Theoretical and Practical Application Sample Clauses

Theoretical and Practical Application. The Black & Scholes problem consists of a differential equation depending on d + 1 independent variables, where d is the number of independent underlying and the additional dependence is on time t. For an extensive introduction to the problem, we refer to [9], Chap 2.9. The equation is of second order and needs to be completed by suitable initial and boundary conditions. We will extensively discuss the model in the next chapter. Apart from special cases that we will discuss later on, the analytical solution of the problem cannot be computed. The best we can do is to prove that there exists a unique solution, depending continuously on the data. This is called well posedness of the problem according to Hadamard (see e.g. [3, 9]). The proof requires using concepts of functional analysis and in particular a result called Lax-Milgram Lemma, which are out of the scope of the present work. The relevant point, here, is that in general we cannot solve explicitly the equations and this prevents a quantitative analysis. Fortunately, it is possible to approximate the differential equation with a system of algebraic equations, see e.g. [5, 2]. This can be accomplished in several