  We develop a completely new and straightforward method for simulating the
joint law of the position and running maximum at a fixed time of a general
L\'{e}vy process with a view to application in insurance and financial
mathematics. Although different, our method takes lessons from Carr's so-called
"Canadization" technique as well as Doney's method of stochastic bounds for
L\'{e}vy processes; see Carr [Rev. Fin. Studies 11 (1998) 597--626] and Doney
[Ann. Probab. 32 (2004) 1545-1552]. We rely fundamentally on the Wiener-Hopf
decomposition for L\'{e}vy processes as well as taking advantage of recent
developments in factorization techniques of the latter theory due to Vigon
[Simplifiez vos L\'{e}vy en titillant la factorization de Wiener-Hopf (2002)
Laboratoire de Math\'{e}matiques de L'INSA de Rouen] and Kuznetsov [Ann. Appl.
Probab. 20 (2010) 1801--1830]. We illustrate our Wiener--Hopf Monte Carlo
method on a number of different processes, including a new family of L\'{e}vy
processes called hypergeometric L\'{e}vy processes. Moreover, we illustrate the
robustness of working with a Wiener--Hopf decomposition with two extensions.
The first extension shows that if one can successfully simulate for a given
L\'{e}vy processes then one can successfully simulate for any independent sum
of the latter process and a compound Poisson process. The second extension
illustrates how one may produce a straightforward approximation for simulating
the two-sided exit problem.
