Approximating Accreting Swaption Prices in the LMM

Summary

This note discusses pricing an accreting swaption in the Libor Market Model when a closed-form solution is unavailable. One response says that when the fixed and floating legs accrue nominal in the same way, the Rebonato approximation for equivalent Black volatility can still be used. Its weighting terms must incorporate the time-dependent notionals, since the forward swap rate remains a weighted sum of forward Libor rates with weights adjusted for nominal amounts.

A second response adds a practical modeling caveat: prices for accreting swaptions are strongly affected by assumptions about correlations within the yield curve. It recommends checking that these assumptions align with other correlation-sensitive products, such as curve options. The material offers an approximation and a calibration consideration rather than a derivation, numerical example, or evidence that the approximation works under broader structures. The equal accrual pattern across both legs is a key condition stated for the approximation.

Key ideas

• When both swap legs accrue nominal in the same way, a Rebonato-style equivalent Black volatility approximation can be retained.
• The approximation weights need to reflect the time-dependent notionals.
• Accreting swaption values can depend heavily on assumed correlations among rates on the curve.
• Correlation assumptions should be checked against other correlation-sensitive markets, including curve options.

Tags

# Accreting swaption


# Accreting swaption












Is there any literature on the maths behind the computation of the price of an accreting swaption in the LMM model (no monte carlo, closed formula or close enough...)?

Thank you!!

## Answer by Antoine Conze (score 0, accepted)

https://quant.stackexchange.com/a/34281

As long as the accretion of nominal is the same on the fixed and on the floating leg the Rebonato approximation for the equivalent Black volatility still works but the weights have to be multiplied (numerator and denominator) by the time dependent nominals. This is because the forward swap rate is still a weighted sum of forward Libors, albeit with the weights multiplied by the nominal.

## Answer by dm63 (score 3)

https://quant.stackexchange.com/a/34287

From a practitioner standpoint, we know the prices of non accreting swaptions. The price of the accreting swaption in any model calibrated to these non accreting swaptions, is heavily dependent on the intra curve correlation assumptions in the model. We check that these correlations are consistent with other correlation dependent markets such as curve options.

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