Let $X,Y$ be two random variables and let the joint distribution of $(X,Y)$ be bivariate normal with mean $[0,0]$ and covariance $\begin{bmatrix} 1 & \rho \\ \rho & 1 \end{bmatrix}$. And let $\Phi_2$ be the CDF of $(X,Y)$. Then, can we prove (or disprove) that $\log{\Phi_2(a,b,\rho)}$ is concave in $\rho$? where $a,b$ are constants and $\log{\Phi_2(a,b,\rho)} = \mathbb{P}(X \leq a, Y \leq b; \rho)$
Here are the first and second derivatives of $\log{\Phi_2(a,b,\rho)}$. Let $f(\sigma) = \log{\Phi_2(a, b; \sigma)}$. Then : \begin{equation} f'(\sigma) = \frac{\phi_2(a,b;\sigma)}{\Phi_2(a,b;\sigma)}, \end{equation} \begin{equation} f''(\sigma) = \frac{\phi_2(a,b;\sigma)}{\Phi_2(a,b;\sigma)}\left[\frac{\sigma(1 - c)}{1 - \sigma^2} + \frac{ab}{1 - \sigma^2} - \frac{\phi_2(a,b;\sigma)}{\Phi_2(a,b;\sigma)}\right], \end{equation} where $c = \frac{a^2 + b^2 - 2\sigma ab}{1 - \sigma^2}$ and $\phi_2$ is the PDF of $(X,Y)$.