A Workout in Computational Finance

6 Boundary, Terminal and Interface Conditions and their Influence

In Chapters 3 and 4, we have learnt about the Black-Scholes differential equation and the differential equations of a number of short rate models. These differential equations describe (in the PDE equivalent of the risk-free measure) the model for the movement of the underlying, which is, in the case of Black-Scholes, the equity, and in the case of the interest rate models of Chapter 4, the short rate. The differential equation itself is not sufficient to valuate a financial instrument; in oder to do so, we additionally need final conditions, boundary conditions and sometimes also interface conditions. All of these depend on the term sheet of the specific instrument. In this chapter, we deal with the formulation of such conditions for specific examples. There are quite a few financial instruments with more or less heavily path-dependent payoffs available, and this path-dependence may be arbitrarily complicated. Therefore, we will concentrate on specific aspects that are fundamental from our point of view. It may be a reasonable advice to avoid instruments for which the formulation of the, say, interface conditions, already cause a headache.

6.1 TERMINAL CONDITIONS FOR EQUITY OPTIONS

In Chapter 2, we have introduced call and put options and their payoffs. Digital options (other names: binary options, cash-or-nothing options) pay a certain amount of cash c if, at expiry T, the underlying S is above the strike price ...