Before revision:  Here, when calculating the price of concrete multiple-expiry exotic options, we do not try to get the formal solution of corresponding initial-boundary problem of the Black-Scholes equation, but explain how to express the expiry payoffs of the exotic option as a combination of the payoffs of some class of higher order binary options.
After revision:  Here, when calculating the price of concrete multiple-expiry exotic options, we do not try to get the formal solution to corresponding initial-boundary problem of the Black-Scholes equation, but explain how to express the expiry payoffs of the exotic option as a combination of the payoffs of some class of higher order binary options.