In terms of initial guess, we looked at the more usual use cases and showed that matching a parabola at the minimum variance point often leads to a decent initial guess if one has an ok estimate of the wings. We will see here that we can do also something a bit better than just a flat slice at-the-money in the case where rho is close to -1.
In general when the asymptotes lead to rho < -1, it means that we can't compute b from the asymptotes as there is in reality only one usable asymptote, the other one having a slope of 0 (rho=-1). The right way is to just recompute b by matching the ATM slope (which can be estimated by fitting a parabola at the money). Then we can try to match the ATM curvature, there are two possibilities to simplify the problem: s >> m or m >> s.
Interestingly, there is some kind of discontinuity at m = 0:
- when m = 0, the at-the-money slope is just b*rho.
- when m != 0 and m >> s, the at-the-money slope is b*(rho-1).