エキゾチック・オプションと偏微分方程式

最近エキゾチック・オプションの種類と値計算を学んでいます。
Exotic options differ from vanilla calls and puts in that they have non-standard payoffs. One example of an exotic option would be an up-and-out barrier call. This option would pay the usual vanilla option payoff, unless the stock exceeds the barrier boundary. At this point, the option becomes worthless.
My curiosity is with regard to cases where numerical formulas are necessary to compute the value of the option (as opposed to an analytic formula like Black-Scholes). However, for path dependent options, Monte Carlo simulation is sometimes used to estimate prices (trying paths through the price tree randomly and evaluating the prices at nodes). My particular interest is in the use of Partial Differential Equations (PDE) and the Finite Difference Method. One case where such a model would be useful would be in valuing an American option (option which can be exercised at any time prior to expiry) using the Black-Scholes Partial Differential Equation.
(dV/dt) + rS(dV/dS) +(1/2)sigma^2*S^2*(d^2V/dV^2)
It is not apparent immediately why use of the above equation is more appropriate than using the analytic Black-Scholes formula, until one substitutes the derivative terms with approximations. We can then use algebra to write V(t,S) in terms of V(t+dt,S+dS), V(t+dt,S), and V(t+dt,S-dS). In other words, we can iterate backwards in a binomial tree from maturity, filling in the nodes using the values in the nodes after them. American options really are not exotic, so I will be interested to read about where PDEs and the finite difference method are useful for pricing something like a barrier option or an Asian option.