凸性

用 Python 进行债券定价与分析

Joshua Mayhew

Options Trader

预测价 vs. 实际价绘图

import numpy as np
import numpy_financial as npf
import pandas as pd
import matplotlib.pyplot as plt

price = -npf.pv(rate=0.05, nper=20, pmt=5, fv=100)
price_up = -npf.pv(rate=0.06, nper=20, pmt=5, fv=100)
price_down = -npf.pv(rate=0.04, nper=20, pmt=5, fv=100)
duration = (price_down - price_up) / (2 * price * 0.01)
dollar_duration = duration * price * 0.01
print("Bond Price (USD): ", price)
print("Dollar Duration (USD): ", dollar_duration)

Bond Price (USD): 100.00
Dollar Duration (USD): 12.53
用 Python 进行债券定价与分析

预测价 vs. 实际价绘图

bond_yields = np.arange(0, 10, 0.1)
bond = pd.DataFrame(bond_yields, columns=['bond_yield'])
bond['price'] = -npf.pv(rate=bond['bond_yield'] / 100, nper=20, pmt=5, fv=100)

    bond_yield       price
0          0.0  200.000000
1          0.1  196.978503
..         ...         ...
98         9.8   58.570780
99         9.9   57.997210

[100 rows x 2 columns]
用 Python 进行债券定价与分析

预测价 vs. 实际价绘图

bond['yield_change'] = bond['bond_yield'] - 5

    bond_yield       price  yield_change
0          0.0  200.000000        -5.0
1          0.1  196.978503        -4.9
2          0.2  194.013231        -4.8
..         ...         ...           ...
97         9.7   59.153044         4.7
98         9.8   58.570780         4.8
99         9.9   57.997210         4.9

[100 rows x 3 columns]
用 Python 进行债券定价与分析

预测价 vs. 实际价绘图

$ \text{Price Change} = -100 \times \text{Dollar Duration} \times \Delta y$

bond['price_change'] = -100 * dollar_duration * bond['yield_change'] / 100

    bond_yield       price  yield_change  price_change
0          0.0  200.000000          -5.0     62.650619
1          0.1  196.978503          -4.9     61.397607
..         ...         ...           ...           ...
98         9.8   58.570780           4.8    -60.144594
99         9.9   57.997210           4.9    -61.397607

[100 rows x 4 columns]
用 Python 进行债券定价与分析

预测价 vs. 实际价绘图

bond['predicted_price'] = price + bond['price_change']

   bond_yield       price  yield_change  price_change  predicted_price
0          0.0  200.000000          -5.0     62.650619       162.650619
1          0.1  196.978503          -4.9     61.397607       161.397607
2          0.2  194.013231          -4.8     60.144594       160.144594
..         ...         ...           ...           ...              ...
97         9.7   59.153044           4.7    -58.891582        41.108418
98         9.8   58.570780           4.8    -60.144594        39.855406
99         9.9   57.997210           4.9    -61.397607        38.602393

[100 rows x 5 columns]
用 Python 进行债券定价与分析

预测价 vs. 实际价绘图

plt.plot(bond['bond_yield'], bond['price'])
plt.plot(bond['bond_yield'], bond['predicted_price'])
plt.xlabel('Yield (%)')
plt.ylabel('Price (USD)')
plt.title("Actual Bond Prices vs. 
    Predicted Prices Using Duration")
plt.legend(["Actual Price", "Predicted Price"])
plt.show()

用 Python 进行债券定价与分析

久期的局限性

  • 久期是线性度量
  • 债券价格呈凸性
  • 久期仅对小幅收益率变动准确

用 Python 进行债券定价与分析

什么是凸性？

  • 衡量债券价格的曲率
  • 用于改进价格预测与风险衡量
  • 凸性越高=价格/收益率曲线越弯

用 Python 进行债券定价与分析

凸性公式

我们使用简化的凸性公式：

 

${\large Convexity = \frac{P_{down}\ +\ P_{up} \ - \ 2\times P}{P\ \times\ (\Delta y)^2}}$

 

  • $P_{down}$ = 收益率低1%时的债券价格
  • $P_{up}$ = 收益率高1%时的债券价格
  • $2 \times P$ = 当前收益率下价格的两倍
  • $(\Delta y)^2$ = 收益率变动的平方（此处用1%^2）
用 Python 进行债券定价与分析

凸性示例

10年期债券，年息5%，到期收益率4%，其凸性是多少？

${ Convexity = \frac{P_{down}\ +\ P_{up} \ - \ 2\times P}{P\ \times\ (\Delta y)^2}}$

price = -npf.pv(rate=0.04, nper=10, pmt=5, fv=100)

price_up = -npf.pv(rate=0.05, nper=10, pmt=5, fv=100)
price_down = -npf.pv(rate=0.03, nper=10, pmt=5, fv=100)

convexity = (price_down + price_up - 2 * price) / (price * 0.01 ** 2)
print("Convexity: ", convexity)

Convexity:  77.56
用 Python 进行债券定价与分析

总结

  • 久期是线性度量
  • 债券价格是曲线的
  • 久期仅对小幅收益率变动准确
  • 凸性衡量这种曲率

${ Convexity = \frac{P_{down}\ +\ P_{up} \ - \ 2\times P}{P\ \times\ (\Delta y)^2}}$

用 Python 进行债券定价与分析

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用 Python 进行债券定价与分析

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