Python for Finance: Analyze Big Financial Data

Finally, let us check whether the end-of-period values are indeed log-normally distributed.

This boils down to a normality test as well, since we only have to transform the data by

applying the log function to it (to then arrive at normally distributed data — or maybe

not). Figure 11-4 plots both the log-normally distributed end-of-period values and the

transformed ones (“log index level”):

In  [ 15 ]: f,  (ax1,   ax2)    =   plt.subplots( 1 ,    2 ,    figsize=( 9 ,    4 ))

ax1.hist(paths[- 1 ],   bins= 30 )

ax1.grid(True)

ax1.set_xlabel(‘index   level’)

ax1.set_ylabel(‘frequency’)

ax1.set_title(‘regular  data’)

ax2.hist(np.log(paths[- 1 ]),   bins= 30 )

ax2.grid(True)

ax2.set_xlabel(‘log index   level’)

ax2.set_title(‘log  data’)

Figure 11-4. Histogram of simulated end-of-period index levels

The statistics for the data set show expected behavior — for example, a mean value close

to 105 and a standard deviation (volatility) close to 20%:

In  [ 16 ]: print_statistics(paths[- 1 ])

Out[16]:                        statistic                                           value

                                    ––––––––––

                                                                            size                250000.00000

                                                                                min                             42.74870

                                                                                max                         233.58435

                                                                            mean                            105.12645

                                                                                std                             21.23174

                                                                            skew                                    0.61116

                                                            kurtosis                                    0.65182

The log index level values also have skew and kurtosis values close to zero:

In  [ 17 ]: print_statistics(np.log(paths[- 1 ]))

Out[17]:                        statistic                                           value

                                    ––––––––––

                                                                            size                250000.00000

                                                                                min                                 3.75534

                                                                                max                                 5.45354

                                                                            mean                                    4.63517

                                                                                std                                 0.19998

                                                                            skew                                -0.00092

                                                            kurtosis                                -0.00327

This data set also shows high p-values, providing strong support for the normal

distribution hypothesis:

In  [ 18 ]: normality_tests(np.log(paths[- 1 ]))