# cdsbootstrap

Bootstrap default probability curve from credit default swap market quotes

## Description

example

[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.

example

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

## Examples

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

Settle = '17-Jul-2009'; % valuation date for the CDS
Spread_Time = [1 2 3 5 7]';
Spread = [140 175 210 265 310]';
Zero_Time = [.5 1 2 3 4 5]';
Zero_Rate = [1.35 1.43 1.9 2.47 2.936 3.311]'/100;
ZeroData = [Zero_Dates Zero_Rate];

format long
[ProbData,HazData] = cdsbootstrap(ZeroData,MarketData,Settle)
ProbData = 5×2
105 ×

7.3434    0.0000
7.3470    0.0000
7.3507    0.0000
7.3580    0.0000
7.3653    0.0000

HazData = 5×2
105 ×

7.3434    0.0000
7.3470    0.0000
7.3507    0.0000
7.3580    0.0000
7.3653    0.0000

## Input Arguments

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Zero rate data, specified as a M-by-2 vector of dates 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

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

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

Data Types: double | char

### Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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.

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

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

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

Data Types: double

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

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

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

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

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

Dates for probability data, specified as the comma-separated pair consisting of 'ProbDates' and a P-by-1 vector of dates, given as serial date numbers or date character vectors.

Data Types: double | char

## Output Arguments

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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.

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:

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

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 0tt1

λ2, for t1 < tt2

λ3, for t2 <t

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

${e}^{-\lambda 1t}$, for 0tt1

${}^{{e}^{-\lambda 1t-\lambda 2\left(t-t1\right)}}$, for t1 < tt2

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

Given n market dates t1,...,tn and corresponding market CDS spreads S1,...,Sn, cdsbootstrap calibrates the parameters λ1,...,λn 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.