Using SABR to Infer Negative-Moneyness Swaption Volatility Quotes
Summary
The document asks how to infer negative-moneyness implied volatilities for interest-rate swaptions when broker data provide at-the-money volatility and payer and receiver spreads only at positive offsets from the at-the-money forward. It considers lognormal, shifted-lognormal, and normal quoting conventions, along with premiums for structures such as strangles and collars.
The response recommends constructing a volatility curve from at-the-money quotes and payer and receiver spreads, then calibrating a SABR model to the market data. It cautions that a smooth smile is not guaranteed by raw quotes; a calibrated model produces its own curve. The same general calibration approach can be applied to normal and shifted-lognormal volatility, with the latter requiring the appropriate displacement adjustment. The document does not provide market data, a calibration procedure, or a worked example, so it does not establish that receiver quotes can simply be relabeled as negative-moneyness payer volatility or specify how to back out missing strikes from premiums alone.
Key ideas
- At-the-money volatility and payer and receiver spreads can be used to construct quoted volatility points around the forward.
- A SABR model can be calibrated to market quotes to generate an implied volatility curve.
- A visually smooth smile is a model output and is not guaranteed by market quotes themselves.
- Normal and shifted-lognormal quotes can use the same broad calibration approach, with a displacement adjustment for the shifted model.
- The response does not validate a direct equivalence between receiver quotes and negative-moneyness payer quotes.
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# Modern market conventions for interpreting interest rate swaptions quotations in a negative interest rate environment # 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: - Lognormal (Black) - Shifted Lognormal (Black with displaced diffusion) - 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 vol spread - Receiver vol spread - Payer premium (Forward and discounted) - Receiver 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. ## Answer by Kiann (score 2, accepted) https://quant.stackexchange.com/a/42692 I assume your underlying pricing model uses a derivative of the standard Hagan's SABR formulation. - 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). - * 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. - 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. - 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. - 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.
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