  Let $(X,d)$ be a geodesic Gromov-hyperbolic space, $o \in X$ a basepoint and
$\mu$ a countably supported non-elementary probability measure on
$\operatorname{Isom}(X)$. Denote by $z_n$ the random walk on $X$ driven by the
probability measure $\mu$. Supposing that $\mu$ has finite exponential moment,
we give a second-order Taylor expansion of the large deviation rate function of
the sequence $\frac{1}{n}d(z_n,o)$ and show that the corresponding coefficient
is expressed by the variance in the central limit theorem satisfied by the
sequence $d(z_n,o)$. This provides a positive answer to a question raised in
\cite{BMSS}. The proof relies on the study of the Laplace transform of
$d(z_n,o)$ at the origin using a martingale decomposition first introduced by
Benoist--Quint together with an exponential submartingale transform and large
deviation estimates for the quadratic variation process of certain martingales.
