  In this article, we obtain sharp conditions for the existence of the high
order derivatives ($k$-th order) of intersection local time $
\widehat{\alpha}^{(k)}(0)$ of two independent d-dimensional fractional Brownian
motions $B^{H_1}_t$ and $\widetilde{B}^{H_2}_s$ with Hurst parameters $H_1$ and
$H_2$, respectively. We also study their exponential integrability.
