最小化使用 Python 查找我的 Heston Nandi 模型参数的对数可能性

问：

我有一个目标函数，我想使用 MLE 来查找模型的参数。这是我到目前为止的代码。此外，我的returnarr是股票的日志返回数组 以下是我的 returnarr 的前 30 个值，仅供参考：

returnarr = filterdata['logreturn'].to_numpy(dtype=np.longdouble)
returnarr[0:30]

array([ 0.00075018, -0.00696353, -0.00080357,  0.00287779, -0.00384807,
       -0.00496853,  0.00608317, -0.00252242, -0.00731938,  0.01879965,
       -0.00673696, -0.00757294, -0.02209292,  0.01152935,  0.00609485,
        0.004072  , -0.01832714, -0.01604845,  0.0059958 ,  0.00251289,
        0.00740885, -0.01594026,  0.0076436 , -0.00298763,  0.00460656,
       -0.00237508,  0.0078305 ,  0.003722  ,  0.00649657])

import numpy as np
from scipy.optimize import minimize

# Define the log-likelihood function
def log_likelihood(parameters,returnarr):
    size=len(returnarr)
    phi = parameters[0]
    w = parameters[1]
    B = parameters[2]
    alpha = parameters[3]
    lambda1=parameters[4]
    epsilon = np.zeros(size, dtype='longdouble')
    variance = np.zeros(size + 1, dtype='longdouble')
    variance[0]=0.001

    for x,val in  enumerate(returnarr):
        epsilon[x] = (val - (rate + (lambda1 * variance[x]))) / np.sqrt(variance[x])
        variance[x+1] = w + (B * variance[x]) + (alpha * np.square(epsilon[x]-(phi*np.sqrt(variance[x]))))
        
    variance=variance[:-1]#remove last term
    log_likelihood_on_return = -0.5 * np.sum(np.log(variance) +np.square(epsilon))
    return log_likelihood_on_return  # Minimize the negative log-likelihood

关于如何使用最小化在此处查找参数的任何建议？

我试过这个：

initial_parameters=[0.01,0.01,0.01,0.01,0.01]
bounds = [(0, 1), (0, 1), (0, 1), (0, 1), (0, 1)]

result = minimize(log_likelihood, initial_parameters, args=(returnarr,),bounds=bounds)

我仍然收到错误，上面写着

RuntimeWarning: overflow encountered in square
  variance[x+1] = w + (B * variance[x]) + (alpha * np.square(epsilon[x]-(phi*np.sqrt(variance[x]))))
RuntimeWarning: invalid value encountered in double_scalars
  epsilon[x] = (val - (rate + (lambda1 * variance[x]))) / np.sqrt(variance[x])```