Fractional Brownian motion and it's application in the Norwegian stock market

Abstract

In this thesis, I investigate the properties of fractional Brownian motion for use in the stock market, and I also look at what types of calculus that should be used when one works with it. I then use the calculus to see what would happen to the market if stock returns followed fractional Brownian motion. The second part of the thesis consists of finding a method to estimate discretized fractional Brownian motion by using ARFIMA models. I apply this theory to stocks in the Oslo Stock Exchange to look for long memory in the returns and volatility by estimating the Hurst coeficient. Would it be easy to make money by using this model? I find that the fractional Brownian motion has several traits we appre- ciate when analysing stocks. However, the market would not be eficient if stock returns could be modelled by fractional Brownian motion, as it allows for arbitrage. In the Norwegian stock market, I find that the main index shows some ev- idence of long memory in the returns. However, this is not much, as the Hurst coeficients estimated are quite close to 1 2 . This means that it is unlikely that one could make much money from trying to find arbitrage opportunities like this. The same is true for the analysed stocks, and the ARFIMA model is not a perfect fit for any of them. I do find evidence of long memory in the volatility of the stock returns, and this may be used to help understand and predict the risk of the stocks better.