FINANCIAL DATA ANALYSIS WITH STABLE LAWS
Data Collection, visual analysis and initial parameter estimation
Here we draw financial data in the form of a stock price series for any stock ("symbol") you choose from the Wolfram webserver and performs a number of numerical and graphic procedures on that data. Enter the stock mutual fund or index symbol and the start date. The program collects data between the start date and present. Change the inputs (symbol and date), select all the cells in the notebook, and evaluate.
Get data from webserver
Produce plots of closing prices and log volume over the selected time period
The code in the cell below graphs the (21 trading day) monthly moving average (click the button to show all the volatility information in that period)
Next we estimate the Stable parameters using Nolan's S1 parameterization for closing data, scaled weekly and monthly data
| α | β | γ | δ | |
| Day Close Data | 1.8397 | -0.439021 | 0.00501237 | 0.000360071 |
| Scaled Week from Day | 1.8397 | -0.439021 | 0.0120221 | 0.00180036 |
| Scaled Month from Day | 1.8397 | -0.439021 | 0.0262275 | 0.0075615 |
Below on the left is the histogram for closing price data. The right shows the Log-Log Normal and Stable fits as empirical (dots) and theoretical. Notice that the theoretical fit from the estimated parameters is wilder than the empirical. This suggests that using Stable parameters for risk management will overestimate risk thus tending toward a conservative error.
Notes on scaled stable probabilities.
The scaling used in the program assumes that the random variables are independent and identically distributed. In fact stock market returns show clustering of extreme returns, what this means for projections is not clear, but it is a good idea to assume that the distributions are not identically distributed, but rather that some of the parameters may be varying. The simplest assumption is that the scale factor, γ, varies. Therefore if you are in a period of high volatility, the scale factor that will you will be experiencing will not be the average over the interval as you have computed above. Also the scaling assumes the probability of the price at the end of the period, it may well be lower or higher before the end of the period. Therefore these probabilities should be used as a rough scale. You might want to consider squaring the probability, i.e. assume you might have two identical events one after another to account for the clustering effect. The same thoughts apply to daily probabilities, if the most recent volatility is not like the average.
| Created by Wolfram Mathematica 6.0 (10 August 2007) | |