Choosing Swaption Settlement for CMS Option Replication
Summary
The document examines whether CMS options should be replicated with cash-settled or physically settled swaptions. The key distinction is the annuity used in settlement. Physical settlement uses a delivery annuity based on discount factors, while cash settlement uses a contractually calculated cash annuity linked to the swap rate. Under the conditions described, cash-settled swaptions make the replication exact for CMS options fixing in arrears, because the cash annuity aligns with the street approximation discussed in the source.
In principle, this supports using cash-settled swaptions for replication. In practice, replication requires a clean volatility cube across a continuous range of swaption expiries and tenors, so market liquidity can determine the instrument set. The response cites cash settlement as more liquid in EUR and physical settlement as more liquid in USD, while qualifying that market observation. The conclusion is therefore conditional on fixing conventions, model assumptions, and available volatility data rather than a universal market rule.
Key ideas
- Cash settlement uses an annuity computed from the swap rate, unlike the delivery annuity for physical settlement.
- For arrears fixing under the stated assumptions, cash-settled swaptions provide exact CMS replication.
- A continuous replication requires suitable swaption volatility data across strikes and maturities.
- Liquidity may lead practitioners to use different settlement types across currencies.
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Full text
# CMS options, cash-settled/physically-settled swaptions
# CMS options, cash-settled/physically-settled swaptions
CMS options are traditionaly replicated using a theoritical "continuous" strip of swaptions (see for instance Hagan's paper "Convexity Conundrums : Pricing CMS Swaps, Caps and Floors"):
- In the paper, Hagan implicitely chooses physically-settled swaptions by using the delivery annuity $L(t) = \sum_{i=1}^{n} \delta_{i} P(t, T_{i})$
- At a point, he makes a modeling hypothesis in order to rewrite the zero coupon bond and the (delivery) annuity only in terms of the swap rate $R$ and ends up having "street-standard" formula which reminds me of the cash-annuity:
$$ \frac{P(t, T)}{L(t)} = \frac{R}{(1+\frac{R}{q})^{\delta}}\frac{1}{1-\frac{1}{(1+\frac{R}{q})^{n}}}$$ where q is the (swap) number of periods per year and $\delta$ some corresponding fraction period: see section 2.1. CMS Caplets in the paper for more details.
my question is the following:
Since we now know that there is a need to correctly model cash-settled and swap-settled swaptions (ICAP quotes for the cash-settled/physically-settled straddles forward premiums are actually non-negligeable, especially for long tenors), what is the market practice for the CMS options replication ? is it done by using cash-settled or physically-settled ?
Thanks for the insight.
## Answer by Antoine Conze (score 3, accepted)
https://quant.stackexchange.com/a/39437
In a cash settled swaption the payoff is settled using the cash annuity contractually computed using the swap rate. Thus is you work out the replication procedure you will find that CMS replication is exact when you replicate on cash settled swaption (at least when $\delta=0$, that is for CMS with fixing in arrears), because Hagan's "street approximation" is no longer an approximation.
So it is in principle best to replicate on cash settled swaptions, but then since clean volatility cubes are required to correctly model a continuous set of replication swaptions one uses the market with highest liquidity which if I am not mistaken would be cash settled for EUR and physically settled for USD.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.