Market Value of a CDS

Question

I need to model the market value of CDS in a portfolio. My current approach is to calculate the present value of the future spread payments - does anybody have a better idea to solve the problem?

Edit: I calculated the spread in the following way (as in Hull-White):

$PV_{surv} = \sum_{i=1}^T {(1−p_d )^i \cdot e^{-y\cdot i }}; $

$PV_{def}=\sum_{i=1}^{t}{p_d \cdot (1-p_d)^{i-1} \cdot (1-R)}$

$s=PV_{def}/PV_{surv}$

2nd edit: I found the following statement: http://www.yieldcurve.com/Mktresearch/files/Abukar_Dissertation_Sep05.pdf "the market value of a cds is the difference between the two legs", leading to:

$MV_{CDS} = s\cdot PV_{surv} - PV_{def}$

Answer 1

There is a much better pricing formula which is an accurate approximation. Anecdotally I believe that the difference between this and the "offical" CDSW calculator on Bloomberg will be within about 0.5% or less of the notional, especially if the CDS curve is flat.

For a \$1 notional of short-protection contract with coupon $C$, market spread $S$ and $T$ years to maturity, where $R$ is the expected recovery rate, and $r$ is the continuously compounded $T$-year swap rate, we have

$$ V= (C-S) \cdot\frac{1- e^{ -gT } }{g} \cdot\frac{365}{360} $$

where

$$ g=r+\frac{S}{1-R} $$

This approximation is exact in the limit of a continuously paying premium leg with a flat credit and interest rate curve. As CDS pay quarterly and as credit curves are often quoted using a flat spread, this formula is a good approximation. Note that the factor of 365/360 corrects for the Actual 360 basis used to calculate CDS premium payments, while $T$ is calculated in calendar years.

To get a more accurate pricing would require you to calculate all of the premium flows correctly. You would also need to have the ability to value the protection leg which requires a time-integral to contract expiry. Finally you would need to fit your model to the term structure of CDS spreads. There is a more detailed description at this link.

Answer 2

I'm not expert. However, it seems clear that you're generating an upper bound on the seller value. You have to model the risk of default, as well as any convenantal terms for structured default, to generate an expected payout rate, and deduct that from the DCFs, to get a more realistic value. If the terms include a swap put model that separately. To set a bid, you need to model counter-party and (ideally) liquidity risk as well. You might want to read the standard: http://www.cdsmodel.com/

Answer 3

A simpler solution I found is to discount the differences between current spread and original spread:

$MV_{CDS}=T \cdot (s_0 - s_t )\cdot \sum_{i=1}^{T}{e^{-r\cdot i }}$