# Basis swap spread pricing and bootstrapping

Here is the expression of a basis floating versus floating swap where the first term is a forward CMS Swap leg and the second one is a forward BOR leg where X is the margin that would make equal both leg?

Suppose X and all other parameters are know except the CA term? How can I express the CA term related to the others?

As far as I can tell, not being an expert in basis swap pricing, this is just algebra -

$$X_{n,c} = \frac{\sum_{i=1}^n \left( S_{i,c}'(0) + {\rm\bf CA}(S_{i,c}'; \delta)\right) P(0,T_i')}{\sum_{i=1}^n P(0,T_i')} - \frac{1 - P(0,T_n')}{\delta \sum_{i=1}^n P(0,T_i')}$$

which rearranges to

$$\sum_{i=1}^n {\rm\bf CA}(S_{i,c}'; \delta) P(0,T_i') = - \sum_{i=1}^n S_{i,c}'(0)P(0,T_i') + \frac{1 - P(0,T_n')}{\delta} + X_{n,c} \sum_{i=1}^n P(0,T_i')$$

You can't reduce it any further, since there are multiple CA terms (one for each $i$) not just one.

As pointed out in the comments, you can bootstrap the curve if you have $X_{n,c}$ for multiple $n$. For example, for $n = 1$ you can derive

$${\rm\bf CA}(S_{1,c}'; \delta) = - S_{1,c}'(0) + \frac{1 - P(0,T_1')}{\delta} + X_{1,c}$$

For any other $n$, assuming that you have already computed ${\rm\bf CA}(S_{k,c}';\delta)$ for $k=1,\dots,n-1$ then then you have

$${\rm\bf CA}(S_{n,c}';\delta) = \frac{ - \sum_{i=1}^n S_{i,c}'(0)P(0,T_i') + \frac{1 - P(0,T_n')}{\delta} + X_{n,c} \sum_{i=1}^n P(0,T_i') - \sum_{i=1}^{n-1} {\rm\bf CA}(S_{i,c}'; \delta) P(0,T_i') }{P(0,T_n') }$$

which allows you to compute all values of ${\rm\bf CA}(S_{n,c}';\delta)$ iteratively.

• pretty sure this is the point of bootstrapping though - you set n=1, that makes the algebra tractable for what you have, solve that, and the sub it into the equation for n=2, and then repeat... – will Jan 18 '17 at 23:38