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 $S_{\tau}(T)$:


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:

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


2 Answers 2


The market standard formula approximation for cash settled swaptions applies Black/shifted Black/Bachelier around the forward swap rate so that with this formula parity between payer and receiver swaptions occurs around the forward swap rate, and in particular the zero wide collar struck at the forward swap rate is worth zero (a zero wide collar is the difference between a payer and a receiver both struck at the same strike).

However the market has started to quote zero wide collar struck at the forward swap rate at a non zero premium, which is incompatible with the standard formula approximation, hence the need for an improved approach and more involved models such as described in Matthias Lutz: Two Collars and a Free Lunch (link) or Raoul Pietersz, Frank Sengers: Cash-settled swaptions: A new pricing model (link).

At a minimum one can use the market quoted premium for zero wide collar struck at the forward swap rate to obtain the expectation of the swap rate under the cash annuity measure - call that the cash forward swap rate - and then use payer/receiver parity around the cash forward swap rate to deduce payer (resp. receiver) ITM from quoted receiver (resp. payer) OTM, which is I believe what the Risk article is referring to.


Another approach is to utilize a full yield curve dynamics such as BGM in order to model the spread between the two types of swaption. This will help determine the zero collar valuation and also will show you that the valuation difference depends on the correlation between different parts of the yield curve : specifically , between the rates used to determine the annuity value, which in cash settle case is the terminal swap rate but in the physical case is a blend of discount rates applying to each cash flow in the annuity.


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