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Let's say we'd like to find a par rate for a 1 month forward starting 20-year interest rate swap. In this case, we'd need to discount cash flows for the payment periods shifted +1 month from standard semiannual or quarterly payments (which we can find by bootstrapping from frequent 1st year values and yearly rates). And annual rates should available well past the final date.

Is there some standard approach to find these discount factors? I assume since par rate is something which defines swaptions' fixing values (its strike), it should be some agreed procedure and not just interpolation.

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No it's just interpolation. In practice there isn't much disagreement among participants for something like one month 20yr forward rate.

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    $\begingroup$ Thank you! So, does it mean that when two parties are trading a swaption like that over-the-counter, they simply specify the agreed par rate so there is no disagreement on what exactly an ATM swaption is? $\endgroup$ – sashkello Jul 27 '16 at 23:36
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    $\begingroup$ Yes. Strike must be agreed at execution. $\endgroup$ – dm63 Jul 28 '16 at 15:45

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