cdsbootstrap

Bootstrap default probability curve from credit default swap market quotes

Syntax

[ProbData,HazData] = cdsbootstrap(ZeroData,MarketData,Settle)

[ProbData,HazData] = cdsbootstrap(___,Name,Value)

Description

[ProbData,HazData] = cdsbootstrap(ZeroData,MarketData,Settle) bootstraps the default probability curve using credit default swap (CDS) market quotes. The market quotes can be expressed as a list of maturity dates and corresponding CDS market spreads, or as a list of maturities and corresponding upfronts and standard spreads for standard CDS contracts. The estimation uses the standard model of the survival probability.

Note

Alternatively, you can use the Financial Instruments Toolbox™ defprobstrip (Financial Instruments Toolbox) function to bootstrap a defprobcurve object from market CDS instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments (Financial Instruments Toolbox).

[ProbData,HazData] = cdsbootstrap(___,Name,Value) adds optional name-value pair arguments.

Examples

Bootstrap Default Probability Curve from Credit Default Swap Market Quotes

This example shows how to use cdsbootstrap with market quotes for CDS contracts to generate ProbData and HazData values.

Settle = datetime(2009,7,17); % valuation date for the CDS
Spread_Time = [1 2 3 5 7]';
Spread = [140 175 210 265 310]';
Market_Dates = daysadd(datenum(Settle),360*Spread_Time,1);
MarketData = [Market_Dates Spread];
Zero_Time = [.5 1 2 3 4 5]';
Zero_Rate = [1.35 1.43 1.9 2.47 2.936 3.311]'/100;
Zero_Dates = daysadd(datenum(Settle),360*Zero_Time,1);
ZeroData = [Zero_Dates Zero_Rate];

format long
[ProbData,HazData] = cdsbootstrap(ZeroData,MarketData,Settle)

ProbData = 5×2
105 ×

   7.343360000000000   0.000000233427859
   7.347010000000000   0.000000575839968
   7.350670000000000   0.000001021397017
   7.357970000000000   0.000002064539982
   7.365280000000000   0.000003234110940

HazData = 5×2
105 ×

   7.343360000000000   0.000000232959886
   7.347010000000000   0.000000352000512
   7.350670000000000   0.000000476383354
   7.357970000000000   0.000000609055766
   7.365280000000000   0.000000785241515

Input Arguments

ZeroData — Zero rate data

vector | IRDataCurve object

Zero rate data, specified as a M-by-2 vector of dates, using a serial date number format, and zero rates or an IRDataCurve object of zero rates.

When ZeroData is an IRDataCurve object, ZeroCompounding and ZeroBasis are implicit in ZeroData and are redundant inside this function. In this case, specify these optional parameters when constructing the IRDataCurve object before using the cdsbootstrap function.

For more information on an IRDataCurve (Financial Instruments Toolbox) object, see Creating an IRDataCurve Object (Financial Instruments Toolbox).

Data Types: double | object

MarketData — Bond market data

matrix

Bond market data, specified as a N-by-2 matrix of dates and corresponding market spreads or N-by-3 matrix of dates, upfronts, and standard spreads of CDS contracts. The dates must be entered as serial date numbers, upfronts must be numeric values between 0 and 1, and spreads must be in basis points.

Data Types: double

Settle — Settlement date

datetime scalar | string scalar | date character vector

Settlement date, specified as a scalar datetime, string, or date character vector. The Settle date must be earlier than or equal to the dates in MarketData

To support existing code, cdsbootstrap also accepts serial date numbers as inputs, but they are not recommended.

Data Types: char | string | datetime

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: [ProbData,HazData] = cdsbootstrap(ZeroData,MarketData,Settle,'RecoveryRate',Recovery,'ZeroCompounding',-1)

Note

Any optional input of size N-by-1 is also acceptable as an array of size 1-by-N, or as a single value applicable to all contracts. Single values are internally expanded to an array of size N-by-1.

RecoveryRate — Recovery rate

0.4 (default) | decimal

Recovery rate, specified as the comma-separated pair consisting of 'RecoveryDate' and a N-by-1 vector of recovery rates, specified as a decimal from 0 to 1.

Data Types: double

Period — Premium payment frequency

4 (default) | numeric with values 1, 2, 3, 4, 6 or 12

Premium payment frequency, specified as the comma-separated pair consisting of 'Period' and a N-by-1 vector with values of 1, 2, 3, 4, 6, or 12.

Data Types: double

Basis — Day-count basis of contract

2 (actual/360) (default) | integers of the set [0...13] | vector of integers of the set [0...13]

Day-count basis of the contract, specified as the comma-separated pair consisting of 'Basis' and a positive integer using a NINST-by-1 vector.

• 0 = actual/actual

• 1 = 30/360 (SIA)

• 2 = actual/360

• 3 = actual/365

• 4 = 30/360 (PSA)

• 5 = 30/360 (ISDA)

• 6 = 30/360 (European)

• 7 = actual/365 (Japanese)

• 8 = actual/actual (ICMA)

• 9 = actual/360 (ICMA)

• 10 = actual/365 (ICMA)

• 11 = 30/360E (ICMA)

• 12 = actual/365 (ISDA)

• 13 = BUS/252

For more information, see Basis.

Data Types: double

BusinessDayConvention — Business day conventions

'actual' (default) | character vector

Business day conventions, specified as the comma-separated pair consisting of 'BusinessDayConvention' and a character vector. The selection for business day convention determines how non-business days are treated. Non-business days are defined as weekends plus any other date that businesses are not open (for example, statutory holidays). Values are:

• 'actual' — Non-business days are effectively ignored. Cash flows that fall on non-business days are assumed to be distributed on the actual date.

• 'follow' — Cash flows that fall on a non-business day are assumed to be distributed on the following business day.

• 'modifiedfollow' — Cash flows that fall on a non-business day are assumed to be distributed on the following business day. However if the following business day is in a different month, the previous business day is adopted instead.

• 'previous' — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day.

• 'modifiedprevious' — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day. However if the previous business day is in a different month, the following business day is adopted instead.

Data Types: char

PayAccruedPremium — Flag for accrued premiums paid upon default

true (default) | integer with value 1 or 0

Flag for accrued premiums paid upon default, specified as the comma-separated pair consisting of 'PayAccruedPremium' and a N-by-1 vector of Boolean flags that is true (default) if accrued premiums are paid upon default, false otherwise.

Data Types: logical

TimeStep — Number of days as time step for numerical integration

10 (days) (default) | nonnegative integer

Number of days to take as time step for the numerical integration, specified as the comma-separated pair consisting of 'TimeStep' and a nonnegative integer.

Data Types: double

ZeroCompounding — Compounding frequency of the zero curve

2 (semiannual) (default) | integer with value of 1,2,3,4,6,12, or –1

Compounding frequency of the zero curve, specified as the comma-separated pair consisting of 'ZeroCompounding' and an integer with values:

• 1 — Annual compounding

• 2 — Semiannual compounding

• 3 — Compounding three times per year

• 4 — Quarterly compounding

• 6 — Bimonthly compounding

• 12 — Monthly compounding

• −1 — Continuous compounding

Data Types: double

ZeroBasis — Basis of the zero curve

0 (actual/actual) (default) | integer with value of 0 to 13

Basis of the zero curve, specified as the comma-separated pair consisting of 'ZeroBasis' and an integer with values that are identical to Basis.

Data Types: double

ProbDates — Dates for probability data

column of dates in MarketData (default) | datetime array | string array | date character vector

Dates for probability data, specified as the comma-separated pair consisting of 'ProbDates' and a P-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, cdsbootstrap also accepts serial date numbers as inputs, but they are not recommended.

Data Types: char | string | datetime

Output Arguments

ProbData — Default probability values

matrix

Default probability values, returned as a P-by-2 matrix with dates and corresponding cumulative default probability values. The dates match those in MarketData, unless the optional input parameter ProbDates is provided.

HazData — Hazard rate values

matrix

Hazard rate values, returned as a N-by-2 matrix with dates and corresponding hazard rate values for the survival probability model. The dates match those in MarketData.

Note

A warning is displayed when non-monotone default probabilities (that is, negative hazard rates) are found.

Algorithms

If the time to default is denoted by τ, the default probability curve, or function, PD(t), and its complement, the survival function Q(t), are given by:

$$PD(t)=P[\tau \le t]=1-P[\tau >t]=1-Q(t)$$

In the standard model, the survival probability is defined in terms of a piecewise constant hazard rate h(t). For example, if h(t) =

λ₁, for 0 ≤t ≤ t₁

λ₂, for t₁ < t ≤ t₂

λ₃, for t₂ <t

then the survival function is given by Q(t) =

$$e^{-\lambda 1t}$$, for 0 ≤ t ≤ t₁

$${}^{{{\displaystyle e}}^{-\lambda 1t-\lambda 2(t-t1)}}$$, for t₁ < t ≤ t₂

$${}^{{{\displaystyle e}}^{-\lambda 1t1-\lambda 2(t2-t1)-\lambda 3(t-t2)}}$$, for t₂ < t

Given n market dates t₁,...,t_n and corresponding market CDS spreads S₁,...,S_n, cdsbootstrap calibrates the parameters λ₁,...,λ_n and evaluates PD(t) on the market dates, or an optional user-defined set of dates.

References

[1] Beumee, J., D. Brigo, D. Schiemert, and G. Stoyle. “Charting a Course Through the CDS Big Bang.” Fitch Solutions, Quantitative Research, Global Special Report. April 7, 2009.

[2] Hull, J., and A. White. “Valuing Credit Default Swaps I: No Counterparty Default Risk.” Journal of Derivatives. Vol. 8, pp. 29–40.

[3] O'Kane, D. and S. Turnbull. “Valuation of Credit Default Swaps.” Lehman Brothers, Fixed Income Quantitative Credit Research, April 2003.

Version History

Introduced in R2010b

R2022b: Serial date numbers not recommended

Although cdsbootstrap supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.