Bootstrapping Adjustment Terms from Basis Swap Spreads

Summary

The document derives an algebraic relationship between a basis swap spread and a set of adjustment terms, denoted CA, applied to forward CMS rates. Rearranging the spread equation expresses a discounted sum of the CA terms using the spread, forward rates, discount factors, and accrual period. The relationship alone does not identify each adjustment separately when several payment dates are involved, because it provides one equation containing multiple unknown terms.

If spreads are available for successive maturities, the answer describes an iterative bootstrap. The first maturity gives a direct equation for its single adjustment term. At each later maturity, previously calculated terms are subtracted from the cumulative relation, allowing the next term to be solved using its discount factor. This is a method for recovering the adjustment curve from sequential market inputs, conditional on the stated pricing expression and known parameters. The source offers an algebraic derivation, not evidence of market calibration accuracy, and assumes that the required spreads, forward rates, and discount factors are available.

Key ideas

• Rearranging the basis swap spread equation gives a discounted sum of adjustment terms.
• A single maturity with multiple payment dates does not identify each adjustment term independently.
• Sequential spreads allow the adjustment terms to be bootstrapped one maturity at a time.
• The first maturity provides the starting adjustment for the iterative procedure.
• The bootstrap depends on known forward rates, discount factors, accrual periods, and market spreads.

Tags

# Basis swap spread pricing and bootstrapping


# 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?

## Answer by Chris Taylor (score 1, accepted)

https://quant.stackexchange.com/a/31945

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.