Narrow Results

We present results of an extension of the market model introduced by Bornholdt to high dimensions. Three and four dimensions are shown to behave similar to two, for suitable parameters.

Aimed at better modeling stock returns and finding robustly optimal investment decisions, a new portfolio selection model is proposed in this paper. The model differs from existing ones in following ways: multiple market frictions are taken into account simultaneously; the adopted multivariate t-distribution can capture the well-recognized fat tails in the return data by adding only one more parameter relative to the normal; the downside loss risk is controlled by a chance constraint which, including VaR as a special case, is flexible in terms of adjusting the threshold return and the loss probability level; one important advantage about the combination of the latter two innovations is that the derived asset allocation model can be transformed into a second-order cone program or a linear program, which can be easily solved in polynomial time. Empirical results based on some S&P 500 component stocks not only demonstrate the practicality of our new model, but show how different model parameters could affect the optimal portfolio selection. This is very useful in guiding investors to choose a correct model and to find the investment strategy most suitable for their specific purpose.

Modifications of the Cont-Bouchaud percolation model for price fluctuations give an asymmetry for time-reversal, an asymmetry between high and low prices, volatility clustering, effective multifractality, correlations between volatility and traded volume, and a power law tail with exponent near 3 for the cumulative distribution function of price changes. Combining them together still gives the same power law. Using Ising-correlated percolation does not change these results. Different modifications give log-periodic oscillations before a crash, arising from nonlinear feedback between random fluctuations.

Longer horizon returns are constructed from data on daily returns. Observed drawbacks of a Lévy process are a sharp decrease in skewness and excess kurtosis. Drawbacks to scaling are a flat term structure of skewness and excess kurtosis. A strategy that combines some exposure to independent increments and some exposure to scaling is developed in the context of self decomposable daily return distributions. Estimations are conducted on 400 stocks and we report that a good strategy for constructing longer horizon returns can be that of accumulating as i.i.d. half the daily return while scaling the remainder at rate one half.