  We develop a novel Monte Carlo algorithm for the vector consisting of the
supremum, the time at which the supremum is attained and the position of an
exponentially tempered L\'{e}vy process. The algorithm, based on the increments
of the process without tempering, converges geometrically fast (as a function
of the computational cost) for discontinuous and locally Lipschitz functions of
the vector. We prove that the corresponding multilevel Monte Carlo estimator
has optimal computational complexity (i.e. of order $\epsilon^{-2}$ if the mean
squared error is at most $\epsilon^{2}$) and provide its central limit theorem
(CLT). Using the CLT we construct confidence intervals for barrier option
prices and various risk measures based on drawdown under the tempered stable
(CGMY) model calibrated/estimated on real-world data. We provide non-asymptotic
and asymptotic comparisons of our algorithm with existing approximations,
leading to rule-of-thumb guidelines for users to the best method for a given
set of parameters, and illustrate its performance with numerical examples.
