Deviations from Normality in a Financial Model without Short-selling

Abstract

We present a variation of the well-known binomial model of asset prices. This variation is based on the interaction of a finite number of investors (grouped in two kinds, bulls and bears) with a single market maker, in an environment with bounds to short selling. We prove a formula for all the moments of the logarithmic returns and from that we derive the corresponding formula for the binomial model. As an application of the model, we show how to compute parameters in order to approximate given moments, enabling the modeling of skewness and excess kurtosis. We provide a concrete example of this procedure, using real data from a stock, and we present two plots revealing its convergence pattern. Finally, we generalize the framework to account for the possibility of more heterogeneous investors, and give the corresponding formula for the moments of the logarithmic returns, and the algorithm for fitting given moments.