 
Summary: Neural Networks for Density Estimation in
Financial Markets
Malik MagdonIsmail 1 and Amir Atiya 2
1 malik@work.caltech.edu
2 amir@work.caltech.edu
Learning Systems Group, California Institute of Technology
13693 Caltech, Pasadena, CA, USA, 91125
Abstract. We introduce two new techniques for density estimation. Our
approach poses the problem as a supervised learning task which can be
performed using Neural Networks. We introduce a stochastic method
for learning the cumulative distribution and an analogous deterministic
technique. We use these techniques to estimate the densities of log stock
price changes, demonstrating that the density is fattailed contrary to
the BlackScholes model which assumes it to be Gaussian.
A majority of problems in science and engineering have to be modeled in a prob
abilistic manner. Even if the underlying phenomena are inherently deterministic,
the complexity of these phenomena often makes a probabilistic formulation the
only feasible approach from the computational point of view. Although quan
tities such as the mean, the variance, and possibly higher order moments of
a random variable have often been sufficient to characterize a particular prob
