Monte Carlo Methods
Simulating risk-neutral paths and averaging discounted payoffs prices the exotics that defeat closed forms, with variance reduction doing the heavy lifting.
When no closed form or tree applies, simulate: draw random paths for the underlying under the risk-neutral measure, evaluate the payoff on each, and average the discounted results. The central limit theorem puts the error on the order of one over the square root of the path count, so variance reduction is the craft. Antithetic variates pair each path with its mirror image to cancel asymmetry, and control variates subtract the known error of a related instrument, often the Black–Scholes value of a vanilla cousin. Monte Carlo is the workhorse for path-dependent and multi-asset derivatives, turning the measure machinery into running code.