153 lines
4.8 KiB
Python
153 lines
4.8 KiB
Python
# encoding: UTF-8
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'''
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Black-Scholes期权定价模型，主要用于标的物为股票的欧式期权的定价
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变量说明
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s：标的物股票价格
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k：行权价
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r：无风险利率
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t：剩余到期时间（年）
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v：隐含波动率
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cp：期权类型，+1/-1对应call/put
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price：期权价格
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出于开发演示的目的，本文件中的希腊值计算基于简单数值差分法，
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运算效率一般，实盘中建议使用更高速的算法。
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本文件中的希腊值计算结果没有采用传统的模型价格数值，而是采用
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了实盘交易中更为实用的百分比变动数值，具体定义如下
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delta：当f变动1%时，price的变动
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gamma：当f变动1%时，delta的变动
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theta：当t变动1天时，price的变动（国内交易日每年240天）
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vega：当v涨跌1个点时，price的变动（如从16%涨到17%）
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'''
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from __future__ import division
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from scipy import stats
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from math import (log, pow, sqrt, exp)
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cdf = stats.norm.cdf
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# 计算希腊值和隐含波动率时用的参数
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STEP_CHANGE = 0.001
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STEP_UP = 1 + STEP_CHANGE
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STEP_DOWN = 1 - STEP_CHANGE
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STEP_DIFF = STEP_CHANGE * 2
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DX_TARGET = 0.00001
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#----------------------------------------------------------------------
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def calculatePrice(s, k, r, t, v, cp):
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"""计算期权价格"""
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# 如果波动率为0，则直接返回期权空间价值
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if v <= 0:
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return max(0, cp * (s - k))
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d1 = (log(s / k) + (r + 0.5 * pow(v, 2)) * t) / (v * sqrt(t))
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d2 = d1 - v * sqrt(t)
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price = cp * (s * cdf(cp * d1) - k * cdf(cp * d2) * exp(-r * t))
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return price
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#----------------------------------------------------------------------
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def calculateDelta(s, k, r, t, v, cp):
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"""计算Delta值"""
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price1 = calculatePrice(s*STEP_UP, k, r, t, v, cp)
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price2 = calculatePrice(s*STEP_DOWN, k, r, t, v, cp)
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delta = (price1 - price2) / (s * STEP_DIFF) * (s * 0.01)
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return delta
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#----------------------------------------------------------------------
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def calculateGamma(s, k, r, t, v, cp):
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"""计算Gamma值"""
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delta1 = calculateDelta(s*STEP_UP, k, r, t, v, cp)
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delta2 = calculateDelta(s*STEP_DOWN, k, r, t, v, cp)
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gamma = (delta1 - delta2) / (s * STEP_DIFF) * pow(s, 2) * 0.0001
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return gamma
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#----------------------------------------------------------------------
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def calculateTheta(s, k, r, t, v, cp):
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"""计算Theta值"""
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price1 = calculatePrice(s, k, r, t*STEP_UP, v, cp)
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price2 = calculatePrice(s, k, r, t*STEP_DOWN, v, cp)
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theta = -(price1 - price2) / (t * STEP_DIFF * 240)
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return theta
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#----------------------------------------------------------------------
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def calculateVega(s, k, r, t, v, cp):
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"""计算Vega值"""
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vega = calculateOriginalVega(s, k, r, t, v, cp) / 100
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return vega
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#----------------------------------------------------------------------
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def calculateOriginalVega(s, k, r, t, v, cp):
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"""计算原始vega值"""
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price1 = calculatePrice(s, k, r, t, v*STEP_UP, cp)
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price2 = calculatePrice(s, k, r, t, v*STEP_DOWN, cp)
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vega = (price1 - price2) / (v * STEP_DIFF)
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return vega
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#----------------------------------------------------------------------
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def calculateGreeks(s, k, r, t, v, cp):
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"""计算期权的价格和希腊值"""
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price = calculatePrice(s, k, r, t, v, cp)
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delta = calculateDelta(s, k, r, t, v, cp)
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gamma = calculateGamma(s, k, r, t, v, cp)
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theta = calculateTheta(s, k, r, t, v, cp)
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vega = calculateVega(s, k, r, t, v, cp)
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return price, delta, gamma, theta, vega
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#----------------------------------------------------------------------
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def calculateImpv(price, s, k, r, t, cp):
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"""计算隐含波动率"""
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# 检查期权价格必须为正数
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if price <= 0:
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return 0
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# 检查期权价格是否满足最小价值（即到期行权价值）
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meet = False
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if cp == 1 and (price > (s - k) * exp(-r * t)):
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meet = True
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elif cp == -1 and (price > k * exp(-r * t) - s):
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meet = True
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# 若不满足最小价值，则直接返回0
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if not meet:
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return 0
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# 采用Newton Raphson方法计算隐含波动率
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v = 0.3 # 初始波动率猜测
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for i in range(50):
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# 计算当前猜测波动率对应的期权价格和vega值
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p = calculatePrice(s, k, r, t, v, cp)
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vega = calculateOriginalVega(s, k, r, t, v, cp)
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# 如果vega过小接近0，则直接返回
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if not vega:
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break
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# 计算误差
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dx = (price - p) / vega
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# 检查误差是否满足要求，若满足则跳出循环
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if abs(dx) < DX_TARGET:
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break
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# 计算新一轮猜测的波动率
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v += dx
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# 检查波动率计算结果非负
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if v <= 0:
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return 0
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# 保留4位小数
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v = round(v, 4)
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return v
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