🐍 Two factor Vasicek model 🐍

A simple model for calculating the nominal interest rates. Used to add inflation to the simulation of interest rates. The model has two sources of randomness (Two correlated Brownian motions)

Problem

When modelling the nominal rate, both the real rate of return and the inflation should be considered. The correlation between them means that one should use a multifactor model as opposed to two independent models. Additionally, there is a robust body of literature showing that both real rates and the inflation are mean-reverting

Solution

The simplest model for modelling real rates and inflation together is the multifactor Vasicek model https://en.wikipedia.org/wiki/Vasicek_model. The Vasicek model is a short rate model describing the evolution of rates. Both the real rate process and the inflation rate process are assumed to follow a Vasicek model. The movement of the two curves is given by a two dimensional correlated Brownian motion

Input

Vasicek model simulator:

• r0 ... Starting annualized real rate and inflation rate. ex. if the annualized real rate is 1.4% and inflation is 6%, then r0 = [0.014, 0.06]
• a ... mean reversion speed for the real and inflation process. ex. if the reversion factor is 0.8 for real rates and 1 for inflation, a = [0.8, 1]
• b ... long-term mean level for the real and inflation process. ex. if the long term real rate is 1% and long term inflation is 1.5%, b = [0.01, 0.015]
• sigma ... instantaneous volatility of the real and inflation process. ex. volatility of the real rate process is 5% and inflation process is 4%, sigma = [0.05, 0.04]
• rho ... correlation between the stochastic noise that generates the two processes. Ex. if the calculated correlation coefficient is 0.t, rho = 0.6
• T ... modelling time horizon. Ex. if time horizon is 25 years, T = 25
• dt ... time increments. ex. time increments are 6 months, dt = 0.5

Vasicek model pricing:

• nScen ... integer, number of simulations of the Monte Carlo method

Vasicek model calibration: TBD

Output

Vasicek model simulator:

• interest_rate_simulation is a pandas dataframe with one sample path generated by the model. One for the real rate process and the other for the nominal rates (real rate + inflation rate)

Vasicek model pricing:

• Price of a zero coupon bond with maturity T based on the model. The technique used is Monte Carlo with 1000 scenarios and numeric integration

Vasicek model calibration: TBD

Getting started

import numpy as np
import pandas as pd
import datetime as dtm

from Vasicek import BrownianMotion
from Pricing import ZeroCouponBond

from IPython.display import display
import matplotlib.pyplot as plt

# Vasicek model simulator
r0 = [0.014, 0.06]   # Starting annual real rate and annual inflation rate
a = [0.8, 1]         # mean reversion speed for real rate and inflation
b = [0.01, 0.015]    # long term trend for real rate and inflation
sigma = [0.05, 0.04] # annualized volatility of real rate and inflation process
rho = 0.6            # correlation
T = 25               # time horizon
dt = 0.5             # time increment

brownian = BrownianMotion()
interest_rate_simulation = brownian.simulate_Vasicek_Two_Factor(r0, a, b, sigma, rho, T, dt)

display(interest_rate_simulation)

interest_rate_simulation.plot(figsize = (15,9), grid = True)
plt.legend()
plt.show()

# Vasicek model pricing of a Zero-Coupon Bond
nScen = 1000
zero_coupon_bond = ZeroCouponBond(1)
zero_coupon_bond.price(r0, a, b, sigma, rho, T, dt, nScen)
print(zero_coupon_bond._price)

# Calibration of the Vasicek model
# Defining a zero curve for the example
# Dates = [[2010,1,1], [2011,1,1], [2013,1,1], [2015,1,1], [2017,1,1], [2020,1,1], [2030,1,1]]
# curveDates = []
# for date in Dates:
#     curveDates.append(dtm.date(date[0],date[1],date[2]))
# 
# zeroRates = np.array([1.0, 1.9, 2.6, 3.1, 3.5, 4.0, 4.3])/100

# plt.figure(figsize = (15,9))
# plt.plot(curveDates,zeroRates)
# plt.title('Yield Curve for ' + str(curveDates[0]))
# plt.xlabel('Date')
# plt.ylabel('Rate')
# plt.show()
# STILL TO DO