Credit Default Swap Options

Overview

A credit default swap option is also known as a credit default swaption. It is an option on a credit default swap (CDS). A CDS option gives its holder the right, but not the obligation, to buy (call) or sell (put) protection on a specified reference entity for a specified future time period for a certain spread. The option is knocked out if the reference entity defaults during the life of the option. This knock-out feature marks the fundamental difference between a CDS option and a vanilla option. Most commonly traded CDS options are European style options.

Similar to the credit default swaps, there are varieties of CDS options:

· CDS options on a single entity with a regular payoff for the default leg;

· CDS options on a single entity with a binary payoff for the default leg;

· CDS options on a basket of entities with regular payoff for the default leg;

· CDS options on a basket of entities with a binary payoff for the default leg.

Generally, the default probability curve and the recovery rate of a reference entity are the most important factors that affect the value of a CDS option. If a CDS option has a basket of reference entities, the default correlations of the reference entities are also important factors that affect the value of a CDS option. CDS values can also be affected significantly by the types of basket defaults. Currently, the most common types of basket defaults are the first-to-default, the n^th-to-default, the first-n-to-default, and all-to-default.

Formulas and Technical Details

Suppose a CDS option gives its buyer the right to buy protection on a credit reference between times and . Let be the forward CDS spread observed at time and the option value at time . Let be the value at time of the CDS premium that pays $1 per year. Then the payoff of the option at time is:

where is a strike spread. Following Jamshidian’s idea there is a risk-neutral probability such that:

where denotes the expectation with respect to the risk-neutral probability. This leads to:

Assuming that conditional on there being no default prior to time , follows a log-normal distribution, and using the well-known Black-Scholes formula we obtain:

where is the cumulative standard normal distribution:

and

For more details see the paper of Hull and White [1].

Calculation of risk statistics

DVOX

1. DVOX on all credit spread curves of the entities in the reference pool

This statistic is an output of a fair value calculation function, e.g., aaCDS_opt. It is defined as the change in the fair value per basis point shift in all the par CDS spread curves of the entities in the reference pool. In more detail, let be the fair value of a CDS option. For every entity in the reference pool, add basis points to its default curve (if it is a default probability curve, derive a par spread curve first) and build a new default curve. Then combine these default curves together to form a basket default curve and at last use this basket default curve as an input to revalue the option. Let be the new fair value. The DVOX is then calculated as follows:

2. DVOX on a single reference entity

This statistic is calculated in functions, e.g., aaCDS_bskt_risk, that calculate risk statistics exclusively. It is defined similarly as above, but only the default curve of the specified entity is shifted basis points.

Delta

The delta on the par CDS spread of a reference entity is defined as:

where DVOX at par of a CDS is the DVOX of a CDS with its premium rate being the par premium rate of the CDS.

Rho of recovery rate

1. Rho of recovery rate of the whole reference pool

This statistic is calculated in a fair value calculation function. It is the change in the fair value of a CDS option per 1% change in the recovery rate. In more detail, let be the fair value of the option when the entity’s recovery rate is . Then:

If the recovery rates of the reference entities differ, the recovery in the above formula should be replaced with a recovery rate vector and to bump it we simply add 0.01 to every component of the vector.

2. Rho of recovery rate of a single reference entity

This statistic is calculated in functions that calculate risk statistics only. It is defined similarly as above, but only the recovery rate of the specified entity is bumped 1%.

Rho of correlation

1. Rho of correlation on the whole correlation matrix

This statistic is an output of functions that take in a correlation matrix and calculate fair values. It is the change in the fair value of a CDS option per 1% change in the correlation. In more detail, let be the fair value of the CDS option with correlation matrix . Then:

where is the correlation matrix getting from by shifting all correlations up with 0.01.

2. The rho of correlation on an entity pair

This statistic is calculated in functions that calculate risk statistics only. It is defined similarly as above, but only the correlation of the specified entity pair is shifted 0.01.

Theta

The theta of a CDS option is the change in the fair value of the option per one day increase of the valuation date. Let be the fair value of the option. Then:

BPV

The BPV (basis point value) of a risk free discount factor curve is the change in the fair value of a CDS option 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.

Note: Above risk statistics for a binary CDS option can be defined similarly, just replacing CDS with binary CDS in above definitions.

FINCAD Functions

aaCDS_opt(d_v, d_exp_u, vlt, swpn, contra_d, cpn_pr, freq_pr, pr_acc_type, acc, d_rul, prem_fix, ref_type, ref_tbl, payoff_type, dp_type, dp_crv, intrp_prob, rate_recover, hl, df_crv_std, intrp, calc_type, stat)

Calculates fair value and risk statistics of a credit default swap option. The default estimation data of the reference name can be a par CDS spread curve or a default probability curve.

aaCDS_bin_opt(d_v, d_exp_u, vlt, swpn, contra_d, cpn_pr, freq_pr, pr_acc_type, acc, d_rul, prem_fix, payoff_type, cash_rate, dp_type, dp_crv, intrp_prob, rate_recover, hl, df_crv_std, intrp, calc_type, stat)

Calculates fair value and risk statistics of a binary credit default swap option. The default estimation data of the reference name can be a par CDS spread curve or a default probability curve.

aaCDS_bskt_opt(d_v, d_exp_u, vlt, swpn, contra_d, rank, protect_type, pr_tbl, pr_fix, ref_type, ref_bskt, corrs, m_corr, p_off, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, calc, stat)

Calculates fair value and risk statistics of an option on a basket credit default swap. The defaults of the reference names can be independent or correlated. Also the default estimation data can be par CDS spread curves or default probability curves.

aaCDS_bskt_bin_opt(d_v, d_exp_u, vlt, swpn, contra_d, rank, protect_type, pr_tbl, pr_fix, corrs, m_corr, p_off, cash_rate, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, calc, stat)

Calculates the fair value, payoff and premium values and other statistics of a binary basket credit default swap. The defaults of the reference names can be independent or correlated. Also the default estimation data can be par CDS spread curves or default probability curves.

aaCDS_bskt_opt_risk(d_v, d_exp_u, vlt, swpn, contra_d, rank, protect_type, pr_tbl, pr_fix, ref_type, ref_bskt, corrs, m_corr, p_off, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, calc, out_type, asset_list)

Calculates the credit spread, recovery rate and correlation sensitivities of an option on a basket credit default swap.

aaCDS_bskt_bin_opt_risk(d_v, d_exp_u, vlt, swpn, contra_d, rank, protect_type, pr_tbl, pr_fix, corrs, m_corr, p_off, cash_rate, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, calc, out_type, asset_list)

Calculates the credit spread, recovery rate and correlation sensitivities of an option on a binary basket credit default swap.

aaCredit_DS_fs_sep_opt(d_v, d_exp_u, vlt, swpn, contra_d, princ_pr, cpn_pr, freq_pr, pr_acc_type, acc, d_rul, prem_fix, loancpn, loanfixed_tbl, acc_fs, payoff_type, prob_crv, intrp_prob, rate_recover, hl, df_crv_std, intrp, calc_type, stat)

Calculates fair value and risk statistics of an option on a credit default swap which is based on a custom-structured bond (or loan)

aaCredit_DS_bskt_fs_sep_opt(d_v, d_exp_u, vlt, swpn, contra_d, rank, protect, prem_tbl, prem_fix, cpn_bskt, loan_bskt, acc_fs, corr_mat, meth_corr, payoff_type, prob_bskt, intrp_prob, recov_tbl, hl, df_crv_std, intrp, calc_type, mc_trial, stat)

Calculates fair value and risk statistics of an option on an nth-to-default basket credit default swap which is based on a custom-structured bond (or loan).

aaCDS_opt_iv(d_v, d_exp_u, price_opt, swpn, contra_d, cpn_pr, freq_pr, pr_acc_type, acc, d_rul, prem_fix, ref_type, ref_tbl, payoff_type, dp_type, dp_crv, intrp_prob, rate_recover, hl, df_crv_std, intrp, calc_type)

Calculates the implied volatility of a credit default swap option. The default estimation data of the reference name can be a par CDS spread curve or a default probability curve.

aaCDS_bin_opt_iv(d_v, d_exp_u, price_opt, swpn, contra_d, cpn_pr, freq_pr, pr_acc_type, acc, d_rul, prem_fix, payoff_type, cash_rate, dp_type, dp_crv, intrp_prob, rate_recover, hl, df_crv_std, intrp, calc_type)

Calculates the implied volatility given a binary CDS option.

aaCDS_bskt_opt_iv(d_v, d_exp_u, price_opt, swpn, contra_d, rank, protect_type, pr_tbl, pr_fix, ref_type, ref_bskt, corrs, m_corr, p_off, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, calc)

Calculates the implied volatility of an option on a basket credit default swap. The defaults of the reference names can be independent or correlated. Also the default estimation data can be par CDS spread curves or default probability curves.

aaCDS_bskt_bin_opt_iv(d_v, d_exp_u, price_opt, swpn, contra_d, rank, protect_type, pr_tbl, pr_fix, corrs, m_corr, p_off, cash_rate, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, calc)

Calculates the implied volatility given a binary basket CDS option.

Description of Inputs

Only a few parameters are shown here. For the description of other parameters see the Credit Default Swaps and Basket Default Swaps FINCAD Math Reference documents, where descriptions for functions on credit default swaps and basket credit default swaps are given.

Input Argument	Description
d_v	Valuation date
d_exp_u	Option expiry date
contra_d	In the functions aaCredit_DS_fs_sep_opt and aaCredit_DS_bskt_fs_sep_opt, it is a an array of 2 to 4 entries (terminating date, effective date, first coupon date, next to last coupon date). The last two entries are optional. Their default values are 0. In all other functions it is an array of 2 to 7 elements. The extra optional 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 and effective date are required. Their default values are: 1 = adjust effective date, 1 = adjust terminating date, and 1 = backward date generation. Note that for most credit default swaps in the market, the effective date and maturity date are not adjusted and the premium cash flow dates are IMM dates. For this case, simply set both the effective date and maturity date adjustment methods (the 5^th and 6^th entries in the table) to 2 (no adjustment) and the date generation method (the 7^th entry in the table) to 3 (IMM date generation method).
vlt	Volatility of the CDS spread
swpn	CDS option type - a switch: 1 = right to receive fixed (sell protection); 2 = right to pay fixed (buy protection)
price_opt	CDS option price
out_type	Output type – a switch: 1 = delta of credit spread; 2 = DVOX of credit spreads; 3 = rho of recovery rate(s); 4 = rho of correlation (matrix)
asset_list	Entity list, any subset of {1,2,…,number of reference entities}

Description of Outputs

Functions valuing CDS options have eight output statistics as follows:

Output Statistics	Description
1	fair value
2	par forward CDS spread
3	theta: the change in fair value per one day increase of the valuation date
4	vega: the change in fair value per 1% change in volatility
5	basis point value of risk-free curve: the change in fair value per one basis point up shift of the risk-free discount factor curve
6	DVOX of CDS spreads: the change in fair value per X basis points up shift in the par CDS spread curve of each entity in the reference pool.
7	rho of recovery rate(s): the change in fair value per 1% up shift in the recovery rate of each entity in the reference pool.

Single entity CDS functions have one more statistic:

Output Statistics	Description
8	delta of credit spread. DVOX of CDS spread/DVOX of CDS at par

For basket CDS functions, there is one different statistic:

Output Statistics	Description
8	rho of correlation matrix. The change in fair value per 1% up shift of all correlations.

Sensitivity functions (aaCDS_bskt_opt_risk, aaCDS_bskt_bin_opt_risk) calculate delta, DVOX, rho of recovery rate and rho of correlation of options on basket CDS related to specific reference entities.

Examples

Context examples are given for a CDS option on a single entity CDS and a CDS option on a basket CDS.

Example 1: A CDS option on a single entity CDS

Consider a CDS option that gives its buyer the right to buy protection on a name with a notional of 1,000,000. The option expires on Dec. 1, 2006. If the option holder exercises the option at expiry, the underlying CDS will be effective immediately. The underlying CDS matures on Dec. 20, 2008. Under the CDS contract the holder pays a quarterly premium coupon of 9% and at the time of default pays the accrued interest. In return, at the time of default the holder receives the difference of the bond principal and its recovery value. The premium accrual method is actual/360 and the premium cash flow dates are adjusted to the next business day except for the effective date and the maturity date which are not adjusted. Suppose the recovery rate of the reference name is 30%, and that the CDS spread curve of the reference name is given as follows:

dp_crv: Default Curve

term	CDS spread
6m	0.05
1y	0.055
2y	0.06
3y	0.065
5y	0.07

Suppose further that the volatility of the par CDS spread is 40%. Today’s date is Dec. 1, 2005. The spot risk-free discount factor curve is

dfstd: Discount Factor Curve – Risk Free

grid 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 CDS option, call the function aaCDS_opt, ignoring holidays for simplicity, to get the following results:

aaCDS_opt

Argument	Description	Example Data	Switch
d_v	value (settlement) date	1-Dec-05
d_exp_u	expiry date of option	1-Dec-06
vlt	volatility	0.4
swpn	swaption type	2	right to pay fixed (buy protection)
contra_d	CDS contract dates	see below
cpn_pr	premium coupon rate	0.09
freq_pr	premium payment frequency	3	quarterly
pr_acc_type	type of premium accrued interest payment	1	pay accrued interest upon default
acc	accrual method	2	actual/360
d_rul	business day convention	2	next business day
pr_fix	upfront fee and fixed premium payment table	0
ref_type	type of reference	1	name (notional)
ref_tbl	reference table	1000000
p_off	payoff type	1	pay at default
dp_type	default curve type	1	CDS spread curve
dp_crv	default curve	see below
intrp_tb	default curve parameter table	see below
rate_recover	recovery rate table	30%
hl	holiday list	0
dfstd	discount factor curve – risk free	see below
intrp	interpolation method	1	linear
calc_para	calculation parameters	see below
stat	stat list	1…8

where the input tables that are not shown above are given in the following:

contra_d: CDS Contract Dates

Terminating date	Effective date	First coupon date	Next to last coupon date	Effective date adjustment	Terminating date adjustment	Date generation method
20-Dec-2008	1-Dec-2006	0	0	2	2	3
				switch: do not adjust effective date	switch: do not adjust terminating date	switch: IMM

intrp_tb: Default Curve Parameter Table

Interpolation of default probability curve	Bootstrapping method	Accrual method	Effective date adjustment	Terminating date adjustment	Date generation method
1	1	4	2	2	3

calc_para: Calculation Parameters

calculation method	bump size of DVOX (basis points)
1	1

Results

Statistics	Description	Value
1	fair value	8014.912
2	par forward CDS rate	0.071326
3	theta	-26.4053
4	vega	386.1224
5	basis point value of risk-free curve	-1.91668
6	DVOX of par CDS spread curve	49.65459
7	rho of recovery rate	-13.6211
8	delta of credit spread	-0.31755

Example 2: A basket CDS option

Suppose in the above example, the option is now to buy protection for the first default of the three reference entities detailed in the following tables:

ref_bskt: Reference Basket Table

notional amount	recovery rate
1000000	40%
1300000	50%
1200000	30%

The CDS spread curves of reference entities and the CDS counterparty are:

dp_bskt: Basket Default Curve

effective date	maturity date	CDS spread of entity 1	CDS spread of entity 2	CDS spread of entity 3	CDS spread of counterparty
1-Dec-2005	1-Jun-2006	0.05	0.05	0.06	0.005
1-Dec-2005	1-Dec-2006	0.055	0.055	0.067	0.0055
1-Dec-2005	1-Dec-2007	0.06	0.06	0.074	0.006
1-Dec-2005	1-Dec-2008	0.065	0.065	0.081	0.0065
1-Dec-2005	1-Dec-2009	0.07	0.07	0.088	0.007

and their correlation matrix is:

corrs: Correlation Matrix

entity 1	entity 2	entity 3	counterparty
1	0.5	0.7	0.4
0.5	1	0.3	0.5
0.7	0.3	1	0.2
0.4	0.5	0.2	1

The premium payment table is as follows:

pr_tbl: Premium Payment Table

principal	coupon	frequency	Premium accrual type	accrual method	Business day convention
3500000	0.1	3	1	2	2

The calculation parameters:

calc: Calculation Parameters

number of random trials	number of time steps	default barrier calculation method	CDS calculation method	basis points (bump size for DVOX)
5000.5	1	1	1	10

Using the same discount factor curve and the fixed premium payment table as in Example 1, the function aaCDS_bskt_opt gives the following results:

aaCDS_bskt_opt

Argument	Description	Example Data	Switch
d_v	value (settlement) date	1-Dec-2005
d_exp_u	expiry date of option	1-Dec-2006
vlt	volatility	0.4
swpn	swaption type	2	right to pay fixed (buy protection)
contra_d	CDS contract dates	see above
rank	default rank	1
protect	basket protection type	1	the nth default only
pr_tbl	premium payment table	see above
pr_fix	fixed premium payment table	see above
ref_type	type of references	1	name (notional)
ref_bskt	reference basket table	see above
corrs	credit index correlation matrix	see above
m_corr	method of handling credit correlation	2	normal copula
p_off	payoff type	1	pay at default
dp_type	default curve type	1	par CDS spread curve
dp_bskt	basket default curve	see above
intrp_tb	default curve parameter table	see Example 1
hl	holiday list	0
dfstd	discount factor curve – risk free	see Example 1
intrp	interpolation method	1	linear
calc	calculation parameters	see above
stat	stat list	1…8

Results

Statistics	Description	Value
1	fair value	2437.886
2	par forward CDS spread	0.052553
3	theta	23.39028
4	vega	300.4097
5	basis point value of risk-free curve	-0.80398
6	DVOX of par CDS spreads	204.4632
7	rho of recovery rates	22.49972
8	rho of correlation matrix	-83.9676

Related Functions

All functions valuing credit default swaps on a single entity and a basket of entities, such as aaCDS and aaCDS_bskt. See the Credit Default Swaps and Basket Default Swaps FINCAD Math Reference documents.

All functions forecasting/bootstrapping default probability curves and other default probability related utility functions, such as aaCredit_dfltProb_DSSprd3, aaCredit_dfltProb_DSSprd2 and aaCredit_interp_dfltProb2. See the Default Probability Curves and Default Correlations FINCAD Math Reference document.

References

[1] Hull, J. and White, A., (Spring 2003), ‘Valuation of credit default swap options’, The Journal of Derivatives, 10, No. 3 , p. 40-50.

[2] Hull, J., (2005), Options, Futures, and Other Derivatives, 6th ed., Prentice Hall.

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