# Modern market conventions for interpreting interest rate swaptions quotations in a negative interest rate environment

I have broker data and I see three sets of swaption vol data:

1. Lognormal (Black)
2. Shifted Lognormal (Black with displaced diffusion)
3. Normal (Bachelier)

The quotes are given by the following key (Date, Currency, Option expiry, Swap tenor, and Moneyness).

Moneyness is given on a fixed scale relative to the at-the-money forward in basis points from 12.5 to 300 ONLY - no data provided specifically for negative moneyness.

I am given quotes for:

• ATM vol
• Payer premium (Forward and discounted)
• Collar premium (Forward and discounted)
• Strangle premium (Forward and discounted)

My question is, how can I extract the implied volatilities for the negative moneyness? I.e. for basis points from -12.5 to -300. Please answer in the context of validating my assumptions defined below

Assumptions

I think the receiver swaption quoted is the negative moneyness payer as it gives a nice smile shape, but I am not an expert. (assumption 1)

I believe there may be some tricks to translate between the vols or prices that I am not aware of. Any input on using strangle/collar vols to correctly back out negative moneyness. (assumption 2)

Goal:

My ultimate goal is to (a) figure out the negative money implied vols via validating assumption 1, then to (b) see how to get a price from either normal or shifted lognormal model for negative moneyness by validating assumption 2.

I assume your underlying pricing model uses a derivative of the standard Hagan's SABR formulation.

1. Then, the lognormal quotes are merely, where the volatilities quoted on the basis of Black-scholes standard lognormal form, and vol(K=strike) = f(alpha = atm_vol, corr_vol, vol_vol, beta, K, f=forward).

2. * Independent * of the black-scholes formula, you should be able to create an implied volatility curve from ATM_Vol, payer_vol_spread, and receiver_vol_spread from 12.5 to 300bp relative-to-ATMF.

3. As you proposed, the shape should be in a nice curve, but this is not necessarily the case in the quoted markets. Only a * calibrated * SABR model and it's generated volatilities will give you a distinctly nice curve shape.

4. Then, you should be able to calibrate out the necessary SABR parameters in (1) by doing a minimisation routine. This will be true for the case of quoted Normal vols and the shifted lognormal vols as well.

5. The big difference for the shifted lognormal vols, is that F = F - z_shift. I presume you must be familiar with the analytical expansion form for Hagan's SABR that allows you to do this. If not, 100% sure your quants will have that.

Hope that helps.

• If i am reading you correctly, we use the Hagan extensions for negative rates, use the broker shifted lognormal quotes for complete surface, minimize to the parameters, then "adapt" them to non-shifted lognormal vol space and use put-call parity to get prices on both sides of ATM? – Michael Lowenstein Nov 19 '18 at 15:52