TABLE OF CONTENTS A contingent claim such as options has pre-determined terminal values. Such terminal values and other possible constraints on the price movements of the claim naturally defines a barrier on the binary tree and conditions on it. In 4.1, we consider the Black-Scholes type partial differential equations derived from the binary tree hypermodel and relate barrier conditions to initial and boundary conditions for the differential equations.
In 4.2, we solve the differential equations using the classical method of Green s function and get the option pricing from the solution.
In 4.3 the method is extended to the barrier option. In both these two sections, we obtain explicit formulas for the option pricing.
We briefly mention the more difficult problem of the American put option in 4.4. Approximation solution to such option will be dealt with in the next chapter.
Quite often one considers a contingent claim with either specified terminal values depending on the terminal stock price, or specified values at other time moments depending on stock price levels at those time moments.
A typical example is the binary tree in Fig. 4.1, where the convex boundary represents what we will call barrier conditions of the claim, i.e. a pre-specified values of the claim, based on the time and stock price level. One is interested in values of the claim for nodes on the concave side of the barrier. Starting from the barrier, one...
