Portfolio Optimization and Statistics in Stochastic Volatility Markets

Abstract: Large financial portfolios often contain hundreds of stocks. The aim of this thesis is to find explicit optimal trading strategies that can be applied to portfolios of that size for different n-stock extensions of the model by Barndorff-Nielsen and Shephard [3]. A main ambition is that the number of parameters in our models do not grow too fast as the number of stocks n grows. This is necessary to obtain stable parameter estimates when we fit the models to data, and n is relatively large. Stability over the parameter estimates is needed to obtain accurate estimates of the optimal strategies. Statistical methods for fitting the models to data are also given. The thesis consists of three papers. Paper I presents an n-stock extension to the model in [3] where the dependence between different stocks lies strictly in the volatility. The model is primarily intended for stocks that are dependent, but not too dependent, such as stocks from different branches of industry. We develop optimal portfolio theory for the model, and indicate how to do the statistical analysis. In Paper II we extend the model in Paper I further, to model stronger dependence. This is done by assuming that the diffusion components of the stocks contain one Brownian motion that is unique for each stock, and a few Brownian motions that all stocks share. We then develop portfolio optimization theory for this extended model. Paper III presents statistical methods to estimate the model in [3] from data. The model in Paper II is also considered. It is shown that we can divide the centered returns by a constant times the daily number of trades to get normalized returns that are i.i.d. and N(0,1). It is a key feature of the Barndorff-Nielsen and Shephard model that the centered returns divided by the volatility are also i.i.d. and N(0,1). This suggests that we identify the daily number of trades with the volatility, and model the number of trades within the framework of Barndorff-Nielsen and Shephard. Our approach is easier to implement than the quadratic variation method, requires much less data, and gives stable parameter estimates. A statistical analysis is done which shows that the model fits the data well.

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