## Negative Rates and Free-Boundary SABR

The rates becoming negative in JPY have brought this subject to my attention. I started the implementation of Free-Boundary SABR (let’s call it FSABR below). I had some practical problems at and around ATM as there are divisions of very small numbers by very small numbers, but this can be resolved by Taylor expansions.

Now I get a pretty good match against Monte-Carlo in the non-correlated case, but things get not so good in the correlated case. And the problem here is that I’m trying to match a complicated and approximated closed-form (only exact for 0 correlation), to a complicated Monte-Carlo with problems reported at 0. Indeed, the FSABR paper explains that a simple Euler scheme can go quite wrong at 0, and that’s what I’ve been using in my tests. So, it’s difficult to say if the mismatches I’m seeing are due to a mistake in the implementation of the correlation map or if it’s the Monte-Carlo that is not accurate enough.

A few questions to the readers:

- Has somebody tried the above and seen similar problems as mine (or other problems)?
- Has somebody tried to compare their implementation against Numerix’s?
- Has somebody used Numerix’s FSABR and how good is it?
- Has somebody implemented the other models suggested by Antonov, i.e. the family of the Mixture SABR?

**Posted in:**
Valuation Models

Hi,

Can we add one more question on the list – as a “Zero number” question?

0. In regard the convexity behavior of the shifts, their very non-linear pricing results from the different shifts – the shifted SABR has larger impacts on swaption pricing; and the larger the shift, the larger convexity effect the shifted SABR has on swaption pricing.

So is someone is aware of any research which investigate the relations between the market data (level of the negative rates) and the shift values?

Thanks,

krastyu