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Converting Normal Volatility to Shifted Black for SABR Calibration

Article Quant Q&A · Author: Natan Kowalski
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Summary

The document discusses converting market normal volatilities into shifted Black volatilities for SABR calibration in a negative-rate setting. Its stated approach is to treat each implied volatility as a way to represent an option price: calculate a price using the normal-volatility convention, then find the shifted Black volatility that reproduces that same price under the shifted model. The shift must accommodate the negative strikes in the example; the quoted setup uses a three-percent shift.

The response highlights two practical details: express volatility inputs in decimal units, since normal and lognormal quotes may use different display conventions, and include the shift in the Black pricing formula. It claims the prices can be matched but does not provide the full conversion calculation or validate every pricing formula. The final implied-volatility extraction is left as an exercise, so this is guidance on the setup rather than a complete calibration procedure.

Key ideas

  • Convert between volatility conventions by matching option prices under the two pricing models.
  • Use a shifted Black model when negative rates or strikes require a shift.
  • The example specifies a three-percent shift to cover strikes below minus two percent.
  • Enter volatilities in decimal form and respect the units used for each quote convention.
  • The response flags omission of the shift from the Black formula as a crucial error.

Tags

OptionsDerivatives pricingVolatilityFixed income
Full text
# From Implied volatility to shifted Black volatility


# From Implied volatility to shifted Black volatility












I don't know who to go from normal to shifted black volatility before calibrating SABR with negative interest rates.

I see: "As we know that implied volatilities have a one-to-one relationship with prices, we can convert the normal volatilities into EUR prices and doing the same for an unknown Shifted Black volatility using the Shifted Black model and setting the equation equal to zero by changing the Shifted volatility. For this instrument, we use a shift parameter of 3%, as we have strikes that go beyond the −2% mark. "

But don't know how to do it.

Market Data is coming from page 82 of thesis: https://research-api.cbs.dk/ws/portalfiles/portal/62188286/818135_Master_Thesis_125476.pdf

## Answer by KevinT (score 1)

https://quant.stackexchange.com/a/71660

Since it would be too long for a comment and you made some effort at least in trying to replicate, I wrap this as an "answer" to your question, while leaving the last and actual part of extracting the implied vol rather than using the given data for you as an exercise.

So, I did not check all your $d$ and pricing formulae, but you can definitely recover the same option price using these implied vols. But you have to take care of the following:

- work in decimals also for your vols - this is important, because normal vol is usually quoted in basis points, while lognormal convention is percentage points

- your Black Scholes misses the shift! this is crucial for negative rates (as in your example)

Here is the "proof" that it should work:

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.