# cdsprice

Determine price for credit default swap

## Description

example

[Price,AccPrem,PaymentDates,PaymentTimes,PaymentCF] = cdsprice(ZeroData,ProbData,Settle,Maturity,ContractSpread) computes the price, or the mark-to-market value for CDS instruments.

example

[Price,AccPrem,PaymentDates,PaymentTimes,PaymentCF] = cdsprice(___,Name,Value) adds optional name-value pair arguments.

## Examples

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This example shows how to use cdsprice to compute the clean price for a CDS contract using the following data.

Settle = '17-Jul-2009'; % valuation date for the CDS
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];

Maturity = '20-Sep-2010';

Price = 1.5461e+04
AccPrem = 10500

## 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 cdsprice function.

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

Data Types: double | struct

Default probability values, specified as a P-by-2 matrix with dates and corresponding cumulative default probability values.

Data Types: double | char

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

Data Types: double | char

Maturity date, specified as a N-by-1 vector of serial date numbers or date character vectors.

Data Types: double | char

Contract spreads, specified as a N-by-1 vector of spreads, expressed in basis points.

Data Types: double

### Name-Value Pair 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.

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 'RecoveryRate' 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 'BusDayConvention' 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

Contract notional values, specified as the comma-separated pair consisting of 'Notional' and a N-by-1vector of integers. Use positive integer values for long positions and negative integer values for short positions.

Data Types: double

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 'ZeroCmpounding' 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

## Output Arguments

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CDS clean prices, returned as a N-by-1 vector.

Accrued premiums, returned as a N-by-1 vector.

Payment dates, returned as a N-by-numCF matrix.

Payment times, returned as a N-by-numCF matrix of accrual fractions.

Payments, returned as a N-by-numCF matrix.

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### CDS Price

The price or mark-to-market (MtM) value of an existing CDS contract.

The CDS price is computed using the following formula:

CDS price = Notional * (Current Spread - Contract Spread) * RPV01

Current Spread is the current breakeven spread for a similar contract, according to current market conditions. RPV01 is the 'risky present value of a basis point,' the present value of the premium payments, considering the default probability. This formula assumes a long position, and the right side is multiplied by -1 for short positions.

## Algorithms

The premium leg is computed as the product of a spread S and the risky present value of a basis point (RPV01). The RPV01 is given by:

$RPV01=\sum _{j=1}^{N}Z\left(tj\right)\Delta \left(tj-1,tj,B\right)Q\left(tj\right)$

when no accrued premiums are paid upon default, and it can be approximated by

$RPV01\approx \frac{1}{2}\sum _{j=1}^{N}Z\left(tj\right)\Delta \left(tj-1,tj,B\right)\left(Q\left(tj-1\right)+Q\left(tj\right)\right)$

when accrued premiums are paid upon default. Here, t0 = 0 is the valuation date, and t1,...,tn = T are the premium payment dates over the life of the contract,T is the maturity of the contract, Z(t) is the discount factor for a payment received at time t, and Δ(tj-1, tj, B) is a day count between dates tj-1 and tj corresponding to a basis B.

The protection leg of a CDS contract is given by the following formula:

$ProtectionLeg={\int }_{0}^{T}Z\left(\tau \right)\left(1-R\right)dPD\left(\tau \right)$

$\approx \left(1-R\right)\sum _{i=1}^{M}Z\left(\tau i\right)\left(PD\left(\tau i\right)-PD\left(\tau i-1\right)\right)$

$=\left(1-R\right)\sum _{i=1}^{M}Z\left(\tau i\right)\left(Q\left(\tau i-1\right)-Q\left(\tau i\right)\right)$

where the integral is approximated with a finite sum over the discretization τ0 = 0,τ1,...,τM = T.

If the spread of an existing CDS contract is SC, and the current breakeven spread for a comparable contract is S0, the current price, or mark-to-market value of the contract is given by:

MtM = Notional (S0SC)RPV01

This assumes a long position from the protection standpoint (protection was bought). For short positions, the sign is reversed.

## 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.