VolatilityRisk1

Outline

• Introduction to Volatility Modelling
• Exploring Financial Time Series
• Autoregressive Integrated Moving Average (ARIMA) Modelling
• Generalized Autoregressive Conditional Heteroscedasticity (GARCH) Modelling
• Comparison of ARIMA and GARCH Models
• Advanced Topics in Volatility Modelling
• Conclusion and Future Directions

Introduction to Volatility Modelling

• Definition of volatility and its importance in financial markets
• Types of volatility (e.g., historical, implied, realized)
• Overview of volatility modeling techniques

Definition

• Volatility is an important concept in financial markets because it is directly related to risk. Higher volatility generally indicates higher risk, as it means that the asset’s price is more likely to move dramatically in either direction, potentially resulting in greater losses or gains for investors.

• Volatility is also important for financial modeling and analysis, as it helps to inform decisions related to risk management, portfolio construction, and asset valuation. Volatility models, such as ARIMA and GARCH models, are used to forecast future volatility based on historical patterns, and this information can be used to inform trading strategies and investment decisions.

Types of Volatility

• Historical Volatility: Historical volatility measures the actual volatility of an asset over a specific historical period, usually calculated as the standard deviation of the asset’s returns over that period.

• Implied Volatility: Implied volatility is a measure of the expected future volatility of an asset, as implied by the current market price of its options contracts. It is calculated using an options pricing model and represents the market’s consensus on how volatile the asset is likely to be in the future.

• Realized Volatility: Realized volatility is a measure of the actual volatility of an asset over a specific recent period, usually calculated as the standard deviation of its returns over that period. It differs from historical volatility in that it uses more recent data to estimate volatility.

Volatility Modelling Techniques

• ARIMA: Autoregressive Integrated Moving Average
• GARCH: Generalized Autoregressive Conditional Heteroskedasticity
• Exponential Smoothing Models
• Machine Learning Techniques
• Implied Volatility Calculation from Option Pricing Models

Important Papers to Ref

• Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. The review of economics and statistics, 542-547.
• Dumas, B., Fleming, J., & Whaley, R. E. (1998). Implied volatility functions: Empirical tests. The Journal of Finance, 53(6), 2059-2106.
• Corsi, F. (2009). A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics, 7(2), 174-196.
• Petropoulos, F., Apiletti, D., Assimakopoulos, V., Babai, M. Z., Barrow, D. K., Taieb, S. B., … & Ziel, F. (2022). Forecasting: theory and practice. International Journal of Forecasting.
• Masini, R. P., Medeiros, M. C., & Mendes, E. F. (2023). Machine learning advances for time series forecasting. Journal of economic surveys, 37(1), 76-111.

Steps in Financial Time Series Forecasting

• Data Retrieval and Data Exploration
• Data Preprocessing
• Modelling and Model Diagnostics
• Model Evaluation: In-sample and Out-sample

Data Retrieval and Data Exploration

#Clear working environment 
#rm(list = ls()) 

#Install packages
#install.packages("quantmod")
library(quantmod)

#getSymbols(c("","",...)) -> obtain historical quotes 
getSymbols(c("AAPL", "TSLA"))

s1p <- AAPL[,6]
s1r <- dailyReturn(s1p)

s2p <- TSLA[,6]
s2r <- dailyReturn(s2p)

Data Retrieval and Data Exploration

##    vars    n  mean    sd median trimmed   mad  min    max  range skew kurtosis
## X1    1 4129 43.66 49.22  23.12   34.63 25.36 2.37 180.43 178.06 1.42     0.63
##      se
## X1 0.77

##    vars    n  mean    sd median trimmed   mad  min    max  range skew kurtosis
## X1    1 3251 62.41 96.26  16.54   41.35 13.19 1.05 409.97 408.92 1.69     1.48
##      se
## X1 1.69

##    vars    n mean   sd median trimmed  mad   min  max range  skew kurtosis se
## X1    1 4129    0 0.02      0       0 0.01 -0.18 0.14  0.32 -0.13      5.8  0

##    vars    n mean   sd median trimmed  mad   min  max range skew kurtosis se
## X1    1 3251    0 0.04      0       0 0.03 -0.21 0.24  0.45 0.32     4.97  0

Data Retrieval and Data Exploration

#install.packages("psych")
library(psych)
describe(s1p)
describe(s2p)
describe(s1r)
describe(s2r)

Data Preprocessing

• You may use Box-Cox transformation
• Taking differences
• Remove duplicates and outliers
• Take care of the missing data
• Our data is quite clean, so I’m not going to do them here (except taking first difference by computing the stock returns)

First Model: Autoregressive Integrated Moving Average (ARIMA)

• Explanation of ARIMA models and their components (AR, I, MA)
• Time series decomposition and identification of ARIMA models
• Parameter estimation and model selection using AIC and BIC
• Interpretation and evaluation of ARIMA models
• Application of ARIMA modeling to financial time series data

Explanation of ARIMA models and their components (AR, I, MA)

• Autoregressive (AR) component: The AR component of an ARIMA model refers to the dependence of the current value of the series on its past values. Order = “p”

• Integrated (I) component: The I component of an ARIMA model refers to the degree of differencing needed to make the time series stationary. Stationary time series have a constant mean and variance over time, making them easier to model and analyze. Order = “I”

• Moving Average (MA) component: The MA component of an ARIMA model refers to the dependence of the current value of the series on the past errors (the difference between the predicted and actual values) of the series. Order = “q”

Notes on Stationarity

• A stationary time series is one whose properties do not depend on the time at which the series is observed. It does not matter when you observe it, it should look much the same at any point in time.

• In practice, we would take the difference until we achieve the stationarity (thus, the I component)

ARIMA Equations

• Once you have achieved a stationary time series, you can apply ARMA equations
• AR(p): \(r_t = \alpha_1 r_{t-1} + \alpha_2 r_{t-2} + ... + \epsilon_t\)
• MA(q): \(r_t = \epsilon_t + \beta_1 \epsilon_{t-1} + ... + \beta_q \epsilon_{t-q}\)
• ARMA(p,q): \(r_t = \alpha_1 r_{t-1} + \alpha_2 r_{t-2} + ... + \epsilon_t + \beta_1 \epsilon_{t-1} + ... + \beta_q \epsilon_{t-q}\)
• We will choose the model with the best fit (low AIC and/or BIC) with significant parameters
• Don’t forget to check the \(\epsilon_t\) after the estimation to see if it is indeed a white noise process

Autocorrelation Function and Partial Autocorrelation Function

• Why: We need to check correlations among past lags of the variable of interest in order to indicate the lag variables in the model.
• ACF plot shows the autocorrelations which measure the relationship between \(y_t\) and \(y_{t-k}\) for different values of k
• Partial Autocorrelation Function measures the relationship between \(y_t\) and \(y_{t-k}\) for different values of k after removing the effects of lags 1,2,…,(k-1)

ACF Plots

par(mfrow = c(2,2))
acf(s1p)
acf(s1r)
acf(s2p)
acf(s2r)

PACF Plots

par(mfrow = c(2,2))
pacf(s1p)
pacf(s1r)
pacf(s2p)
pacf(s2r)

Parameter estimation and model selection using AIC and BIC

• One common method for estimating ARIMA model parameters is maximum likelihood estimation (MLE), which involves selecting the parameters that maximize the likelihood function of the observed data.

• One approach to model selection is to use information criteria such as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).

• \(AIC = -2*log(likelihood) + 2*k\)

• where k is the number of parameters in the model.

• \(BIC = -2*log(likelihood) + log(n)*k\)

• where n is the number of observations in the time series.

• In general, lower values of AIC and BIC indicate a better model fit.

Model Estimation

# ARIMA
res1 <- arima(s1p, order = c(1,0,0))
res1

res2 <- arima(s1p, order = c(1,1,1))
res2

res3 <- arima(s1p, order = c(2,1,1))
res3

# AUTO.ARIMA
# install.packages("forecast")
library(forecast)
res4 <- auto.arima(s1p)
res4

Evaluation of ARIMA models

• Interpret the model parameters and their significance.

• Measures of forecasting accuracy, such as mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), or mean absolute percentage error (MAPE)

• The residuals should be normally distributed and uncorrelated, with no remaining patterns or trends. Apply Ljung-Box Test, Augmented Dickey-Fuller Test, QQ Plot, Histogram, Jacque-Bera Test etc. on the residuals.

Residuals Diagnostics

# We need tseries library for some test statistics
# install.packages("tseries")
library(tseries)

# Obtain residuals and run some diagnostics 
err4 <- res4$residuals
fit4 <- res4$fitted

# Calculate model accuracy measures
describe(err4)
mse4 <- mean(err4^2)
mad4 <- mean(abs(err4))
mape4 <- mean(abs(err4/fit4))

Residuals Diagnostics: What the error should look like?

Residuals Diagnostics: What the error should look like?

Residuals Diagnostics

# Run a few more tests 
# Ljung-Box Test -> H1: Error is not stationary 
Box.test(err4, lag = 16, type = "Ljung")

# ADF Test -> H1: Error has no unit root 
adf.test(err4)

# Jarque-Bera Test -> H1: Error is not normal 
jarque.bera.test(err4)

Residuals Diagnostics: What the test results should be

# Run a few more tests 
# Ljung-Box Test -> H1: Error is not stationary 
Box.test(err, lag = 16, type = "Ljung")

# ADF Test -> H1: Error has no unit root 
adf.test(err)

# Jarque-Bera Test -> H1: Error is not normal 
jarque.bera.test(err)

Applcation: Stock Price and Stock Returns Forecasting in R

Second Model: GARCH

• Explanation of GARCH models and their components (ARCH, GARCH)
• Parameter estimation and model selection using AIC and BIC
• Interpretation and evaluation of GARCH models
• Application of GARCH modeling to financial time series data

Data Exploration: Returns

Data Exploration: Squared Returns

Explanation of GARCH models and their components (ARCH, GARCH)

• You can think of the ARIMA and GARCH models as a building block like this:

$$

\[\begin{aligned} r_t &= `ARIMA' + \sigma_t\epsilon_t \\ \sigma_t &= `GARCH' \\ \epsilon_t &\rightarrow `Distribution' \end{aligned}\]

$$

Explanation of GARCH models and their components (ARCH, GARCH)

• GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models are a type of time series model that are commonly used to analyze and model volatility in financial markets. The GARCH model consists of two components: the ARCH component and the GARCH component.

• The ARCH component is defined as:

\[ \mathrm{ARCH}(p) : \sigma_t^2 = \omega + \sum_{i=1}^p \alpha_i \epsilon_{t-i}^2 \]

where \(\sigma_t^2\) is the conditional variance at time \(t\), \(\omega\) is a constant, \(\alpha_i\) are the ARCH parameters, and \(\epsilon_{t-i}^2\) are the squared residuals at time \(t-i\).

Explanation of GARCH models and their components (ARCH, GARCH)

The GARCH component is defined as:

\[ \mathrm{GARCH}(q) : \sigma_t^2 = \omega + \sum_{i=1}^p \alpha_i \epsilon_{t-i}^2 + \sum_{j=1}^q \beta_j \sigma_{t-j}^2 \]

where \(\beta_j\) are the GARCH parameters, and \(\sigma_{t-j}^2\) are the past conditional variances.

Together, the ARCH and GARCH components form the GARCH(p, q) model, which can be expressed as:

\[ \mathrm{GARCH}(p, q) : \sigma_t^2 = \omega + \sum_{i=1}^p \alpha_i \epsilon_{t-i}^2 + \sum_{j=1}^q \beta_j \sigma_{t-j}^2 \]

Parameter estimation and model selection using AIC and BIC

• You need ugarchspec and ugarchfit functions

# install.packages("rugarch")
library(rugarch)

gspec1 <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
                    mean.model = list(armaOrder = c(0, 0)),
                    distribution.model = "std")

# You can use different variance, mean and distribution models

gfit1 <- ugarchfit(gspec1, s2r)
gfit1

Residuals Diagnostics

Residuals Diagnostics

# Run a few more tests 
# Ljung-Box Test -> H1: Error is not stationary 
Box.test(gerr1, lag = 16, type = "Ljung")

# ADF Test -> H1: Error has no unit root 
adf.test(gerr1)

# Jarque-Bera Test -> H1: Error is not normal 
jarque.bera.test(gerr1)

Application: Returns Forecasting

• You need ugarchforecast function

Application: Returns Simulation

• You need ugarchsim function: n.sim = steps ahead, m.sim = number of simulated paths

Extension of GARCH

• Exponential GARCH (EGARCH) models: These models allow for asymmetric volatility, where negative and positive shocks have different impacts on volatility.

• Threshold GARCH (TGARCH) models: These models allow for changes in the volatility structure based on a threshold value. When the time series values exceed the threshold, the volatility structure changes.

• Component GARCH (CGARCH) models: These models decompose the volatility of a time series into two components: a long-term component and a short-term component.

• GARCH-in-mean (GARCH-M) models: These models allow the conditional variance to enter the conditional mean equation.

• Markov-switching GARCH (MS-GARCH) models: These models allow for changes in the volatility structure based on a hidden Markov process.

DIY

• Try ARIMA and/or GARCH with different stocks prices and returns i.e. TSLA
• Try ARIMA and/or GARCH with exchange rates i.e. THBUSD=X
• Try ARIMA and/or GARCH with cryptocurrencies i.e. BTC-USD

getSymbols('TSLA')
getSymbols('BTC-USD')
getSymbols('USDTHB=X')