我试着用准随机标准正态数进行蒙特卡罗模拟。我知道我们可以用Sobol序列生成一致数,然后用概率积分变换把它们转换成标准正态数。我的代码给出了模拟资产路径的不切实际的值:
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]])
像8.11596295e-02这样的值不应该生成,因此我认为代码中存在错误。如果我使用来自numpy
库rand = np.random.standard_normal(NoOfPaths)
的标准正常绘制,则价格与Black-Scholes价格匹配。所以我认为问题出在随机数发生器上。值8.11596295e-02
是指路径中的一个价格,它不太可能从100(初始价格)降到8.11596295e-02
。在
似乎
sobol_seq
中有一个bug。Anaconda,python 3.7,64位,Windows 10 x64,通过pip安装sobol_seq
简单代码
^{pr2}$产出
来自http://people.sc.fsu.edu/~jburkardt/py_src/sobol/sobol.html的代码,sobol_库.py行为合理(好吧,除了第一点)。在
好吧,封闭的代码看起来可以工作,保持种子与采样数组在一起。但是很慢。。。在
这是图片
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