market, the chances of a 401(k) allowing for retirement on a target income can be calculated. As appropriate, the worker in question can then take greater risks with the retirement portfolio or start saving more money. Discrete event simulation can be used in evaluating a proposed capital investment's impact on existing operations. Here, a "current state" model is constructed. Once operating correctly, having been tested and validated against historical data, the simulation is altered to reflect the proposed capital investment. This "future state" model is then used to assess the investment, by evaluating the improvement in performance (i.e. return) relative to the cost (via histogram as above); it may also be used in stress testing the design. See . Although Monte Carlo methods provide flexibility, and can handle multiple sources of uncertainty, the use of these techniques is nevertheless not always appropriate. In general, simulation methods are preferred to other valuation techniques only when there are several state variables (i.e. several sources of uncertainty). These techniques are also of limited use in valuing American style derivatives. See below. Applicability Level of complexity Many problems in mathematical finance entail the computation of a particular integral (for instance the problem of finding the arbitrage-free value of a particular derivative). In many cases these integrals can be valued analytically, and in still more cases they can be valued using numerical integration, or computed using a partial differential equation (PDE). However, when the number of dimensions (or degrees of freedom) in the problem is large, PDEs and numerical integrals become intractable, and in these cases Monte Carlo methods often give better results. For more than three or four state variables, formulae such as Black–Scholes (i.e. analytic solutions) do not exist, while other numerical methods such as the Binomial options pricing model and finite difference methods face several difficulties and are not practical. In these cases, Monte Carlo methods converge to the solution more quickly than numerical methods, require less memory and are easier to program. For simpler situations, however, simulation is not the better solution because it is very time-consuming and computationally intensive. Monte Carlo methods can deal with derivatives which have path dependent payoffs in a fairly straightforward manner. On the other hand, Finite Difference (PDE) solvers struggle with path dependence. American options Monte-Carlo methods are harder to use with American options. This is because, in contrast to a partial differential equation, the