Example of the stationary bootstrap algorithm in practice

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib notebook

Goal

The goal of this workbook is to demonstrate the use of the stationary bootstrap algorithm. In this example, 3 new samples are generated.

nSample = 3 # Number of generated samples

EURO denominated 6M interest-rate-swap rate obtained from https://www.teleborsa.it/Quotazioni/Tassi/Eurirs for date 12/11/2021 and interpolated using the Smith & Wilson algorithm for missing durations.

swap = pd.DataFrame({'Swap rate': [-0.00497999999999998,
-0.00336999999999998,
-0.00219000000000003,
-0.00137999999999994,
-0.000839999999999952,
-0.000349999999999961,
0.000140000000000029,
0.000669999999999948,
0.00116999999999989,
0.00165999999999999,
0.00208999999999993,
0.00245999999999991,
0.00277690952371801,
0.00303541131903184,
0.00323000000000007,
0.00336225372994803,
0.00345212702349018,
0.00351682745133952,
0.00356726077755898,
0.00360999999999989,
0.00364393467821089,
0.00365001038370538,
0.00360991273768496,
0.00350996855727703,
0.00333999999999990,
0.00310968998405592,
0.00288888447388769,
0.00274107227669318,
0.00271077735186531,
0.00283000000000011]})

swap.head()

	Swap rate
0	-0.00498
1	-0.00337
2	-0.00219
3	-0.00138
4	-0.00084

sampleLen = swap.size; # Length of the sample will be the same as the input.

The swap rates cann ot be stacked in a sampling algorithm therefore, they are used to Vector with the 1-year forward curve. Convert rates into 1-year forward curve

Note that you loose 1 degree of freedom when converting from rates to 1-year-forward rates using the formula

\[fwd(t) = \frac{(1 + r_{SWAP}(0, t+1))^{t+1}}{(1 + r_{SWAP}(0, t))^{t}}\]

forwards = np.zeros((sampleLen-1,1))

for iTime in range(0,sampleLen-1):
    forwards[iTime][0] = np.power(1 + swap["Swap rate"][iTime+1],iTime+1) / np.power(1+swap["Swap rate"][iTime], iTime) - 1

forwards = pd.DataFrame(forwards, columns={'Forward rate'})

forwards.head()

	Forward rate
0	-0.003370
1	-0.001009
2	0.000242
3	0.000782
4	0.001612

Calculated using the method presented in 2004 paper by Politis & White.

m = 3.8299

Generate Stationary bootstraped samples

In this step, the samples are generated by repeatedly calling th StationaryBootstrap function. These samples are then saved into the Samples array

from StationaryBootstrap import StationaryBootstrap
for iSample in range(0, nSample):
    if iSample == 0:
        Samples = np.array(StationaryBootstrap(forwards.values,m,sampleLen))
    else:
        Samples = np.append(Samples, np.array(StationaryBootstrap(forwards.values,m,sampleLen)),axis=1)

Samples = pd.DataFrame(Samples)
Samples.head()

	0	1	2
0	0.002768	0.006270	-0.000561
1	-0.002402	0.006143	-0.002402
2	0.006270	0.005763	-0.002615
3	-0.001009	0.005216	-0.001094
4	0.004553	0.004801	0.004380

plot_data = Samples.copy().to_numpy()
fig, ax = plt.subplots(figsize=(10, 3))
ax.plot(plot_data)
ax.set_title('Forward rate samples')
fig.tight_layout()