CMS Spread Option Convexity, Dependence, and Measure Changes

Summary

The document asks how to model convexity adjustments for a payoff on the spread between two constant maturity swap rates when the rates are dependent. It notes that some approaches model each rate through an annuity mapping function and then use a copula to represent the joint distribution, while questioning whether the same mapping remains appropriate when the rates are not independent.

The brief answer separates the problem into modeling the marginal distributions of the two rates, which are relevant as hedging instruments, and specifying their dependence through a copula. It also identifies the change of measure from the annuity measure to a forward measure as a source of the required adjustment, suggesting a simple Hull–White setting. The exchange offers only a high-level direction: it gives no derivation, calibration procedure, or numerical evidence, and the response explicitly allows that it may have misunderstood the question.

Key ideas

• CMS spread option pricing requires both rate marginals and a model of their dependence.
• A copula can be used to connect modeled marginal distributions.
• Convexity adjustment involves changing from an annuity measure to a forward measure.
• A Hull–White framework is suggested as one setting for that measure change.
• The response is conceptual and does not provide calibration or pricing details.

Tags

# Is there a way to get convexity adjustements for any CMS-payoffs?


# Is there a way to get convexity adjustements for any CMS-payoffs?












In the litterature we specify a dynamic for $\frac{P(T,T_{p})}{A(T)} = G(S(T))$ for each Swap rate S(T) , and there are supposed independant so that we can obtain some value using copulas for calculing the CMS spread with payoff $( S_{1}(T)-S_{2}(T) - K)_{+}$. But in reality the swap rates are not independant so that we cant'suppose the same G(S(.)).How do we account for this ? Is there a way to model those " annuity mapping function" consistent with swaps rates dependance?

## Answer by Arshdeep (score 0, accepted)

https://quant.stackexchange.com/a/55111

To the extent that I understood your question:

- You want to capture the marginals of the 2 rates perfectly, since they will be your hedging instruments. You can correlate them with a copula.

- You need to model a change of measure from the annuity to the T-forward measure. This can be done in a simple hull white setting.

Pricing is then trivial.

Let me know if I misunderstood the question.

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