金融分析与风险管理——期权中的希腊字母

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#BSM模型 #希腊字母 #Python #delta #gamma

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本文详细介绍了金融分析中的期权希腊字母，包括Delta（期权价格与标的物价格的关系）、Gamma（Delta对标的物价格的敏感性）、Theta（期权价值随时间流逝的变化率）、Vega（期权对标的物波动率的敏感性）和Rho（期权对无风险利率的敏感性）。通过BSM模型，解析了各种希腊字母的数学表达式，并用Python实现了相关函数计算欧式期权的这些参数。

1. 期权的Delta

期权的Delta：期权价格变动与标的物价格变动的比率，即期权价格与标的物价格之间关系曲线的切线斜率，公式如下：

$\Delta = \frac{\partial \prod}{\partial S}$

$\prod$ 表示期权价格，S 表示标的物价格。

Delta = 0.6 表示：当标的物价格变化一个很小的金额时，相应的期权价格变化约为标的物价格变化的60%。

利用BSM模型的期权定价公式，欧式看涨、看跌的分布如下，具体的过程可以参照:金融分析与风险管理——期权BSM模型。

欧式看涨期权的定价公式：

$c = SN(d_1) - Ke^{-r(T-t)}N(d_2)$

欧式看跌期权的定价公式：

$p = Ke^{-r(T-t)}N(-d_2) - SN(-d_1)$

其中, $T$ 是期权的有效期限， $t$ 是流失的时间。

$\int\limits \frac{1}{\sqrt{2 \pi}}e^{-x^2/2}\mathrm{d}x$

$d_1 = \frac{ln(\frac{S}{K}) + (r+\sigma^2/2)(T-t)}{\sigma \sqrt{T-t}}$

$d_2 = \frac{ln(\frac{S}{K}) + (r-\sigma^2/2)(T-t)}{\sigma \sqrt{T-t}} = d_1 - \sigma \sqrt{T-t}$

接下来以欧式看涨期权的多头为例对其进行推导：

$N^{'}(x) = \frac{1}{\sqrt{2 \pi}}e^{-x^2/2}$

$d_2 = \frac{ln(\frac{S}{K}) + (r-\sigma^2/2)(T-t)}{\sigma \sqrt{T-t}} ==> d_2\sigma \sqrt{T-t} + \sigma^2(T-t)/2 = ln(\frac{S}{K}*e^{r(T-t)})$

$N^{'}(d_1) = N^{'}(d_2 + \sigma \sqrt{T-t}) = \\[10pt] \frac{1}{\sqrt{2 \pi}}e^{-(d_2 + \sigma \sqrt{T-t})^2/2} = \\[10pt] \frac{1}{\sqrt{2 \pi}}e^{-d_2^2/2 - d_2 \sigma \sqrt{T-t}-\sigma^2(T-t)/2} = \\[10pt] \frac{1}{\sqrt{2 \pi}}e^{-d_2^2/2}* e^{- d_2 \sigma \sqrt{T-t}-\sigma^2(T-t)/2} = \\[10pt] N^{'}(d_2)e^{- d_2 \sigma \sqrt{T-t}-\sigma^2(T-t)/2} = \\[10pt] N^{'}(d_2)*\frac{K}{S}e^{-r(T-t)} ==> SN^{'}(d_1) = Ke^{-r(T-t)}N^{'}(d_2) ---->(1)$

$\frac{\partial d_1}{\partial S} = \frac{\partial \frac{ln(\frac{S}{K}) + (r+\sigma^2/2)(T-t)}{\sigma \sqrt{T-t}}}{\partial S} = \frac{1}{S \sigma \sqrt{T-t}}$

$\frac{\partial d_2}{\partial S} = \frac{\partial \frac{ln(\frac{S}{K}) + (r-\sigma^2/2)(T-t)}{\sigma \sqrt{T-t}}}{\partial S} = \frac{1}{S \sigma \sqrt{T-t}} ==>\frac{\partial d_1}{\partial S} = \frac{\partial d_2}{\partial S} ---->(2)$

$\Delta = \frac{\partial c}{\partial S} = \frac{\partial (SN(d_1) - Ke^{-r(T-t)}N(d_2))}{\partial S} = \\[10pt] N(d_1) + SN^{'}(d_1)\frac{\partial d_1}{\partial S} - Ke^{-r(T-t)}N^{'}(d_2)\frac{\partial d_2}{\partial S} \\[10pt] 由（1）与（2）可推出==>\Delta = \frac{\partial c}{\partial S} = N(d_1)$

其Delta的具体表达式如下：

期权类型	方向	Delta表达式
欧式看涨期权	多头	$\Delta = N(d_1)$
欧式看涨期权	空头	$\Delta = -N(d_1)$
欧式看跌期权	多头	$\Delta = N(d_1) - 1$
欧式看跌期权	空头	$\Delta = 1 - N(d_1)$

利用Python构建欧式看涨、看跌期权 Delta 的函数：

#期权的Delta
def delta_option(S,K,sigma,r,T,optype,positype):
    '''
    T:期权的剩余期限
    optype:表示期权类型，call看涨、put看跌
    positype:表示期权头寸，long多头、short空头
    '''
    import numpy as np
    from scipy.stats import norm
    d1 = (np.log(S/K)+(r+sigma**2/2)*T)/(sigma*np.sqrt(T))
    if optype == 'call':
        if positype == 'long':
            delta = norm.cdf(d1)
        else:
            delta = -norm.cdf(d1)
    else:
        if positype == 'long':
            delta = norm.cdf(d1) - 1
        else:
            delta = 1 - norm.cdf(d1)
    return delta

#标的价格与delta的关系
S_list = np.linspace(4,8,100)
Delta_call = delta_option(S = S_list,K=6,sigma=0.24,r=0.04,T=0.5,optype='call',positype='long')
Delta_put = delta_option(S = S_list,K=6,sigma=0.24,r=0.04,T=0.5,optype='put',positype='long')

plt.figure(figsize=(12,6))
plt.subplot(1,2,1)
plt.plot(S_list,Delta_call,label='看涨期权多头')
plt.plot(S_list,Delta_put,label='看跌期权多头')
plt.legend()
plt.grid('True')

#期权期限与delta的关系
T_list = np.linspace(0.1,5,100)
Delta_call1 = delta_option(S = 7,K=6,sigma=0.24,r=0.04,T=T_list,optype='call',positype='long')
Delta_call2 = delta_option(S = 6,K=6,sigma=0.24,r=0.04,T=T_list,optype='call',positype='long')
Delta_call3 = delta_option(S = 5,K=6,sigma=0.24,r=0.04,T=T_list,optype='call',positype='long')

plt.subplot(1,2,2)
plt.plot(T_list,Delta_call1,label='实值看涨期权多头')
plt.plot(T_list,Delta_call2,label='平价看涨期权多头')
plt.plot(T_list,Delta_call3,label='虚值看涨期权多头')
plt.legend()
plt.grid('True')

在这里插入图片描述

2. 期权的Gamma

期权的Gamma：期权Delta变动与标的物价格变动的比率，即Gamma是期权价格关于标的物价格的二阶偏导，公式如下：

$\Gamma = \frac{\partial^2 \prod}{\partial S^2}$

欧式看涨期权的表达式推导如下：

$\Gamma = \frac{\partial^2 c}{\partial S^2} = N^{'}(d_1) \frac{\partial d_1}{\partial S} = \frac{N^{'}(d_1)}{S \sigma \sqrt{T-t}}$

欧式看跌期权的表达式推导如下：

$\Gamma = \frac{\partial^2 p}{\partial S^2} = \frac{\partial (N(d_1) - 1)}{\partial S} = N^{'}(d_1) \frac{\partial d_1}{\partial S} = \frac{N^{'}(d_1)}{S \sigma \sqrt{T-t}}$

利用Python构建欧式看涨、看跌期权Gamma的函数：

#期权的Gamma
def gamma_option(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))
    
    return np.exp(-d1**2/2)/(S*sigma*np.sqrt(2*np.pi*T))
    
#标的价格与Gamma的关系
S_list = np.linspace(4,8,100)
gamma_list = gamma_option(S=S_list, K=6, sigma=0.24, r=0.04, T=0.5) 


plt.figure(figsize=(12,6))
plt.subplot(1,2,1)
plt.plot(S_list,gamma_list)
plt.grid('True')

#期限与Gamma的关系
T_list = np.linspace(0.1,5,100)
gamma1 = gamma_option(S=7, K=6, sigma=0.24, r=0.04, T=T_list)
gamma2 = gamma_option(S=6, K=6, sigma=0.24, r=0.04, T=T_list)
gamma3 = gamma_option(S=5, K=6, sigma=0.24, r=0.04, T=T_list)

plt.subplot(1,2,2)
plt.plot(T_list,gamma1,label='实值看涨期权')
plt.plot(T_list,gamma2,label='平价看涨期权')
plt.plot(T_list,gamma3,label='虚值看涨期权')
plt.legend()
plt.grid('True')

在这里插入图片描述

3. 期权的Theta

期权的Theta：期权价格变化与时间变化(期权剩余有效期)的比率，公式如下：

$\Theta = \frac{\partial \prod}{\partial T}$

欧式看涨期权的表达式推导如下：

$d_1 - d_2 = \sigma \sqrt{T - t} ===> (d_1 - d_2)^{'} = \frac {\partial \sigma \sqrt{T - t}}{\partial t} = - \frac{\sigma}{2 \sqrt {T-t}} ----->(3) \\[10pt] \frac{\partial c}{\partial t} = \frac{\partial (SN(d_1) - Ke^{-r(T-t)}N(d_2))}{\partial t} = \\[10pt] SN^{'}(d_1)\frac{\partial d_1}{\partial t} - Ke^{-r(T-t)}N^{'}(d_2)\frac{\partial d_2}{\partial t} - rKe^{-r(T-t)}N(d_2) \\[10pt] 由（1）与（3）可推出==> \\[10pt] \Theta_{call} = \frac{\partial c}{\partial t} = SN^{'}(d_1)(\frac{\partial d_1}{\partial t} - \frac{\partial d_2}{\partial t}) - rKe^{-r(T-t)}N(d_2) = \\[10pt] -\frac{SN^{'}(d_1) \sigma}{2 \sqrt {T-t}} - rKe^{-r(T-t)}N(d_2) = \\[10pt] -\frac{S \sigma e^{-d_1^2/2}}{2 \sqrt {2 \pi (T-t)}} - rKe^{-r(T-t)}N(d_2)$
欧式看跌期权的表达式推导如下：
$\Theta_{put} = \frac{\partial p}{\partial t} = \frac{\partial (Ke^{-r(T-t)}N(-d_2) - SN(-d_1))}{\partial t} = \\[10pt] \frac{\partial [Ke^{-r(T-t)}(1-N(d_2)) - S(1-N(d_1))]}{\partial t} = \\[10pt] rKe^{-r(T-t)}(1-N(d_2)) - Ke^{-r(T-t)}N^{'}(d_2)\frac{\partial d_2}{\partial t} + SN^{'}(d_1)\frac{\partial d_1}{\partial t} = \\[10pt] SN^{'}(d_1)(\frac{\partial d_1}{\partial t} - \frac{\partial d_2}{\partial t}) + rKe^{-r(T-t)}(1-N(d_2)) = \\[10pt] -\frac{SN^{'}(d_1) \sigma}{2 \sqrt {T-t}} + rKe^{-r(T-t)}N(-d_2) = \\[10pt] \Theta_{call} + rKe^{-r(T-t)}$

在日常的使用中，时间是以天为单位，但是在 BSM 模型中，时间则是以年为单位的，使用该模型计算时，需要把天转化成年，其中“每日历天”是按365天计算的；“每交易日”是按252天计算的。

利用Python构建欧式看涨、看跌期权Theta的函数：

#期权的Theta
def theta_option(S,K,sigma,r,T,optype):
    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)
    theta_call = -(S*sigma*np.exp(-d1**2/2))/(2*np.sqrt(2*np.pi*T)) - r*K*np.exp(-r*T)*norm.cdf(d2)
    
    if optype == 'call':
        theta = theta_call
    else:
        theta = theta_call + r*K*np.exp(-r*T)
    
    return theta

#标的价格与Theta的关系
S_list = np.linspace(1,11,100)
theta_list1 = theta_option(S=S_list,K=6, sigma=0.24, r=0.04, T=0.5,optype='call') 
theta_list2 = theta_option(S=S_list,K=6, sigma=0.24, r=0.04, T=0.5,optype='put') 


plt.figure(figsize=(12,6))
plt.subplot(1,2,1)
plt.plot(S_list,theta_list1,label='看涨期权')
plt.plot(S_list,theta_list2,label='看跌期权')
plt.legend()
plt.grid('True')

#期限与Theta的关系
T_list = np.linspace(0.1,5,100)
theta1 = theta_option(S=7, K=6, sigma=0.24, r=0.04, T=T_list,optype='call') 
theta2 = theta_option(S=6, K=6, sigma=0.24, r=0.04, T=T_list,optype='call') 
theta3 = theta_option(S=5, K=6, sigma=0.24, r=0.04, T=T_list,optype='call') 


plt.subplot(1,2,2)
plt.plot(T_list,theta1,label='实值看涨期权')
plt.plot(T_list,theta2,label='平价看涨期权')
plt.plot(T_list,theta3,label='虚值看涨期权')
plt.legend()
plt.grid('True')

在这里插入图片描述

4. 期权的Vega

期权的Vega：期权价格变化与标的波动率变化的比率，公式如下：

$\frac{\partial \prod}{\partial \sigma}$

欧式看涨的表达式推导如下：

$d_1 - d_2 = \sigma \sqrt{T - t} ===> (d_1 - d_2)^{'} = \frac {\partial \sigma \sqrt{T - t}}{\partial \sigma} = \sqrt {T-t} ---->(4) \\[10pt] \frac{\partial c}{\partial \sigma} = \frac{\partial (SN(d_1) - Ke^{-r(T-t)}N(d_2))}{\partial \sigma} = \\[10pt] SN^{'}(d_1)\frac{\partial d_1}{\partial \sigma} - Ke^{-r(T-t)}N^{'}(d_2)\frac{\partial d_2}{\partial \sigma} \\[10pt] 由（1）与（4）可推出==>V = \frac{\partial c}{\partial \sigma} = S\sqrt{T-t}N^{'}(d_1) = \frac{S\sqrt{T-t}e^{-d_1^2/2}}{\sqrt{2\pi}}$

欧式看跌的表达式推导如下：

$\frac{\partial p}{\partial \sigma} = \frac{\partial (Ke^{-r(T-t)}N(-d_2) - SN(-d_1))}{\partial \sigma} = \\[10pt] \frac{\partial [Ke^{-r(T-t)}(1-N(d_2)) - S(1-N(d_1))]}{\partial \sigma} = \\[10pt] -Ke^{-r(T-t)}N^{'}(d_2)\frac{\partial d_2}{\partial \sigma} + SN^{'}(d_1)\frac{\partial d_1}{\partial \sigma} = \\[10pt] \frac{\partial c}{\partial \sigma}$

利用Python构建欧式看涨Vega的函数：

#期权的Vega
def vega_option(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))
  
    return S*np.sqrt(T)*np.exp(-d1**2/2)/np.sqrt(2*np.pi)

vega = vega_option(S=5.8, K=6, sigma=0.24, r=0.04, T=0.5)

#标的价格与Vega
S_list = np.linspace(3,10,100)
vega_list = vega_option(S=S_list, K=6, sigma=0.24, r=0.04, T=0.5)

plt.figure(figsize=(12,6))
plt.subplot(1,2,1)
plt.plot(S_list,vega_list)
plt.grid('True')

#期限与Vega
T_list = np.linspace(0.1,5,100)
vega1 = vega_option(S=8, K=6, sigma=0.24, r=0.04, T=T_list)
vega2 = vega_option(S=6, K=6, sigma=0.24, r=0.04, T=T_list)
vega3 = vega_option(S=4, K=6, sigma=0.24, r=0.04, T=T_list)

plt.subplot(1,2,2)
plt.plot(T_list,vega1,label='实值看涨期权')
plt.plot(T_list,vega2,label='平价看涨期权')
plt.plot(T_list,vega3,label='虚值看涨期权')
plt.legend()
plt.grid('True')

在这里插入图片描述

5. 期权的Rho

期权的Rho：期权价格变化与无风险收益率变化的比率，公式如下：

$\frac{\partial \prod}{\partial r}$

欧式看涨的表达式推导如下：

$d_1 - d_2 = \sigma \sqrt{T - t} ===> (d_1 - d_2)^{'} = \frac {\partial \sigma \sqrt{T - t}}{\partial r} = 0 ---->(5) \\[10pt] \frac{\partial c}{\partial r} = \frac{\partial (SN(d_1) - Ke^{-r(T-t)}N(d_2))}{\partial r} = \\[10pt] SN^{'}(d_1)\frac{\partial d_1}{\partial r} + (T-t)Ke^{-r(T-t)}N(d_2) - Ke^{-r(T-t)}N^{'}(d_2)\frac{\partial d_2}{\partial r} = \\[10pt] SN^{'}(d_1)(\frac{\partial d_1}{\partial t} - \frac{\partial d_2}{\partial t}) + (T-t)Ke^{-r(T-t)}N(d_2) \\[10pt] 由（1）与（5）可推出==> Rho = (T-t)Ke^{-r(T-t)}N(d_2)>=0$

欧式看跌期权的表达式推导如下：

$\frac{\partial p}{\partial r} = \frac{\partial (Ke^{-r(T-t)}N(-d_2) - SN(-d_1))}{\partial r} = \\[10pt] \frac{\partial [Ke^{-r(T-t)}(1-N(d_2)) - S(1-N(d_1))]}{\partial r} = \\[10pt] -(T-t)Ke^{-r(T-t)}(1-N(d_2)) - Ke^{-r(T-t)}(N^{'}(d_2)\frac{\partial d_2}{\partial r} + SN^{'}(d_1)\frac{\partial d_1}{\partial r} = \\[10pt] SN^{'}(d_1)(\frac{\partial d_1}{\partial t} - \frac{\partial d_2}{\partial t}) - (T-t)Ke^{-r(T-t)}(1-N(d_2)) =\\[10pt] -(T-t)Ke^{-r(T-t)}(1-N(d_2)) = -(T-t)Ke^{-r(T-t)}N(-d_2) <= 0$

利用Python构建欧式看涨期权 Rho 的函数：

#期权的Rho
def rho_option(S,K,sigma,r,T,optype):
    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)
    
    if optype == 'call':
        rho = K*T*np.exp(-r*T)*norm.cdf(d2)
    else:
        rho = -K*T*np.exp(-r*T)*norm.cdf(-d2)
    
    return rho

#标的价格与rho
S_list = np.linspace(3,10,100)
rho_list1 = rho_option(S=S_list, K=6, sigma=0.24, r=0.04, T=0.5,optype='call')
rho_list2 = rho_option(S=S_list, K=6, sigma=0.24, r=0.04, T=0.5,optype='put')

plt.figure(figsize=(12,6))
plt.subplot(1,2,1)
plt.plot(S_list,rho_list1,label='看涨期权')
plt.plot(S_list,rho_list2,label='看跌期权')
plt.legend()
plt.grid('True')

#期限与rho
T_list = np.linspace(0.1,5,100)
rho1 = rho_option(S=8, K=6, sigma=0.24, r=0.04, T=T_list,optype='call')
rho2 = rho_option(S=6, K=6, sigma=0.24, r=0.04, T=T_list,optype='call')
rho3 = rho_option(S=4, K=6, sigma=0.24, r=0.04, T=T_list,optype='call')

plt.subplot(1,2,2)
plt.plot(T_list,rho1,label='实值看涨期权')
plt.plot(T_list,rho2,label='平价看涨期权')
plt.plot(T_list,rho3,label='虚值看涨期权')
plt.legend()
plt.grid('True')