  In this article, our focus lies on computing the quantiles of a random
variable $f(X)$, where $X$ is a $[0,1]^d$-valued random variable, $d \in
\mathbb{N}^{\ast}$, and $f:[0,1]^d\to \mathbb{R}$ is a deterministic Lipschitz
function. We are particularly interested in scenarios where the cost of a
single function call is high, while assuming the law of $X$ is known. In this
context, we propose a deterministic algorithm to obtain exact deterministic
lower and upper bounds for the quantile of $f(X)$ at a given level $\alpha \in
(0,1)$. With a fixed budget of $N$ function calls, we demonstrate that our
algorithm achieves exponential convergence rate for $d=1$ ($\sim \rho^N$ with
$\rho \in (0,1)$) and polynomial convergence rate for $d>1$ ($\sim
N^{-\frac{1}{d-1}}$) and show the optimality of those rates within the class of
deterministic algorithms. Furthermore, we design two algorithms based on
whether the Lipschitz constant of $f$ is known or unknown.
