Calculating the present value of CDS contracts¶

The model¶

The contingent leg¶

The underlying integral which models the price of the contingent part of the CDS contract is:

$V_{\rm P}=-(1-R)\int_{T_{0}}^{T_{\rm F}} dt Z(t) dS/dt$

where:

$V_{\rm P}$ is the value at the present time of the contingent leg of the CDS contract
$R$ is the recovery fraction
$Z(t)$ is the discount function, as defined by the zero curve
$S(t)$ is the survival probability
$T_{0}$ is the time at beginning of risk
$T_{\rm F}$ is the time at end of risk

The value of the above integral is evaluated using the approximation that the combination of the spread and zero curves are piecewise flat. In that approximation, the integral is evaluated exactly in the usual way using the sum:

$V_{\rm P}=(1-R)\sum_i \frac{\lambda_i}{\lambda_{i}+f_{i}} \left[1 - \exp\left(-\Delta t_i(\lambda_i+f_i)\right) \right] s_{i} d_i$

where

$\lambda_{i}=\log(s_i/s_{i+1})/\Delta t_i$
$f_i=\log(d_i/d_{i+1})/\Delta t_i$
$s_i$ is discount factor at time $t_i$ computed from the spread curve
$d_i$ is the discount factor at time $t_i$ computed from the zero curve $Z(t)$
$\Delta t_{i}=t_{i+1}-t_{i}$

The set of time points, $\{t_{i}\}$ used in this sum are the union of:

All dates appearing in the zero (or discount) curve
All dates appearing in the spread curve

The fee leg¶

There are two conventions for determining if the fee leg should be payable in case of a default of the reference names and the choice of these naturally affect the valuation of the present value. The two possibilities are:

The fixed rate coupon is only paid if there is no default by the day the full coupon amount has accrued (the accrual end date is not necessarily the same as the payment date)
Or, the coupon is paid in full if there is no default by the end-of-accrual date and the fractional amount of the coupon accrued by the default date is paid if a default does occur

The fee leg of contract adopting the first (no accrual payment on default) convention is valued very simply as discounted value of the coupon payment times the survival probability:

$V_{\rm P,1}= \sum_i C_{i} Z(T_{i}) S(T_{i})$

where as before $Z(t)$ is the discount function and $S(t)$ is the survival probability, and $C_{i}$ is the amount of the $i$ -th coupon payment.

In the case of the second convention noted above (i.e., if a default occurs before the end of the accrual of the coupon, the fractional accrued amount is paid), the value $V_{\rm P,1}$ needs to be increased by the expected value of the fractional accrued amount. Therefore the value of the coupon payments when the contract are made with the second convention can be written as:

$V_{\rm P,2}= V_{\rm P,1} + \sum_{i} \frac{C_i}{T_{i,\rm F}-T_{i,0}}\int_{T_{i,0}}^{T_{i,\rm F}} Z(t) dS/dt$

The integral within the sum over coupon payments can be evaluated in a way similar to the integral for the contingent leg, leading to an exact result in the case that the rates are flat-forward.

Bojan Nikolic / BN Algorithms Ltd

Calculating the present value of CDS contracts¶

The model¶

The contingent leg¶

The fee leg¶

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Bojan Nikolic / BN Algorithms Ltd

Calculating the present value of CDS contracts¶

The model¶

The contingent leg¶

The fee leg¶

Table Of Contents

Previous topic

Next topic

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