# 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

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 $$P^A(S_T < K) = \frac{1}{A(0)} \frac{\partial \text{receiver swaption premium}}{\partial K}$$

• Dear Antoine, many thanks for your response. May I please request you to give me a worked out example using the numbers I have put and your formula. Apologies, I don’t have a quant background and have high maths handicap!!! – Vinit Vaid Mar 15 '18 at 14:08