Question

我是一名计算机科学家，试图更多地了解量化金融。我有一个计算Black-Scholes模型中欧式看涨期权价值的程序，我正在尝试添加一种计算隐含波动率的方法。

import math
import numpy as np
import pdb
from scipy.stats import norm

class BlackScholes(object):
  '''Class wrapper for methods.'''

  def __init__(self, s, k, t, r, sigma):
    '''Initialize a model with the given parameters.
       @param s: initial stock price
       @param k: strike price
       @param t: time to maturity (in years)
       @param r: Constant, riskless short rate (1 equals 100%)
       @param sigma: Guess for volatility. (1 equals 100%)
    '''
    self.s = s
    self.k = k
    self.t = t
    self.r = r
    self.sigma = sigma
    self.d = self.factors()

  def euro_call(self):
    ''' Calculate the value of a European call option
        using Black-Scholes. No dividends.
        @return: The value for an option with the given parameters.'''
    return norm.cdf(self.d[0]) * self.s - (norm.cdf(self.d[1]) * self.k *
                                     np.exp(-self.r * self.t))

  def factors(self):
    '''
      Calculates the d1 and d2 factors used in a large
      number of Black Scholes equations.
    '''
    d1 = (1.0 / (self.sigma * np.sqrt(self.t)) * (math.log(self.s / self.k)
                    + (self.r + self.sigma ** 2 / 2) * self.t))
    d2 = (1.0 / (self.sigma * np.sqrt(self.t)) * (math.log(self.s / self.k)
                    + (self.r - self.sigma ** 2 / 2) * self.t))
    if math.isnan(d1):
      pdb.set_trace()
    assert(not math.isnan(d1))
    assert(not math.isnan(d2))
    return (d1, d2) 

  def imp_vol(self, C0):
    ''' Calculate the implied volatility of a call option,
        where sigma is interpretered as a best guess.
        Updates sigma as a side effect.
        @rtype: float
        @return: Implied volatility.'''
    for i in range(128):
      self.sigma -= (self.euro_call() - C0) / self.vega()
      assert(self.sigma != -float("inf"))
      assert(self.sigma != float("inf"))
      self.d = self.factors()
    print(C0,
      BlackScholes(self.s, self.k, self.t, self.r, self.sigma).euro_call())
    return self.sigma

  def vega(self):
    ''' Returns vega, which is the derivative of the
        option value with respect to the asset's volatility.
        It is the same for both calls and puts.
        @rtype: float
        @return: vega'''
    v = self.s * norm.pdf(self.d[0]) * np.sqrt(self.t)
    assert(not math.isnan(v))
    return v

以下是我目前的两个测试用例：

print(BlackScholes(17.6639, 1.0, 1.0, .01, 2.0).imp_vol(16.85))
print(BlackScholes(17.6639, 1.0, .049, .01, 2.0).imp_vol(16.85))

最高的一个打印出1.94，这相当接近http://www.option-price.com/implied-volatility.php给出的195.21％的值。然而，底部的一个（如果你删除断言语句）打印出＆＃39; nan＆＃39;和以下警告信息。使用断言语句，self.vega()在imp_vol方法中返回零，然后在assert(self.sigma != -float("inf"))中返回。

so.py:51: RuntimeWarning: divide by zero encountered in double_scalars
  self.sigma -= (self.euro_call() - C0) / self.vega()
so.py:37: RuntimeWarning: invalid value encountered in double_scalars
  + (self.r + self.sigma ** 2 / 2) * self.t))
so.py:39: RuntimeWarning: invalid value encountered in double_scalars
  + (self.r - self.sigma ** 2 / 2) * self.t))

Answer 1

您在做什么并没有多大意义。您正试图撤消大量短期期权的隐含波动。此选项上的vega有效为0，因此您得到的隐含卷号将毫无意义。浮点舍入为您提供了无限的音量，这一点也不足为奇。

Answer 2

如果你使用 vega 来估计隐含波动率，你可能正在做一些牛顿梯度搜索的变体，它不会在所有情况下收敛到一个解决方案，我用 R 或 VBA 编程，所以只能提供一个解决方案你来翻译一下，二分搜索方法简单稳健并且总是收敛，从写完整的期权定价模型书的人这里是 Espen Haugs 算法，用于二分搜索以找到隐含波动率；

Newton-Raphson 方法需要部分知识关于波动率的期权定价公式的衍生 (vega) 在搜索隐含波动率时。对于某些选项（特别是异国情调和美式期权），vega 不为人所知裂解性地。二分法是一种更简单的估计方法 vega 未知时的隐含波动率。二分法需要两个初始波动率估计值（种子值）：

隐含波动率的“低”估计值 al，对应于一个选项值，CL
“高”波动率估计值 aH，对应于期权值，CH 期权市场价格 Cm 介于 CL 和 cH 之间。平分- 重估估计作为两者之间的线性插值给出估计：如果 c(cr, + ) < c m ，则将 al 替换为 0 i+1 ，否则将 aH 替换为 i+ 1 如果 C(7,+) > cm 直到 lcm — c(cri+i)i < E，此时 cr, ±1 是隐含的波动率和 E 是所需的准确度。计算机算法此函数返回欧洲平原的隐含波动率香草看涨期权或看跌期权。通过小的修改，该功能可以也可用于寻找美国和异国情调的隐含波动率选项。变量计数器跟踪有多少循环已经完成。如果在 100 次循环内未找到指定准确度 E 的隐含波动率，则算法停止并返回“NA”（不是可用）。

      Function  GBlackScholesImpVolBisection(CallPutFlag 
       As String, S As Double,
       X As Double, T As Double, r As Double, _
       b As Double, cm As Double) As Variant
       Dim vLow As Double, vHigh As Double, vi As Double
       Dim cLow As Double, cHigh As Double, epsilon As 
   Double
   Dim counter As Integer
   vLow = 0.005
   vHigh = 4
   epsilon = le-08
   cLow = GBlackScholes ( CallPutFlag , S, X, T, r, b, vLow)
   cHigh = GBlackScholes ( CallPutFlag , S, X, T, r, b, vHigh)
   counter = 0
   vi = vLow + (cm — cLow ) * (vHigh — vLow) / ( cHigh — cLow)
   While Abs(cm — GBlackScholes ( CallPutFlag , S, X, T, r, b, vi )) > epsilon
   counter = counter + 1
   If counter = 100 Then
   GBlackScholesImpVolBisection
   Exit Function
   End If
   If GBlackScholes ( CallPutFlag , S, X, T, r, b, vi ) < cm Then
   vLow = vi
   Else
   vHigh = vi
   End If
   cLow = GBlackScholes ( CallPutFlag , S, X, T, r, b, vLow)
   cHigh = GBlackScholes ( CallPutFlag , S, X, T, r, b, vHigh )
   vi = vLow + (cm — cLow ) * (vHigh — vLow) / ( cHigh — cLow)
   Wend
   GBlackScholesImpVolBisection = vi
   End Function```

隐含波动率计算器是错误的

2 个答案: