I am trying to perform Monte Carlo Simulations using quasi random standard normal numbers. I understand that we can use sobol sequences to generate uniform numbers, and then use probability integral transform to convert them to standard normal numbers. My code gives unrealistic values of the simulated asset path:
import sobol_seq
import numpy as np
from scipy.stats import norm
def i4_sobol_generate_std_normal(dim_num, n, skip=1):
"""
Generates multivariate standard normal quasi-random variables.
Parameters:
Input, integer dim_num, the spatial dimension.
Input, integer n, the number of points to generate.
Input, integer SKIP, the number of initial points to skip.
Output, real np array of shape (n, dim_num).
"""
sobols = sobol_seq.i4_sobol_generate(dim_num, n, skip)
normals = norm.ppf(sobols)
return normals
def GBM(Ttm, TradingDaysInAYear, NoOfPaths, UnderlyingPrice, RiskFreeRate, Volatility):
dt = float(Ttm) / TradingDaysInAYear
paths = np.zeros((TradingDaysInAYear + 1, NoOfPaths), np.float64)
paths[0] = UnderlyingPrice
for t in range(1, TradingDaysInAYear + 1):
rand = i4_sobol_generate_std_normal(1, NoOfPaths)
lRand = []
for i in range(len(rand)):
a = rand[i][0]
lRand.append(a)
rand = np.array(lRand)
paths[t] = paths[t - 1] * np.exp((RiskFreeRate - 0.5 * Volatility ** 2) * dt + Volatility * np.sqrt(dt) * rand)
return paths
GBM(1, 252, 8, 100., 0.05, 0.5)
array([[1.00000000e+02, 1.00000000e+02, 1.00000000e+02, ...,
1.00000000e+02, 1.00000000e+02, 1.00000000e+02],
[9.99702425e+01, 1.02116774e+02, 9.78688323e+01, ...,
1.00978615e+02, 9.64128959e+01, 9.72154915e+01],
[9.99404939e+01, 1.04278354e+02, 9.57830834e+01, ...,
1.01966807e+02, 9.29544649e+01, 9.45085180e+01],
...,
[9.28295879e+01, 1.88049044e+04, 4.58249200e-01, ...,
1.14117599e+03, 1.08089096e-02, 8.58754653e-02],
[9.28019642e+01, 1.92029616e+04, 4.48483141e-01, ...,
1.15234371e+03, 1.04211828e-02, 8.34842557e-02],
[9.27743486e+01, 1.96094448e+04, 4.38925214e-01, ...,
1.16362072e+03, 1.00473641e-02, 8.11596295e-02]])
Values like 8.11596295e-02 should not be generated, hence I think there is something wrong in the code.
References: https://stats.stackexchange.com/questions/27450/best-method-for-transforming-low-discrepancy-sequence-into-normal-distribution, https://stackoverflow.com/questions/9412339/recommendations-for-low-discrepancy-e-g-sobol-quasi-random-sequences-in-pytho, https://github.com/naught101/sobol_seq
Any help is appreciated.