CMS Coupon Convexity Adjustments Under Negative Rates

Summary

The document raises a pricing question about bonds with constant-maturity swap (CMS) coupons in euros, where interest rates may be negative. It asks whether the convexity adjustment from Hull, which assumes lognormal rates, can be adapted to a normal-rate model, and whether that adjustment can be related to formulas in a review of interest-rate convexity methods.

No formula, derivation, or solution is provided; the text is a request for practical guidance. It notes that swaption volatility data is unavailable to the questioner, which limits direct use of volatility-based adjustments. Any usable approach would therefore need to specify its rate model and required inputs, and the document itself gives no evidence or recommendation for choosing among methods.

Key ideas

• CMS-linked bond coupons may require a convexity adjustment to forward swap rates.
• The question challenges use of a lognormal-rate formula when rates can be negative.
• It asks whether a normal-rate counterpart can be related to established convexity methods.
• The document provides no answer or calibration guidance, and reports that swaption volatilities are unavailable.

Tags

# CMS Convexity adjustment with negative interest rates


# CMS Convexity adjustment with negative interest rates












I need to price bonds with CMS-linked coupons. In order to determine the convexity adjustment to apply to the forward rates, I would use the formula that appears in Hull's Futures, Options and other derivatives:

The bonds are denominated in EUR whose curve presents negative interest rates. I understand Hull's adjustment implies a log-normal process of the interest rate so that the above formula should not be used (swaption vols are not available anyway). Is there an equivalent formula to be used assuming normal process?

I've been reading some papers like "Convexity Adjustments Made Easy -A Review of Convexity Adjustment Methodologies and Formulae in Interest Rate Markets" by Nicholas Burgess but I find hard to understand and apply in practice, specially because I don't see the relation between the lognomal adjustment there and Hull's:

Your help is really appreciated.

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