1. 隐含波动率的计算方法
在期权市场上,可以直接观察到标的物当前价格(S)、期权的执行价格(K)、合约期限(T)以及无风险收益率(r),但是波动率不能直接观察。当然,我们可以通过标的物历史价格来估计波动率,但是在实际的应用中,通常会使用隐含波动率,隐含波动率可通过期权的市场价格、运用 BSM 模型进行计算。由于期权的定价公式非常复杂,不能直接通过反解公式求解(定价公式可参考金融分析与风险管理——期权BSM模型),在数学上可以通过无限逼近的方法来求解,常用的求解方法有:牛顿迭代法、二分查找法
1.1 牛顿迭代法
由金融分析与风险管理——期权BSM模型可知:波动率与期权价格成正比。
计算步骤如下:
- 1 输入一个初始隐含波动率
- 2 建立迭代关系:
- 2.1 若初始隐含波动率获得的期权价格较高,则需要减去一个标量
- 2.2 若初始隐含波动率获得的期权价格较低,则需要加上一个标量
- 3 迭代临界值控制,隐含波动率计算的期权价格与期权的市场价格的绝对值要小于等于设定的临界值,若满足条件,则隐含波动率为返回的最新 。
其 Python 代码如下:
#期权的隐含波动率
#牛顿迭代法求解隐含波动率
#call的计算
def impvol_call_Newton(C,S,K,r,T,sigma0 = 0.2):
def call_BS(S,K,sigma,r,T):
import numpy as np
from scipy.stats import norm
d1 = (np.log(S/K)+(r+sigma**2/2)*T)/(sigma*np.sqrt(T))
d2 = d1 - sigma*np.sqrt(T)
return S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
diff = C - call_BS(S, K, sigma0, r, T)
i = 0.0001
while abs(diff) > 0.0001:
diff = C - call_BS(S, K, sigma0, r, T)
if diff > 0:
sigma0 += i
else:
sigma0 -= i
return sigma0
#put的计算
def impvol_put_Newton(P,S,K,r,T,sigma0 = 0.2):
def put_BS(S,K,sigma,r,T):
import numpy as np
from scipy.stats import norm
d1 = (np.log(S/K)+(r+sigma**2/2)*T)/(sigma*np.sqrt(T))
d2 = d1 - sigma*np.sqrt(T)
return K*np.exp(-r*T)*norm.cdf(-d2) - S*norm.cdf(-d1)
diff = P - put_BS(S, K, sigma0, r, T)
i = 0.0001
while abs(diff) > 0.0001:
diff = P - put_BS(S, K, sigma0, r, T)
if diff > 0:
sigma0 += i
else:
sigma0 -= i
return sigma0
impvol_call = impvol_call_Newton(C=0.1566,S=5.29,K=6,r=0.04,T=0.5,sigma0 = 0.2)
impvol_put = impvol_put_Newton(P=0.7503,S=5.29,K=6,r=0.04,T=0.5,sigma0 = 0.2)
print('牛顿法计算的看涨期权隐含波动率:',round(impvol_call,4))
print('牛顿法计算的看跌期权隐含波动率:',round(impvol_put,4))
1.2 二分查找法
计算步骤如下:
- 1 输入初始较小隐含波动率 、较大隐含波动率 ,并计算波动率均值
- 2 建立迭代关系:
- 2.1 若期权的市场价格小于较小隐含波动率计算的期权价格或者大于较大隐含波动率计算的期权价格,则需要重新设定初始输入的两个隐含波动率;
- 2.2 若期权的市场价格介于较小隐含波动率计算的期权价格与较大隐含波动率计算的期权价格之间,则使用 替换 ,例如:期权市场价格介于 。
- 3 迭代临界值控制,隐含波动率计算的期权价格与期权的市场价格的绝对值要小于等于设定的临界值,若满足条件,则隐含波动率为返回的最新 。
#二分法求解隐含波动率
#call的隐含波动率
def impvol_call_Binary(C,S,K,r,T,sigma_min = 0.001,sigma_max = 1.00):
#call的计算
def call_BS(S,K,sigma,r,T):
import numpy as np
from scipy.stats import norm
d1 = (np.log(S/K)+(r+sigma**2/2)*T)/(sigma*np.sqrt(T))
d2 = d1 - sigma*np.sqrt(T)
return S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
sigma_mid = (sigma_min + sigma_max)/2
call_min = call_BS(S, K, sigma_min, r, T)
call_max = call_BS(S, K, sigma_max, r, T)
call_mid = call_BS(S, K, sigma_mid, r, T)
diff = C - call_mid
if C < call_min:
print('Error:请重新输入更小的sigma_min')
if C > call_max:
print('Error:请重新输入更大的sigma_max')
while abs(diff) > 0.000001:
diff = C - call_BS(S, K, sigma_mid, r, T)
sigma_mid = (sigma_min + sigma_max)/2
call_mid = call_BS(S, K, sigma_mid, r, T)
if C > call_mid:
sigma_min = sigma_mid
else:
sigma_max = sigma_mid
return sigma_mid
#put的隐含波动率
def impvol_put_Binary(P,S,K,r,T,sigma_min = 0.001,sigma_max = 1.00):
#put的计算
def put_BS(S,K,sigma,r,T):
import numpy as np
from scipy.stats import norm
d1 = (np.log(S/K)+(r+sigma**2/2)*T)/(sigma*np.sqrt(T))
d2 = d1 - sigma*np.sqrt(T)
return K*np.exp(-r*T)*norm.cdf(-d2) - S*norm.cdf(-d1)
sigma_mid = (sigma_min + sigma_max)/2
put_min = put_BS(S, K, sigma_min, r, T)
put_max = put_BS(S, K, sigma_max, r, T)
put_mid = put_BS(S, K, sigma_mid, r, T)
diff = P - put_mid
if P < put_min:
print('Error:请重新输入更小的sigma_min')
if P > put_max:
print('Error:请重新输入更大的sigma_max')
while abs(diff) > 0.000001:
diff = P - put_BS(S, K, sigma_mid, r, T)
sigma_mid = (sigma_min + sigma_max)/2
put_mid = put_BS(S, K, sigma_mid, r, T)
if P > put_mid:
sigma_min = sigma_mid
else:
sigma_max = sigma_mid
return sigma_mid
impvol_call_Binary = impvol_call_Binary(C=0.1566,S=5.29,K=6,r=0.04,T=0.5,sigma_min = 0.001,sigma_max = 1)
impvol_put_Binary = impvol_put_Binary(P=0.7503,S=5.29,K=6,r=0.04,T=0.5,sigma_min = 0.001,sigma_max = 1)
print('二分法计算的看涨期权隐含波动率:',round(impvol_call_Binary,4))
print('二分法计算的看跌期权隐含波动率:',round(impvol_put_Binary,4))
2. 波动率微笑与斜偏
2.1 波动率微笑
波动率微笑:描述期权隐含波动率与执行价格之间的图形关系,即:对同一标的物,具有相同到期日、不同执行价格的一系列期权,当执行价格偏离现货价格越远时,期权的隐含波动率就越大,类似于微笑曲线。
本文以2018/6/27到期的、不同执行价格的上证50ETF期权合约在2017/12/29日的收盘价为例,其Python代码如下:
#波动率微笑
#2018/6/27到期的、不同执行价格的上证50ETF期权合约在2017/12/29日的收盘价为例
import datetime as dt
T1 = dt.datetime(2017, 12, 29) #计算隐含波动率时的日期
T2 = dt.datetime(2018, 6, 27) #期权到期日
T_delta = (T2 - T1).days/365 # 期权的剩余期限并转化成年
S0 = 2.859 # 当日的上证50ETF基金净值
#认购(call)期权收盘价
call_list = np.array([0.2841,0.2486,0.2139,0.1846,0.1586,0.1369,0.1177])
#认沽(put)期权收盘价
put_list = np.array([0.0464,0.0589,0.0750,0.0947,0.1183,0.1441,0.1756])
#期权执行价格
K_list = np.array([2.7,2.75,2.8,2.85,2.9,2.95,3.0])
shibor = 0.048823 # 计算隐含波动率当日6个月期的shibor利率
sigma_clist = np.zeros_like(call_list)
#使用牛顿法计算隐含波动率
for i in np.arange(len(call_list)):
sigma_clist[i] = impvol_call_Newton(C=call_list[i], S=S0, K=K_list[i], r=shibor, T=T_delta,sigma0 = 0.2)
print('call的隐含波动率:',sigma_clist)
sigma_plist = np.zeros_like(put_list)
for i in np.arange(len(put_list)):
sigma_plist[i] = impvol_put_Newton(P=put_list[i], S=S0, K=K_list[i], r=shibor, T=T_delta,sigma0 = 0.2)
print('put的隐含波动率:',sigma_plist)
plt.figure(figsize=(8,6))
plt.plot(K_list,sigma_clist,label='50ETF认购期权')
plt.plot(K_list,sigma_plist,label='50ETF认沽期权')
plt.legend()
plt.grid('True')
2.2 波动率斜偏
在多数交易日,期权的波动率并不是微笑的,而是表现为波动率斜偏。波动率斜偏通常是指当期权的执行价格由小变大时,期权的隐含波动率则是由大变小的,即隐含波动率是执行价格的减函数。
本文以 2019/6/26 到期的、不同执行价格的上证50ETF期权合约在 2018/12/28 日的收盘价为例,其Python代码如下:
#波动率偏斜
#2019/6/26到期的、不同执行价格的上证50ETF期权合约在2018/12/28日的收盘价为例
import datetime as dt
T1 = dt.datetime(2018, 12, 28) #计算隐含波动率时的日期
T2 = dt.datetime(2019, 6, 26) #期权到期日
T_delta = (T2 - T1).days/365 #期权的剩余期限并转化成年
S0 = 2.289 #当日的上证50ETF基金净值
#认购(call)期权收盘价
call_list = np.array([0.2866,0.2525,0.2189,0.1912,0.1645,0.1401,0.1191,0.0996,0.0834,0.0690,0.0566,0.0464,0.0375])
#认沽(put)期权收盘价
put_list = np.array([0.0540,0.0689,0.0866,0.1061,0.1294,0.1531,0.1814,0.2122,0.2447,0.2759,0.3162,0.3562,0.3899])
#期权执行价格
K_list = np.array([2.1,2.15,2.2,2.25,2.3,2.35,2.4,2.45,2.5,2.55,2.6,2.65,2.7])
shibor = 0.03297 #计算隐含波动率当日6个月期的shibor利率
sigma_clist = np.zeros_like(call_list)
#使用牛顿法计算隐含波动率
for i in np.arange(len(call_list)):
sigma_clist[i] = impvol_call_Newton(C=call_list[i], S=S0, K=K_list[i], r=shibor, T=T_delta,sigma0 = 0.2)
print('call的隐含波动率:',sigma_clist)
sigma_plist = np.zeros_like(put_list)
for i in np.arange(len(put_list)):
sigma_plist[i] = impvol_put_Newton(P=put_list[i], S=S0, K=K_list[i], r=shibor, T=T_delta,sigma0 = 0.2)
print('put的隐含波动率:',sigma_plist)
plt.figure(figsize=(8,6))
plt.plot(K_list,sigma_clist,label='50ETF认购期权')
plt.plot(K_list,sigma_plist,label='50ETF认沽期权')
plt.legend()
plt.grid('True')