I would like to ask a question on Marc Henrard’s quantitative research paper, specifically about Theorem 2 (the exact European swaption price).

As far as I can tell, the exact European swaption price relies on the same Jamshidian-style decomposition of the payoff as in the single-curve case.

However, as Marc states directly after the theorem, “the original proof required that d_i > 0, while here we have coupon equivalent like 1 - \beta with usually \beta > 1.”

Why does this theorem *not* require that all d_i > 0?

If we value a swaption in OG with a few negative coupons (which would usually be the case), is this exact formula still exact?

I wasn’t able to find an explicit derivation of the theorem for the dual-curve case, and it seems to me that payoff decomposition is not valid when not all of the coupons are positive.

Does it become an approximation in the presence of negative coupons?