Option Pricing Applications of Quadratic Volatility Models
Srimantoorao S. Appadoo1*, Aerambamoorthy Thavaneswaran2, Saman Muthukumarana2
1Department of Supply Chain Management, University of Manitoba, Winnipeg, Canada 2Department of Statistics, University of Manitoba, Winnipeg, Canada
Email: *[email protected]
Received January 20,2012; revised March 1, 2012; accepted March 16, 2012
ABSTRACT
Recently there has been a surge of interest in higher order moment properties of time varying volatility models. Various GARCH-type models have been developed and successfully applied in empirical finance. Moment properties are im-portant because the existence of moments permit verification of how well theoretical models match stylized facts such as fat tails in most financial data. In this paper, we consider various types of random coefficient autoregressive (RCA) models with quadratic generalized autoregressive conditional heteroscedasticity (GARCH) errors and study the mo-ments, mean, variance and kurtosis. We also consider the Black-Scholes model with RCA GARCH volatility and show that these moments can be used to evaluate the call price for European options.
Keywords: Garch Processes; RCA Models; Garch Models; Time Varying Volatility; Kurtosis
1. Introduction
It is well-known that many financial time series such as stock returns exhibit leptokurtosis and time-varying vola-tility [1]. The generalized autoregressive conditional het-eroscedasticity (GARCH) and the random coefficient autoregressive (RCA) models have been extensively used to capture the time-varying behaviour of the volatility. Studies using GARCH models commonly assume that the time series is conditionally normally distributed; how- ever, the kurtosis implied by the normal GARCH tends to be lower than the sample kurtosis observed in many time series Bollerslev [1]. Thavaneswaran et al. [2] use an ARMA representation to derive the kurtosis of various classes of GARCH models such as power GARCH, non- Gaussian GARCH, non-stationary and random coeffi-cient GARCH. Recently, Thavaneswaran et al. [3], Ap-padoo et al. [4] have extended the results to stationary RCA processes with GARCH errors and Paseka et al. [5] further extended the results to RCA processes with sto-chastic volatility (SV) errors.
Leptokurtosis is commonly observed in financial time series, as well as in currency and commodity markets. The opening and closure of the markets, time-of-the-day and day-of-the-week effects, weekends and vacation pe-riods cause changes in the trading volume that translates into regular changes in price variability. Financial, cur-rency, and commodity data also respond to new informa-tion entering into the market, which usually have large kurtosis. Recently, there has been growing interest in
using volatility models [3,4]. Most of the studies use GARCH models with dummy variables in the volatility equation, and a few of them have been extended to a more flexible form such as the RCA GARCH. However, even though much research has been performed on vola-tility models applied to market data such as stock returns, more general specifications accounting for RCA with GARCH errors have been little explored. First we derive the kurtosis of a simple time series model with behaviour in the mean. Then we introduce various classes of RCA GARCH models and study the moments and discuss ap-plications in option pricing. We extend the results for RCA GARCH volatility models to RCA quadratic GARCH models. The RCA GARCH model is appropriate for time series where significant autocorrelation exists. Option pricing with RCA model with quadratic GARCH errors is also discussed in some detail. The moments derived for the RCA GARCH volatility models provide more ac- curate estimates of market data behaviour and help in-vestors, decision makers, and other market participants develop improved trading strategies. The rest of the pa-per is organized as follows. In rest of Section 1, we pre-sent results on standard GARCH models. These results are interesting for their own sake. In Section 2, we derive the higher order moments of some RCA models with GARCH errors, and in Section 3 we discuss some option pricing applications with RCA models with GARCH errors.
GARCH Models
Consider the general class of GARCH (P,Q) model for
the time series yt, where
,
t h Zt
t
t j
h
(1.1)
2
1 1
Q P
t i t i j
i j
h h
(1.2) where Zt is a sequence of independent, normallydistrib-uted random variables with zero mean, unit variance. Let
t be the martingale difference and let
2
t t
u 2
u
be the variance of ut, (2.12) and (1.2) could be written as:
2 2
1 1
Q P
t t i t i j t j
i j
u h
j t
u
(1.3)
2
1 1 1
1 P i Q j Q
i j t j
i j j
B B B
, (1.4)
2
.t
B B
ut
j
B
(1.5) where
1
1 R i, , 1 Q
i i i i j
i
B B B
1 j
and R = max(P,Q).We shall make the following station-arity assumptions for 2
t
which has an ARMA(R,Q) representation.All the zeroes of the polynomial Φ(B) lie outside of the unit circle. 2 where the
0 i
i
isare obtained from the relation
B B with
1
1 i
i i
B
B . The assumption sensure that thet are uncorrelated with zero mean and finite variance
and that the
u s
2
t
process is weakly stationary. In this case, the autocorrelation function of 2
t
will be exactly the same as that for astationary ARMA(R,Q) model. For any random variable ε with finite fourth moments, the kurtosis defined by
4 2
E Var
and if the process {Zt}
is normal then the process {εt}defined by Equations (1.3)
and (1.4) is called a normal GARCH (p, q) process. The kurtosis ofthe GARCHprocess is denoted by K when it exists.
2. Random Coefficient Volatility Models
Consider the class of random coefficient autoregressive (RCA) models defined by allowing random additive per-turbations of the autoregressive (AR) coefficients of or-dinary AR models. That is, we assume that the process yt
is given by,
1
p
t i i t i
i
y b t y
where the parameters θi, i = 2, ···, p, are assumed to be
known, et and b ti
are zero mean square integrableindependent processes and the variances are denoted by
2
e
and 2
b
. b ti
1,2,ı
s are independent of t and t ie y and may be thought of as incorporating
structural changes. In order to motivate nonlinear fore- casts for nonlinear models, we consider a class of esti- mating functions of the form
2
n
n t i
i t
g b h
[6], where1
1
p y
t t t t t yt i
i
h y E y F y bt i
and is a func- tion of y y1, , ,2 yt i, ,
and possibly the known parame-ters 1p
1 n
y
(i.e. We assume that the fitted model is available). If we restrict ourselves to a class of estimating functions of the above form then we can forecast the fu-ture value of based on the observed values
1, , ,2 n
y y y as yˆ 1n
E y n1 y yn, n1,. That is,whether we have an AR(p) model or RCA(p) model we will get the same linear predictor of n1. However, for the RCA model under consideration, we have
y
1 1
p y
t t t i
i
E y F y
and 1 2 21
var y p
t t t i b
i
y F y 2
.Thus, the conditional variance is a nonlinear function and hence the RCA model may be viewed as a non-linear time series model. Nicholls and Quinn [7] studied linear as well as some nonlinear (proposed) forecast by fitting a nonlinear (RCA) model for the classical lynx cycle data. Using heuristic reasoning they proposed a nonlinear forecast and
1 2 2 2 2 1 1 1
ˆn sgn ˆn ˆn
y y y theyshowed
empirically that the forecast n1 is a better predictor
(having smalller forecast errors when compared with the actual observations) than the linear forecast for the lynx data. It is of interest to note that by defining
y
2 2
1 y
t t t t
h y E y F , the optimal forecast for yn1 can be obtained as
1
1* 2 2 2 2 2 2 2
1 1
1 y sgn
n n
n t t
y E y F y .
That is, the estimating function method can be used to obtain a nonlinear forecast for a nonlinear models by considering a class of elementary martingale estimating functions generated by nonlinear functions of the obser-vations. Using a similar argument we could also obtain forecasts for various class of GARCH models, see Tha-vaneswaran and Heyde [6] for details. The main message is RCA models could be used to improve the forecasting performance of stocha ic volatility models.
t
e (2.1)
st
Lemma 2.1. When Zt is a standard normal random
1 ,t t t t
y b y t Z
2
2
. 2 π
t
E Z
1 Now if
Z
2~ 0,
t
Z N then
where Z denotes the set of integers and
2 2 21 . 2
π
Z t
E Z
1)
2 2
0 0
~ ,
0 0
t b
t
b
,
As a general case for even
2) 2 2 1.
b
<
powers we have
2 2 2 1 .2
π
E Z
The sequences {bt}and {εt}respectively, are the errors
in the model.
Theorem 2.1. Let {yt}be a modified RCA (1) time
se-ries with an absolute value random coefficient satisfying conditions (1) and (2). The modified RCA (1) model is given by
2.1. RCA Models
Random coefficient autoregressive time series were in-troduced by Nicholls and Quinn [10] and some of their properties have been studied recently by Thavaneswaran [3]. RCA models exhibiting long memory properties have been considered in Leipus and Sugailis [8]. A se-quence of random variables {yt} is called an RCA (1)
time series if it satisfies the equations
1t t t
y b y t (2.2)
2
~ 0, , ~ 0, .
t N bt N
2
b Then we have the
fol-lowing
1)
2 2
2 2
0,
2
1 2
π
t t
b b
E y E y
2)
4 2 2
4
3
4 4 3 2 2 2 2
2
3 1 2
π
,
8 2 2 2
1 3 4 6 1 2
π π
π
b b
t
b
b b b b b
E y
3)
2
2 2
3
4 4 3 2 2 2
3 1 2
π
.
8 2 2
1 3 4 6
π π
b b
y
b
b b b
K
The autocovariance and the autocorrelation functions are given by
2 21
2 1
2
π
b
k k
and
2 2
2 1
. 2
π
k
b k
p
Proof:
2 2
2
2
2
2
21 1 1
0, 2
t t t t t t t
E y E y E y E y E b E y E b E t
Thus, we have
22
2 2 2
1 2
π
t
b b
E y
and we have
44
3
4 4 3 2 2
4 2 2
3
4 4 3 2 2 2 2
4 2 2
3
4 4 3
3
8 2 2
1 3 4 6
π π
2
6 2
π
8 2 2 2
1 3 | 4 6 1 2
π π
π
2
3 1 2
π
8 2 2
1 3 4
π π
t
b
b b b
b b
b
b b b b b
b b
b
b b
E y
6 2 2 1 2 2 2 2
π
b b
b
2
2 2
4
2 3
2
4 4 3 2 2
2
3 1 2
π
8 2 2
1 3 4 6
π π
b b
t y
b t
b b b
E y K
E y
(2.4)
When , the kurtosis of the process yt converge
to K(y) = 3. Thus, the autocorrelation function is given by
2 0
b
1
2 22 1
2
π
k
b
k E bt k k
where we use the fact that ρ0 = 1.
Theorem 2.2. Suppose yt is an Random Coefficient
Moving average process model of the form
21 1
yt b yt t t t (2.5)
where bt is an uncorrelated Gaussian process with zero
mean and with variance 2
b
. εt is an uncorrelated
Gaus-sian process with zero mean and with variance 2
. Then, we have the following
Proof:
21 1 0
t t t t
E y E b y t (2.6)
and
2 2 2 2
1 1
2 2 6
1
2 2
π 3
t t t b
t b
E y E y E y
E y
We have
2 62 2
3 2
1 2
π
t
b b
E y
(2.7)
and
4 2 2 4 2 2 4 2 2 8 4 1
4 1
4 3 2 2 3 4
6 2 6 2 6 2 12 1
4 3 2 2 3 4
2
12 3 6 3 6 3 105 3
π
2 2
1 4 2 6 4 3
π π
2
36 18 18 315
π
2 2
1 8 6 4 3
π π
b b t
t
b b b b
b b t
b b b b
E y E y
E y
Thus we have
6 2 6 2 6
6 4
1
4 3 2 2 3 4 2
12
4 3 2 2 3 4
2
36 18 18
π 3
2 2
1 8 6 4 3 1 2
π π
315
2 2
1 8 6 4 3
π π
b b
t
b b b b b b
b b b b
E y
2
2 π
The kurtosis of the process is given by
6 2 6 2 6 2 2
4 3 2 2 3 4 6
2
2 2
4 3 2 2 3 4
2 2
36 18 18 1 2
π π
2 2
1 8 6 4 3 3
π π
2 35 1 2
π
2 2
1 8 6 4 3
π π
b b b
y
b b b b
b b
b b b b
K
b
(2.8)
When b2 0, the kurtosis of the process yt converge
to
2 2
35 29 1
y
K
2 0
and when 0, and
b
the kurtosis of the process yt turns out to be 35.
Theorem 2.3. Suppose yt is an Random Coefficient
Moving average process model of the form
1t t t t
y b (2.9) where bt is an uncorrelated noise process with zero mean
and with variance 2
b
. εt is an uncorrelated noise
proc-ess with zero mean and with variance 2
. Then, we have the following relationships
2 2 2 2
4 4 4 4 2 2 2 2 2 2
4 4 2 2 2 2 2 2
2 2
2
1 2 ,
π
2
3 3 1 12 1 2 6 3
π
2
3 3 1 12 1 2 6 3
π .
2
1 2
π
t b b
t b b b b
b b b b b
y
b b
E y
E y
K
,
b
and the autocorrelation functions are given by
2 2
1 0
2
π 1
2
1 2
π
0 2
b k
b b
k
k
k
,3,
Proof:
2 2 2 2 2 2 21 1 1 1 1
2 2 2 2 2 2 2 2 2
2 2 2 ,
2 2
2 1 2
π π
t t t t t t t t t t t t
b b b
E y E b b b
b.
4 4 4 4 2 2 3 2 2 3
4 4 4 2 2 2 2 2 2
2 2
3 9 3 12 6 6 12 18 24
π π
2
3 3 1 12 1 2 6 3
π
t b b b b b
b b b b b
E y
2
π b
Thus,
4 4
2 2
2 2 2 2
4
2 2
2 2
2
3 3 1 12 1 2 6 3
π
2
1 2
π
b b b b
y t
t
b b
E y K
Var y
b
The autocorrelation function is given by
0 1
2 2
2
π
1, , 0 2,3, 4,
2
1 2
π
b
k
b b
k
Theorem 2.4. Let {yt} be a Sign RCA-GARCH (1,1)
time series satisfying conditions (i) and (ii) given by
1t t t t
y b s y t (2.11)
where
,
t h Zt
t
j t j
(2.12)
2
1 1
p q
t i t i
i j
h h
(2.13)where Zt and bt are sequences of independent, identically
distributed random variables with zeromean, variance given by 2
Z
and 2
b
respectively,
1 if 0
0 if 0
1 if 0
t
t t
t
y
s y
y
ω, α1, β1 and Φ are real parameters, satisfying the
fol-lowing conditions,ω > 0,α1≥0,β1 ≥0. |Φx| ≤ω. Note:
2 1 tE s , and in order to calculate the kurtosis, we observe that
4 1t
E s . Then, we have the following moment properties
2 2
2 2 2 2 2
1 2 1
π
Z t
b b Z
E y
4
4 2
1
4 4 2 2 2 2 2 3 2 2 2
1
4 2 2 2 2 2 2
2
4 4 2 2 2 2 2 3 2 2 2
2
1 6 12 4 8 3 2
π
2 2
6 2 1 2
π π
2
1 6 12 4 8 3 2
π
t
t t
b b b b b b
Z b b b b
t
b b b b b b
E Z
E y E h
E h
2 4
2 2 2
4 4 2
1
4 4 4 2 2 2 2 3 2 2 2
2 2 2 2 2 2
4 4 2 2 2
2
1 2
π
1 2
1 6 12 8 3 2
π
2 2
6 2 1 2
π π
1 6
t
b b
t t j
j y
Z b b b b
b b b b
b
E Z
E Z E Z
K
b
2 2 3
2
2 2
2 12 4 8 3 2
π b b b b b
(2.14)
Proof:
1
1
,
0
t t t t t
t t t t t
y b s y
E y E b s y
2 2
1
2 2
2 2 2 2 2 2 2 2 2
1 2 1 2 1
π π
t t t t t
t Z Z
b b b b Z
E y E b s y
E h
4 2
1
4 4 2 2 2 2 2 3 2 2 2
1
4 2 2 2 2 2 2
2
4 4 2 2 2 2 2 3 2 2 2
2
1 6 12 4 8 3 2
π
2 2
6 2 1 2
π π
2
1 6 12 4 8 3 2
π
t
t t
b b b b b b
Z b b b b
t
b b b b b b
E Z
E y E h
E h
4
4
2 2t
1
t t
E y AE h BE h
where
4
2 2 2 2 2 3 2 2 2 2 3 4 4 4
1 6 12 4 6 8 3
π π π
t
b b b b b b
E Z A
1
b
4 2 2 2 2 2 2
4 4 2 2 2 2 2 3 2 2 2
2 2
6 2 1 2
π π
2
1 6 12 4 8 3 2
π
Z b b b b
b b b b b
B
2
2 2 2
4
2 2
2 2 4
2 2
2 2 2 2 2
2
4 2 4
2
1 2
π
2 2
1 2 1 2
π π
b b
y t
t t
t Z
t
b b b b
t
Z t Z
E y
K AE h BE h
E h Var y
E h
A B
E h
2
Using the facts that
2 2
4 4
1 1
1
t
t
t t
j
E h
E h E Z E Z 2
j
2 2
2 2 2 2 2
2
4 2 2
2
2 2 2
4
4 4 2
1
2 2
1 2 1 2
π π
2
1 2
π 1
1
b b b b
t y
Z t Z
b b
Z
t t j
j
E h
K A B
E h
A
E Z E Z
2
2
2 2 2
4 2
1 2
π b b
Z
B
Thus, we have the following expression for the Kurtosis of the process.
2 4
2 2 2
4 4 2
1
4 4 4 2 2 2 2 2 3 2 2 2
2 2 2 2 2 2
4 4 4
2
1 2
π
1
2
1 6 12 4 8 3 2
π
2 2
6 2 1 2
π π
1 6
t
b b
t t j
j y
Z b b b b b
b b b b
Z
E Z
E Z E Z K
b
2 2 2 2 12 2 4 2 8 3 3 2 2 2 2
π
b b b b b
b
2 2
2 2 2
2 2
2 2 2 2 3 2 2 3 4 4
2 2 2 2 2 2
2 2 2 2
1 2
3 1 2
π 1 2
2
1 6 12 4 6 8 3
π
2 2
6 2 1 2
π π
1 6 12
b b
y
b b b b b
b b b b
b b
K
2 4 3 2 6 2 2 8 2 3 3 4 4
π b π b π b b
4
(2.15)
Note, that when 2 0, 0, and
b
Zt~N
0,1 in (2.15), the kurtosis of the process converges toTheorem 2.5. Suppose yt is a modified RCA model
with GARCH (p, q) innovations of the form
12
1 1
t t t t
t t t
p q
t i t i
i j
y b y
h Z
h h
2
2 2 1 1
2 2
2 2
1 1 1 1
6 1
3 1
1 1 2
y
K
(2.16) j t j
When 0 in (2.16),the kurtosis of the process con- where b
t is an uncorrelated noise process with zero mean
and with variance 2
b
and Zt is an uncorrelatednoise
process with zero mean and with variance 2
Z
. Then, we have the following relationship
verge to
2
2 2
3 1
3
1 2
y
K
2 2
2 2 2
1 2
π
Z t
b b
E y E h
t
(2.17)
4
4 2
3 2 2 3 4 4
2 2 2 2 2 2
2
3 2 2 3 4 4 2 2
3
2 2
1 4 6 8 3
π π
2
12 6 6
π
2 2 2
1 4 6 8 3 1 2
π π π
Z
t t
b b b b
b Z b Z Z Z
t
b b b b b b
E y E h
E h
(2.18)
2
2 2
3 2 2 3 4 4 4 4
1
2 2 2 2
3 2 2 3 4 4
2
3 1 2
π
2 2
1 4 6 8 3 1
π π
2 2
1 2 2
π π
6
2 2
1 4 6 8 3
π π
b b
y
b b b b t t
j
b b b b
b b b b
K
E Z E Z
2
j
Proof: Let, utt2ht
t
B u
2 2 2
1 1 1 1 1
1 1
p q p q q
i j j
t t i t i j t j i j t j
i j i j j
u h B B
2
1 1
2 4
2
4 4 2 4 4
1 1
, 1 ,
1 , max( , )
1 ,
1 1
r i
t t i i i i
i q
j j j
t t
t
t t j t t
j j
B B u B B
B B r p q
E h E Z
K
E h E Z E Z E Z E Z
2
j
Now we have
2 2
2
2
2 2
21 1 1
2
2 ,
π
t t t b t b t Z
E y E y E y E y E h
Thus,
2 2
2
2
2 2
21 1 1
2
2 ,
π
t t t b t b t Z
E y E y E y E y E h
and
2 2 2 2
1 2
π b b E yt E ht
2 Z
2 2
2 2 2
1 2
π
Z t
b b
E y E h
t
(2.20)
2
2 4 2 2 2 2 2 2 2
1 1 1
4
2 3 4
3 2 4
4 2 2 2 2 2 2 2 2
1 1 1
3 2 2 3 4 4
4
12 6 6
1 4 6 4
2
3 12 6 6
π
2 2
1 4 6 8 3
π π
3
1
t t t t t t t t t t t t t
t
t t t t
2
Z t b Z t t b Z t t t t
b b b b
Z
E h Z E y b h X E y b h Z E y h Z E y
E b E b E b E b
E h E y E h E y E h E y E h
Z
2
3 2 2 3 4 4
2 2 2 2 2
2 1
3 2 2 3 4 4
2 2
4 6 8 3
π π
2
12 6 6
π
2 2
1 4 6 8 3
π π
t
b b b b
b Z b Z Z
t t
b b b b
E h
E y E h
Using the fact that
2 2
2 2 2
1 2
π
Z
t t
b b
E y E h
, we have
4
4 2
3 2 2 3 4 4
2 2 2 2 2 2
2
3 2 2 3 4 4 2 2
3
2 2
1 4 6 8 3
π π
2
12 6 6
π
2 2 2
1 4 6 8 3 1 2
π π π
Z
t t
b b b b
b Z b Z Z Z
t
b b b b b b
E y E h
E h
t
For convenience let
4
2
2 where,t t
E y AE h BE h
4
3 2 2 3 4 4
2 2 2 2 2 2
3 2 2 3 4 4 2
3
2 2
1 4 6 8 3
π π
2
12 6 6
π
2 2 2
1 4 6 8 3 1 2
π π π
Z
b b b b
b Z b Z Z Z
b b b b b
A
B
2 b
The Kurtosis of the process is given by
2 2
4
2 2
2 2
2
2 2
2 2
2 4
2 2
2 2 2 2
2
4 2 4
2
1 2
π
2 2
1 2 1 2
π π
t t
t y
t t
b b
t t
Z t
b b b b
t
Z t Z
AE h BE h E y
K
E y E y
AE h BE h
E h
E h
A B
E h
Using the fact that
2 2
4 4
1 1
1
t
t
t t
j
E h
E h E Z E Z 2
j
2
2 2
3 2 2 3 4 4 4 4
1
2 2 2 2
3 2 2 3 4 4 2
3 1 2
π
2 2
1 4 6 8 3 1
π π
2 2
1 2 2
π π
6
2
1 4 6 8 3
π
b b
y
b b b b t t
j
b b b b
b b b b
K
E Z E Z
2
j
(2.21)
2.2. Quadratic GARCH Model This model reduces to the GARCH (1,1) model when the shift parameters δ3 = 0. The QGARC Hmodel can
improve upon the standard GARCH since they can cope with positive (or negative) skewness.
Besides having excess kurtosis market returns may dis-play seriously skewed distributions. Linear GARCH models cannot cope with such skewness, and therefore we can expect forecast of linear GARCH model to be biased for skewed time series. To deal with this problem non-linear GARCH models are introduced, which take into account skewed distributions. The QGARCH model differs from model the classical GARCH model by
Theorem 2.6. Consider the general class of RCA
QGARCH (1,1) Volatility Models for the time series yt,
where
t t
y h Zt (2.24)
2
20 2 3 1 1 2 2 3 1 2 1
t t t
h h y at yt1
t t
y h Zt
1 t
y
(2.22)
(2.25)
2
0 1 1 2 1 3
2 2
0 2 3 1 1 2 3 1 2
( ) 2
t t t
t t
h h y
h y
(2.23) where
2
~ 0,
t Z
Z N and ~
0, 2t
a N a
. Then, we have the following moment properties
2 0 2 3 2 2 1 2
2 2 2 4
2 0 2 3 0 2 3
1
4 2 4 2 4 2 2 2
2 1 2 1 2 1
2 2 2 2 2 2 2 2
2 3 0 0 2 0 1 2 3 1 2 3 2
1
π
2
2 2
1 6 3 3 2 2
π π
2 2 3
π
t
Z Z a
t
Z a Z Z a Z Z a
Z Z a Z Z a
E h
E h
14 2 4 2 4 2 2 2
2 1 2 1 2 1
2
π
2 2
1 6 3 3 2 2
π π
t
Z a Z Z a Z Z a
E h
2
2 2 2 4 2 2
0 2 3 0 2 3 1 2
2
4 2 4 2 4 2 2 2 2
2 1 2 1 2 1 0 2 3
2 2 2 2 2 2 2
2 3 0 2 0 2 3 0 2 3 1 1 2
2
3 2 1
π
2 2
1 6 3 3 2 2
π π
2
6 3 1
π
Z Z a
y
Z a Z Z a Z Z a
Z Z a Z
K
4 2 4 2 4 2 2 2 2
2 1 2 1 2 1 0 2 3
2 π
2 2
1 6 3 3 2 2
π π
a
Z a Z Z a Z Z a
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