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PyCurve - Python Yield Curve Toolkit

What is it ?

PyCurve is a Python package that provides to user high level yield curve usefull tool. For example you can istanciate a Curve and get a d_rate, a discount factor, even forward d_rate given multiple methodology from Linear Interpolation to parametrization methods as Nelson Siegel or Bjork-Christenssen. PyCurve is also able to provide solutions in order to build yield curve or price Interest rates derivatives via Vasicek or Hull and White.

Features

Below this is the features that this package tackle :

  • Curve Smoothing:
    • Create Curve Object with two numpy array (t,rt)
    • Linear interpolation given a Curve
    • Cubic interpolation given a Curve
    • Nelson Siegel and Svensson model creation and components plotting
    • Nelson Siegel and Svensson calibration given a Curve
    • Bjork Christensen and Augmented (6 factors) model creation and components plotting
  • Stochastic Modelling:
    • Vasicek Model Simulation
    • Hull and White one factor Model Simulation

How to install

From pypi

pip install PyCurve

From pypi specific version

pip install PyCurve==0.0.5

From Git

git clone https://github.com/ahgperrin/PyCurve.git
pip install -e .

Objects

Curve Object

This object consists in a simple yield curve encapsulation. This object is used by others class to encapsulate results or in order to directly create a curve with data obbserved in the market.

Attributes Type Description
rt Private Interest rates as float in a numpy.ndarray
t Private Time as float or int in a numpy.ndarray
Methods Type Description Return
get_rate Public rt getter _rt
get_time Public rt getter _t
set_rate Public rt getter None
set_time Public rt getter None
is_valid_attr(attr) Private Check attributes validity attribute
plot_curve() Public Plot Yield curve None

Example

from PyCurve.curve import Curve
time = np.array([0.25, 0.5, 0.75, 1., 2., 
        3., 4., 5., 10., 15., 
        20.,25.,30.])
rate = np.array([-0.63171, -0.650322, -0.664493, -0.674608, -0.681294,
        -0.647593, -0.587828, -0.51251, -0.101804,  0.182851,
        0.32962,0.392117,  0.412151])
curve = Curve(rate,time)
curve.plot_curve()
print(curve.get_rate)
print(curve.get_time)
[ 0.25  0.5   0.75  1.    2.    3.    4.    5.   10.   15.   20.   25.
 30.  ]
  
[-0.63171  -0.650322 -0.664493 -0.674608 -0.681294 -0.647593 -0.587828
 -0.51251  -0.101804  0.182851  0.32962   0.392117  0.412151]

Simulation Object

This object consists in a simple simulation encapsulation. This object is used by others class to encapsulate results of monte carlo simulation. This Object has build in method that could perform the conversion from a simulation to a yield curve or to a discount factor curve.

Attributes Type Description
sim Private Simulated paths matrix numpy.ndarray
dt Private delta_time as float or int in a numpy.ndarray
Methods Type Description & Params Return
get_sim Public sim getter _rt
get_nb_sim Public nb_sim getter sim.shape[0]
get_steps Public steps getter sim.shape[1]
get_dt Public dt getter _dt
is_valid_attr(attr) Private Check attributes validity attribute
yield_curve() Public Create a yield curve from simulated paths Curve
discount_factor() Public Convert d_rate simulation to discount factor np.ndarray
plot_discount_curve(average) Public Plot discount factor (average :bool False plot all paths True Plot estimate) None
plot_simulation() Public Plot Yield curve None
plot_yield_curve() Public Plot Yield curve None
plot_model() Public Plot Yield curve None

Example

Using Vasicek to Simulate

from PyCurve.vasicek import Vasicek
vasicek_model = Vasicek(0.02, 0.04, 0.001, -0.004, 50, 30 / 365)
simulation = vasicek_model.simulate_paths(2000) #Return a Simulation and then we can apply Simulation Methods
simulation.plot_yield_curve()

simulation.plot_model()

Yield Curve Construction Tools

This section is the description with examples of what you can do with this package Please note that for all the examples in this section curve is referring to the curve below you can see example regarding Curve Object in the dedicated section.

from PyCurve import Curve
time = np.array([0.25, 0.5, 0.75, 1., 2., 
        3., 4., 5., 10., 15., 
        20.,25.,30.])
d_rate = np.array([-0.63171, -0.650322, -0.664493, -0.674608, -0.681294,
        -0.647593, -0.587828, -0.51251, -0.101804,  0.182851,
        0.32962,0.392117,  0.412151])
curve = Curve(time,d_rate)

linear

Interpolate any d_rate from a yield curve using linear interpolation. THis module is build using scipy.interpolate

Attributes Type Description
curve Private Curve Object to be intepolated
func_rate Private interp1d Object used to interpolate
Methods Type Description & Params Return
d_rate(t) Public d_rate interpolation t: float, array,int float
df_t(t) Public discount factor interpolation t: float, array,int float
forward(t1,t2) Public forward d_rate between t1 and t2 t1,t2: float, array,int float
create_curve(t_array) Public create a Curve object for t values t:array Curve
is_valid_attr(attr) Private Check attributes validity attribute

Example

from PyCurve.linear import LinearCurve
linear_curve = LinearCurve(curve)
print("7.5-year d_rate : "+str(linear_curve.d_rate(7.5)))
print("7.5-year discount d_rate : "+str(linear_curve.df_t(7.5)))
print("Forward d_rate between 7.5 and 12.5 years : "+str(linear_curve.forward(7.5,12.5)))
7.5-year d_rate : -0.307157
7.5-year discount d_rate : 1.0233404498400862
Forward d_rate between 7.5 and 12.5 years : 0.5620442499999999

cubic

Interpolate any d_rate from a yield curve using linear interpolation. THis module is build using scipy.interpolate

Attributes Type Description
curve Private Curve Object to be intepolated
func_rate Private PPoly Object used to interpolate
Methods Type Description & Params Return
d_rate(t) Public d_rate interpolation t: float, array,int float
df_t(t) Public discount factor interpolation t: float, array,int float
forward(t1,t2) Public forward d_rate between t1 and t2 t1,t2: float, array,int float
create_curve(t_array) Public create a Curve object for t values t:array Curve
is_valid_attr(attr) Private Check attributes validity attribute

Example

from PyCurve.cubic import CubicCurve
cubic_curve = CubicCurve(curve)
print("10-year d_rate : "+str(cubic_curve.d_rate(7.5)))
print("10-year discount d_rate : "+str(cubic_curve.df_t(7.5)))
print("Forward d_rate between 10 and 20 years : "+str(cubic_curve.forward(7.5,12.5)))
10-year d_rate : -0.3036366057950627
10-year discount d_rate : 1.0230694659050514
Forward d_rate between 10 and 20 years : 0.6078001168478189

Nelson-Siegel

Attributes Type Description
beta0 Private Model Coefficient Beta0
beta1 Private Model Coefficient Beta1
beta2 Private Model Coefficient Beta2
tau Private Model Coefficient tau
attr_list Private Coefficient list
Methods Type Description & Params Return
get_attr(str(attr)) Public attributes getter attribute
set_attr(attr) Public attributes setter None
print_model() Public print the Ns model set None
_calibration_func(x,curve) Private Private method used for calibration method float:sqr_err
_is_positive_attr(attr) Private Check attributes positivity (beta0 and tau attribute
_is_valid_curve(curve) Private Check if the curve given for calibration is a Curve Object Curve
_print_fitting() Private Print the result after the calibration None
calibrate(curve) Public Minimize _calibration_func(x,curve) sco.OptimizeResult
_time_decay(t) Private Compute the time decay part of the model t (float or array) float,array
_hump(t) Private Compute the hump part of the model given t (float or array) float,array
d_rate(t) Public Get d_rate from the model for a given time t (float or array) float,array
plot_calibrated() Public Plot Model curve against Curve None
plot_model_params() Public Plot Model parameters None
plot_model() Public Plot Model Components None
df_t(t) Public Get the discount factor from the model for a given time t (float or array) float,array
cdf_t(t) Public Get the continuous df from the model for a given time t (float or array) float,array
forward_rate(t1,t2) Public Get the forward d_rate for a given time t1,t2 (float or array) float,array

Example

Creation of a model and calibration

from PyCurve.nelson_siegel import NelsonSiegel
ns = NelsonSiegel(0.3,0.4,12,1)
ns.calibrate(curve)

Nelson Siegel Model
============================
beta0 = 0.751506062319988
beta1 = -1.3304971868997248
beta2 = -2.2203178895179176
tau = 2.5493056203052005
____________________________
============================
Calibration Results
============================
CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
Mean Squared Error 0.0042367306926415285
Number of Iterations 20
____________________________
Out[19]:
      fun: 0.0042367306926415285
 hess_inv: <4x4 LbfgsInvHessProduct with dtype=float64>
      jac: array([-2.40077054e-06,  9.51322360e-07, -2.33927462e-07,  7.97278914e-07])
  message: 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL'
     nfev: 105
      nit: 20
     njev: 21
   status: 0
  success: True
        x: array([ 0.75150606, -1.33049719, -2.22031789,  2.54930562])

Plotting and analyse

ns.plot_calibrated()


ns.plot_model_params()


ns.plot_model()

nelson-siegel-svensson

Attributes Type Description
beta0 Private Model Coefficient Beta0
beta1 Private Model Coefficient Beta1
beta2 Private Model Coefficient Beta2
beta3 Private Model Coefficient Beta3
tau Private Model Coefficient tau
tau2 Private Model Coefficient tau2
attr_list Private Coefficient list
Methods Type Description & Params Return
get_attr(str(attr)) Public attributes getter attribute
set_attr(attr) Public attributes setter None
print_model() Public print the Ns model set None
_calibration_func(x,curve) Private Private method used for calibration method float:sqr_err
_is_positive_attr(attr) Private Check attributes positivity (beta0 and tau attribute
_is_valid_curve(curve) Private Check if the curve given for calibration is a Curve Object Curve
_print_fitting() Private Print the result after the calibration None
calibrate(curve) Public Minimize _calibration_func(x,curve) sco.OptimizeResult
_time_decay(t) Private Compute the time decay part of the model t (float or array) float,array
_hump(t) Private Compute the hump part of the model given t (float or array) float,array
_second_hump(t) Private Compute the second hump part of the model given t (float or array) float,array
d_rate(t) Public Get d_rate from the model for a given time t (float or array) float,array
plot_calibrated() Public Plot Model curve against Curve None
plot_model_params() Public Plot Model parameters None
plot_model() Public Plot Model Components None
df_t(t) Public Get the discount factor from the model for a given time t (float or array) float,array
cdf_t(t) Public Get the continuous df from the model for a given time t (float or array) float,array
forward_rate(t1,t2) Public Get the forward d_rate for a given time t1,t2 (float or array) float,array

Example

Creation of a model and calibration

from PyCurve.svensson_nelson_siegel import NelsonSiegelAugmented
nss = NelsonSiegelAugmented(0.3,0.4,12,12,1,1)
nss.calibrate(curve)

Augmented Nelson Siegel Model
============================
beta0 = 0.7515069899513361
beta1 = -1.3304984652740972
beta2 = -1.3582175270153745
beta3 = -0.8621237370245594
tau = 2.5492666085730384
tau2 = 2.5493745447283485
____________________________
============================
Calibration Results
============================
CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
Mean Squared Error 0.004236730702075479
Number of Iterations 31
____________________________
Out[31]:
      fun: 0.004236730702075479
 hess_inv: <6x6 LbfgsInvHessProduct with dtype=float64>
      jac: array([-8.70041881e-06, -3.48375844e-06, -1.71824361e-06, -1.71911096e-06,
        1.00535900e-06,  3.23178986e-07])
  message: 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL'
     nfev: 245
      nit: 31
     njev: 35
   status: 0
  success: True
        x: array([ 0.75150699, -1.33049847, -1.35821753, -0.86212374,  2.54926661,
        2.54937454])

Plotting possibilities are the same as for the Nelson-Siegel model for example

nss.plot_model_params()

bjork-christensen

Attributes Type Description
beta0 Private Model Coefficient Beta0
beta1 Private Model Coefficient Beta1
beta2 Private Model Coefficient Beta2
beta3 Private Model Coefficient Beta3
tau Private Model Coefficient tau
attr_list Private Coefficient list
Methods Type Description & Params Return
get_attr(str(attr)) Public attributes getter attribute
set_attr(attr) Public attributes setter None
print_model() Public print the Ns model set None
_calibration_func(x,curve) Private Private method used for calibration method float:sqr_err
_is_positive_attr(attr) Private Check attributes positivity (beta0 and tau attribute
_is_valid_curve(curve) Private Check if the curve given for calibration is a Curve Object Curve
_print_fitting() Private Print the result after the calibration None
calibrate(curve) Public Minimize _calibration_func(x,curve) sco.OptimizeResult
_time_decay(t) Private Compute the time decay part of the model t (float or array) float,array
_hump(t) Private Compute the hump part of the model given t (float or array) float,array
_second_hump(t) Private Compute the second hump part of the model given t (float or array) float,array
d_rate(t) Public Get d_rate from the model for a given time t (float or array) float,array
plot_calibrated() Public Plot Model curve against Curve None
plot_model_params() Public Plot Model parameters None
plot_model() Public Plot Model Components None
df_t(t) Public Get the discount factor from the model for a given time t (float or array) float,array
cdf_t(t) Public Get the continuous df from the model for a given time t (float or array) float,array
forward_rate(t1,t2) Public Get the forward d_rate for a given time t1,t2 (float or array) float,array

Example

Creation of a model and calibration

from PyCurve.bjork_christensen import BjorkChristensen
bjc = BjorkChristensen(0.3,0.4,12,12,1)
bjc.calibrate(curve)

Bjork & Christensen Model
============================
beta0 = 0.7241026361747042
beta1 = 1.2630412759302045
beta2 = -4.075775255903699
beta3 = -2.61578890758314
tau = 2.0454907238894267
____________________________
============================
Calibration Results
============================
CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
Mean Squared Error 0.002575936865445517
Number of Iterations 37
____________________________
Out[36]:
      fun: 0.002575936865445517
 hess_inv: <5x5 LbfgsInvHessProduct with dtype=float64>
      jac: array([-1.34584183e-06,  7.22165387e-07, -9.63335320e-07,  1.34501786e-06,
        4.57750160e-07])
  message: 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL'
     nfev: 252
      nit: 37
     njev: 42
   status: 0
  success: True
        x: array([ 0.72410264,  1.26304128, -4.07577526, -2.61578891,  2.04549072])

Plotting possibilities are the same as for the Nelson-Siegel model for example

bjc.plot_model()

bjork-christensen-augmented

Attributes Type Description
beta0 Private Model Coefficient Beta0
beta1 Private Model Coefficient Beta1
beta2 Private Model Coefficient Beta2
beta3 Private Model Coefficient Beta3
beta4 Private Model Coefficient Beta4
tau Private Model Coefficient tau
attr_list Private Coefficient list
Methods Type Description & Params Return
get_attr(str(attr)) Public attributes getter attribute
set_attr(attr) Public attributes setter None
print_model() Public print the Ns model set None
_calibration_func(x,curve) Private Private method used for calibration method float:sqr_err
_is_positive_attr(attr) Private Check attributes positivity (beta0 and tau attribute
_is_valid_curve(curve) Private Check if the curve given for calibration is a Curve Object Curve
_print_fitting() Private Print the result after the calibration None
calibrate(curve) Public Minimize _calibration_func(x,curve) sco.OptimizeResult
_time_decay(t) Private Compute the time decay part of the model t (float or array) float,array
_hump(t) Private Compute the hump part of the model given t (float or array) float,array
_second_hump(t) Private Compute the second hump part of the model given t (float or array) float,array
_third_hump(t) Private Compute the third hump part of the model given t (float or array) float,array
d_rate(t) Public Get d_rate from the model for a given time t (float or array) float,array
plot_calibrated() Public Plot Model curve against Curve None
plot_model_params() Public Plot Model parameters None
plot_model() Public Plot Model Components None
df_t(t) Public Get the discount factor from the model for a given time t (float or array) float,array
cdf_t(t) Public Get the continuous df from the model for a given time t (float or array) float,array
forward_rate(t1,t2) Public Get the forward d_rate for a given time t1,t2 (float or array) float,array

Example

Creation of a model and calibration

from PyCurve.bjork_christensen_augmented import BjorkChristensenAugmented
bjc_a = BjorkChristensenAugmented(0.3,0.4,12,12,12,1)
bjc_a.calibrate(curve)

Bjork & Christensen Augmented Model
============================
beta0 = 1.5954945516202643
beta1 = -0.1362673420894012
beta2 = -1.921347491829477
beta3 = -3.100138400789165
beta4 = -0.2790540854856497
tau = 3.3831338085688625
____________________________
============================
Calibration Results
============================
CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH
Mean Squared Error 4.6222147406189135e-05
Number of Iterations 45
____________________________
Out[39]:
      fun: 4.6222147406189135e-05
 hess_inv: <6x6 LbfgsInvHessProduct with dtype=float64>
      jac: array([-3.13922277e-05, -1.14224797e-04, -3.47444433e-05,  2.07803821e-05,
        7.91378953e-06,  6.50288949e-06])
  message: 'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
     nfev: 357
      nit: 45
     njev: 51
   status: 0
  success: True
        x: array([ 1.59549455, -0.13626734, -1.92134749, -3.1001384 , -0.27905409,
        3.38313381])

Plotting possibilities are the same as for the Nelson-Siegel model for example

bjc_a.plot_calibrated(curve)

Stochastic Tools

vasicek

Attributes Type Description
alpha Private Model Coefficient alpha (mean reverting speed)
beta Private Model Coefficient Beta (long term mean)
sigma Private Short rate Volatility
rt Private Initial Short Rate
time Params Time in years
dt Private time for each period
steps Private calculated with dt & time as time/dt
Methods Type Description & Params Return
get_attr(str(attr)) Public attributes getter attribute
sigma_part(n) Private compute n sigma part float
mu_dt(rt) Private compute drift part float
simulate_paths(n) Public Simulate n Short rate paths np.ndarray
plot_calibrated(simul,curve) Public Plot yield curve against simulate curve None 
time = np.array([0.25, 0.5, 0.75, 1., 2.,
                 3., 4., 5., 10., 15.,
                 20., 25., 30.])
rate = np.array([-0.0063171, -0.00650322, -0.00664493, -0.00674608, -0.00681294,
                 -0.00647593, -0.00587828, -0.0051251, -0.00101804, 0.00182851,
                 0.0032962, 0.0030092117, 0.00412151])
curve = Curve(time,rate)
vasicek_model = Vasicek(0.5, 0.0040, 0.001, -0.0067, 30, 1 / 365)
simulation = vasicek_model.simulate_paths(200)
vasicek_model.plot_calibrated(simulation,curve)

All the tools for graphing from simulation could be applied to vasicek simulation results.

hull & white

Attributes Type Description
alpha Private Model Coefficient alpha (mean reverting speed)
sigma Private Short rate Volatility
rt Private Initial Short Rate
time Params Time in years
dt Private time for each period
steps Private calculated with dt & time as time/dt
f_curve Private Curve : Initial instantaneous forward structure
method Private method used in order to interpolate f_curve
Methods Type Description & Params Return
get_attr(str(attr)) Public attributes getter attribute
_is_valid_curve(curve) Private Check if the curve given for calibration is a Curve Object Curve
sigma_part(n) Private compute n sigma part float
interp_forward(t) Private interpolate forward curve for maturity t float
theta_part(t) Private compute theta(t) float
mu_dt(rt,t) Private compute drift part at time t float
simulate_paths(n) Public Simulate n Short rate paths np.ndarray
plot_calibrated(simul) Public Plot yield curve against simulate curve None

Example

from PyCurve.bjork_christensen_augmented import BjorkChristensenAugmented
from PyCurve.hull_white import HullWhite
import numpy as np
from PyCurve.curve import Curve

# Instance of curve : Spot Rates
time = np.array([0.25, 0.5, 0.75, 1., 2.,
                 3., 4., 5., 10., 15.,
                 20., 25., 30.])
rate = np.array([-0.0063171, -0.00650322, -0.00664493, -0.00674608, -0.00681294,
                 -0.00647593, -0.00587828, -0.0051251, -0.00101804, 0.00182851,
                 0.0032962, 0.00392117, 0.00412151])
curve = Curve(time, rate)

# Deduce Forward rate via Bjork Christensen (as example but you can directly create an instance of Curve with values)

bjc_a = BjorkChristensenAugmented(0.3, 0.4, 12, 12, 12, 1)
bjc_a.calibrate(curve)
forward_curve = [-0.006301821217413436379]
forward_curve_t = [0]
for i in range(12):
    forward_curve.append(bjc_a.forward_rate(time[i], time[i + 1]))
    forward_curve_t.append(time[i])
instantaneous_forward = Curve(forward_curve_t, forward_curve)


# Hull and white model  with High Volatility
hull_white_model = HullWhite(1, 0.02, -0.0063, 25, 1 / 365, instantaneous_forward, 'linear')
simulation = hull_white_model.simulate_paths(1000)
hull_white_model.plot_calibrated(simulation,curve)

Bjork & Christensen Augmented Model
============================
beta0 = 0.0003242320890548229
beta1 = 0.00042283628067360974
beta2 = 0.014729859086888815
beta3 = -0.03083749691652102
beta4 = -0.020626731632810553
tau = 1.137911384276111
____________________________
============================
Calibration Results
============================
CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH
Mean Squared Error 3.084012924460394e-06
Number of Iterations 24
____________________________

All the tools for graphing from simulation could be applied to Hull-White simulation results.

# Hull and white model  with High Volatility
hull_white_model_low_vol = HullWhite(1, 0.00002, -0.0063, 25, 1 / 365, instantaneous_forward, 'cubic')
simulation = hull_white_model_low_vol.simulate_paths(1000)
simulation.plot_model()
hull_white_model_low_vol.plot_calibrated(simulation,curve)

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