# 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

• What have you tried so far? As some starting points: are you familiar with the option pricing formula of Bachelier? Using the normal vol, you can (as indicated) get the option price in EUR. Using this option price and assuming a shifted Black Scholes (sometimes also called Displaced Diffusion model) with shift parameter of 3%, you would back-solve what the equivalent shifted BS implied vol would need to be in order to recover the above EUR prices. For this "inversion" excercise you'll need some sort of solver generally. This paper is maybe a good intro. Jul 18, 2022 at 6:56
• @KevinT thanks for suggestion so I looked on below thesis: research-api.cbs.dk/ws/portalfiles/portal/62188286/… and tested for 1y5Y swpation and firstly instead of getting shifted vol I tested with already given numbers but unfortunately not get satisfying results. Maybe you can look on my code and spot something wrong Jul 20, 2022 at 21:10
• I added a code into original post Jul 20, 2022 at 21:16

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: