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Why Payer Swaption Corridor Payoffs Depend on the Discount Curve

Article Quant Q&A · Author: jake_r
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Summary

The discussion explains why the payoff of a payer swaption cannot generally be inferred from the terminal swap rate alone. If the swaption finishes in the money, its value is the present value of the fixed leg annuity, whose payments must each be discounted using the corresponding zero-coupon rates.

A single 10-year swap rate does not specify those discount factors. Assuming the entire yield curve shifts in parallel can provide an estimate, but that is an additional modeling assumption. The answer therefore cautions that there is no simple formula mapping how far the swaption is in the money to its value. The response addresses an individual payer swaption example rather than fully deriving the combined payoff of a corridor spread, and offers no numerical valuation or empirical evidence.

Key ideas

  • An in-the-money payer swaption has value linked to the present value of its fixed-leg annuity.
  • Each annuity payment is discounted using the relevant zero-coupon rate.
  • The terminal swap rate alone does not determine the full discount curve.
  • A parallel curve shift is a modeling assumption, not a general payoff rule.

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OptionsFixed incomeDerivatives pricing
Full text
# Swaption Corridor Payoff Diagram


# Swaption Corridor Payoff Diagram












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.

## Answer by dm63 (score 0)

https://quant.stackexchange.com/a/30601

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.