?原文鏈接：http://tecdat.cn/?p=11803?

Nelson-Siegel- [Svensson]模型是擬合收益曲線(xiàn)的常用方法。它的優(yōu)點(diǎn)是其參數(shù)的經(jīng)濟(jì)可解釋性，被銀行廣泛使用。但它不一定在所有情況下都有效：模型參數(shù)有時(shí)非常不穩(wěn)定，無(wú)法收斂。

納爾遜（Nelson）和西格爾（Siegel）在其原始論文中從遠(yuǎn)期利率入手，然后推導(dǎo)了收益率至到期曲線(xiàn)的公式.

Nelson-Siegel模型是簡(jiǎn)約的，可以生成豐富的收益曲線(xiàn)。

但是，由于簡(jiǎn)單地使用它，它通常失去了經(jīng)濟(jì)上的可解釋性，甚至無(wú)法收斂。

上圖顯示了這種情況。

1. 


2. plot(MATURITY_BASES, oldYields

3. lines(MATURITY_BASES, oldYields)

4. points(newMats, newYields, col="blue")

5. lines(newMats, newYields, col="blue")

此代碼模仿了一個(gè)頻繁使用的案例，當(dāng)前的收益曲線(xiàn)與昨天的曲線(xiàn)進(jìn)行了比較。從某種意義上講，這是一個(gè)簡(jiǎn)單示例，因?yàn)閷?duì)于給定的到期日，我們已經(jīng)具有零收益率。實(shí)際上，我們通常與票息債券有關(guān)，這會(huì)使事情變得更加復(fù)雜。

您可能會(huì)認(rèn)為，由于軟件的實(shí)施而導(dǎo)致收斂失敗。我要講的不是不好的實(shí)現(xiàn)，而是要高度依賴(lài)所使用的數(shù)值方法，如下面的更實(shí)際的示例所示。

提供更逼真的建模

1. #include <ql/qldefines.hpp>

2. #ifdef BOOST_MSVC

3. #? include <ql/auto_link.hpp>

4. #endif

5. #include <ql/termstructures/yield/fittedbonddiscountcurve.hpp>

6. #include <ql/termstructures/yield/piecewiseyieldcurve.hpp>

7. #include <ql/termstructures/yield/flatforward.hpp>

8. #include <ql/termstructures/yield/bondhelpers.hpp>

9. #include <ql/termstructures/yield/nonlinearfittingmethods.hpp>

10. 


11. 


12. using namespace QuantLib;

13. 


14. int main(int, char*[]) {

15. ????try {

16. ????????Calendar calendar = NullCalendar();

17. ????????Date today = Date(18, December, 2017);

18. ????????Settings::instance().evaluationDate() = today;

19. 


20. ????????//市場(chǎng)數(shù)據(jù)

21. ????????double cleanPrices1[] = { 107.96, 135.88, 110.6,?? 133.46, 135.8,? 142.155, 121.045, 134.97, 117.04,

22. ????????????101.61, 128.67, 106.615, 106.36, 99.515, 101.21,? 105.655, 114.828 };

23. ????????double cleanPrices2[] = { 107.9,? 134.965, 110.37,? 132.89, 135.62,140.845, 120.585, 133.995, 116.745,

24. ????????????101.58, 128.115,105.985, 105.395,99.385, 100.79,104.955, 114.7985 };

25. ????????double cleanPrices3[] = { 107.96, 134.625, 110.58, 132.65, 135.145, 140.585, 120.385, 133.735, 116.635,

26. ????????????101.62, 127.925, 105.6, 105.085, 99.29, 100.6, 104.945, 114.7415 };

27. ????????double cleanPrices4[] = { 107.78, 134.39, 110.175, 132.445, 134.905, 139.515, 120.115, 133.475, 116.455,

28. ????????????101.58, 127.845, 105.53,104.805, 99.07, 100.46, 104.885, 114.6225 };

29. 


30. 


31. ????????std::vector<boost::shared_ptr<BondHelper> > bondHelpersA;

32. ????????std::vector< boost::shared_ptr<SimpleQuote> > quoteA;

33. ????????std::vector<boost::shared_ptr<BondHelper> > bondHelpersB;

34. 


35. ????????for (Size i = 0; i < numberOfBonds; i++) {

36. ????????????boost::shared_ptr<SimpleQuote> cp1(new SimpleQuote(cleanPrices1<em class="d4pbbc-italic"></em>));

37. ????????????quoteA.push_back(cp1);

38. ????????????boost::shared_ptr<SimpleQuote> cp2(new SimpleQuote(cleanPrices2<em class="d4pbbc-italic"></em>));

39. ????????????quoteB.push_back(cp2);

40. ????????????boost::shared_ptr<SimpleQuote> cp3(new SimpleQuote(cleanPrices3<em class="d4pbbc-italic"></em>));

41. ????????????quoteC.push_back(cp3);

42. ????????????boost::shared_ptr<SimpleQuote> cp4(new SimpleQuote(cleanPrices4<em class="d4pbbc-italic"></em>));

43. ????????????quoteD.push_back(cp4);

44. ????????}

45. 


46. ????????RelinkableHandle<Quote> quoteHandleA[numberOfBonds];

47. 


48. 


49. 


50. ????????//Nelson-Siegel模型擬合

51. ????????Real tolerance = 1.0e-14;

52. ????????Size max = 10000;

53. 


54. ????????boost::shared_ptr<FittedBondDiscountCurve> tsA(

55. ????????????new FittedBondDiscountCurve(curveSettlementDays,

56. ????????????????calendar,

57. ????????????????instrumentsA,

58. ????????????????ActualActual(),

59. ????????????????NelsonSiegelFitting(),

60. ????????????????tolerance,

61. ????????????????max));

62. 


63. 


64. ????????boost::shared_ptr<FittedBondDiscountCurve> tsB(

65. ????????????new FittedBondDiscountCurve(curveSettlementDays,

66. ????????????????calendar,

67. ????????????????instrumentsB,

68. ????????????????ActualActual(),

69. ????????????????NelsonSiegelFitting(),

70. ????????????????tolerance,

71. ????????????????max));

72. 


73. ????????boost::shared_ptr<FittedBondDiscountCurve> tsC(

74. ????????????new FittedBondDiscountCurve(curveSettlementDays,

75. ????????????????calendar,

76. ????????????????instrumentsC,

77. ????????????????ActualActual(),

78. ????????????????NelsonSiegelFitting(),

79. ????????????????tolerance,

80. ????????????????max));

81. 


82. ????????boost::shared_ptr<FittedBondDiscountCurve> tsD(

83. ????????????new FittedBondDiscountCurve(curveSettlementDays,

84. ????????????????calendar,

85. ????????????????instrumentsD,

86. ????????????????ActualActual(),

87. ????????????????NelsonSiegelFitting(),

88. ????????????????tolerance,

89. ????????????????max));

90. 


91. ????????std::cout << tsA->fitResults().numberOfIterations() << std::endl;

92. ????????std::cout << tsB->fitResults().numberOfIterations() << std::endl;

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正式而言，收益曲線(xiàn)每天的變化并不顯著，但是模型參數(shù)卻可以：

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Nelson-Siegel意識(shí)到了這些問(wèn)題，并提供了解決這些問(wèn)題的方法。特別是，他們考慮了Taus的時(shí)間序列，并確定了Taus的最佳擬合值的中值和合理范圍。
但是，與往常一樣，原始論文被引用的次數(shù)可能多于閱讀次數(shù)。此外，如果需要按時(shí)間順序排列的收益率數(shù)據(jù)，可能會(huì)感到困惑，而不是僅僅考慮相關(guān)日期的數(shù)據(jù)。即使處理時(shí)間序列不是問(wèn)題，Nelson和Siegel也沒(méi)有指定正式的算法來(lái)選擇的最佳值。

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