Python for Finance: Analyze Big Financial Data

As before, translation into Python and NumPy code is straightforward:

In  [ 17 ]: I   =    10000

M   =    50

dt  =   T   /   M

S   =   np.zeros((M +    1 ,    I))

S[ 0 ]  =   S0

for t in range( 1 , M   +    1 ):

S[t]    =   S[t -    1 ]    *   np.exp((r   -   0.5 *   sigma   **   2 )    *   dt

                                                                                    +   sigma   *   np.sqrt(dt) *   npr.standard_normal(I))

The resulting end values for the index level are log-normally distributed again, as

Figure 10-5 illustrates:

In  [ 18 ]: plt.hist(S[- 1 ],   bins= 50 )

plt.xlabel(‘index   level’)

plt.ylabel(‘frequency’)

plt.grid(True)

Figure 10-5. Simulated geometric Brownian motion at maturity

The first four moments are also quite close to those resulting from the static simulation

approach:

In  [ 19 ]: print_statistics(S[- 1 ],   ST2)

Out[19]:                        statistic                   data    set 1                   data    set 2

                                    –––––––––––––––

                                                                            size                        10000.000                       10000.000

                                                                                min                                 25.531                                  27.266

                                                                                max                             425.051                             358.997

                                                                            mean                                110.900                             110.528

                                                                                std                                 40.135                                  40.894

                                                                            skew                                        1.086                                       1.150

                                                            kurtosis                                        2.224                                       2.273

Figure 10-6 shows the first 10 simulated paths:

In  [ 20 ]: plt.plot(S[:,   : 10 ], lw=1.5)

plt.xlabel(‘time’)

plt.ylabel(‘index   level’)

plt.grid(True)