Neural Networks for Density Estimation in Financial Markets

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 fat-tailed contrary to the Black-Scholes 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 quantities such as the mean, the variance, and possibly higher order moments of a random variable have often been suucient to characterize a particular problem , the quest for higher modeling accuracy, and for more realistic assumptions drives us towards modeling the available random variables using their probability density. This of course leads us to the problem of density estimation (see 6]). According to the Black-Scholes model for option pricing 1], the log stock price changes are assumed Gaussian. What if the true distribution is not Gaus-sian? The option prices can be computed (numerically) using the true distribution with a possibility of detecting mispricing. This could have large nancial implications. Traditional density estimation methods can be grouped into two broad categories (see 6]). The rst of these is the parametric approach. It assumes that the density has a speciic functional form, such as a Gaussian or a mixture of Gaussians. The unknown density is estimated by using the data to obtain estimates for the parameters of the functional form. Typically, the parameters are estimated using maximum likelihood or Bayesian techniques. The drawback of the parametric approach is that the functional form of the density is rarely known beforehand, and the commonly assumed Gaussian or mixture of Gaussian models rarely t densities that are encountered in practice. The more common approach for density estimation is the non parametric approach where the density is determined according to a formula involving the data points available.