Option Pricing with Cointegrated Assets
Notebook by Marco Tavora
Introduction
A simple model for option hedging based on cointegrated vectors is:
The inclusion of a stationary process z presupposes that in the long-run, stationarity is eventually achieved. The volatility is a simple GARCH(1,1) process. To control for growth and violation of mean reverting behavior, we added a deterministic term to z process.
Conceptual Bird's Eye Review
Vector Error Correction Model VECM
The VECM model is given by:
where the components of the y vector are the SPY and the SHY series. If both series are cointegrated, this information is included in the model via error correction terms.
# install.packages('ggplot2')
# install.packages('xts')
# install.packages('quantmod')
# install.packages('broom')
# install.packages('tseries')
# install.packages("kableExtra")
# install.packages("knitr")
# install.packages("vars")
# install.packages("urca")
To load the data we will use the library quantmod which contains the function getSymbols.
From the documents
getSymbols is a wrapper to load data from various sources, local or remote.
In our case we will load data from Yahoo Finance.
rm(list=ls())
library(tseries)
library(dynlm)
library(vars)
library(nlWaldTest)
library(lmtest)
library(broom)
library(car)
library(sandwich)
library(knitr)
library(forecast)
library(quantmod)
setSymbolLookup(SHY='yahoo',SPY='yahoo')
getSymbols(c('SHY','SPY'))
Defining y1 and y2 as the adjusted prices and joining them:
y1 <- SPY$SPY.Adjusted
y2 <- SHY$SHY.Adjusted
time_series <- cbind(y1, y2)
print('Our dataframe is:')
colnames(time_series) <- c('SHY', 'SPY')
head(time_series)
We can check for stationary of the series individually:
adf.test(time_series$SHY)
adf.test(time_series$SPY)
The output is:
Augmented Dickey-Fuller Test
data: time_series$SHY
Dickey-Fuller = -1.5363, Lag order = 14, p-value = 0.7747
alternative hypothesis: stationary
Augmented Dickey-Fuller Test
data: time_series$SPY
Dickey-Fuller = -3.7478, Lag order = 14, p-value = 0.02158
alternative hypothesis: stationary
The p-values are obtained using:
print('p-values:')
adf.test(time_series$SHY)$p.value
adf.test(time_series$SPY)$p.value
giving:
[1] "p-values:"
0.774665887814869
0.0215848302364735
Plotting the series we obtain:
par(mfrow = c(2,1))
plot.ts(time_series$SHY,
type='l')
plot.ts(time_series$SPY,
type='l')
Johansen Test for Cointegration
The most well known cointegration test is the Johansen test which estimates the VECM parameters and determines whether the determinant of
br>
is zero or not. If the determinant is not zero, the series are cointegrated.
library(urca)
johansentest <- ca.jo(time_series, type = "trace", ecdet = "const", K = 3)
summary(johansentest)
The result is:
######################
# Johansen-Procedure #
######################
Test type: trace statistic , without linear trend and constant in cointegration
Eigenvalues (lambda):
[1] 2.248927e-02 3.620157e-03 4.437196e-18
Values of teststatistic and critical values of test:
test 10pct 5pct 1pct
r <= 1 | 10.65 7.52 9.24 12.97
r = 0 | 77.46 17.85 19.96 24.60
Eigenvectors, normalised to first column:
(These are the cointegration relations)
SHY.l3 SPY.l3 constant
SHY.l3 1.00000 1.0000 1.00000
SPY.l3 -84.59851 163.3147 -8.78908
constant 6927.78683 -12622.5372 543.67420
Weights W:
(This is the loading matrix)
SHY.l3 SPY.l3 constant
SHY.d 2.147229e-05 1.049741e-04 -1.531138e-17
SPY.d 2.088562e-05 -1.870590e-06 1.148836e-17
The lines r=0 and r<= 1 are the actual results of the test. More specifically:
- line r=0: these are the results of the hypothesis test with null hypothesis
$r=0$ . More concretely, this test checks if the matrix has zero rank. In the present case the hypothesis is rejected since the test variable is well above the$1%$ value; - line r<=1: these are the results of the hypothesis test
$r\le 1$ . Now since the test value is below the$1%$ value value we fail to reject the null hypothesis. Hence we conclude that the rank of$\alpha \beta$ is 1 and therefore the two series are cointegrated and we can use the VECM model.
Note that if hypotheses were reject we would have r=2 corresponding to two stationary series.
Fitting the VECM
t <- cajorls(johansentest, r = 1)
t
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