Dollar convexity 與債券價格預測

使用 Python 進行債券評價與分析

Joshua Mayhew

Options Trader

Dollar convexity

  • Convexity＝殖利率變動 1% 時，存續期間的％變動

  • Dollar convexity＝殖利率變動 1% 時，存續期間的金額變動：

$ \text{Dollar Convexity} = \text{Convexity} \times \text{Bond Price} \times 0.01^2$

使用 Python 進行債券評價與分析

Dollar convexity 範例

  • 10 年期債券，票面利率 3%、殖利率 5%、面額 USD 100，其 dollar convexity 為何？
price = -npf.pv(rate=0.05, nper=10, pmt=3, fv=100)
price_up = -npf.pv(rate=0.06, nper=10, pmt=3, fv=100)
price_down = -npf.pv(rate=0.04, nper=10, pmt=3, fv=100)
convexity = (price_down + price_up - 2 * price) / (price * 0.01 ** 2)
dollar_convexity = convexity * price * 0.01 ** 2
print("Dollar Convexity: ", dollar_convexity)

Dollar Convexity:  0.69
使用 Python 進行債券評價與分析

Convexity 調整

  • 使用 convexity 可提升債券價格預測
  • Convexity 調整＝因 convexity 導致的價格變動量

$ \text{Convexity Adjustment} = 0.5 \times \text{Dollar Convexity} \times 100^2 \times (\Delta y)^2$

使用 Python 進行債券評價與分析

Convexity 調整範例

  • 10 年期債券，票面利率 3%、殖利率 5%、面額 USD 100
  • 其 convexity 調整為何？
price = -npf.pv(rate=0.05, nper=10, pmt=3, fv=100)
price_up = -npf.pv(rate=0.06, nper=10, pmt=3, fv=100)
price_down = -npf.pv(rate=0.04, nper=10, pmt=3, fv=100)
convexity = (price_down + price_up - 2 * price) / (price * 0.01 ** 2)

dollar_convexity = convexity * price * 0.01 ** 2

convexity_adjustment = 0.5 * dollar_convexity * 100 ** 2 * 0.01 ** 2
print("Convexity Adjustment: ", convexity_adjustment)

Convexity Adjustment:  0.35
使用 Python 進行債券評價與分析

結合存續期間與 convexity

  • 僅用存續期間預測價格變動：

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

  • 同時用存續期間與 convexity 預測價格變動：

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

$ = -100 \times \text{Dollar Duration} \times \Delta y \ + \ 0.5 \times \text{Dollar Convexity} \times 100^2 \times (\Delta y)^2$

  • 結合存續期間與 convexity 可提升估價準確度
使用 Python 進行債券評價與分析

存續期間與 convexity 範例

  • 10 年期債券，票面利率 3%、殖利率 5%、面額 USD 100：
price = -npf.pv(rate=0.05, nper=10, pmt=3, fv=100)
price_up = -npf.pv(rate=0.06, nper=10, pmt=3, fv=100)
price_down = -npf.pv(rate=0.04, nper=10, pmt=3, fv=100)

duration = (price_down - price_up) / (2 * price * 0.01)
dollar_duration = duration * price * 0.01

convexity = (price_down + price_up - 2 * price) / (price * 0.01 ** 2)
dollar_convexity = convexity * price * 0.01
convexity_adjustment = dollar_convexity * 100 ** 2 * 0.01 ** 2

combined_prediction = -100 * dollar_duration * 0.01 + convexity_adjustment
print("Predicted Price Change: ", combined_prediction)

Predicted Price Change:  -6.64
使用 Python 進行債券評價與分析

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使用 Python 進行債券評價與分析

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