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For a swaption, the "Pricing And Hedging Of Swaptions" paper by Akume et al (2003) says:

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I get that he's just taking the derivative of the swaption valuation formula (which is N * A * BLACKSCHOLES), but he's assuming A (the annuity) is independent of the underlying swap rate. Surely if we live in a world where swap rates triple, then discount rates are also affected, so A and S aren't independent. Or am I wrong?

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Yes, you are actually wrong. The pricing of swaptions is done in the ‘swaption measure’, or ‘annuity measure’, where the unit under consideration is a 1% per annum annuity matching the life of the underlying swap. Because of this, there is no need to consider the correlation you mention. This is quite hard to believe initially, but it is standard swaption pricing analytics.

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  • $\begingroup$ how would PnL of a swaption explained then ? say you have a swaption and two dates d1 and d2, and pv1 and pv2 are the values of the swaption on d1 and d2. pv1 clearly would be priced with annuity1 and pv2 would be priced with annuity2. using Black's greeks would not be able to explain the part of PnL due to the the changes of annuity between these two days. $\endgroup$
    – Peaceful
    Nov 8, 2023 at 19:31

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