# Put-call parity for cash settled swaptions

The Euro swaption market is changing from cash to physical settlement quotation in July 2018 $$-$$ see e.g. "Euro swaptions market prepares for pricign revamp (Risk, 2018)". When describing the issues around cash settled swaptions valuation and trading, at one point the aforementioned article states the following (my emphasis):

"Valuation of in-the-money [cash] swaptions was split between market participants that used models, and those that used the principle of put-call parity to infer the price of swaptions from so-called zero-wide collars $$-$$ a receiver and a payer swaption both struck at-the-money.

As volatility rose and rates fell [after the ECB lowered rates at the end of 2014] , swaptions valuation become more difficult, also making it harder to obtain reliable prices for zero-wide collars."

The payoff of a cash settled (payer) swaption is a function $$h$$ of the swap rate $$S_{\tau}(T)$$:

$$h\left(S_{\tau}(T)\right)=A^c(S_{\tau}(T))(S_{\tau}(T)-K)^+$$

where the cash annuity is defined as:

$$A^c(S_{\tau}(T))=\sum_{i=1}^n\prod_{j=1}^i\frac{\delta_i}{(1+\delta_j S_{\tau}(T))^j}$$

I am assuming the model valuation method consists on Black's approximation:

$$\text{Swaption}_{\ \tau}^{\text{Pay}}(t)\approx A^c(S_{\tau}(t))E_t^{A^{\phi}}\left[(S_{\tau}(T)-K)^+\right]$$

Is anyone familiar with the zero-wide collar pricing method mentioned in the article? Is the parity relationship related to the physical annuity:

$$A^{\phi}(S_{\tau}(T))=\sum_{i=1}^n\delta_iP(T,T_i), T \leq T_1, \dots, T_n \text{ ?}$$