration, the payoff of the payer swaption
is:
Equation 2
Note that in either Equation 1
or Equation 2,
there are more terms in the payoff formula than in the payoff formula of a
single-entity CDS option. This makes the calculation of the present value of a
CD index swaption much more difficult than a single-entity CD swaption. In the paper of Liu and Jaeckel [1],
an approximation is used to simplify the calculation as is shown below:
where
is the par CDIS rate
at time 
is the time 
value of $1 at time and
are constants that can
be determined from the underlying CDIS.
If we further assume that the par CDIS spread follows a lognormal
distribution, as we do for a single-entity CD swaption, or more generally, a
displacement diffusion model, we can calculate the present value of the payoff given
in Equation 1.
To calculate the value of the CD index
swaption we can simply multiply the present value of Equation 1
by the probability of exactly defaults and then sum
up the results over 
To calculate the probability of exactly defaults the
correlation of the reference credits must be considered. To estimate correlation one method is to assume
a one factor Gaussian copula model and use the base correlation of a synthetic
CDO (for the one-factor model and the base correlation see the Synthetic CDO Valuation Using Quasi-Analytic
Methods FINCAD Math Reference document). Since the reference pool is homogeneous, the
required probabilities can be calculated by counting the defaulted and survived
entities. For details and results of the
calculation of default probabilities and option valuation, see the paper of Liu
and Jaeckel [1].
Remark 1:
In the above it is assumed that the par CD index swap spread follows a
displacement diffusion model. This model
says that the behavior of a CD index swap spread is like something between a
lognormal model and a normal model:
Equation 3
where
is a volatility
parameter and
is a displacement
parameter. When
, a displacement diffusion model is simply a lognormal model.
When
, it is a normal model. A general displacement diffusion
model can fit the market data better, but the side effect is that the par CD
index spread can become negative with a small but positive probability.
Remark 2: In the above it is assumed that all entities
in the pool have the same default probability curve. If this is not the case, a par CDIS spread
curve can be derived from the basket default probability curve and then a
default probability curve can be bootstrapped from the par spread curve. This probability curve can be used to value
the CD index swaption.
The definitions and calculation formulas of the sensitivity
statistics shown above are given below:
Definitions and formulas of risk statistics
DVOX
The DVOX of a CD index swaption is defined as the change
in the fair value per X basis point upshift in the par CDS spread curve of each
entity in the reference pool. In more
detail, let 
be the fair value of
the CD index swaption. For each entity
in the reference pool, add X basis points to its default curve and bootstrap it
to build a new default curve. Then
revalue the CD index swaption to get a new fair value 
. The DVOX is then
calculated as follows:
Delta
The delta is defined with respect to the par CDIS spread:
where DVOX of CDIS at par is the DVOX of the
underlying CDIS when the premium rate is the par premium rate of the swap.
Theta
The theta of a CD index swaption is the change in the fair
value of the swaption per one day increase of the valuation date. Let 
be the fair value of
the swaption. Then:
BPV
The BPV (basis point value) on a risk free curve is the
change in the fair value of the CD index swaption when the risk-free discount
factor curve is shifted up one basis point. To shift up a discount factor curve simply add
one basis point to every point of the corresponding spot rate curve of the
discount factor curve.
Vega
The vega of a CD index swaption is the change in the fair
value per 1% change in volatility. In
more detail, let 
be the fair value of
the CD index swaption at the CDS spread volatility . Then:
Rho
The rho of recovery rate of a CD index swaption is the
change in the fair value of the CD index swaption per 1% change in the recovery
rate. In more detail, let 
be the fair value of
the
CD index swaption at the recovery rate . Then:
Rho
The rho of correlation of a CD index swaption is the
change in the fair value of the CD index swaption per 1% change in the correlation.
In more detail, let 
be the fair value of
the
CD index swaption at the correlation
Then:
FINCAD Functions
aaCDS_index_std_opt(d_v,
d_exp_u, swpn, d_CDS, ref_npa, num_ref_ini, num_dflt, loss_cum, npa_CDS, cpn_fix,
cpn_ex, freq_pr, acc, d_rul, dp_type, dp_bskt, intrp_tb, rate_recover, modl_para,
hl, dfstd, intrp, calc_para, stat)
Calculates the fair value and risk statistics of a
standardized credit default index swap option.
aaCDS_index_std_opt_iv(d_v,
d_exp_u, swpn, d_CDS, ref_npa, num_ref_ini, num_dflt, loss_cum, npa_CDS, cpn_fix,
cpn_ex, freq_pr, price_option, acc, d_rul, dp_type, dp_bskt, intrp_tb, rate_recover,
corr_displace, hl, dfstd, intrp, calc_para)
Given an option price, a displacement parameter and a
correlation calculates the implied volatility of a credit default index
swaption.
Description of Inputs
|
Input Argument |
Description |
|
d_v |
Valuation date |
|
d_exp_u |
Option expiry date |
|
swpn |
swaption type - switch, 1 = right to receive fixed (sell protection); 2 = right to pay fixed (buy protection) |
|
d_CDS |
It is the
contract dates table of the underlying credit default index swap. It can have 2 to 7 entries. The first four entries hold in sequence the
terminating date, effective date, odd first coupon date and
next-to-terminating coupon date for premium payments. The last three entries hold switch values in
sequence the effective date adjustment, terminating date adjustment and date
generation method for premium payments. Only the terminating date is required. The third and fourth entries default to 0
and the last three entries default to 1. In this release the effective date is always
assumed to be the option's expiry date d_exp_u and hence the value in the second entry is
ignored. |
|
ref_npa |
Notional of a reference entity |
|
num_ref_ini |
Number of reference entities in the initial pool |
|
num_dflt |
Number of defaulted reference entities |
|
loss_cum |
Accumulated loss of the pool. If it has a value of 0, it will be
calculated from the number of defaulted entities. Otherwise this number is used as the
accumulated loss of the reference pool and the number of defaulted entities
is ignored for this purpose. |
|
npa_CDS |
Premium notional.
This parameter can be user specified or calculated. If it is positive, it will be recognized as
a user-specified notional. If it is 0,
it will be calculated as the total notional of the initial reference
pool. A negative value is not allowed. |
|
cpn_fix |
Fixed premium coupon rate of the underlying credit default
index swap. |
|
cpn_ex |
Strike spread |
|
freq_pr |
Premium payment frequency - a switch |
|
acc |
Accrual method of premium – a switch |
|
d_rul |
Business date convention for premium payments - a switch |
|
dp_type |
Curve type – a switch, 1 = a CDS spread curve, 2 = a default probability curve |
|
dp_bskt |
The default curve dp_bskt can be a single-entity default
curve or a multi-entity default curve.
If it is a single-entity default curve, all entities in the reference
pool are assumed to have the same default curve. If it is a multi-entity default curve, the
number of columns of CDS spreads or default probability values must equal the
number of the outstanding (survived) reference entities. For descriptions of an m-entity default
curve see the following: |
|
A default curve, either a default probability or a CDS
spread curve, can be time-based (the first column >0 and <1000) or
date-based (the first column >=1000).
If it is a default probability curve or a time-based CDS spread curve,
the first column has time (in years) or dates. If it is a date-based CDS spread curve, the
first two columns have the effective and terminating dates of the CDS
spreads. The next m columns have
default probabilities/CDS spreads, where m is the number of reference
entities. These columns must be
ordered in the same way as in the reference data table. Times and dates (the terminating dates for
a CDS spread curve) must be increasing.
For a time-based curve, the time-accrual method can be put in the
third entry of the table intrp_tb.
Probabilities must be nonnegative, <=1 and nondecreasing. CDS
spreads must be nonnegative. For a
date-based default probability curve, the probabilities in the first row must
be 0 and the first date must be the valuation date if there is one. Excel users may pass term strings to
represent time intervals. Term strings
must be in the format kB, kD, kW, kM, or kY where k is an integer and D, W, M
and Y stands for calendar day, week, month and year respectively, e.g., 3M =
3 months. |
|
|
intrp_tb |
A default curve
parameter table. The array intrp_tb can have one to six entries. The first
entry has the default probability curve interpolation method. The second entry has the default probability
curve bootstrapping method (see switch sw_1040 in aaCredit_dfltprob_DSSpred2). It is used only if the default curve type is
"par CDS spread curve" (dp_type =1). If this entry is missing and dp_type=1, the bootstrapping method will be set
to method 1 (assuming constant default density). The third entry stores the time accrual (day
counting) method if the default curve is time-based. If this entry is missing and the default
curve is time-based, the time accrual method will be set to 30/360. If the default curve is not time based, this
entry will be ignored. The fourth to
sixth entries store the effective and terminating date adjustment methods and
the date generation method of a CDS curve, respectively. These entries are used only when dp_type = 1. Their default values are 1. |
|
rate_recover |
Recovery rate - any number between 0 and 1. |
|
modl_para |
Model parameter table.
It can be a 2- or 3-entry table.
The first two entries are the volatility and the correlation
parameters, respectively. The third
entry is optional. It is the displacement
parameter of the displacement diffusion model, i.e., the parameter |
|
hl |
|
|
dfstd |
The discount factor curve may be input as a 2-column,
multi-row table (col 1 = date, col 2 = discount factor), or as a single cell
containing a rate. If input as a
single rate, there are three format choices:
1. a rate (1 row, 1
column); or 2. a rate and a
rate quotation basis (1 row, 2 columns); or 3. a rate, a rate
quotation basis, and an accrual method (1 row, 3 columns). If the basis or accrual are not provided, they are assumed
to be annual, actual/365. A 2-column
flat-rate discount factor curve is constructed internally using the rate,
basis, and accrual method. The last
date in the discount factor curve >= last "date" in the default
curve and the terminating date of the swap. |
|
intrp |
Interpolation method of the discount factor curve |
|
calc_para |
A calculation parameter table. It can have 1 or 2
entries. The first entry is a
calculation method used in CDS valuation (see switch sw_624 in the function
aaCDS). The second entry is
optional. For the function aaCDS_index_std_opt
it is the number of basis points used in DVOX calculation. Note that negative values or very large
positive values of the second entry can cause the function to fail. For the function aaCDS_index_std_opt_iv,
it is a level of accuracy. There are
four accuracy levels, corresponding to the convergence tolerance levels
0.0001, 0.000001, 0.0000001 and 0.00000001, respectively. The optional second entry is defaulted to 1
when missing. |
|
stat |
Statistics: any subset of {1,2,…,12} |
|
corr_displace |
Correlation and displacement parameter table. See the descriptions of the table modl_para. |
|
price_option |
An option price table.
It can have one to three entries.
The first entry is the price of the option. The optional second and third entries are,
respectively, the lower and upper bounds of the implied volatility to be
searched. They are not required, but
their inclusion can help reduce the calculation time of the function. Note also that for any set of parameters,
there is a limited range of possible (and reasonable) prices. Inputting a
price outside of this range will lead to an error being returned by the
function. |
|
price_option |
An option price table.
It can have one to three entries.
The first entry is the price of the option. The optional second and third entries are
any estimates of the lower and upper bounds of the implied volatility that is
to be searched, respectively. Including a proper bound or both bounds in the
input list can significantly reduce the calculation time, but a solution
found may not necessarily stay within the given bounds. Note also that for any set of parameters,
there is a limited range of possible (and reasonable) prices. Inputting a price outside of this range
will lead to an error or an inaccurate result being returned by the function. |
Description of Outputs
The following table lists the output statistics:
aaCDS_index_std_opt
|
Output Statistics |
Description |
|
1 |
Fair value |
|
2 |
Value of defaulted entities. This is the present value of
the historical loss of the reference pool. |
|
3 |
Forward par credit default index swap spread |
|
4 |
Delta with respect
to the par credit default index swap spread |
|
5 |
DVOX |
|
6 |
Theta |
|
7 |
Vega of volatility |
|
8 |
BPV on risk-free curve |
|
9 |
|
|
10 |
|
|
11 |
Value of the payoff leg of the underlying credit default
index swap |
|
12 |
Value of the premium leg of the underlying credit default
index swap |
aaCDS_index_std_opt_iv: its output is a table of two elements
|
implied volatility |
number of internal function calls |
Examples
Example 1
Consider a CD index swaption on a standardized CD index
swap that is linked to a reference pool of 125 companies. Each company in the pool has a notional of
1,000,000 and a recovery rate of 40%. Up
to the valuation date, no company has defaulted yet. The option will expire on June 1, 2006. The underlying CD index swap will become
effective immediately after the option is exercised. The fixed annual coupon rate of the default
swap is 3%. According to the contract of
the CD index swaption, no upfront payment will be made to the protection seller
of the swap when the option is exercised and hence the strike spread can be set
to the fixed coupon rate. The index has
been traded in the market for a few years and its historical spread data shows
that its par spread roughly follows a lognormal distribution with a volatility
of 30%. Moreover, synthetic CDOs based
on the reference pool have been actively traded and the implied base
correlation of the equity tranche has been around 0.5. This correlation can be used as the
correlation of the companies when the Gaussian copula model is used to model
credit correlation. These and further
detailed information of the swaption that is required in its valuation is given
in the following four tables.
aaCDS_index_std_opt
|
Argument |
Description |
Example Data |
Switch |
|
d_v |
valuation date |
1-Dec-2005 |
|
|
d_exp_u |
option expiry date |
1-Jun-2006 |
|
|
swpn |
swaption type |
2 |
right to pay fixed (buy protection) |
|
d_CDS |
CDIS contract dates |
see below |
|
|
ref_npa |
notional of each entity |
1000000 |
|
|
num_ref_ini |
number of entities in the initial reference pool |
125 |
|
|
num_dflt |
number of defaulted entities |
0 |
|
|
loss_cum |
accumulated loss of the pool |
0 |
|
|
npa_CDS |
premium notional - to be calculated |
0 |
|
|
cpn_fix |
fixed coupon |
0.03 |
|
|
cpn_ex |
strike spread |
0.03 |
|
|
freq_pr |
premium payment frequency |
3 |
quarterly |
|
acc |
accrual method of premium |
2 |
actual/360 |
|
d_rul |
business date convention for premium payments |
2 |
next business day |
|
dp_type |
default curve type |
1 |
par CDS spread curve |
|
dp_bskt |
default curve |
see below |
|
|
intrp_tb |
default curve parameter table |
see below |
|
|
rate_recover |
recovery rate |
0.4 |
|
|
modl_para |
model parameter table |
see below |
|
|
hl |
holiday list |
0 |
|
|
dfstd |
discount factor curve |
see below |
|
|
intrp |
Interpolation method of the discount factor curve |
1 |
linear |
|
calc_para |
calculation parameter table |
1 |
|
|
stat |
stat list |
{1,2,…,12} |
|
CDIS contract dates (d_CDS)
|
Terminating date |
Effective date |
First coupon date |
Next to last coupon date |
Effective date adjustment |
Terminating date adjustment |
Date generation method |
|
20-Dec-08 |
0 |
0 |
0 |
2 |
2 |
3 |
|
|
|
|
|
switch: do not adjust effective date |
switch: do not adjust terminating date |
switch: IMM |
CDS spread curve of any name (dp_bskt)
|
time (in years) |
CDS spread of credit |
|
0.5 |
0.02 |
|
1 |
0.021 |
|
2 |
0.023 |
|
3 |
0.024 |
|
4 |
0.026 |
|
5 |
0.027 |
|
7 |
0.029 |
|
10 |
0.035 |
Default curve parameter table (intrp_tb)
|
Interpolation of
default probability curve |
Bootstrapping method |
Accrual method |
Effective date
adjustment |
Terminating date
adjustment |
Date generation
method |
|
1 |
1 |
4 |
2 |
2 |
3 |
Model parameter table (modl_para)
|
spread volatility |
correlation of
reference credits |
|
0.3 |
0.5 |
Risk-free discount factor curve (dfstd)
|
Date |
Discount Factor |
|
1-Dec-2005 |
1 |
|
1-Jun-2006 |
0.971285862 |
|
1-Dec-2006 |
0.943396226 |
|
1-Dec-2007 |
0.88999644 |
|
1-Dec-2008 |
0.839619283 |
|
1-Dec-2010 |
0.747258173 |
|
1-Dec-2015 |
0.558394777 |
|
1-Dec-2020 |
0.417265061 |
To value the CD index swaption call the function aaCDS_index_std_opt
to get the following results:
Results
|
Output Statistics |
Description |
Value |
|
1 |
fair value |
1050771 |
|
2 |
value of defaulted entities. |
0 |
|
3 |
forward par CD index swap spread |
0.024907 |
|
4 |
CD index swap spread delta |
0.160032 |
|
5 |
DVOX |
14755.16 |
|
6 |
theta |
-6884.19 |
|
7 |
vega of volatility |
13910.12 |
|
8 |
BPV on risk-free curve |
-34.0727 |
|
9 |
rho of recovery rate |
-1447.153 |
|
10 |
rho of correlation |
6950.884 |
|
11 |
value of the payoff leg of the underlying CD index swap |
6762618 |
|
12 |
value of the premium leg of the underlying CD index swap |
-8145320 |
Example 2
Suppose in Example 1 three companies have defaulted before
the valuation date and the actual loss from these defaults can be calculated
from the notional and the recovery rate. Suppose all other information of the CD index
swaption is the same in Example 1. Then,
to value the swaption, we only need to change the value of the input parameter num_dflt
to 3. The following table shows the results
of the function call to aaCDS_index_std_opt.
Results
|
Output Statistics |
Description |
Value |
|
1 |
fair value |
1878141 |
|
2 |
value of defaulted entities. This is the present value of
the historical loss of the reference pool. |
1748315 |
|
3 |
forward par CD index swap spread |
0.024907 |
|
4 |
CD index swap spread delta |
0.277445 |
|
5 |
DVOX |
25212.48 |
|
6 |
theta |
-8062.06 |
|
7 |
vega of volatility |
13704.45 |
|
8 |
BPV on risk-free curve |
-3.98774 |
|
9 |
rho of recovery rate |
-19733.2 |
|
10 |
rho of correlation |
2215.094 |
|
11 |
value of the payoff leg of the underlying CD index swap |
6505270 |
|
12 |
value of the premium leg of the underlying CD index swap |
-7835355 |
Example 3: Implied Volatility
Suppose in example 1 the volatility is unknown, but the
option is traded at the price of 1100000.
Given that all other input parameters are the same as in the example, to
back out the volatility from the option’s price call the function aaCDS_index_std_opt_iv. The following tables show the inputs of the
function:
aaCDS_index_std_opt_iv
|
Argument |
Description |
Example Data |
Switch |
|
d_v |
valuation date |
1-Dec-2005 |
|
|
d_exp_u |
option expiry date |
1-Jun-2006 |
|
|
swpn |
swaption type |
2 |
right to pay fixed (buy protection) |
|
d_CDS |
CDS contract dates |
see Example 1 |
|
|
ref_npa |
notional of each entity |
1000000 |
|
|
num_ref_ini |
number of entities in the initial reference pool |
125 |
|
|
num_dflt |
number of defaulted entities |
0 |
|
|
loss_cum |
accumulated loss of the pool |
0 |
|
|
npa_CDS |
premium notional - to be calculated |
0 |
|
|
price_option |
option price table |
see below |
|
|
cpn_fix |
fixed coupon |
0.03 |
|
|
cpn_ex |
strike spread |
0.03 |
|
|
freq_pr |
premium payment frequency |
3 |
quarterly |
|
acc |
accrual method of premium |
2 |
actual/360 |
|
d_rul |
business day convention for premium payments |
2 |
next business day |
|
dp_type |
default curve type |
1 |
par CDS spread curve |
|
dp_bskt |
default curve |
see Example 1 |
|
|
intrp_tb |
default curve parameter table |
see Example 1 |
|
|
rate_recover |
recovery rate |
0.4 |
|
|
corr_displace |
correlation and displacement parameter table |
see below |
|
|
hl |
holiday list |
0 |
|
|
dfstd |
discount factor curve |
see Example 1 |
|
|
intrp |
interpolation method of the discount factor curve |
1 |
linear |
|
calc_para |
calculation parameter table |
see below |
|
Option Price Table
|
option price |
lower bound of
volatility |
upper bound of
volatility |
|
1100000 |
0.1 |
0.5 |
Correlation and Displacement Parameter Table
|
correlation of
reference credits |
|
0.5 |
Calculation Parameter Table
|
calculation method |
accuracy level |
|
1 |
3 |
The output of the function call is given as
follows:
|
implied volatility |
number of internal
function calls |
|
0.334633 |
8 |
References
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