Hi. I'm new to the world of swaptions and I'm having some trouble with the following:Consider a payer swaption, struck at K and maturing at T, on a swap starting at t0 (>= T), with payment dates t1,...,tn.Let's denote this swap's rate observed at any time t by , and the annuity by .The payoff of the swaption is . This can be written as. Is the latter equivalent to a put option on a coupon-bearing bond that starts at t0, pays K every payment date and has a notional of 1? Is the strike of the put equal to P(T, t0)?If so, I find a problem when trying to price it using the analytical formulae of HW model along with the conversion of Jamshidian (the one that converts an option on a coupon-bearing bond in a portfolio of options on zero-coupon bonds modifying the strike), because you need to find a certain rK that makes the price of the bond equal to the strike of the option (see for example Hull 5th ed.-p541), but at the same time the strike P(T, t0) depends on r_K.

you can find r_K using Newton-Raphson or any other iterative methodsP.

Yes, Excel's Solver for example. But this is not the point. The point is that we have to find rK, the short-rate observed at T, such that the bond price equals the strike of the option. If T=t0, then the strike is 1 (= P(T,t0)), independent of rK. But when T<t0, what doesn't make sense to me is that we want to find rK such that BondPrice(rK) = Strike(rK), but both terms are dependent of rK. Of course, we can find such rK. But I don't see the financial meaning, because what we call "strike" is supposed to be a fixed and known amount.

QuoteOriginally posted by: manolom But I don't see the financial meaning, because what we call "strike" is supposed to be a fixed and known amount.I don't see the point. When you price a forward start option the strike is stochastic.In this case you're pricing a forward start coupon bond. If you want, the strike is 1 "a fixed and known amount" (the par value of the bond).However what really matters is P(T,t0) ( the present value at time T of this fixed and known amount)P.

What I don't know for sure is I can use the method described in paper, because the strike in this paper is known and independent of any short-rate r.

Yes, you can still use a similar method even if T<t0.In this case, as you correctly pointed out, the strike P(r(T),T,t0) of the coupon bond put (payer swaption) is stochastic, because it depends on the short rate value r(T) at time T, but you can still find r* such that P(r*,T,t0) - P(r*,T,tn) - K \sum_{i=1}^n\delta_i P(r*,T, t_i) = 0and proceed accordingly