Modelling of stock prices by the Markov chain Montecarlo method

Abstract

This paper presents a universal approach to modelling stock prices. The technique involves Markov Chain Monte Carlo (MCMC) sampling from piecewise-uniform distribution. Today’s fi nancial models are based on assumptions which make them inadequate in many cases. One of the most important issues is determining the distribution of a stock price, its return or other financial mean. The approach proposed in this paper removes almost all presumptions from a distribution of a stock price. The probability density must be evaluated using some nonparametric estimates. The kernel density estimate (KDE) suits well for that purpose. It gives a smooth and presentable estimate. MCMC was chosen due to its versatility and is applied to KDE using piecewise-linear distribution as proposal density. The proposal density is constructed according to the KDE. Such link between the piecewise-linear distribution’s simplicity and relative massiveness of KDE balances together. Involving the kernel density estimate and the methodology to sample from it makes the technique universal for modelling any real stochastic system while having empirical data only and barely any assumptions about the distribution of it.