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Cash-Settled Swaption Payoffs and the Cash Annuity Approximation

Article Quant Q&A · Author: Shannon
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Summary

A cash-settled swaption pays a function of the market swap rate, commonly the intrinsic value multiplied by a cash annuity. The annuity converts the rate payoff into a cash amount. Because the swap rate is observable, this convention can make the payoff depend on one market variable, unlike a physically settled swaption, whose value depends on the physical swap annuity derived from discount factors.

The cash annuity is described as an approximation that assumes a flat curve at the swap rate and no funding basis spread. The discussion cautions that this assumption may no longer fit current markets, and that quoted volatility can differ between cash-settled and physically settled swaptions. It offers conceptual motivation and a limitation, but no quantitative comparison or detailed valuation method.

Key ideas

  • A cash-settled swaption payoff is based on the swap rate and a cash annuity.
  • The cash annuity approximates the physical annuity under a flat-curve assumption tied to the swap rate.
  • Physical swaption valuation uses discount factors to calculate the underlying swap annuity.
  • The approximation may be inaccurate in current markets, and market volatility quotes can differ by settlement type.

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# Cash-settled swaptions


# Cash-settled swaptions












I was wondering, what is the motivation behind the payoff of the cash swaptions being multiplied by the swap annuity? $$c(S_{\theta, T})=\sum_{i=\theta+1}^{T}\tau_i\frac{1}{{(1+S_{\theta,T}(\theta))}^{\tau_{\theta,i}}}$$ Why not using the classic one: $$A_{t} = \sum_{i=1}^{T} P_{t,T_i}\tau_i$$

Thank you in advance for your answer!

Cheers

S.

## Answer by AFK (score 6)

https://quant.stackexchange.com/a/35857

The advantage of cash-settled swaptions is that the payoff only depends on one variable: the corresponding swap rate which is directly observable in the market: $$ \mathrm{Payoff}(T) = f(S_T) = A^{\mathrm{Cash}}(S_T)\max(S_T - K,0) $$

The payoff of a physical swaption on the other hand depends on the physical annuity which is not directly observable. You typically have to bootstrap the discount curve to get all the discount factors and sum those to get the value of the annuity. The cash-settled annuity is the approximation corresponding to a flat curve with zero rate $S_T$ (and zero funding basis spread).

## Answer by user32151 (score 1)

https://quant.stackexchange.com/a/38656

The cash-settled annuity is the approximation corresponding to a flat curve with zero rate $S_T$ (and zero funding basis spread). This approximation is not anymore valid in present market. ICAP quotes the volatility for cash-settled and physically settled swaption differently.

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.