For $X_j$'s : $\log(X_1)\sim \mathcal{N}(0,1),\log(X_2)\sim \mathcal{N}(0,\sigma^2)$

I'm trying to calculate $\rho_{\max}$ and $\rho_{\min}$ for $\rho (X_1,X_2)$. One step in the given solution I have is:

$\rho_{\max}=\rho (\exp(Z),\exp(\sigma Z))$ and $\rho_{\min}=\rho (\exp(Z),\exp(-\sigma Z))$

but they've made some references to comonotonicity and countercomonotonicity. I was hoping someone help me understand how they're relevant. (I know how to get this from the general expression but want to know specifically what the comonotonicity parts were saying.)

Consider the lognormal random variables $X_1$ and $X_2$ with $\log(X_1)\sim \mathcal{N}(0,1)$, and $\log(X_2)\sim \mathcal{N}(0,\sigma^2)$.

I'm trying to calculate $\rho_{\max}$ and $\rho_{\min}$ for $\rho (X_1,X_2)$. One step in the given solution I have is:

$\rho_{\max}=\rho (\exp(Z),\exp(\sigma Z))$ and $\rho_{\min}=\rho (\exp(Z),\exp(-\sigma Z))$,

but they've made some references to comonotonicity and countercomonotonicity. I was hoping someone help me understand how they're relevant. (I know how to get this from the general expression but want to know specifically what the comonotonicity parts were saying.)

For $X_j$'s : $log(X_1)\sim N(0,1),log(X_2)\sim N(0,\sigma^2)$

I'm trying to calculate $\rho_{max}$ and $\rho_{min}$ for $\rho (X_1,X_2)$. One step in the given solution I have is:

$\rho_{max}=\rho (exp(Z),exp(\sigma Z))$ and $\rho_{min}=\rho (exp(Z),exp(-\sigma Z))$

but they've made some references to comonotonicity and countercomonotonicity. I was hoping someone help me understand how they're relevant. (I know how to get this from the general expression but want to know specifically what the comonotonicity parts were saying.)

For $X_j$'s : $\log(X_1)\sim \mathcal{N}(0,1),\log(X_2)\sim \mathcal{N}(0,\sigma^2)$

I'm trying to calculate $\rho_{\max}$ and $\rho_{\min}$ for $\rho (X_1,X_2)$. One step in the given solution I have is:

$\rho_{\max}=\rho (\exp(Z),\exp(\sigma Z))$ and $\rho_{\min}=\rho (\exp(Z),\exp(-\sigma Z))$

but they've made some references to comonotonicity and countercomonotonicity. I was hoping someone help me understand how they're relevant. (I know how to get this from the general expression but want to know specifically what the comonotonicity parts were saying.)

For $X_j$'s : $log(X_1)\sim N(0,1),log(X_2)\sim N(0,\sigma^2)$

I'm trying to calculate $\rho_{max}$ and $\rho_{min}$ for $\rho (X_1,X_2)$. One step in the given solution I have is:

$\rho_{max}=\rho (exp(Z),exp(\sigma Z))$ and $\rho_{min}=\rho (exp(Z),exp(-\sigma Z))$

but they've made some references to comonotonicity and countercomonotonicity. I was hoping someone help me understand how they're relevant. (I know that how to get this from the genereal expression but want to know specifically what the comonotonicity parts were saying)

Thanks

For $X_j$'s : $log(X_1)\sim N(0,1),log(X_2)\sim N(0,\sigma^2)$

I'm trying to calculate $\rho_{max}$ and $\rho_{min}$ for $\rho (X_1,X_2)$. One step in the given solution I have is:

$\rho_{max}=\rho (exp(Z),exp(\sigma Z))$ and $\rho_{min}=\rho (exp(Z),exp(-\sigma Z))$

but they've made some references to comonotonicity and countercomonotonicity. I was hoping someone help me understand how they're relevant. (I know how to get this from the general expression but want to know specifically what the comonotonicity parts were saying.)