Newton method not converging after iteration (Python - scipy)

问题

I am trying to solve the kmv merton model for default prediction (based on the black scholes model) in Python. I referred to the following code as a jump off point for my code.

My code is as follows:

# -*- coding: utf-8 -*-
"""
Created on Mon Sep 10 15:42:24 2018

@author: CAFRAL
"""
from io import StringIO
import time, boto3
import pandas as pd
import numpy as np
import scipy.optimize as sco
from scipy.stats import norm
import datetime


#generating random values to create a database for testing the code

r = np.random.uniform(low=0.2, high=0.9, size=(252,))
px =np.random.uniform(low=10, high=20, size=(252,))
cso = np.random.uniform(low=10000, high=40000, size=(252,))
cud = np.random.uniform(low=200000, high=800000, size=(252,))
ltd = np.random.uniform(low=400000, high=900000, size=(252,))

d = pd.DataFrame({'r':r,'px':px,'cso':cso,'cud':cud,'ltd':ltd})

#formatting the vaules as required
d["cud"] = d["cud"].astype(float)
d["ltd"] = d["ltd"].astype(float)
d["px"] = d["px"].astype(float)
d["cso"] = d["cso"].astype(float)
#d['year'] = pd.DatetimeIndex(d['date']).year

#computating of key input variable for the model

d["f"] = d["cud"].add(d["ltd"]*0.5)
d['e'] = abs(d['px'])*d['cso']
d['a'] = d["f"].add(d["e"])


#defining a function for the black Scholes equation

def bseqn(a, debug=False):
    d1 = (np.log(a/f) + (r + 0.5*sigma_a**2)*T)/(sigma_a*np.sqrt(T))
    d2 = d1 - sigma_a*np.sqrt(T)
    y1 = e - (a*norm.cdf(d1) - np.exp(-r*T)*f*norm.cdf(d2))

    if debug:
        print("d1 = {:.6f}".format(d1))
        print("d2 = {:.6f}".format(d2))
        print("Error = {:.6f}".format('y1'))

    return y1

#Solving the model

time_horizon=[1]
timesteps = range(1, len(d))

results = np.empty((d.shape[0],len(time_horizon)))

#looping to solve for each row

for i, years in enumerate(time_horizon):
    T = 252
    results[:,i] = d.loc[:,'a'] 
    for i_t, t in enumerate(timesteps):  
        a = results[t-10:t,i]
        ra =np.log(a/np.roll(a,1))
        sigma_a = np.nanstd(ra) #gives initial value of sigma_a

        if i_t == 0:
            subset_timesteps = range(t-1, t+1)
            print(subset_timesteps)
        else:
            subset_timesteps = [t]

        n_its = 0
        while n_its < 10:
            n_its += 1
            for t_sub in subset_timesteps:
                r = d.iloc[t_sub]['r'] 
                f = d.iloc[t_sub]['f']
                e = d.iloc[t_sub]['e']
                sol = sco.newton(bseqn, results[t_sub,i]) #if I replace newton with fsolve the code works properly
                results[t_sub,i] = sol # stores the new values of a 

            # Update sigma_a based on new values of a
            last_sigma_a = sigma_a
            a = results[t-10:t,i]
            ra = np.log(a/np.roll(a,1))
            sigma_a = np.nanstd(ra) #new val of sigma
            diff = last_sigma_a - sigma_a
            if abs(diff) < 1e-3:
                d.loc[t_sub, 'sigma_a'] = sigma_a
                break 
        else:

            pass

When I run my code using the fsolve method, the code runs just fine. However, once i use the Newton method, I get the following error.

 C:\Users\CAFRAL\Anaconda3\lib\site-packages\numpy\lib\nanfunctions.py:1434: RuntimeWarning: Degrees of freedom <= 0 for slice.
  keepdims=keepdims)
C:\Users\CAFRAL\Anaconda3\lib\site-packages\scipy\stats\_distn_infrastructure.py:879: RuntimeWarning: invalid value encountered in greater
  return (self.a < x) & (x < self.b)
C:\Users\CAFRAL\Anaconda3\lib\site-packages\scipy\stats\_distn_infrastructure.py:879: RuntimeWarning: invalid value encountered in less
  return (self.a < x) & (x < self.b)
C:\Users\CAFRAL\Anaconda3\lib\site-packages\scipy\stats\_distn_infrastructure.py:1738: RuntimeWarning: invalid value encountered in greater_equal
  cond2 = (x >= self.b) & cond0
Traceback (most recent call last):

  File "<ipython-input-70-8382063cd297>", line 72, in <module>
    sol = sco.newton(bseqn, results[t_sub,i])#results[t_sub,i] is the initial guess for a

  File "C:\Users\CAFRAL\Anaconda3\lib\site-packages\scipy\optimize\zeros.py", line 206, in newton
    raise RuntimeError(msg)

RuntimeError: Failed to converge after 50 iterations, value is nan

Could someone please help me understand the following: a) Why does the Newton method not converge? b) Is it possible to fix the convergence error in the Newton method? If yes, how?

来源：https://stackoverflow.com/questions/52270610/newton-method-not-converging-after-iteration-python-scipy