# 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"):

1. In the paper, Hagan implicitely chooses physically-settled swaptions by using the delivery annuity $L(t) = \sum_{i=1}^{n} \delta_{i} P(t, T_{i})$
2. 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.

• have an eye on the paper of Henrard wich investigates the difference in price between the two types of swaptions, papers.ssrn.com/sol3/… – Beer4All Dec 5 '19 at 9:09

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.