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What does the payoff diagram look like for a long payer swaption corridor?

For example, suppose that I am looking at a long-payer $1 \times 10$-year swaption with 10Y swaps as the underlying. If I am buying a 2.0% strike and selling a 2.5% strike, I'm trying to plot the payoffs at various future potential 10Y swap rates in one year (e.g. 1.5%, 2.0%, 2.5%, 3.0%, $...$). I haven't found a good example online (and am having trouble calculating in excel) and am concerned that the convexity of the bonds will make the payoff nonlinear as the 10Y market swap rate in one year increases above my high strike (unlike IR caps). If it is nonlinear, is there a quick, intuitive explanation?

Any thoughts/guidance would be appreciated. Thanks.

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  • $\begingroup$ What is the swap on? Is it on the same index (and cncy) you're collateralized in? $\endgroup$
    – will
    Jul 22, 2016 at 8:52
  • $\begingroup$ Specifically looking at a swaption on 10Y USD swaps (semi-annual) vs 3m LIBOR. For example, if I buy the right to enter into a pay-2%, 10Y swap and the market swap rate on 10Y vs 3m LIBOR in one year is 4%, then I calculate the payoff in excel as: $$notional - PV(4\% / 2, 2 \times 10, notional \times 2\% / 2, -notional)$$ where $PV$ is the excel function to calc the present value difference in the payments of 2% at a 4% d.f. twice a year. I'm then unsure if I can do this same methodology in reverse for selling a 3% long-payer strike (or if I have to look at a short receiver swaption). $\endgroup$
    – jake_r
    Jul 22, 2016 at 14:57
  • $\begingroup$ I'm not sure I understand what you mean. If can price the swaps properly (ie including correlated stochastic rates, as it will make a difference here), then you can just plot the pv for each potential payoff. $\endgroup$
    – will
    Jul 24, 2016 at 10:54
  • $\begingroup$ Thanks, yes. So if I'm following, the correlated stochastic rates drive the nonlinearity? $\endgroup$
    – jake_r
    Jul 25, 2016 at 14:15
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    $\begingroup$ Yes. When you price a normal option, it is normally ignored since the correlation is often very small - or the notionals too small for it to be of consequence. When the correlation becomes high though, it's really quite important (draw yourself the discounted payoff of an option on the discount rate (i.e. 100% correlation) in excel and you'll see what happens). $\endgroup$
    – will
    Jul 26, 2016 at 10:43

1 Answer 1

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Let's say a 2% payer swaption expires with the 10 yr rate equal to 4%. The value of this payoff is the present value of a 2% 10 year annuity. However it is not appropriate to use the 4% 10yr rate to discount this annuity. Each payment of the annuity must be discounted at its appropriate zero coupon rate. You can make some assumptions (for example, that these rates move in parallel to the 10 year swap rate), but those are just assumptions. Hence there is no nice formula that tells you what the swaption will be worth given how far in the money it expires.

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