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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.

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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).

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    $\begingroup$ I agree with the @AFK answer but would add that the cash settled payoff is a highly unnatural object, because it is not economically equivalent to an option on a swap. A payer/receiver combination does not obey put/call parity , for example. $\endgroup$ – dm63 Aug 31 '17 at 0:01
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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.

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