Boundary Conditions of Options: A Demonstration Based on the Stochastic Discount

The aim of this paper is to provide a proof of the generally accepted boundary conditions of (call and put) financial options from a novel point of view. To do this, we will use an auxiliary discounting function which will be defined in this work. However, the financial options are derivative instruments whose function is risk hedging in contexts of uncertainty, whereby the employed discount function will be necessarily stochastic. More specifically, we will apply the classic properties of the magnitude “discount” to the so-defined discount function to obtain, in a natural way, the noteworthy boundary conditions of financial options. It is well-known that financial options (belonging to the field of stochastic finance) have been studied without any relation with the magnitude “discount” (more characteristic of the classic Financial Mathematics). Consequently, the principal contribution of this work is the construction of a stochastic discount function as a bridge connecting its associated discount and the financial options, being demonstrated that their properties can be mutually derived.