QuantLib 金融计算——收益率曲线之构建曲线（4）

如果未做特别说明，文中的程序都是 C++11 代码。

QuantLib 金融计算——收益率曲线之构建曲线（4）

本文代码对应的 QuantLib 版本是 1.15。相关源代码可以在 QuantLibEx 找到。

概述

QuantLib 中提供了用三次 B 样条函数拟合期限结构的功能，但是，并未提供使用三次样条函数拟合期限结构的功能。本文展示了如何在 QuantLib 的框架内实现三次样条函数，并拟合期限结构。

示例所用的样本券交易数据来自专门进行期限结构分析的 R 包——termstrc。具体来说是数据集 govbonds 中的 GERMANY 部分，包含 2008-01-30 这一天德国市场上 52 只固息债的成交数据。

注意：为了适配 QuantLib，实际计算中删除了两只债券的数据，以保证所有样本券的到期时间均不相同。样本券数据在《收益率曲线之构建曲线（3）》的附录中列出。

三次样条函数与期限结构

用三次样条函数拟合期限结构，实质上是用若干三次样条函数的组合近似贴现因子曲线的形状，

\[ d(t,\beta) = 1 + \sum_{l=1}^n \beta_l c_l(t) \]

贴现因子 \(d(t,\beta)\) 表示为三次样条函数的线性组合，\(\beta_l\) 是最优化计算需要估计出的参数。

三次样条函数 \(c_l(t)\) 的形式基于文献 (Ferstl and Hayden, 2010)，

\[ \begin{cases} & \text{ if } l=n, c_l(t)=t\\ & \text{ else }, c_{l}\left(t\right)= \left\{\begin{array}{ll} 0 & {t

对于有 \(n\) 个参数的贴现因子曲线，用户需要提供 \(n-1\) 个 knots \(k_i(1\le i\lt n)\)，并令 \(k_0 = 0\) 以及 \(k_n = k_{n-1}\)。

knots 的选择

knots 的选择基于文献 (McCulloch, 1975)，也可以参考文献 (Ferstl and Hayden, 2010)，

\[ \begin{cases} & \text{ if } l=1, k_l = 0\\ & \text{ else if } l=n-1,k_l=m_N \\ & \text{ else }, k_l = m_h + \theta(m_{h+1} - m_h) \end{cases} \]

其中，\(h=\left\lceil\frac{(l-1) k}{n-2}\right\rceil\)，\(\theta=\frac{(l-1) k}{n-2}-h\)，\(n = \left\lfloor\sqrt{k}+0.5 \right\rfloor\)，\(m_i(1 \le i\le N)\) 是升序排列后样本券的剩余期限。

实现三次样条函数

三次样条函数类 CubicSpline 的实现仿照已存在的 BSpline 类，

CubicSpline.hpp

class CubicSpline {
  public:
    CubicSpline(const std::vector& knots);
    ~CubicSpline();
    Real operator()(Natural i, Real x) const;

  private:
    Size n_;
    std::vector knots_ex_;
};

CubicSpline.cpp

CubicSpline::CubicSpline(const std::vector& knots)
    : n_(knots.size() + 1), knots_ex_(knots) {
    knots_ex_.insert(knots_ex_.begin(), 0.0);
    knots_ex_.insert(knots_ex_.end(), knots.back());
}

CubicSpline::~CubicSpline() {
}

Real CubicSpline::operator()(Natural i, Real x) const {
    using namespace std;

    if (i < n_) {
        Real q = knots_ex_[i], q_minus = knots_ex_[i - 1], q_plus = knots_ex_[i + 1];

        if (x < q_minus) {
            return 0.0;
        } else if (q_minus <= x and x < q) {
            return pow(x - q_minus, 3) / (6.0 * (q - q_minus));
        } else if (q <= x and x < q_plus) {
            return pow(q - q_minus, 2) / 6.0
                   + (q - q_minus) * (x - q) / 2.0
                   + pow(x - q, 2) / 2.0
                   - pow(x - q, 3) / (6.0 * (q_plus - q));
        } else {
            return (q_plus - q_minus)
                   * ((2.0 * q_plus - q - q_minus) / 6.0
                      + (x - q_plus) / 2.0);
        }
    } else {
        return x;
    }
}

实现拟合方法

拟合方法 CubicSplinesFitting 的实现仿照已存在的 CubicBSplinesFitting 类，两者均是 FittedBondDiscountCurve::FittingMethod 的派生类，

CubicSplinesFitting.hpp

class CubicSplinesFitting
    : public FittedBondDiscountCurve::FittingMethod {
  public:
    CubicSplinesFitting(const std::vector& knotVector,
                        const Array& weights = Array(),
                        ext::shared_ptr
                            optimizationMethod = ext::shared_ptr(),
                        const Array& l2 = Array());
    CubicSplinesFitting(const std::vector& knotVector,
                        const Array& weights,
                        const Array& l2);
    //! cubic spline basis functions
    Real basisFunction(Integer i, Time t) const;
    static std::vector autoKnots(const std::vector& maturities);
#if defined(QL_USE_STD_UNIQUE_PTR)
    std::unique_ptr clone() const;
#else
    std::auto_ptr clone() const;
#endif
  private:
    Size size() const;
    DiscountFactor discountFunction(const Array& x, Time t) const;
    CubicSpline splines_;
    Size size_;
    //! N_th basis function coefficient to solve for when d(0)=1
    Natural N_;
};

CubicSplinesFitting.cpp

CubicSplinesFitting::CubicSplinesFitting(const std::vector& knots,
                                         const Array& weights,
                                         ext::shared_ptr optimizationMethod,
                                         const Array& l2)
    : FittedBondDiscountCurve::FittingMethod(
          false, weights, optimizationMethod, l2),
      splines_(knots) {

    Size basisFunctions = knots.size() + 1;

    size_ = basisFunctions;
    N_ = 0;
}

CubicSplinesFitting::CubicSplinesFitting(const std::vector& knots,
                                         const Array& weights,
                                         const Array& l2)
    : FittedBondDiscountCurve::FittingMethod(
          false, weights, ext::shared_ptr(), l2),
      splines_(knots) {

    Size basisFunctions = knots.size() + 1;

    size_ = basisFunctions;
    N_ = 0;
}

Real CubicSplinesFitting::basisFunction(Integer i,
                                        Time t) const {
    return splines_(i, t);
}

QL_UNIQUE_OR_AUTO_PTR CubicSplinesFitting::clone() const {
    return QL_UNIQUE_OR_AUTO_PTR(
        new CubicSplinesFitting(*this));
}

Size CubicSplinesFitting::size() const {
    return size_;
}

DiscountFactor CubicSplinesFitting::discountFunction(const Array& x,
                                                     Time t) const {
    DiscountFactor d = 1.0;

    for (Size i = 0; i < size_; ++i) {
        d += x[i] * splines_(i + 1, t);
    }

    return d;
}

std::vector CubicSplinesFitting::autoKnots(const std::vector& maturities) {
    using namespace std;

    vector m(maturities);
    sort(m.begin(), m.end());

    Size k = m.size();
    Size n(floor(sqrt(k) + 0.5));

    vector knots(n - 1);

    knots[0] = 0.0;
    knots[n - 1] = m.back();

    for (Size l = 1; l < n - 1; ++l) {
        Size h(ceil(Real(l * k) / Real(n - 2)));
        Real theta = Real(l * k) / Real(n - 2) - h;
        knots[l] = m[h - 1] + theta * (m[h] - m[h - 1]);
    }

    return knots;
}

测试

用上述两个类拟合样本券的期限结构，并和 termstrc 的结果做比较。

辅助函数 CubicSplineSpotRate 用于将样条函数表示的贴现因子转换成即期利率。

QuantLib::Real CubicSplineSpotRate(const std::vector& knots,
                                   const QuantLib::Array& weights,
                                   const QuantLib::Time& t) {
    using namespace std;
    using namespace QuantLib;

    CubicSpline spline(knots);
    Size s = weights.size();
    Real d = 1.0, r;

    for (Size i = 0; i < s; ++i) {
        d += weights[i] * spline(i + 1, t);
    }

    r = -std::log(d) / t;

    return r;
}

测试函数

void TestCubicSplineFitting() {

    using namespace std;
    using namespace QuantLib;

    // 样本券数据，以及相关配置

    Size bondNum = 50;

    vector cleanPrice = {
        100.002, 99.92, 99.805, 99.75, 100.305, 99.76, 99.75, 99.975, 100.0416, 100.0574,
        99.5049, 101.0971, 101.137, 100.7199, 99.8883, 100.908, 103.3553, 99.5034, 103.913, 97.4229,
        104.5636, 99.7527, 104.3708, 99.6051, 104.8603, 101.3415, 105.29, 102.4969, 103.7602, 100.2803,
        102.6046, 102.5291, 99.4748, 95.9702, 97.1815, 114.2849, 100.2847, 112.23, 98.397, 102.0235,
        99.8483, 121.2711, 125.9157, 114.5791, 103.2202, 123.4668, 113.4694, 103.1873, 91.5603, 95.4441};

    vector> priceHandle(bondNum);

    for (Size i = 0; i < bondNum; ++i) {
        ext::shared_ptr q(
            new SimpleQuote(cleanPrice[i]));
        Handle hq(q);
        priceHandle[i] = hq;
    }

    vector issueYear = {
        2002, 2006, 2003, 2006, 1998, 2006, 2003, 2006, 1999, 2007,
        2004, 2007, 1999, 2007, 2004, 2007, 1999, 2005, 2000, 2005,
        2000, 2006, 2001, 2006, 2001, 2007, 2002, 2007, 2002, 2003,
        2003, 2004, 2004, 2005, 2005, 1986, 2006, 1986, 2006, 2007,
        2007, 1993, 1997, 1998, 1998, 2000, 2000, 2003, 2004, 2006};

    vector issueMonth = {
        Aug, Mar, Apr, May, Jul, Aug, Sep, Nov, Jan, Feb,
        Feb, May, Jul, Aug, Aug, Sep, Oct, Feb, May, Aug,
        Sep, Feb, May, Aug, Dec, Feb, Jun, Aug, Dec, Jun,
        Oct, Apr, Oct, Apr, Oct, Jun, Apr, Sep, Oct, Apr,
        Sep, Dec, Jul, Jan, Oct, Jan, Oct, Jan, Dec, Dec};

    vector issueDay = {
        14, 8, 11, 30, 4, 30, 25, 30, 4, 28, 2, 30, 4, 24, 25, 21, 22,
        24, 5, 26, 29, 26, 23, 30, 28, 28, 26, 24, 31, 24, 21, 25, 27, 28,
        30, 20, 26, 20, 31, 27, 21, 29, 3, 4, 7, 4, 27, 22, 24, 28};

    vector maturityYear = {
        2008, 2008, 2008, 2008, 2008, 2008, 2008, 2008, 2009, 2009,
        2009, 2009, 2009, 2009, 2009, 2009, 2010, 2010, 2010, 2010,
        2011, 2011, 2011, 2011, 2012, 2012, 2012, 2012, 2013, 2013,
        2014, 2014, 2015, 2015, 2016, 2016, 2016, 2016, 2017, 2017,
        2018, 2024, 2027, 2028, 2028, 2030, 2031, 2034, 2037, 2039};

    vector maturityMonth = {
        Feb, Mar, Apr, Jun, Jul, Sep, Oct, Dec, Jan, Mar,
        Apr, Jun, Jul, Sep, Oct, Dec, Jan, Apr, Jul, Oct,
        Jan, Apr, Jul, Oct, Jan, Apr, Jul, Oct, Jan, Jul,
        Jan, Jul, Jan, Jul, Jan, Jun, Jul, Sep, Jan, Jul,
        Jan, Jan, Jul, Jan, Jul, Jan, Jan, Jul, Jan, Jul};

    vector maturityDay = {
        15, 14, 11, 13, 4, 12, 10, 12, 4, 13, 17, 12, 4, 11, 9, 11,
        4, 9, 4, 8, 4, 8, 4, 14, 4, 13, 4, 12, 4, 4, 4, 4, 4, 4, 4,
        20, 4, 20, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4};

    vector issueDate(bondNum), maturityDate(bondNum);

    for (Size i = 0; i < bondNum; ++i) {
        Date idate(issueDay[i], issueMonth[i], issueYear[i]);
        Date mdate(maturityDay[i], maturityMonth[i], maturityYear[i]);
        issueDate[i] = idate;
        maturityDate[i] = mdate;
    }

    vector couponRate = {
        0.0425, 0.03, 0.03, 0.0325, 0.0475, 0.035, 0.035, 0.0375, 0.0375, 0.0375,
        0.0325, 0.045, 0.045, 0.04, 0.035, 0.04, 0.05375, 0.0325, 0.0525, 0.025,
        0.0525, 0.035, 0.05, 0.035, 0.05, 0.04, 0.05, 0.0425, 0.045, 0.0375, 0.0425,
        0.0425, 0.0375, 0.0325, 0.035, 0.06, 0.04, 0.05625, 0.0375, 0.0425, 0.04,
        0.0625, 0.065, 0.05625, 0.0475, 0.0625, 0.055, 0.0475, 0.04, 0.0425};

    Frequency frequency = Annual;
    Actual365Fixed dayCounter(Actual365Fixed::Standard);
    BusinessDayConvention paymentConv = Unadjusted;
    BusinessDayConvention terminationDateConv = Unadjusted;
    BusinessDayConvention convention = Unadjusted;
    Real redemption = 100.0;
    Real faceAmount = 100.0;
    Germany calendar(Germany::Eurex);

    Date today = calendar.adjust(Date(30, Jan, 2008));
    Settings::instance().evaluationDate() = today;

    Natural bondSettlementDays = 0;
    Date bondSettlementDate = calendar.advance(
        today,
        Period(bondSettlementDays, Days));

    vector> instruments(bondNum);
    vector maturity(bondNum);

    // 配置 helper

    for (Size i = 0; i < bondNum; ++i) {

        vector bondCoupon = {couponRate[i]};

        Schedule schedule(
            issueDate[i],
            maturityDate[i],
            Period(frequency),
            calendar,
            convention,
            terminationDateConv,
            DateGeneration::Backward,
            false);

        ext::shared_ptr helper(
            new FixedRateBondHelper(
                priceHandle[i],
                bondSettlementDays,
                faceAmount,
                schedule,
                bondCoupon,
                dayCounter,
                paymentConv,
                redemption));

        maturity[i] = dayCounter.yearFraction(
            bondSettlementDate, helper->maturityDate());

        instruments[i] = helper;
    }

    Real tolerance = 1.0e-6;
    Natural max = 5000;

    ext::shared_ptr optMethod(
        new LevenbergMarquardt());

    vector knots = CubicSplinesFitting::autoKnots(maturity);
    vector termstrcKnotes = {
        0.000000, 1.006027, 2.380274, 5.033425, 9.234521, 31.446575};

    cout << "QuantLib knots:\t";
    for (auto v : knots) {
        cout << setprecision(6) << fixed << v << ", ";
    }
    cout << endl;

    cout << "termstrc knots:\t";
    for (auto v : termstrcKnotes) {
        cout << setprecision(6) << fixed << v << ", ";
    }
    cout << endl;

    cout << endl;

    CubicSplinesFitting csf(
        knots, Array(), optMethod);

    FittedBondDiscountCurve tsCubicSplines(
        bondSettlementDate,
        instruments, dayCounter,
        csf, tolerance, max);

    Array weights = tsCubicSplines.fitResults().solution();
    Array termstrcWeights(7);
    termstrcWeights[0] = 1.9320e-02, termstrcWeights[1] = -8.4936e-05,
    termstrcWeights[2] = -3.2009e-04, termstrcWeights[3] = -3.7101e-04,
    termstrcWeights[4] = 7.2921e-04, termstrcWeights[5] = 2.0159e-03,
    termstrcWeights[6] = -4.1632e-02;

    cout << "QuantLib weights: \t" << weights << endl;
    cout << "termstrc weights: \t" << termstrcWeights << endl;

    cout << endl;

    cout << "QuantLib final cost value:\t"
         << tsCubicSplines.fitResults().minimumCostValue() << endl;

    cout << endl;

    // 比较 QuantLib 和 termstrc 的结果

    Real spotRate, termstrcSpot;

    for (Size i = 0; i < bondNum; ++i) {

        Time t = dayCounter.yearFraction(
            bondSettlementDate, maturityDate[i]);

        spotRate =
            tsCubicSplines.zeroRate(t, Compounding::Continuous, frequency).rate() * 100.0;
        termstrcSpot =
            CubicSplineSpotRate(termstrcKnotes, termstrcWeights, t) * 100.0;

        cout << setprecision(3) << fixed
             << t << ",\t"
             << spotRate << ",\t"
             << termstrcSpot << ",\t"
             << spotRate - termstrcSpot << endl;
    }
}

部分结果：

QuantLib knots: 0.000000, 1.117808, 2.690411, 5.430137, 9.432877, 31.446575,
termstrc knots: 0.000000, 1.006027, 2.380274, 5.033425, 9.234521, 31.446575,

QuantLib weights:   [ 0.005281; 0.004565; -0.002934; 0.000804; 0.000652; 0.001886; -0.038316 ]
termstrc weights:   [ 0.019320; -0.000085; -0.000320; -0.000371; 0.000729; 0.002016; -0.041632 ]

QuantLib final cost value:  0.000338

0.044,  3.823,  4.125,  -0.302
0.121,  3.809,  4.061,  -0.253
0.197,  3.794,  4.001,  -0.207
0.370,  3.761,  3.878,  -0.116
...
..
.

图 1：结果对比

注意：尽管以 termstrc 的结果作为基准，并不意味着基准就是正确答案。

由于样本券的数量不同（termstrc 使用了 52 只券），两者的 knots 差异较大。同时，因为优化方法的不同（termstrc 使用 OLS，QuantLib 使用 Levenberg-Marquardt 算法），估计出的参数也有差异。最终导致两个期限结构在两端差异较大。

不过，考虑到最终的 cost value，QuantLib 的结果可能更好一些。

参考文献

1. Ferstl.R, Hayden.J (2010). "Zero-Coupon Yield Curve Estimation with the Package termstrc." Journal of Statistical Software, Volume 36, Issue 1.
2. McCulloch JH (1975). "The Tax-Adjusted Yield Curve." The Journal of Finance, 30(3), 811–830.