APR 2018 - Lecture for the course of Applied Statistics for Finance at Università Cattolica del Sacro Cuore of Milan.

1. Part 5: Advanced Topics
Applied Statistics for Finance
Professor Asmerilda Hitaj
asmerilda.hitaj1@unimib.it
April 10, 2018
A. Hitaj ASFF - Part 5 Spring 2018 1 / 27

2. ARCH Processes
Applied Statistics for Finance
Francesco Bianchi
francesco.bianchi04@icatt.it
April 10, 2018
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3. Volatility
Most popular option pricing models, such as Black-Scholes-Merton,
assume that the volatility of the underlying asset is constant.
In practice, the volatility of an asset, like the asset’s price, is a stochastic
variable. Unlike the asset price, it is not directly observable.
”Volatility is a statistical measure of the dispersion of returns for
a given security or market index. Volatility can either be
measured by using the standard deviation or variance between
returns from that same security or market index. Commonly, the
higher the volatility, the riskier the security.”
How historical data can be used to produce estimates of the current and
future levels of volatilities (and correlations).
ARCH and GARCH processes
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4. ARCH and GARCH Model: Returns Dependence
Models:
ARCH: autoregressive conditional heteroscedasticity
GARCH: generalized autoregressive conditional heteroscedasticity
⇒ The distinctive feature of the models is that they recognize that
volatilities and correlations are not constant.
noncostant variances conditional on the past: V (ut|ut−1)
constant unconditional variances: V (ut)
The recent past gives information about the one-period forecast variance.
The main idea behind the ARCH/GARCH model is that the log-returns rt
are usually uncorrelated but there is still dependence.
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5. Volatility (2)
Return: ui = ln
Si
Si−1
= ln
pi
pi−1
Variance rate: σ2
n =
1
m − 1
m
i=1
(un−i − ¯u)2
where ¯u = 1
m un−i
Volatility: σ2
n = σn
⇓
Return (% change): ui =
Si − Si−1
Si−1
=
pi − pi−1
pi−1
Variance rate: σ2
n =
1
m
m
i=1
u2
n−i (5)
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6. Volatility: Weighting Scheme
We want to give more weights to recent data, hence equation (5) becomes:
σ2
n =
m
i=1
αi u2
n−i
where
α is positive: α > 0
less weight is given to older observations: αi < αj when i > j
sum to unity: m
i=1 αi = 1
Further assume that there is a long-run average variance rate
σ2
n = γVL +
m
i=1
αi u2
n−i with γ +
m
i=1
αi = 1
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7. The ARCH(m) Model
The Autoregressive Conditional Heteroskedasticity (ARCH) model was
first developed by Engle in 1982.
The estimate of the variance is based on a long-run average variance and
m observations. The older an observation, the less weight it is given.
ARCH(1): σ2
t = ω + α1 u2
t−1 where ω = γVL
Generalizing: 


ut = σt t
σ2
t = ω +
p
i=1
αi u2
t−i
ARCH term
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8. The GARCH(p, q) Model
The Generalized Autoregressive Conditional Heteroskedasticity (GARCH)
model was first introduced by Bollerslev in 1986.
The simplest version of the model is the GARCH(1,1) one, where the
variance rate is calculated from a long-run average variance rate, VL, as
well as from σn−1 and un−1. Defined as:
σ2
t = γVL + α1u2
t−1 + β1σ2
t−1 = ω + α1u2
t−1 + β1σ2
t−1
where α1 + β1 < 1 in order to ensure stability of the process.
A special case of the GARCH(1,1) model is the Exponentially weighted
moving average (EWMA) model, where γ = 0, α1 = 1 − λ and β1 = λ.
σ2
t = λσ2
t−1 + (1 − λ)u2
t−1
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9. The GARCH(p, q) Model (Cont.)
The Generalized Autoregressive Conditional Heteroscedasticity
(GARCH(p,q)) model is defined by the following system of equations:



ut = σt t
σ2
t = ω +
p
i=1
αi u2
t−i
ARCH term
+
q
j=1
βj σ2
t−j
GARCH term
where ω > 0, αi ≥ 0 and βj ≥ 0 and αi + βi < 1 in order to ensure the
finiteness of the unconditional variance.
How can we estimate ω, α and β? ⇒ Maximum Likelihood Estimation
Basically, choosing values for the parameters that maximize the chance (or
likelihood) of the data occurring.
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10. The GARCH(p, q) Model (Cont.)
The GARCH(p, q) process is strictly related to the ARMA process since
the squared of residual process u2
t is an ARMA(q, p − 1) process:
u2
t = α0 +
max (p,q)
i=1
(αi + βi )u2
t−i + ηt
q
j=1
βj ηt−j
where ηt = u2
t − σ2
t . The ηt is a martingale difference series (i.e.
E(|ηt|) < +∞ and E(ηt|Ft−1) = 0).
Mean Reversion: The GARCH (p, q) model recognizes that over time
the variance tends to get pulled back to a long-run average level of VL.
The process is equivalent to a model where the variance V follows the
stochastic process
dV = α(VL − V )dt + ξV dz
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11. Exercises
Exercise 1
Suppose that a GARCH(1, 1) model is estimated from daily data as
σ2
n = 0.000002 + 0.13 u2
n−1 + 0.86 σ2
n−1
Find the value of the long-run variance rate (VL).
Remember that ω = γVL. ⇒ γ = 1 − α − β and VL =
ω
γ
Solution:
γ = 1 − α − β = 1 − 0.13 − 0.86 = 0.01
VL =
ω
1 − α − β
=
ω
γ
=
0.000002
0.01
= 0.002
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12. FTSEMIB.MI (2010-2017)
2010 2012 2014 2016 2018
12000140001600018000200002200024000
FTSEMIB.MI
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13. FTSEMIB.MI: Log Returns
2010 2012 2014 2016 2018
-0.10-0.050.000.050.10
FTSEMIB.MILogReturns
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14. FTSEMIB.MI: R Code
install.packages("tseries")
library(tseries)
library(PerformanceAnalytics)
start <- "2010-01-01"
end <- "2017-12-31"
FTSEMIB.MI <- get.hist.quote("FTSEMIB.MI", quote="Close", start, end)
View(FTSEMIB.MI)
log_returns_FTSEMIB.MI <- apply(log(FTSEMIB.MI), 2, diff)
plot(FTSEMIB.MI, main="The level series")
plot(log_returns_FTSEMIB.MI, main="The return series", type="l")
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15. Autocorrelation in Returns
There is usually a certain form of heteroskedasticity in a series of
returns.
High volatility today can lead to high volatility tomorrow.
Variances today and tomorrow are somehow related.
This form of heteroskedasticity implies that there will be
autocorrelation in squared returns. → ARCH Effect
To check the ARCH effect we use the R package FinTS. Two tests are
available in the package: Ljung-Box test and Lagrange Multiplier test.
Further packages can be used to implement this tests: stats and fGarch.
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16. Returns Autocorrelation in R
We study the autocorrelation between log returns
start <- "2010-01-01"
end <- "2017-12-31"
FTSEMIB.MI <- get.hist.quote("FTSEMIB.MI", quote="Close", start, end)
log_returns_FTSEMIB.MI <- apply(log(FTSEMIB.MI), 2, diff)
log_returns_FTSEMIB.MI <- na.omit(log_returns_FTSEMIB.MI)
num_log_returns_FTSEMIB.MI <- as.numeric(log_returns_FTSEMIB.MI)
acf(num_log_returns_FTSEMIB.MI, lag.max = 6)
acf(num_log_returns_FTSEMIB.MI^2, lag.max = 6)
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17. Returns Autocorrelation in R: ACF plot
0 1 2 3 4 5 6
0.00.20.40.60.81.0
ACF
Figure: Uncorrelated FTSEMIB.MI Returns
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18. Returns Autocorrelation in R: ACF plot (2)
0 1 2 3 4 5 6
0.00.20.40.60.81.0
ACF
Figure: Correlated FTSEMIB.MI Squared Returns
→ ARCH Effect
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19. ARCH Effect removal
library(fGarch)
Garch11 <- garchFit( ~ garch(1,1), trace = F,
data = num_log_returns_FTSEMIB.MI)
class(Garch11)
summary(Garch11)
residuals_Garch11 <- residuals(Garch11)
plot(residuals_Garch11)
vol_Garch11 <- volatility(Garch11)
plot(vol_Garch11)
stand.res <- residuals_Garch11/vol_Garch11
acf(stand.res, lag.max = 6)
acf(stand.res^2, lag.max = 6)
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20. Daily Volatility of FTSEMIB.MI
2010 2012 2014 2016 2018
0.010.020.030.04
Volatility
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21. ARCH Effect removal (1)
0 1 2 3 4 5 6
0.00.20.40.60.81.0
ACF
Figure: Uncorrelated FTSEMIB.MI Returns
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22. ARCH Effect removal (2)
0 1 2 3 4 5 6
0.00.20.40.60.81.0
ACF
Figure: Uncorrelated FTSEMIB.MI Squared Returns
→ ARCH Effect removed!
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23. fGarch Package: Results
The results obtained from the analysis can be used to:
Risk quantification: VaR and ES
Correlations play a key role in the calculation of VaR
covn = ω + α xn−1 yn−1 + β covn−1
Portfolio Selection: multivariate GARCH
Option Pricing: need to identify an equivalent Martingale measure
(see Duan (1997))
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24. The COGARCH Model
A further implementation of the GARCH Model is the Continuos GARCH
Model (COGARCH) firstly introduced by Kl¨uppelberg in 2004.
Let Lt be a pure jump L´evy process with finite variation. We define Gt as
a COGARCH(p, q) process with q ≥ p if it satisfies the following system
of stochastic differential equations:



dGt =
√
VtdLt withG0 = 0
Vt = a0 + a Yt−
dYt = BYt−dt + a0 + a Yt− d [L, L]
(d)
t
Where (Vt)t≥0 is a CARMA(q, p − 1) process driven by the discrete part
of the quadratic variation of the L´evy process (Lt)t≥0.
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25. COGARCH Model Key Features
Why choosing a continuous GARCH model?
As in GARCH models:
ARCH Effect
Heavy tails
Moreover:
High frequency and irregularly spaced data management
No missing values approximation
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26. COGARCH.rm Package
Dataset Download
COGgetdata
Market Data
Leader Selection
COGleader
Univariate Risk
Measure Analysis
estCOGuniv
SimBoot
univariateRM
Portfolio
Optimization
portfolioCOG
Univariate
Out-of sample
Forecast
forecastCOG
Multivariate
Out-of-sample
Forecast
forecastCOG
Dependencies: Yuima, fastICA, quantmod, cluster, rugarch
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27. References
[1] Engle R. (1982) ”Autoregressive Conditional Heteroskedasticity with
Estimates of the Variance of UK Inflation”. Econometrica, 50: 987:1008.
[2] Bollerslev T. (1986). ”Generalized Autoregressive Condtional
Heteroskedasticity”. Journal of Econometrics, 31: 307:327.
[3] Duan, J. (1997). ”Augmented GARCH (p,q) process and its diffusion limit”.
Journal of Econometrics, 79, issue 1, p. 97-127.
[4] Kl¨uppelberg C., Maller R. and Lindner A. (2004). ”A continuous time garch
process driven by a L´evy process: stationarity and second-order behaviour.
Journal of Applied Probability.”
[5] Bianchi F., Mercuri L. and Rroji E. (2016). ”Measuring Risk with
Continuous Time Generalized Autoregressive Conditional Heteroscedasticity
models”. SSRN.
[6] Bianchi F., Mercuri L. and Rroji, E. (2017). COGARCH.rm: Portfolio
selection with Multivariate COGARCH(p,q) models. R package version 0.1.0.
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