Question about derivation of SABR volatility formula in original paper 'Managing Smile Risk' by Hagan et al

Question

I have a question regarding the starting point of the derivation of SABR volatilities formulas in the appendix of the famous paper 'Managing Smile Risk' by Hagan et al.

To derive SABR volatility formulas one need to :

1. solve a differential equation for the joint probability density on the values of the underlying and underlying's volatility

2. compute the price of a european option integrating the payoff times the probability density found at point 1

3. equate the price found at point 2 with the price of the same option computed assuming Black (or Bachelier) dynamics and solving for the constant volatility of the latter.

My question concerns the integration mentioned at point 2 above:

In the original paper Hagan et al. perform such integration from $K$ to $\infty$ in the underlying domain (and this is fine) but integrate from $-\infty$ to $\infty$ in the volatility domain and this seems quite odd to me.

Moreover they use this fact to let some terms of the integral go to $0$ (see after equation A6).

Volatility is a positive number by definition so shouldn't integration of a probability density over negative values of volatility be forbidden? Did anyone have the same doubt in reading Hagan's et al. paper?

Answer 1

Apparently Hagan et al original formula has a problem, and strictly speaking is incorrect. This was later corrected by Obloj 2008 using the BBF paper giving the implied volatilities directly from the Dupire PDE:

https://arxiv.org/pdf/0708.0998.pdf

I did not check your claims against the original paper but it is a possibility that you have precisely bumped into that error.