I want to show that the following expression:
$$\left | \frac{ 1-\rho } {1-\rho e^{i\mu -\sigma^2/2} } \right |^2 $$
Is very well approximated by:
$$\frac{ 1 } {\big (1-\frac{\sigma^2}{2\ln{\rho}} \big)^2+\big (\frac{\mu}{\ln{\rho}} \big)^2 } $$
where $\sigma,\mu \in \mathbb{R}$ smaller than $\pi$ ($\rho$ is smaller but close to 1). Would you have any hint?
I could get to the second form by using $$1-\rho \approx -\ln{\rho}$$ and $$1-\rho e^{i\mu -\sigma^2/2}\approx -\ln{(\rho e^{i\mu -\sigma^2/2})} \approx -\ln{\rho}-i\mu +\sigma^2/2$$ The second approximation is wrong, but somehow it gives a good result when combined with the first one.