Today’s post will be a short one about calculation of discount curves from swap rates. I’ve discussed both swaps and discount curves in previous posts, you should read those before this one or it might not make much sense!
Although bonds can be used to calculate discount bond prices, typically swaps are the most liquid products on the market and will go to the longest expiry times (often 80+ years for major currencies), so these are used to calculate many of the points on the discount curve [and often both of these can be done simultaneously to give better reliability].
In the previous post on swaps, I calculated the swap rate 
that makes swaps zero-valued at the current time
![\[X = {Z(0,t_0) - Z(0,t_N) \over \sum_{n=0}^{N-1} \tau \cdot Z(0,t_{n+1})}\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-a6d073e8e8e0ed3f6385850cda92a345_l3.png "Rendered by QuickLaTeX.com")
where the 
‘s here represent the fixing dates of the swap (although payment is made at the beginning of the following period, so the 
‘th period is received at 
.
Consider the sequence of times 
for which a sequence of swaps are quoted on the markets, with swap rates 
for the swap running from 
up to 
. We can back out the discount factor at each time as follows:
![\[X_0 = {Z(0,0) - Z(0,t_1) \over \tau \cdot Z(0,t_1)}\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-40b35366fe4287dd7512d715f7c9cc5c_l3.png "Rendered by QuickLaTeX.com")
![\[\Rightarrow Z(0,t_1) = {1 \over 1 + \tau X_0 }\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-55459d8bf73203a680b1fb27ab6e6813_l3.png "Rendered by QuickLaTeX.com")
![\[X_1 = {Z(0,0) - Z(0,t_2) \over \tau \big(Z(0,t_1) + Z(0,t_2)\big) }\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-6ba1bba40801088364971c2df84df1c1_l3.png "Rendered by QuickLaTeX.com")
![\[\Rightarrow Z(0,t_2) = {1 - \tau \cdot X_1 \cdot Z(0,t_1) \over 1 + \tau X_1}\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-8a91e248313a76c41382e6954be45deb_l3.png "Rendered by QuickLaTeX.com")
and we can see from this the general procedure, calculating another ZCB from each successive swap rate using the expression
![\[Z(0,t_i) = {1 - \sum_{i=1}^{i-1} \big(\tau \cdot X_i \cdot Z(0,t_i) \big) \over 1 + \tau X_{i-1}}\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-592be81fd94c0844c21f1a647ee3bbd5_l3.png "Rendered by QuickLaTeX.com")
These swaps and ZCBs are called co-initial because they both started at the same time .
Now imagine that instead the swaps have the first fixing at time and their final fixing at time 
for 
– such swaps are called co-terminal swaps as they start at different times but finish at the same one. Once again we can calculate the discount factors up to a constant factor, this time by working backwards:
![\[X_{n-1} = {Z(0,t_{n-1}) - Z(0,t_n) \over \tau \cdot Z(0,t_n)}\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-ef7589117556d869b3ad70ae191fd140_l3.png "Rendered by QuickLaTeX.com")
![\[\Rightarrow Z(0,t_{n-1}) = Z(0,t_n)\cdot\big( { 1 + \tau X_{n-1} }\big)\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-734f713619832d7cae81141af753c420_l3.png "Rendered by QuickLaTeX.com")
![\[X_{n-2} = {Z(0,t_{n-2}) - Z(0,t_n) \over \tau \big(Z(0,t_{n-1}) + Z(0,t_n)\big) }\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-fe5a6e2b9c786e883bf8630e72c136fc_l3.png "Rendered by QuickLaTeX.com")
![\[\Rightarrow Z(0,t_{n-2}) = Z(0,t_{n})\big( 1 + \tau X_{n-2} ( 1 + \tau X_{n-1}) \big)\]](http://www.quantopia.net/wp-content/ql-cache/quicklatex.com-1dbeccbb43935fe0d736e40b79a6f205_l3.png "Rendered by QuickLaTeX.com")
and so on, the dcfs can be backed out.
To specify the exact values of co-terminal swaps, we need to know at least one dcf exactly. In general the co-initial case will also require this – I implicitly assumed that they started fixing at 
where we know 
, but for general co-initial swaps we would also have this issue.
What does tau represent?
Hi George – here tau is the year-fraction coming from the swap. For example, if the swap ‘fixes’ (ie. payments are made between the contract holders) twice annually, then tau will be 0.5. See my article on swaps for full info – http://www.quantopia.net/forward-rate-agreements-and-swaps/.
Can discount factors be lower than zero if the swap curve has a great enough slope?