Financial Time Series Volatility 201 (Excel)

NumXL Tips and Hints – Volatility 201 ‐1‐ © Spider Financial Corp, 2012

Volatility201This is the third entry in our ongoing series on volatility modeling. In an earlier issue, we introduced the

broad concept of volatility in financial time series, defined its general characteristics (e.g. clustering,

mean‐reversion) and identified important volatility terms in financial time series. We explored holding

periods, volatility scaling, the serial correlation assumption, multi‐period volatility (i.e. term structure)

and discussed a few non‐parametric methods to estimate volatility using historical data.

In this issue, we’ll take the prior discussion further and develop an understanding of ARCH volatility

modeling.

Whyshouldyoucare?Once again, the concepts discussed here are pivotal to a solid understanding of financial time series

volatility.

BackgroundLet’s consider the dependent asset’s log return { }tr time series. First, let’s model the return as the sum

of two components:

t t tr a

Where:

t is the conditional mean (non‐stochastic) component

ta is the innovation, error‐term or shock (stochastic) component

E[ ] 0ta

2Var[ ] Var[ ]t t tr a

Cov[ , ] 0t t ka a

To compute the multi‐period volatility:

1

1 1 1 1

Var[ ] Var[ ] Var[ ] 2 Cov( , )k k k k

t t k t i t i t i t ji i i j i

r r r r r

And

2( , ) [( ) ( )] [ ] [ ] [ ]t i t j t i t j t i t j t i t iCov r r E r r E r r E r E r

Where:


is the unconditional (long‐run) mean of the time series

But,

[ ] [( )( )]t i t j t i t i t j t j t i t jE r r E a a

Thus:

2( , ) ( )

( , ) ( )( )

t i t j t i t j t i t j

t i t j t i t j

Cov r r

Cov r r

In sum, the covariance is a function of the conditional mean and unconditional mean.

The multi‐period volatility (term structure) depends not only on the conditional volatility of each period,

but on the conditional mean as well.

Assumption 1: Let’s assume 1 2 ...t t t t kk

'

t t

t t t

r a

r r a

And

2 2 2Var[ ] [( - ) ]=E[ ]t t t tr E r a

Note: The assumption is not contrary to what we see in financial time series, as the majority of time

series don’t possess significant mean or any serial correlation; at the same time, they all exhibit time‐

varying volatility.

Next, let’s assume the innovation, error term or shock term can be represented as follows:

t t ta

Where:

t is the conditional volatility (scalar) at time t

t is the random variable with zero mean ( [ ] 0tE ) and unit variance ( 2Var[ ] [ ] 1t tE )

t is serially uncorrelated, but still dependent (higher‐order).


Assumption 2: let’s assume the random variable ( t ) is identically distributed over time (not necessarily

Gaussian).

Definition: Let’s define tZ as the squared of the mean‐adjusted asset’s returns:

2 2( )t t tZ r a

And

2[ ]t tE Z

Let’s now examine the methods used in earlier issues to estimate volatility:

Equal weighted moving standard deviation:

2

2 2 21

1 1

( )1 1

1 1 1

m

t i m mi

t t i t ii i

ra Z

m m m

Exponential weighted moving average (EWMA):

2 2 2 21

0 0

(1 ) (1 ) (1 )i it t t t i t i

i i

a a Z

ProbabilityDistributionof tZ

So far, we have not assumed any functional form for the probability distribution of tZ , but here’s a few

observations about the candidate distribution function:

1. ( 0) 0P Z (i.e. 0tZ )

2. ( )P Z is asymmetric

3. ( )P Z is positively (aka right) skewed

4. ( )P Z is the distribution of the squared innovations or shocks

Derivation

~ (.)ta

Let (.) be the probability density function with zero mean and 2t variance.


( ) ( ) (1 ( ))

( ) ( )( )

2

t tP Z z H a z H a z

P z zp z

z z

Assume (.) is a symmetrical distribution.

( )

( )z

p zz

Case1: Let’s use the standardized residuals of a Gaussian distribution.

2

2

2 221

1~ (0,1)

2

1~ (.) (.)

2

t

t

t

Z

t t

t

e

Z eZ

The distribution of the squared values of a Gaussian distributed random variable is Chi‐square with one

degree of freedom.

2 2 2 2 2

2 4 2 4 4

[ ] [ ]

[ ] [ ] 2 2

t t t

t t t t t

t t t t t

a

E a E

Var a Var

Case2: Let’s use the standardized residuals of the student’s t distribution.

12 2

2

1

22

2

1( ) ( 1)2( ) 1

( )2

1( ) ( 1)2( ) 1

( )2

tt

tt t

t

p

Zp Z

Z

Alright, the equation is getting a bit complicated here, but the general principles are applicable to how

we related the conditional probability distribution of the squared time series with the original

conditional probability distribution.


2{ }ta ModelingSo far, we have examined the squared innovations properties (i.e. at a single time instance) at a given

time instance (i.e. t), but how do we describe the 2{ }t tZ a evolution over time?

Similar to ARMA/ARIMA modeling, we examine the ACF/PACF correlogram for the squared time series in

an attempt to identify a dependency between lagged time series, and propose a model.

There are two main categories of statistical volatility models: (1)

‐ Deterministic form ‐ exact function to govern the evolution of volatility (e.g. ARCH, GARCH,

EGARCH, etc.)

2 21( | )t t tf F

‐ Stochastic form – use of a stochastic equation, i.e. allowing a innovation/shock term in the

volatility equation (e.g. stochastic volatility model)

2 21( | )t t t tf F

The conditional volatility values are computed indirectly, not directly observed, which further

complicates the process.

ARCHModelThe first model that provides a systematic framework for volatility modeling is Engle’s ARCH model

(1982). This is a good model to start with due to its simplicity and relevance to other models.

2 2 2 21 1 1

...

~ i.i.d~ (0,1)

t t t t

m

t o t m t m o i t ii

t

r a

a a a

The ARCH model is an AR(p) for 2{ }t tZ a times series, but without the error terms or the shocks.

Alternatively, we can view the ARCH model as a weighted moving average of the squared time series

(WMA) with a constant.

The coefficient’s value must meet some regulatory requirement to ensure that (1) conditional variance

is always positive, and (2) the unconditional variance is finite and positive.


2 2

1

m

t o i t iia

Volatility clustering: the ARCH model captures the volatility clustering observed in assets returns: a large

past‐squared shock 1{ }mt i ia implies a large conditional variance ( 2

t ) for the mean‐corrected return ta .

Consequently, ta tends to be followed by a large value (in absolute terms) due to the large variance,

and vice versa for smaller shocks.

ARCH(1)

2 21 1

2 2 21 1 1

2 2 2 2 22 1 1 1 1 1 1

2 2 3 23 1 1 1

2 2 3 1 21 1 1 1 1

2

1

( )

...

...

1

t o t

t o t o t

t o t o o t o o t

t o o o t

k kt k o o o o o t

ot kk

a

E a

E a

Thus, for a positive conditional variance and a finite unconditional variance, then 0o and 10 1 .

ModelparametersIn the earlier issue (volatility 101), we did not assume any distribution for the time series and thus used

the root mean square error as our utility function, searching for a set of parameters’ values that

minimize the RMSE .

In the ARCH model, the mean‐corrected returns ( ta ) are inter‐dependent (e.g. clustering) and are not

identically distributed, so how do we go about estimating an efficient set of values for its parameters?

~ (0,1)

~ i.i.d

tt

t

t

aN

And

2

221 1 1 2

1

2 22

1 1 1 2 21 1

1( , ,..., | , ,..., )

2

1 1( , ,..., | , ,..., ) ln(2 ) ln(2 ) ln

2 2 2

t

t

aT

T T o mt m t

T Tt t

T T o m tt m t mt t

LF a a a e

a aLLF a a a


For an initial set of 1{ , ,..., }o m , we recursively compute the conditional volatility values and revise

the alpha values in an effort to maximize the overall likelihood.

ModelCheckingThe ARCH model does not assume i.i.d assumption among the mean‐corrected returns{ }ta , but the

standardized residuals { }t are i.i.d.

~ (0,1)

~ i.i.d

tt

t

t

aN

In short, we need to examine the standardized residuals { }t for independence (e.g. the white‐noise

test and arch effect test) and the normality distribution assumption.

ModelExtensionIn some applications, it is more appropriate to assume that standardized residuals { }t follow a heavy –

tailed distribution such as the student’s t distribution or the generalized error distribution (GED).

This extension affects the computation of the log‐likelihood function (LLF) (using the alternative

probability density function), and the interpretation of conditional volatility.

To illustrate, let’s take the student’s t‐distribution for{ }t

~ i.i.d ~ (0,1)t

Where:

‐ (0,1)t is the standardized student’s t distribution (zero mean and unit variance)

‐ is the degrees of freedom of the student’s distribution ( 2 )

To yield a standardized t‐distribution, with zero skew and finite excess kurtosis,:

1. 4

2. 2

t

The standardized t‐distribution exhibit a fat‐tail with excess kurtosis =6 / ( 4)

ForecastingFor the first p‐steps out‐of‐sample, the forecast formula includes a mix of squared residuals 2{ }ta and

estimated variances 2t

.


2 2 2 21 1 2 1

2 2 2 2 2 2 22 1 1 2 1 1 1 2 1

2 2 2 2 2 22 1 1 2 1 2 3 1 1 1 1 1 1

2 23 1 2 2

ˆ ...

ˆ ˆ... ...

ˆ ( 1) ( ) ( ) ...( )

ˆ

t o t t p t p

t o t t p t p o t t p t p

t o t t p p t p p t p

t o t

a a a

a a a a a

a a a a

a

2 2 2 2 2 21 2 1 2 2 1 3 2

2 2 2 2 21 1 2 2 3 3 1

2 2 2 2 21 1 2 1 3 2 2

ˆ ˆ... ...

ˆ ˆ ˆ ˆ ˆ...

ˆ ˆ ˆ ˆ ˆ...

t p t p o t t t p t p

t p o t p t p t p p t

t p o t p t p t p p t

a a a a

For a longer forecast horizon, the estimated conditional volatilities converge to a long term value

determined by the model parameters; for instance ARCH(1) has the following long‐run variance:

2

11o

t kk

ARCHEffectAn ARCH effect is a characteristic used to describe whether a given time series exhibits correlation

among its squared data point values.

The original test conducted by Engle (1982) is using the LaGrange multiplier (LM) and ordinary least

squares regression.

Alternatively, we can use the Ljung‐Box test on the squared (mean‐adjusted) time series, compute

modified Q(m) and test whether the data exhibits a significant serial correlation or not.

ConclusionIn this paper, we built upon several bedrock lessons from earlier issues and constructed a general

framework for volatility modeling. In the beginning, we looked into correlated returns and derived a

relationship between term structure (i.e. multi‐period) volatility and conditional means.

Next, assuming a mean‐adjusted asset’s returns time series, we proceed with our analysis to volatility.

In practice, the models describing volatility evolution over time are categorized into two groups: (1)

deterministic functional form based models (e.g. ARCH, GARCH, etc.), and (2) stochastic models which

permit the volatility model to include a shock/innovation term.

Finally, we examined in depth a rather important model – the ARCH model (Engle 1980); a building

block mode for many models (e.g. GARCH, EGARCH, etc.), which we will cover in future issues. The

ARCH can be thought of as a weighted moving average of the squared time series, but the weights are

relatively constrained to yield a positive variance and (existent) finite long‐run variance. Nevertheless, it

does not provide an insight into the volatility process and treats positive and negative shock

indiscriminately, which is contrary to what has been observed/documented in financial time series.

Financial Time Series Volatility 201 (Excel)

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