Quantlib InterpolatedDiscountCurve 端點處的零遠期利率

我使用 QuantLib Python 根據插值折扣因子 ( https://github.com/lballabio/QuantLib-SWIG/blob/master/SWIG/discountcurve.i ) 校準曲線。對貼現因子使用 LogLinear 插值會產生良好的遠期利率：

（黑線是零利率，虛線是一維遠期曲線）

但是，使用 MonotonicLogCubic 或 SplineCubic，我的遠期匯率表現不佳 - 在終點，遠期為 0：

現在我明白了，使用這些插值方法，端點的二階導數設置為零，但這並不一定意味著前向在端點中為零。是什麼賦予了？如何克服這個問題？

這些是我的輸入 DF：

February 27th, 2020: 1.0000102501050636
February 28th, 2020: 1.0000205003151919
March 10th, 2020: 1.000135036504439
March 17th, 2020: 1.0002140481753308
April 1st, 2020: 1.0003903064472135
April 30th, 2020: 1.0007383708059454
June 1st, 2020: 1.0011864659905814
July 1st, 2020: 1.001606452340883
July 30th, 2020: 1.0020206702099752
September 1st, 2020: 1.0025224673306345
September 30th, 2020: 1.003007005031276
October 30th, 2020: 1.0034783631568704
December 2nd, 2020: 1.0040034616057292
December 30th, 2020: 1.0045093968882473
February 1st, 2021: 1.0050604130179728
March 3rd, 2021: 1.0055474482943993
September 1st, 2021: 1.0087481016987652
March 2nd, 2022: 1.01191304422858
August 31st, 2022: 1.0148211625629888
March 2nd, 2023: 1.0174479377825414
March 1st, 2024: 1.0215623647797287
March 4th, 2025: 1.023909696966405
March 4th, 2026: 1.0242610376714252
March 3rd, 2027: 1.02262864674708
March 1st, 2028: 1.0191710493588495
March 2nd, 2029: 1.013740853380904
March 4th, 2030: 1.0061949692591952
March 4th, 2031: 0.998108130543049
March 3rd, 2032: 0.9884922702699311
March 2nd, 2035: 0.961448481843742
March 1st, 2040: 0.9138611360550781
March 2nd, 2045: 0.8771539172898629
March 2nd, 2050: 0.8489994312646763

我這樣構造實例：

ql.NaturalCubicDiscountCurve（日期，折扣因子，ql.Actual365Fixed（），ql.UnitedStates（））

其中日期和折扣因子是基於上述的列表。

為了得到遠期，我在貼現曲線上呼叫 forwardRate(date, date+1, ql.Actual360(), ql.Simple()).rate()

我似乎無法複製您的問題…

import pandas as pd
import QuantLib as ql
import matplotlib.pyplot as plt

dates =  [
   '2020-02-26',  '2020-02-27',  '2020-02-28',  '2020-03-10',  '2020-03-17',  '2020-04-01',  '2020-04-30',  '2020-06-01',  '2020-07-01',  '2020-07-30',  '2020-09-01',
   '2020-09-30',  '2020-10-30',  '2020-12-01',  '2020-12-30',  '2021-02-01',  '2021-03-01',  '2021-09-01',  '2022-03-01',  '2022-08-31',  '2023-03-01',  '2024-03-01',
   '2025-03-01',  '2026-03-01',  '2027-03-01',  '2028-03-01',  '2029-03-01',  '2030-03-01',  '2031-03-01',  '2032-03-01',  '2035-03-01',  '2040-03-01',  '2045-03-01',
   '2050-03-01']

dfs = [1,  1.0000102501050636,  1.000020500315192,  1.000135036504439,  1.0002140481753308,  1.0003903064472135,  1.0007383708059454,  1.0011864659905814,  1.001606452340883,
1.0020206702099752,  1.0025224673306343,  1.0030070050312758,  1.0034783631568704,  1.0040034616057292,  1.0045093968882473,  1.0050604130179728,  1.0055474482943991,
1.0087481016987652, 1.01191304422858, 1.0148211625629888, 1.0174479377825414, 1.0215623647797287, 1.023909696966405, 1.0242610376714252, 1.02262864674708, 1.0191710493588495,
1.013740853380904, 1.0061949692591952, 0.9981081305430491, 0.9884922702699313, 0.9614484818437421, 0.913861136055078, 0.8771539172898629, 0.8489994312646763]

qlDates = [ql.Date(dt, '%Y-%m-%d') for dt in dates]

params = [qlDates, dfs, ql.Actual365Fixed(),ql.UnitedStates()]
curves = {
   'DiscountCurve': ql.DiscountCurve(*params),
   'NaturalCubicDiscountCurve': ql.NaturalCubicDiscountCurve(*params),
   'MonotonicLogCubicDiscountCurve': ql.MonotonicLogCubicDiscountCurve(*params)   
}
plt.figure(figsize=(10,5))
for key in curves:
   crv = curves[key]
   crv.enableExtrapolation()
   times = crv.times()
   zeros = [crv.zeroRate(date, ql.Actual365Fixed(), ql.Continuous).rate() for date in crv.dates()]
   plt.plot(times, zeros, label=f"Spot {key}")
   fwds = [crv.forwardRate(date, date + ql.Period('1d'), ql.Actual360(), ql.Simple).rate() for date in crv.dates()]
   plt.plot(times, fwds, label=f"Fwd {key}")

plt.legend();

這將導致：

引用自：https://quant.stackexchange.com/questions/51366