Inferring Swap Rate Probabilities from Receiver Swaption Prices

Summary

The document explains how prices of European receiver swaptions relate to the risk-neutral distribution of the underlying swap rate. It corrects the idea that an out-of-the-money premium, compared directly with an at-the-money premium, gives the probability of finishing below that strike. Instead, the receiver price is expressed as the annuity value multiplied by the expected discounted intrinsic payoff under the annuity measure. The probability that the exercise-date swap rate is below a chosen strike is obtained from the strike derivative of the receiver premium, divided by the current annuity.

This relationship follows the general result for European vanilla options: the slope of option value across strikes reveals a cumulative distribution under the relevant pricing measure. It does not directly provide a real-world forecast probability. In practice, the derivative must be inferred from prices across strikes, and the document does not discuss quote noise, interpolation, market conventions, or numerical differentiation. Its example motivates the question but does not calculate a probability from the cited premiums.

Key ideas

• Receiver swaption prices encode distribution information about the exercise-date swap rate.
• The relevant pricing measure is the annuity measure, with the swap annuity as numeraire.
• The receiver premium’s strike derivative, scaled by the current annuity, gives the pricing-measure probability below that strike.
• A premium ratio between an out-of-the-money and at-the-money swaption is not itself that probability.
• The inferred probability is under the pricing measure rather than necessarily a real-world probability.

Tags

# Can you determine USD swap rate movement probability from OTM swaption premiums?


# Can you determine USD swap rate movement probability from OTM swaption premiums?












E.g., the USD 1y x 4y swap rate is currently 2.84%. ATM receiver swaption , European exercise is currently at ATM premium of 1.15% while swaption premium at strike 1.5% is 0.15% or about 90% lower than ATM premium. Can we infer that there is only a 10% chance that 4Y rates will be lower than 1.5% in 1 years time? Is there any other way to use OTM Swaption premiums to determine rate movement probability

## Answer by Antoine Conze (score 1, accepted)

https://quant.stackexchange.com/a/38740

As for any European vanilla option you can infer the cumulative distribution function under the pricing measure by taking the derivative w.r.t. strike.

In the case of European swaptions the natural numeraire is the annuity $A(t)$, the pricing measure is the annuity probability measure $P^A$, and $$ \text{receiver swaption premium} = A(0) E^A[(K - S_T)^+] $$ where $S_T$ is the swap rate on exercise, therefore $$ P^A(S_T < K) = \frac{1}{A(0)} \frac{\partial \text{receiver swaption premium}}{\partial K} $$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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