Introduction

The Hull-White model is financial modeling in Python. It is an ideal of future interest rates in financial mathematics. It is right to the class of no-arbitrage models. Those are capable of appropriate to the latest term structure of interest rates in its best generic development.

The Hull-White model is comparatively direct to translate the mathematical description of the progress of future interest rates onto a tree or frame. Therefore, the interest rate derivatives for example Bermudan swaptions may be valued in the model.

The first Hull-White model was labeled by John C. Hull and Alan White in 1990. That is quite widespread in the market nowadays.

In this article, we will understand the Hull-White model and also do Simulations with QuantLib Python.

Description

We can define the Hull-White Short Rate Model as:

There is a degree of uncertainty among practitioners about exactly that parameters in the model are time-dependent. Similarly, what name to spread over to the model in each case? The most usually known naming convention is the following:

•  has t (time) dependence that is the Hull-White model.
•  And  are both time-dependent — the long Vasicek model.

We use QuantLib to display how to simulate the Hull-White model and examine some of the properties. We import the libraries and set things up as described below:

import
 QuantLib 
as
 ql
import
 matplotlib.pyplot 
as
 plt
import
 numpy 
as
 np
%
 matplotlib inline

• We use the constant for this instance is all well-defined as described below.
• Variables sigma and are the constants that define the Hull-White model.
• We discretize the time span of length thirty years into 360 intervals.
• This is defined by the timestep variable in the simulation.
• We would use a constant forward rate term structure as an input for ease.
• It is the right way to swap with another term structure here.

sigma 
=
 0.1
a 
=
 0.1
timestep 
=
 360
length 
=
 30 
# in years
forward_rate 
=
 0.05
day_count 
=
 ql
.
Thirty360()
todays_date 
=
 ql
.
Date(15, 1, 2015)

ql
.
Settings
.
instance()
.
evaluationDate 
=
 todays_date
spot_curve 
=
 ql
.
FlatForward(todays_date, ql
.
QuoteHandle(ql
.
SimpleQuote(forward_rate)), day_count)
spot_curve_handle 
=
 ql
.
YieldTermStructureHandle(spot_curve)

hw_process 
=
 ql
.
HullWhiteProcess(spot_curve_handle, a, sigma)
rng 
=
 ql
.
GaussianRandomSequenceGenerator(ql
.
UniformRandomSequenceGenerator(timestep, ql
.
UniformRandomGenerator()))
seq 
=
 ql
.
GaussianPathGenerator(hw_process, length, timestep, rng, False)

• The Hull-White process is built bypassing the term structure, a and sigma.
• One has to make available a random sequence generator along with other simulation inputs for example timestep and `length to create the path generator.
• A function to make paths may be written as demonstrated below:

def
 
generate_paths
(num_paths, timestep):
    arr 
=
 np
.
zeros((num_paths, timestep
+
1))
    
for
 i 
in
 range(num_paths):
        sample_path 
=
 seq
.
next()
        path 
=
 sample_path
.
value()
        time 
=
 [path
.
time(j) 
for
 j 
in
 range(len(path))]
        value 
=
 [path[j] 
for
 j 
in
 range(len(path))]
        arr[i, :] 
=
 np
.
array(value)
    
return
 np
.
array(time), arr

• The simulation of the short rates appearance is as follows:

num_paths 
=
 10
time, paths 
=
 generate_paths(num_paths, timestep)
for
 i 
in
 range(num_paths):
    plt
.
plot(time, paths[i, :], lw
=
0.8, alpha
=
0.6)
plt
.
title("Hull-White Short Rate Simulation")
plt
.
show()

Monte-Carlo simulation

• On the other hand, valuing vanilla instruments for example caps and swaptions is valuable mainly for calibration.
• The actual use of the model is to value rather more exotic derivatives for example Bermudan swaptions on a lattice.
• Also, other derivatives in a multi-currency context for example Quanto Constant Maturity Swaps.
• These are explained for instance in Brigo and Mercurio (2001).
• The well-organized and precise Monte-Carlo simulation of the Hull-White model with time-dependent parameters may be easily performed.

For more details visit:https://www.technologiesinindustry4.com/2022/01/the-hull-white-model.html