バリア・オプションとカンパウンド・オプション

最近ハル教授の教科書の問題を解こうとしています。カンパウンド・オプションはオプションです。The formula for compound options is interesting, being expressed in terms of a bivariate normal cumulative distribution function. バリア・オプションの値を計算する数式の由来は熱伝導方程式と関係があります。The heat equation, which describes the temperature at an offset x along a metal bar at time t, is:
du/dt = d^2u/dx^2
The derivation of the down-and-out option value based on the heat equation, provided by Turner of The University of Chicago, is somewhat puzzling. It requires that constraints be placed on the coefficients of variables u and du/dx in the Black-Scholes PDE which do not seem reasonable. Also, the reason that the heat equation makes the valuation more tractable than a standard Black Scholes (cumulative distribution function) approach might is unclear.
代数学的に正しい方程式が出ますが論理的にちょっとわかりにくいです。　　Basically, this "heat equation" approach is used to enable the price of a down-and-out call to be written in terms of the prices of two vanilla calls, based on the fact that the solution to the heat equation for the barrier call can be written in terms of the two solutions to the heat equation U(x) and U(-x) of a vanilla call. However, the down-and-out option should be the value of a vanilla call minus the value of a vanilla call when the price will be at or below the barrier, at any given time between the present and maturity. It would seem that using this premise as a starting point might simplify the derivation