2020_CurveBuilding.ipynb

Kernel: Python 2 (system-wide)

Восстановление кривой из цены свопа

$$S_{2Y} = \frac{1 - D_{24M}}{0.5 (D_{6M} + D_{12M} + D_{18M} + D_{24M}) }$$

import numpy as np
import collections
import matplotlib.pyplot as plt
import scipy.interpolate
import scipy.optimize

# Дискаунт фактор из процентной ставки
# для простоты капитализация процентов не учитывается
def DF(L,m):
    return 1.0 / (1.0 + L*0.01*m/12)
    
# Процентная ставка из дискаунт фактора 
# zero coupon кривая
def rate(D,m):
    return (1/D - 1)*100*12/m

global tenors
tenors = [3, 6, 9, 12, 15, 18, 21, 24]
L3 = 1.120
L6 = 1.430
L12 = 1.814
S2 = 1.684

# получить по 4м значениям LIBOR интерполированный промежуточные точки
def getLibors(L3, L6, L12, L24, method):
    if (method =='loglinear'):
        D18 = np.sqrt(DF(L24,24)*DF(L12,12))
        L18 = rate(D18,18)
        D9 = np.sqrt(DF(L12,12)*DF(L6,6))
        L9 = rate(D9,9)
        D15 = np.sqrt(DF(L12,12)*DF(L18,18))
        L15 = rate(D15,15)
        D21 = np.sqrt(DF(L18,18)*DF(L24,24))
        L21 = rate(D21,21)
        return [L3, L6, L9, L12, L15, L18, L21, L24.tolist()]
    elif method in ('cubic', 'linear', 'quadratic'):
        knownMonths = [3, 6, 12, 24]
        knownY = [L3, L6, L12, L24]
        f = scipy.interpolate.interp1d(knownMonths, knownY, method)
        return f(tenors) 

# посчитаем с кривой будем 3х месячные депозиты
def get3M(libor, step=3):
    monthly = []
    for i in xrange(len(libors)-1):
        df1=DF(libor[i],(i+1)*step)
        df2=DF(libor[i+1],(i+2)*step)
        r = (df1 / df2 - 1)/step*12*100
        monthly.append(r)
    return monthly

def showPlot(tenors, libors):
    plt.title('USD yield curve')
    plt.xlabel('Month')
    plt.ylabel('USD_rate')
    plt.plot(tenors, libors)
    plt.plot(tenors[0:-1], get3M(libors))

Лог-линейная интерполяция

$$D_{18M} = \sqrt{D_{24M}D_{12M}}$$

# x - LIBOR 24M
def F(x, S2, L6, L12):
    S2 = 0.01*S2
    D6 = DF(L6,6)
    D12 = DF(L12,12)
    D24 = DF(x,24)
    D18 = np.sqrt(D12*D24)
    return 0.5*S2*(D6 + D12 + D18 + D24) - 1 + D24

L24 = scipy.optimize.broyden1(lambda x: F(x,S2,L6,L12), 1.0, f_tol=1e-6)
D18 = np.sqrt(DF(L12,12)*DF(L24,24))

libors=getLibors(L3, L6, L12, L24, 'loglinear')

showPlot(tenors, libors)

1.705143874804221

Кубические сплайны

def F(x, S2, L3, L6, L12):
    S2 = 0.01*S2
    D6 = DF(L6,6)
    D12 = DF(L12,12)
    f = scipy.interpolate.interp1d([3,6,12,24], [L3,L6,L12,x], 'cubic')
    L18 = f(18)
    D18 = DF(L18, 18)
    D24 = DF(x, 24)
    return 0.5*S2*(D6 + D12 + D18 + D24) - 1 + D24

L24 = scipy.optimize.broyden1(lambda x: F(x,S2,L3,L6,L12), 1.0, f_tol=1e-6)
libors=getLibors(L3, L6, L12, L24, 'cubic')
showPlot(tenors, libors)

Квадратичная интерполяция

def F(x, S2, L3, L6, L12):
    S2 = 0.01*S2
    D6 = DF(L6,6)
    D12 = DF(L12,12)
    f = scipy.interpolate.interp1d([3,6,12,24], [L3,L6,L12,x], 'quadratic')
    L18 = f(18)
    D18 = DF(L18, 18)
    D24 = DF(x, 24)
    return 0.5*S2*(D6 + D12 + D18 + D24) - 1 + D24

L24 = scipy.optimize.broyden1(lambda x: F(x,S2,L3,L6,L12), 1.0, f_tol=1e-6)
libors = getLibors(L3, L6, L12, L24, 'quadratic')

showPlot(tenors, libors)