An Introduction to Volatility Models with Indices

S. Peiris and A. Thavaneswaran^{1} School of Mathematics and Statistics The University of Sydney, N.S.W. 2006, Australia

Abstract

This paper considers a class of volatility models generated by autoregressive (AR) type models with indices. Some results associated with the autocorrelation function (acf) of this class are given and the spectral density is obtained in terms of the kurtosis of the error distribution and the model parameters. Key words: Time series, Frequency, Spectrum, Autoregression, Correlation, Index, Moving average, Kurtosis, Moments, ARCH, GARCH AMS 2000 Subject Classification: Primary 62M10; Secondary 60G10, 91B84.

1 Introduction

It is well known that time series have their own frequency behaviour. This is a very common phenomenon in practice, especially in financial time series data (see, for instance, Priestley (1981), Peiris (2003) and Peiris et al (2005) for details) and the series cannot be identified using the existing standard time series techniques. In other words the acf, the pacf, and the spectrum are similar for many series and one may propose the same classical model for all of these cases. Obviously, this may lead to poor forecast values leading to serious consequences in managerial decisions. One way of handling this problem in practice is to introduce a new class of time series models with an additional parameter (or an index) δ(> 0). Therefore, this paper considers a class of time series models satisfying

(I − αB)^{δ}X_{t} = e_{t}; −1 < α < 1; δ > 0 , (1.1) where e_{t} is a white noise sequence and B is the backshift operator such that B^{j}X_{t} = X_{t−j}; j ≥ 0 with B^{0}X_{t} = X_{t}.

1Visiting from the Department of Statistics, The University of Manitoba, Winnipeg, Manitoba, Canada R3T2N2.

This class of models covers the traditional AR(1) family when δ = 1. Peiris (2003) and Peiris et. al. (2005) have discussed some useful properties of (1.1). It is clear that when α = 1 and 0 < δ < \frac{1}{2} , (1.1) reduces to the well-known class of fractionally integrated white noise processes (see, for example, Hosking (1981), Beran (1994) and Chen et al (1994) for details). Therefore, (1.1) constitutes a new family of AR(1) type models and can be applied many standard time series in practice. This class of time series models generated by (1.1) is called ‘Power Integrated AR(1)’ or ‘PIAR(1)’.

This paper attempts to generalise the class in (1.1) to incorporate ARCH and GARCH type models. With that view in mind Section 2 reviews the class of of GARCH process and give some examples of calculating the kurtosis.

2 GARCH(p, q) Process

Consider the general class of GARCH (p, q) models for the time series y_{t} satisfying

y_{t} = \sqrt{h_{t}}Z_{t},  (2.1)
h_{t} = ω + ∑_{i=1}^{p} α_{i}y_{t−i}^{2} + ∑_{j=1}^{q} β_{j}h_{t−j},  (2.2)

where ω > 0; α_{i} ≥ 0; β_{j} ≥ 0 and Z_{t} is a sequence of independent and identically distributed (iid) random variables with zero mean and unit variance.

It is well known that u_{t} = y_{t}^{2} − h_{t} is a martingale difference and let σ_{u}^{2} = var(u_{t}). Now (2.1) and (2.2) can be written as

ϕ(B)y_{t}^{2} = ω + β(B)u_{t}, (2.3)

where ϕ(B) = 1 − ∑_{i=1}^{r} ϕ_{i}B^{i} , ϕ_{i} = (α_{i} + β_{i}) , β(B) = 1 − ∑_{j=1}^{q} β_{j}B^{j}, r = max(p, q) and

μ′ = E(y_{t}^{2}) = \frac{ω}{1−ϕ_{1}−/phi_{2}−···−phi_{r}}. Suppose that the following assumptions are hold: (A.1). all the zeroes of the polynomial ϕ(B) lie outside of the unit circle.

(A.2). ∑_{i=0}^{∞} ψ_{i}^{2} < ∞, where the ψ′_{i}s are obtained from the relation ψ(B) ϕ(B) = β(B) satisfying ψ(B) = 1+∑_{i=1}^{∞} ψ_{i}B^{i}. Let K^{(X)} be the kurtosis of any random variable X given by

K^{(X)} = \frac{E[(X − μ)^{4}]}{[V ar(X)]^{2}},

where μ = E(X).

Below we give two examples of calculating the kurtosis based on two specific cases: Example 2.1

For the GARCH (1,1) model (1 − ϕ_{1}B)y_{t}^{2} = ω + (1 − β_{1}B)u_{t}, (ϕ_{1} = α_{1} + β_{1}) and ψ_{1} = α_{1}, ψ_{2} = α_{1}(α_{1} + β_{1}), ψ_{3} = α_{1}(α_{1} + β_{1})^{2}, … ψ_{j} = α_{1}(α_{1} + β_{1})^{j−1}. Clearly,

∑_{j=1}^{∞} ψ_{j}^{2} = α_{1}^{2} + α_{1}^{2}(α_{1} + β_{1})^{2} + … = \frac{α_{1}^{2}}{1 − (α_{1} + β_{1})^{2}} . The kurtosis K^{(y)} of \{y_{t}\} is

K^{(y)} = \frac{3}{1 − 2 ∑_{j=1}^{∞} ψ_{j}^{2}} = \frac{3}{1 − \frac{2α_{1}^{2}}{1 − (α_{1} + β_{1})^{2}}} = \frac{3[1 − (α_{1} + β_{1})^{2}]}{1 − (α_{1} + β_{1})^{2} − 2α_{1}^{2}} ,

and it turns out to be the same as the one given in Bollerslev (1986). Moreover,

σ_{u}^{2} = \frac{μ^{2}(K^{(y)} − 1)}{1 + \frac{α_{1}^{2}}{1−(α_{1}+β_{1})^{2}}} .

Example 2.2

For the ARCH (1) model of the form y_{t} = \sqrt{h_{t}}Z_{t}, h_{t} = ω + αy_{t−1}^{2} can be obtained by setting β_{1} = 0 in Example 2.1. The corresponding K^{(y)} turns out to be \frac{3(1 − α_{1}^{2})}{1 − 3α_{1}^{2}} and σ_{u}^{2} = μ^{2}(K^{(y)} − 1)(1 − α_{1}^{2}).

Although ARCH and GARCH type models are very popular in volatility modelling in finance, we still have room to accomodate additional components in order to understand the volatility process without violating the priciple of parsimony. With that view in mind, Section 3 introduces a class of volatility models with indices.

3 AR Type Volatility Models with Indices

Suppose that {y_{t}^{2}} is an AR(p) process as in (2.3) with h_{t} = ω + ∑_{i=1}^{p} α_{i}y_{t−i}^{2}, where p is a large positive integer. In this case {y_{t}^{2}} follows a model of the form

y_{t}^{2} = ω + ∑_{i=1}^{p} α_{i}y_{t−i}^{2} + u_{t}, (3.1)

where u_{t} is as defined before.

Since the model (3.1) involves p number of parameters α_{1}, α_{2}, ⋯, α_{p}, we consider the corresponding analog of (1.1) for {y_{t}^{2}} is given by

(I − αB)^{δ}y_{t}^{2} = ω + u_{t} , (3.2)

where |α| < 1, δ > 0 and u_{t} is a suitable martingale difference sequence.

Clearly when δ = 1, (3.2) reduces to (3.1) with p = 1. However, when δ > 0 equation (3.2) can be approximated by a p^{th} order polynomial.

Let

I − αB)^{δ} = ∑_{j=0}^{∞} π_{j}α^{j}B^{j} , (3.3)

where B^{0} = I, π_{0} = 1 and

π_{j} = (−1)^{j}\binom{δ}{j} = \frac{(−δ)(−δ + 1) ⋯ (−δ + j − 1)}{j!}; j ≥ 1.

Notes:

1. If δ is a positive integer, then π_{j} = 0 for j ≥ δ + 1 . For any non-integer δ > 0, it is known that,

π_{j} = \frac{Γ(j − δ)}{Γ(j + 1)Γ(−δ)}, (3.4)

where Γ(·) is the gamma function.

2. The model in (3.2) can be thought of as a model incorporating all the past volatilities in a parsimonius way using only one additional parameter δ. This model can be used to forecast future volatilities using most of the available information than any other ARCH model. In practice we may use the approximation

(I − αB)^{δ} ≈ ∑_{j=0}^{m} π_{j}α^{j}B^{j}

where m is a suitably chosen large integer.

It is easy to see that the series ∑_{j=0}^{∞} π_{j}α^{j} converges for all δ since |α| < 1. Thus y_{t}^{2} in (3.2) has a valid AR representation of the form

∑_{j=0}^{∞} π_{j}α^{j}y_{t−j}^{2} = ω + u_{t} (3.5)

with ∑ |π_{j}α^{j}|^{2} < ∞. Now we state and prove the following theorem for a stationary solution of (3.2).

Theorem 3.1: For all δ > 0 and |α| < 1, the infinite series.

y_{t}^{2} − μ′ = ∑_{j=0}^{∞} Ψ_{j}α^{j}u_{t−j} (3.6)

converges absolutely with probability one provided E(u_{t}^{2}) < C, C > 0, where ψ_{j} =

\frac{Γ(j + δ)}{Γ(j + 1) Γ(δ)} and μ′ = E(y_{t}^{2}).

Proof: Let (I − αB)^{−δ} = ∑_{j=0}^{∞} Ψ_{j}α^{j}B^{j} , where

ψ_{j} = (−1)^{j}\binom{−δ}{j} = \frac{Γ(j + δ)}{Γ(j + 1) Γ(δ)}; j ≥ 0.

Now E (∑_{j=0}^{∞} |Ψ_{j}α^{j}u_{t−j}|)^{2} = ∑_{j=0}^{∞} |Ψ_{j}α^{j}|^{2} E{|u_{t−j}|^{2}}.

Since ∑_{j=0}^{∞} |Ψ_{j}α^{j}|^{2} < ∞, the result follows. Thus (3.6) gives a stationary solution for the process in (3.2). For α = 1, (3.6) converges for all 0 < δ < 1/2.

Let γ_{k} = Cov(y_{t}^{2}, y_{t−k}^{2}) be the autocovariance function at lag k of {y_{t}^{2}} satisfying the conditions of Theorem 3.1. It is clear from (3.5) that {γ_{k}} satisfy a Yule-Walker type recursion

∑_{j=0}^{∞} π_{j} α^{j} γ_{k−j} = 0 ; k > 0 (3.7)

and the corresponding autocorrelation function (acf), ρ_{k}, at lag k is given by

∑_{j=0}^{∞} π_{j} α^{j}ρ_{k−j} = 0 ; k > 0 . (3.8)

It is interesting to note that ρ_{k} = α^{k} is a solution of (3.8), since ∑_{j=0}^{∞} π_{j} = 0 for any δ > 0 . However, the general solution for ρ_{k} may be expressed as

ρ_{k} = g(k, α, δ)α^{k},

where g(.) is a suitably chosen function of k, α, and δ. To find this function g, we use the following approach: The spectrum of {y_{t}^{2}} in (3.2) is

f_{y_{t}^{2}}(ω) = |1 − α e^{−iω}|^{−2δ} \frac{σ_{u}^{2}}{2π}; −π ≤ ω ≤ π
= (1 − 2α Cos ω + α^{2})^{−δ} \frac{σ_{u}^{2}}{2π} .  (3.9)

In a neighbourhood of ω = 0, f_{y_{t}^{2}}^{g} ∼ \frac{σ_{u}^{2}}{2π} (1 − α)^{−2δ}, where g stands for generalized process in (3.2). Now the exact form of γ_{k} (or ρ_{k}) can be obtained from

γ_{k} = ∫_{−π}^{π} e^{ikω} f_{y_{t}^{2}}(ω)dω
= \frac{σ_{u}^{2}}{π} ∫_{0}^{π} \frac{Cos(kω)}{(1 − 2α Cos ω + α^{2})^{δ}} dω .  (3.10)

In order to obtain the variance of the volatility process y_{t}^{2}, we evaluate the integral in (3.10) for k = 0. Section 4 reports this result.

4 Main Results

This section is devoted to report some results associated with AR type process given in (3.2). Since the variance plays a significant role in financial modelling, the following theorem gives an expression for γ_{k}. Theorem 4.1: For the process given in (3.2),

(a) the variance,

γ_{0} = V ar(y_{t}^{2}) = σ_{u}^{2}F (δ, δ; 1; α^{2}), (4.1)

(b) the autocovariance function of y_{t}^{2} is

γ_{k} = \frac{σ_{u}^{2} α^{k}Γ(k + δ) F (δ, k + δ; k + 1; α^{2})}{Γ(δ) Γ(k + 1)}; k ≥ 0 , (4.2)

where F (θ_{1}, θ_{2}; θ_{3}; θ) = ∑_{j=0}^{∞} \frac{Γ(θ_{1}+j) Γ(θ_{2}+j)Γ(θ_{3})θ^{j}}{Γ(θ_{1})Γ(θ_{2})Γ(θ_{3}+j)Γ(j+1)} is the hypogeometric function (see, Gradsteyn and Ryzhik (1980) (GR), p. 1039) for details).

Proof: We first evaluate (3.10) at k = 0 and then for any k. From GR p.384, we have

γ_{0} = \frac{σ_{u}^{2}}{π} ∫_{0}^{π} \frac{dω}{(1 − 2α Cosω + α^{2})^{δ}} = B(\frac{1}{2}, \frac{1}{2})F (δ, δ; 1; α^{2}), (4.3)

where B(x, y) = \frac{Γ(x) Γ(y)}{Γ(x+y)} is the Beta function and B(\frac{1}{2}, \frac{1}{2}) = π, and hence (a) follows.

(b) follows by writing y_{t}^{2} − μ′ = ∑_{j=0}^{∞} ψ_{j} α^{j} u_{t−j} and using

γ_{k} = σ_{u}^{2} α^{k} ∑_{j=0}^{∞} ψ_{j} ψ_{j+k} α^{2j} = σ_{u}^{2}α^{k} ∑_{j=0}^{∞} \frac{Γ(j + δ) Γ(j + k + δ)(α^{2})^{j}}{Γ^{2}(δ)Γ(j + 1)Γ(j + k + 1)} ,

where μ′ = E(y_{t}^{2}) = \frac{ω}{(1−α)^{δ}} . From p.556 of Abramovitz & Stegun (1965), we have

∑_{j=0}^{∞} \frac{Γ(δ + j) Γ(k + δ + j)(α^{2})^{j}}{Γ(k + 1 + j) Γ(j + 1)} = \frac{Γ(δ) Γ(k + δ) F (δ, k + δ; k + 1; α^{2})}{Γ(k + 1)}

and hence (4.2) follows.

Note: Using the properties of F (), it is easy to see that for δ = 1 one has F (1, 1; 1; α^{2}) = (1 − α^{2})^{−1} ( Compare with GR, p.1040). That is (4.1) turns out to be the variance of an ARCH(1) process:

γ_{0} = V ar(y_{t}^{2}) = \frac{σ_{u}^{2}}{1 − α^{2}} for |α| < 1.

It is known that for θ_{3} − θ_{1} − θ_{2} > 0, F (θ_{1}, θ_{2}; θ_{3}; 1) = \frac{Γ(θ_{3}) Γ(θ_{3}−θ_{1}−θ_{2})}{Γ(θ_{3}−θ_{1}) Γ(θ_{3}−θ_{2})} and hence part (a) of the Theorem 4.1 reduces to the V ar(y_{t}^{2}) for a fractionally differenced (long memory) volatility process satisfying (I − B)^{δ}y_{t}^{2} = u_{t} when 0 < δ < \frac{1}{2}. That is, when α = 1 and 0 < δ < \frac{1}{2} , (4.1) gives γ_{0} = \frac{σ_{u}^{2}Γ(1−2δ_{1})}{Γ^{2}(1−δ)}.

5 Conclusions

We have introduced a new class of models for modelling volatility and obtained the autocorrelation function of the underlying process. It is shown that the proposed class of models provides a noval approach to incorporate additional components in volatility modelling. The new result in (4.1) and (4.2) are particularly useful in theoretical developments of power integrated ARMA (PIARMA) and power integrated GARCH (PIGARCH) processes with indices and thse will be discussed in a future paper. Moreover the forecasting with this type of volatility models will be discussed in a forthcoming paper following Thavaneswaran et al (2005).

Acknowledgements

We thank the anonymous referee and the editor of the journal for their suggestions to improve the quality of the paper. The second author thanks the School of Mathematics and Statistics at University of Sydney for the support during his visit.

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