Python for Finance: Analyze Big Financial Data

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                                                                            size                        10000.000                       10000.000

                                                                                min                                     0.004                                       0.005

                                                                                max                                     0.049                                       0.050

                                                                            mean                                        0.020                                       0.020

                                                                                std                                     0.006                                       0.006

                                                                            skew                                        0.529                                       0.572

                                                            kurtosis                                        0.418                                       0.503

However, a major difference can be observed in terms of execution speed, since sampling

from the noncentral chi-square distribution is more computationally demanding than from

the standard normal distribution. To illustrate this point, consider a larger number of paths

to be simulated:

In  [ 29 ]: I   =    250000

                                    %time x1    =   srd_euler()

Out[29]:    CPU times:  user    1.02    s,  sys:    84  ms, total:  1.11    s

                                    Wall    time:   1.11    s

In  [ 30 ]: %time x2    =   srd_exact()

Out[30]:    CPU times:  user    2.26    s,  sys:    32  ms, total:  2.3 s

                                    Wall    time:   2.3 s

The exact scheme takes roughly twice as much time for virtually the same results as with

the Euler scheme:

In  [ 31 ]: print_statistics(x1[- 1 ],  x2[- 1 ])

x1  =   0.0;    x2  =   0.0

Out[31]:                        statistic                   data    set 1                   data    set 2

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                                                                            size                    250000.000                  250000.000

                                                                                min                                     0.003                                       0.004

                                                                                max                                     0.069                                       0.060

                                                                            mean                                        0.020                                       0.020

                                                                                std                                     0.006                                       0.006

                                                                            skew                                        0.554                                       0.578

                                                            kurtosis                                        0.488                                       0.502

Stochastic volatility

One of the major simplifying assumptions of the Black-Scholes-Merton model is the

constant volatility. However, volatility in general is neither constant nor deterministic; it is

stochastic. Therefore, a major advancement with regard to financial modeling was

achieved in the early 1990s with the introduction of so-called stochastic volatility models.

One of the most popular models that fall into that category is that of Heston (1993), which

is presented in Equation 10-7.

Equation 10-7. Stochastic differential equations for Heston stochastic volatility model