Correlation of log-normal random variables

Question

Given $X_1$ and $X_2$ normal random variables with correlation coefficient $\rho$, how do I find the correlation between following lognormal random variables $Y_1$ and $Y_2$?

$Y_1 = a_1 \exp(\mu_1 T + \sqrt{T}X_1)$

$Y_2 = a_2 \exp(\mu_2 T + \sqrt{T}X_2)$

Now, if $X_1 = \sigma_1 Z_1$ and $X_2 = \sigma_1 Z_2$, where $Z_1$ and $Z_2$ are standard normals, from the linear transformation property, we get:

$Y_1 = a_1 \exp(\mu_1 T + \sqrt{T}\sigma_1 Z_1)$

$Y_2 = a_2 \exp(\mu_2 T + \sqrt{T}\sigma_2 (\rho Z_1 + \sqrt{1-\rho^2}Z_2)$

Now, how to go from here to compute correlation between $Y_1$ and $Y_2$?

Answer 1

I assume that $X_1\sim N(0,\sigma_1^2)$ and $X_2\sim N(0,\sigma_2^2)$. Denote $Z_i=\exp(\sqrt{T}X_i)$. Then

\begin{align} \log(Z_i)\sim N(0,T\sigma_i^2) \end{align} so $Z_i$ are log-normal. Thus

\begin{align} EZ_i&=\exp\left(\frac{T\sigma_i^2}{2}\right)\\ var(Z_i)&=(\exp(T\sigma_i^2)-1)\exp(T\sigma_i^2) \end{align} and \begin{align} EY_i&=a_i\exp(\mu_iT)EZ_i\\ var(Y_i)&=a_i^2\exp(2\mu_iT)var(Z_i) \end{align}

Then using the formula for m.g.f of multivariate normal we have

\begin{align} EY_1Y_2&=a_1a_2\exp((\mu_1+\mu_2)T)E\exp(\sqrt{T}X_1+\sqrt{T}X_2)\\ &=a_1a_2\exp((\mu_1+\mu_2)T)\exp\left(\frac{1}{2}T(\sigma_1^2+2\rho\sigma_1\sigma_2+\sigma_2^2)\right) \end{align} So \begin{align} cov(Y_1,Y_2)&=EY_1Y_2-EY_1EY_2\\ &=a_1a_2\exp((\mu_1+\mu_2)T)\exp\left(\frac{T}{2}(\sigma_1^2+\sigma_2^2)\right)(\exp(\rho\sigma_1\sigma_2T)-1) \end{align}

Now the correlation of $Y_1$ and $Y_2$ is covariance divided by square roots of variances:

\begin{align} \rho_{Y_1Y_2}=\frac{\exp(\rho\sigma_1\sigma_2T)-1}{\sqrt{\left(\exp(\sigma_1^2T)-1\right)\left(\exp(\sigma_2^2T)-1\right)}} \end{align}