11. Interest Rate Swaps

Very nascent.

Interest rate swaps are a first step towards including rate-sensitive instruments in the modeling and valuation spectrum of DX Analytics. The model used in the following is the square-root diffusion process by Cox-Ingersoll-Ross (1985). Data used are UK/London OIS and Libor rates.

[1]:

import dx
import time
import numpy as np
import pandas as pd
import datetime as dt
from pylab import plt
plt.style.use('seaborn')
%matplotlib inline

11.1. OIS Data & Discounting

We start by importing OIS term structure data (source: http://www.bankofengland.co.uk) for risk-free discounting. We also adjust the data structure somewhat for our purposes.

[2]:

# UK OIS Spot Rates Yield Curve
oiss = pd.read_excel('data/ukois09.xls', 'oiss')
# use years as index
oiss = oiss.set_index('years')
# del oiss['years']

[3]:

# only date information for columns, no time
oiss.columns = [d.date() for d in oiss.columns]

[4]:

oiss.tail()

[4]:
2014-10-01 2014-10-02 2014-10-03 2014-10-06 2014-10-07 2014-10-08 2014-10-09 2014-10-10 2014-10-13 2014-10-14 2014-10-15 2014-10-16
years
4.666667 1.550947 1.525838 1.570203 1.533901 1.479044 1.445234 1.436023 1.409962 1.337567 1.283932 1.063205 1.234125
4.750000 1.563661 1.538616 1.583198 1.546459 1.491152 1.457593 1.448627 1.422015 1.349782 1.296052 1.074773 1.246277
4.833333 1.576142 1.551158 1.595952 1.558789 1.503050 1.469768 1.461040 1.433887 1.361832 1.308025 1.086254 1.258307
4.916667 1.588400 1.563474 1.608472 1.570898 1.514746 1.481764 1.473269 1.445585 1.373724 1.319859 1.097650 1.270219
5.000000 1.600442 1.575572 1.620768 1.582794 1.526247 1.493588 1.485320 1.457116 1.385461 1.331557 1.108962 1.282014

Next we replace the year fraction index by a DatetimeIndex.

[5]:

# generate time index given input data
# starting date + 59 months
date = oiss.columns[-1]
index = pd.date_range(date, periods=60, freq='M') # , tz='GMT')
index = [d.replace(day=date.day) for d in index]
index = pd.DatetimeIndex(index)
oiss.index = index

Let us have a look at the most current data, i.e. the term structure, of the data set.

[6]:

oiss.iloc[:, -1].plot(figsize=(10, 6))

[6]:

<matplotlib.axes._subplots.AxesSubplot at 0x10ba062b0>
_images/10_dx_interest_rate_swaps_13_1.png

This data is used to instantiate a deterministic_short_rate model for risk-neutral discounting purposes.

[7]:

# generate deterministic short rate model based on UK OIS curve
ois = dx.deterministic_short_rate('ois', list(zip(oiss.index, oiss.iloc[:, -1].values / 100)))

[8]:

# example dates and corresponding discount factors
dr = pd.date_range('2015-1', periods=4, freq='6m').to_pydatetime()
ois.get_discount_factors(dr)[::-1]

[8]:

([0.989825148692371, 0.9922754901627036, 0.9956237774158325, 1.0],
 array([datetime.datetime(2015, 1, 31, 0, 0),
        datetime.datetime(2015, 7, 31, 0, 0),
        datetime.datetime(2016, 1, 31, 0, 0),
        datetime.datetime(2016, 7, 31, 0, 0)], dtype=object))

11.2. Libor Market Data

We want to model a 3 month Libor-based interest rate swap. To this end, we need Libor term structure data, i.e. forward rates in this case (source: http://www.bankofengland.co.uk), to calibrate the valuation to. The data importing and adjustments are the same as before.

[9]:

# UK Libor foward rates
libf = pd.read_excel('data/ukblc05.xls', 'fwds')
# use years as index
libf = libf.set_index('years')

[10]:

# only date information for columns, no time
libf.columns = [d.date() for d in libf.columns]

[11]:

libf.tail()

[11]:
2014-10-01 2014-10-02 2014-10-03 2014-10-06 2014-10-07 2014-10-08 2014-10-09 2014-10-10 2014-10-13 2014-10-14 2014-10-15 2014-10-16
years
4.666667 2.722915 2.686731 2.678433 2.681385 2.639166 2.542012 2.528768 2.489063 2.470154 2.349732 2.372435 2.157431
4.750000 2.733588 2.697887 2.690366 2.692824 2.650841 2.554583 2.543066 2.502357 2.483842 2.366052 2.387671 2.176186
4.833333 2.744126 2.708848 2.702083 2.704147 2.662380 2.567071 2.557216 2.515531 2.497446 2.382298 2.402829 2.194784
4.916667 2.754540 2.719634 2.713604 2.715368 2.673794 2.579486 2.571229 2.528592 2.510973 2.398468 2.417915 2.213221
5.000000 2.764842 2.730264 2.724949 2.726501 2.685095 2.591835 2.585117 2.541548 2.524432 2.414562 2.432939 2.231493 
[12]:

# generate time index given input data
# starting date + 59 months
date = libf.columns[-1]
index = pd.date_range(date, periods=60, freq='M')  # , tz='GMT')
index = [d.replace(day=date.day) for d in index]
index = pd.DatetimeIndex(index)
libf.index = index

And the short end of the Libor term sturcture visualized.

[13]:

libf.iloc[:, -1].plot(figsize=(10, 6))

[13]:

<matplotlib.axes._subplots.AxesSubplot at 0x10bfe8198>
_images/10_dx_interest_rate_swaps_24_1.png

11.3. Model Calibration

Next, equipped with the Libor data, we calibrate the square-root diffusion short rate model. A bit of data preparation:

[14]:

t = libf.index.to_pydatetime()
f = libf.iloc[:, -1].values / 100
initial_value = 0.005

A mean-squared error (MSE) function to be minimized during calibration.

[15]:

def srd_forward_error(p0):
    global initial_value, f, t
    if p0[0] < 0 or p0[1] < 0 or p0[2] < 0:
        return 100
    f_model = dx.srd_forwards(initial_value, p0, t)
    MSE = np.sum((f - f_model) ** 2) / len(f)
    return MSE

And the calibration itself.

[16]:

from scipy.optimize import fmin

[17]:

opt = fmin(srd_forward_error, (1.0, 0.7, 0.2),
           maxiter=1000, maxfun=1000)

Optimization terminated successfully.
         Current function value: 0.000000
         Iterations: 371
         Function evaluations: 649

The optimal parameters (kappa, theta, sigma) are:

[18]:

opt

[18]:

array([0.00544441, 1.80697228, 0.23689443])

The model fit is not too bad in this case.

[19]:

plt.figure(figsize=(10, 6))
plt.plot(t, f, label='market forward rates')
plt.plot(t, dx.srd_forwards(initial_value, opt, t), 'r.', label='model forward rates')
plt.gcf().autofmt_xdate(); plt.legend(loc=0)

[19]:

<matplotlib.legend.Legend at 0x10bfbcef0>
_images/10_dx_interest_rate_swaps_36_1.png

11.4. Floating Rate Modeling

The optimal parameters from the calibration are used to model the floating rate (3m Libor rate).

[20]:

# market environment
me_srd = dx.market_environment('me_srd', dt.datetime(2014, 10, 16))

[21]:

# square-root diffusion
me_srd.add_constant('initial_value', 0.02)
me_srd.add_constant('kappa', opt[0])
me_srd.add_constant('theta', opt[1])
me_srd.add_constant('volatility', opt[2])
me_srd.add_curve('discount_curve', ois)
  # OIS discounting object
me_srd.add_constant('currency', 'EUR')
me_srd.add_constant('paths', 10000)
me_srd.add_constant('frequency', 'w')
me_srd.add_constant('starting_date', me_srd.pricing_date)
me_srd.add_constant('final_date', dt.datetime(2020, 12, 31))

[22]:

srd = dx.square_root_diffusion('srd', me_srd)

Let us have a look at some simulated rate paths.

[23]:

paths = srd.get_instrument_values()

[24]:

plt.figure(figsize=(10, 6))
plt.plot(srd.time_grid, paths[:, :6])

[24]:

[<matplotlib.lines.Line2D at 0x10ba7de10>,
 <matplotlib.lines.Line2D at 0x10c2f6400>,
 <matplotlib.lines.Line2D at 0x10c2f6668>,
 <matplotlib.lines.Line2D at 0x10c2f6898>,
 <matplotlib.lines.Line2D at 0x10c2f6ac8>,
 <matplotlib.lines.Line2D at 0x10c2f6cf8>]
_images/10_dx_interest_rate_swaps_44_1.png

11.5. Interest Rate Swap

Finally, we can model the interest rate swap itself.

11.5.1. Modeling

First, the market environment with all the parameters needed.

[25]:

# market environment for the IRS
me_irs = dx.market_environment('irs', me_srd.pricing_date)
me_irs.add_constant('fixed_rate', 0.01)
me_irs.add_constant('trade_date', me_srd.pricing_date)
me_irs.add_constant('effective_date', me_srd.pricing_date)
me_irs.add_constant('payment_date', dt.datetime(2014, 12, 27))
me_irs.add_constant('payment_day', 27)
me_irs.add_constant('termination_date', me_srd.get_constant('final_date'))
me_irs.add_constant('currency', 'EUR')
me_irs.add_constant('notional', 1000000)
me_irs.add_constant('tenor', '6m')
me_irs.add_constant('counting', 'ACT/360')
# discount curve from mar_env of floating rate

The instantiation of the valuation object.

[26]:

irs = dx.interest_rate_swap('irs', srd, me_irs)

11.5.2. Valuation

The present value of the interest rate swap given the assumption, in particular, of the fixed rate.

[27]:

%time irs.present_value(fixed_seed=True)

CPU times: user 198 ms, sys: 82 ms, total: 280 ms
Wall time: 299 ms

[27]:

482804.6280483039

You can also generate a full output of all present values per simulation path.

[28]:

irs.present_value(full=True).iloc[:, :6]

[28]:
0 1 2 3 4 5
2014-12-27 20919.908406 3765.545880 -10000.000000 20447.662984 15671.171542 -461.905549
2015-06-27 27796.391566 67649.371268 -7952.865109 21069.238749 6547.642190 56254.997722
2015-12-27 72797.267752 65085.851729 -9495.534310 7184.517383 -6249.109421 72603.409748
2016-06-27 146587.773739 60725.900678 -9450.695252 10630.874710 1457.279233 96390.826927
2016-12-27 112006.856512 11685.574401 -9365.011855 36777.060898 40752.233786 42437.807055
2017-06-27 89091.008350 19143.973791 -8652.442452 10394.933700 81112.588234 49316.809090
2017-12-27 153802.099096 37433.554707 -2231.031239 3046.024441 81218.496071 89235.982850
2018-06-27 114033.046751 -4200.273599 8204.181410 86020.232903 92146.494366 44010.774119
2018-12-27 165327.788708 -7186.102024 -8829.883386 46785.300398 171037.865434 14615.688086
2019-06-27 101110.008932 -9038.246291 -6883.724709 29794.380064 183168.704902 -4599.638746
2019-12-27 57890.028467 -8998.424023 -8573.907447 12491.846129 151080.678798 -8960.333669
2020-06-27 95516.235053 -8968.295497 -8963.127240 15234.994497 67282.503413 -7763.697572
2020-12-27 4651.947620 -4500.468014 -6316.280222 39899.010233 107971.761933 -8568.679338

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© Dr. Yves J. Hilpisch | The Python Quants GmbH

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