A neural network is a non-linear system that converts a series of
real input values
over several intermediary
steps to one or more terminal variables
. It
represents a function
:
Figure 18.2 displays a feedback network, in that there is feedback among the nodes of the two hidden layers. In the following we will concentrate on the feed forward network.
Neural network are used in financial statistics to represent
functions, which, for example, can represent the default
probability of a credit, the forecast of an exchange rate or the
volatility of a stock. Here the emphasis is on non-parametric
applications, which in comparison to the local smoothing function
discussed in Chapter 13 require an advanced modelling
and can be quite involved to calculate. On the other hand it is
still practical when numerous variables need to be considered in
forecasts or quantifying risk, i.e., when the dimension of the
function arguments is large.
Since neural networks are still relatively unknown tools in
statistics, in the first section we will give an elementary
introduction in the structure of a neural network. It allows for
the construction of complex functions using simple elements. In
the second section we describe the popular numerical application
for fitting neural networks to the data, before we conclude with
various applications to financial problems and introduce the
underlying assumptions.