Fuzzy coefficient volatility (FCV) models with applications
Mathematical and Computer Modelling 45 (2007) 777–786 www.elsevier.com/locate/mcm
Fuzzy coefficient volatility (FCV) models with applications A. Thavaneswaran a,∗ , K. Thiagarajah b , S.S. Appadoo c a Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada b Department of Mathematics, Illinois State University, Normal, IL, USA c Department of Supply Chain Management, University of Manitoba, Winnipeg, Manitoba, Canada
Received 30 November 2005; received in revised form 14 July 2006; accepted 28 July 2006
Abstract Recently, Carlsson and Fuller [C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001) 315–326] have introduced possibilistic mean, variance and covariance of fuzzy numbers and Fuller and Majlender [R. Fuller, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363–374] have introduced the notion of crisp weighted possibilistic moments of fuzzy numbers. In this paper, we propose a class of FCV (Fuzzy Coefficient Volatility) models and study the moment properties. The method used here is very similar to the method used in Appadoo et al. [S.S. Appadoo, M. Ghahramani, A. Thavaneswaran, Moment properties of some time series models, Math. Sci. 30 (1) (2005) 50–63]. The proposed models incorporate fuzziness, subjectivity, arbitrariness and uncertainty observed in most financial time series. The usual forecasting method does not incorporate parameter variability. Fuzzy numbers are used to model the parameters to incorporate parameter variability. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Possibilistic mean; Possibilistic variance; Fuzzy coefficient autoregressive model
1. Introduction The sophisticated analysis used by the financial industry has lent increasing importance to time series modeling. Recently, there has been growing interest in using fuzzy numbers in finance and economics. Many financial series, such as returns on stocks and foreign exchange rates, exhibit leptokurtosis and volatility varying in time. Many decision making problems exhibit some level of arbitrariness, vagueness and fuzziness (Carlsson and Fuller [4] and Fuller and Majlender [2]). These features have been the subject of extensive studies ever since Zadeh [5] has reported them. Decision-making problems, in general, involved uncertainty as their model parameters are not precisely known. As a result there has been growing interest in using fuzzy algebra in such models. Historically, probability theory is presented as forming theoretical foundations for reasoning and decision making in situations involving uncertainty. However, often one is faced with the situations in which decisions are required to be made on the basis of ill-defined variables and imprecise (vague) data. Fuzzy algebra is a simple but potentially a useful way to study impreciseness
∗ Corresponding author. Tel.: +1 204 474 8984; fax: +1 204 474 7621.
E-mail address: [email protected] (A. Thavaneswaran). c 2006 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2006.07.019
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through a cascade of calculations. As an alternative, fuzzy models have been used to model systems that are hard to define precisely, see for example Dubois and Prade [6] for details. The use of fuzzy set theory as a methodology for modeling and analyzing certain financial problems is of particular interest to a number of researchers due to fuzzy set theory’s ability to quantitatively and qualitatively model those problems which involve vagueness and imprecision. Recent studies has shown that a fuzzy random variable can be considered as a measurable mapping from a probability space to a set of fuzzy variables. Fuzzy time series models provide a new avenue to deal subjectivity observed in most financial time series models. We summarize the preliminaries in Section 1. In Section 2, RCA and GARCH models are given and the corresponding FCV models are introduced. Following Carlsson and Fuller [4,7], higher order moments of fuzzy numbers are defined. We also derive the moments and possibilistic kurtosis of the proposed FCV models. Section 3 concludes with an illustrative numerical example. 1.1. Preliminaries and notations Let a set X be a subset of a universal set U , and let R ⊂ X be the set of real numbers. Definition 1.1. Fuzzy set A in X is a set of ordered pairs A = {(x, µ(x)) : x ∈ X }, where µ(x) maps x ∈ X on the real interval [0, 1] and is known as the membership function of x ∈ X . Definition 1.2. A fuzzy set A in R n is said to be a convex fuzzy set if its α-cuts Aα are (crisp) convex sets for all α ∈ [0, 1]. Alternatively, a fuzzy set A in R n is a convex fuzzy set if and only if for all x1 , x2 ∈ R n and 0 ≤ λ ≤ 1, µ A (λx1 + (1 − λ)x2 ) ≥ Min(µ A (x1 ), µ A (x2 )). Definition 1.3. A fuzzy number A˜ ∈ F, is called a trapezoidal fuzzy number (Tr.F.N.) with core [a, b] left width α and right width β if its membership function has the following form a−t 1 − if a − α ≤ t ≤ a α 1 if a ≤ t ≤ b µ(x) = (1.1) t −b 1− if b ≤ t ≤ b + β β 0 otherwise and we use the notation A˜ = (a, b, α, β). It can easily be shown that Aγ = [a1 (γ ), a2 (γ )] = [a − (1 − γ )α, b + (1 − γ )β]
∀γ ∈ [0, 1].
(1.2)
The support of A˜ is (a − α, b − β). Moreover, for any fuzzy number A˜ and a positive real number C, the following relationship holds A˜ ≤ C ⇐⇒
1
Z
(a1 (γ ) + a2 (γ ))γ dγ ≤ C.
(1.3)
0
1.2. Moments Following Carlsson and Fuller [1], crisp possibilistic mean and possibilistic variance of continuous possibility distributions are given below. 1
Z M(A) = 0
(a1 (γ ) + a2 (γ ))γ dγ ,
(1.4)
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where M(A) is the level-weighted average of the arithmetic means of all γ -level sets. 2 ! Z 1 a1 (γ ) + a2 (γ ) Pos[A ≤ a1 ] Var(A) = − a1 (γ ) dγ 2 0 2 ! Z 1 a1 (γ ) + a2 (γ ) − a2 (γ ) dγ Pos[A ≥ a2 ] + 2 0 Z 1 1 (a2 (γ ) − a1 (γ ))2 γ dγ . = 0 2
779
(1.5)
Carlsson and Fuller [1] defined the variance of a fuzzy number A as the expected value of the squared deviations between the arithmetic mean and the endpoints of its level sets. Let A and B ∈ F be fuzzy numbers with A(γ ) = [a1 (γ ), a2 (γ )] and B(γ ) = [b1 (γ ), b2 (γ )], γ ∈ [0, 1]. Goetschel and Voxman [8] introduced a method for ranking fuzzy numbers as Z 1 Z 1 A ≤ B ⇐⇒ (a1 (γ ) + a2 (γ ))γ dγ ≤ (b1 (γ ) + b2 (γ ))γ dγ . (1.6) 0
0
As pointed out by Goetschel and Voxman [8] the definition given in (1.6) for ordering fuzzy numbers was motivated by the desire to give less importance to the lower levels of fuzzy numbers. For a Tr.F.N A the possibilistic mean value is given by Z 1 Z 1 E(A) = γ (a1 (γ ) + a2 (γ ))dγ = γ [a − (1 − γ )α + b + (1 − γ )β]dγ 0
0 1
Z =
γ [a − (1 − γ )α + b + (1 − γ )β]dγ =
0
=
Z
1
[(a − α + b + β)γ + (α − β)γ 2 ]dγ
0
a+b β −α + . 2 6
(1.7)
The possibilistic variance is given by Z Z 1 1 1 1 Var(A) = γ (a2 (γ ) − a1 (γ ))2 dγ = γ [(b + (1 − γ )β) − (a − (1 − γ )α)]2 dγ 2 0 2 0 (b − a)2 (b − a)(α + β) (α + β)2 = + + . 4 6 24 On the other hand if A˜ is a triangular fuzzy number then the possibilistic mean and possibilistic variance are given by (α + β)2 (β − α) and Var(A) = . E(A) = a + 6 24 The definition of the nth moment of a fuzzy number is partly motivated to give a meaningful definition of possibilistic kurtosis for fuzzy numbers. The kurtosis measures the peakedness of unimodal distributions. Following Carlsson and Fuller [1], we define the possibilistic nth moment of a fuzzy random variable A, E(An ), as Z 1 n n E(A ) = a2 (γ ) + a1n (γ ) γ dγ . (1.8) 0
Now, we compute different moments of Tr.F.N’s. Z 1h Z 1 i E(A2 ) = a22 (γ ) + a12 (γ ) γ dγ = [(a − (1 − γ )α)2 + (b + (1 − γ )β)2 ]γ dγ 0
=
(a 2
0
+ b2 ) 2
2 (bβ − aα) (α + β 2 ) + + 3 12
(1.9)
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and E(A ) = 3
1
Z 0
=
[a23 (γ ) + a13 (γ )]γ dγ
Z
1
=
[(a − (1 − γ )α)3 + (b + (1 − γ )β)3 ]γ dγ
0
2 2 β 3 − α3 α a + β 2b βb − αa 2 + a 3 + b3 + + . 20 4 2
(1.10)
Similarly, E(A4 ) =
1
Z 0
=
[a24 (γ ) + a14 (γ )]γ dγ =
1
Z
((a − (1 − γ )α)4 + (b + (1 − γ )β)4 )γ dγ
0
3 2 2 α4 + β 4 β b − α3a β b + α 2 a 2 + a 4 + b4 2βb3 − 2αa 3 + + + . 30 5 2 3
(1.11)
The kurtosis is an important statistical measure of peakness. For any random variable X with finite fourth moments, −µ)4 . Analogously, we define the possibilistic kurtosis of a fuzzy number A the possibilistic kurtosis is given by E(X [Var(X )]2 as K (A) =
E[A − E(A)]4 . [Var(A)]2
(1.12)
2. Moments of FCV models 2.1. RCA and FCA models Random coefficient autoregressive time series were introduced by Nicholls and Quinn [9] and some of their properties have been studied recently in Appadoo et al. [3]. A sequence of random variables {yt } is called an RCA(1) time series if it satisfies the equations yt = (φ + bt )yt−1 + et
(2.1)
t ∈ T,
where T denotes the set of integers and 2 (i) bett ∼ 00 , σ0b σ02 , e
(ii) φ 2 + σb2 < 1. The sequences {bt } and {et } respectively, are the errors in the model. According to Nicholls and Quinn [9], (ii) is a necessary and sufficient condition for the second order stationarity of {yt }. For the RCA model in (2.1), the following theorem and proof are given in Appadoo et al. [3]. Theorem 2.1. Let {yt } be an RCA(1) time series as in (2.2) satisfying as in conditions (i) and (ii), and let γ y (k) be its covariance function. Then, (a) E(yt ) = 0, E(yt2 ) =
σe2 φ k σe2 , the kth lag autocovariance for yt is given by γ y (k) = and the 2 2 1−φ −σb 1−φ 2 −σb2 ρk = φ k for all k ∈ T . That means the usual AR(1) process has same autocorrelation as
autocorrelation for yt is the RCA(1). (b) If {bt } and {et } are normally distributed random variables then the kurtosis K (y) of the RCA process {yt } is given by K (y) =
3[1 − (σb2 + φ 2 )2 ] [1 − (φ 4 + 6φ 2 σb2 + 3σb4 )]
.
For an AR(1) process, K (y) reduces to 3. y2
y2
(c) The autocorrelation of yt2 is given by ρk = (φ 2 + σb2 )k . For an AR(1) process, it turns out to be ρk = φ 2k .
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Corollary 2.1. Let {yt } be an RCA(1) time series of the form yt = θt yt−1 + et satisfying the stationarity conditions, y and let ρk denote its correlation function. Then, (a) when θt = φ + bt , E(yt ) = 0, and E(yt2 ) = y ρk
)}k
φk .
σe2 , 1−φ 2 −σb2
the kth lag autocorrelation for yt is given by
= {E(φ + bt = y (b) when θt = sgn(bt ), bt ∼ N (0, σb2 ) then ρk = [1 − 2F(0)]k , where F is the cumulative distribution function of bt . (i.e.) when the coefficient θt is driven by a binary random variable {bt }, it takes the values −1 and +1. (c) when θt = (φ + |bt |α ), bt ∼ N (µ, σb2 ), the autocorrelation for yt is given by α k (2σ 2 ) 2 y (d) ρk = φ + √bπ 0 α+1 , 2 where 0(.) is the Gamma function. In analogy with RCA models, a sequence of random variables {yt } is called FCA(1) time series if it satisfies the equation yt = Φ yt−1 + et
t ∈ T,
(2.2)
where T denotes the set of integers and Φ(γ ) = [Φ1 (γ ), Φ2 (γ )] is the γ -level set if a fuzzy number. We also assume that (i) et ∼ N (0, σe2 ) R1 (ii) 0 (Φ12 (γ ) + Φ22 (γ ))γ dγ < 1. Note that the expression given in (2.2) for a FCA(1) model is a fuzzified version of the classical AR(1) model and hence subjectivity, arbitrariness, randomness and fuzziness have been incorporated in the model. The sequence {et } is white noise and Φ is the fuzzy coefficient. For an FCA(1) model, (ii) is a necessary and sufficient condition for the second order stationarity of {yt }. So, together with (i), it also ensures strict stationarity. We prove the following theorem for the FCA(1) model given in (2.2). Theorem 2.2. Let {yt } be an FCA(1) time series satisfying conditions (i) and (ii), and let γ y denote its autocovariance function. Then, i h [E(Φ )]k σ 2 σ2 (a) If E(yt ) = 0, E(yt2 ) = 1−E(eΦ 2 ) then the kth lag autocovariance for yt is given by γ y (k) = 1−E(Φ 2 )e , and the autocorrelation for yt is ρk = [E(Φ)]k for all k ∈ T . h i Φ 2 ))2 (b) If {et } is a normally distributed random variable then the kurtosis K (y) of {yt } is given by K (y) = 3 1−(E( , 4 1−E(Φ ) and for an AR(1) process it reduces to 3. y2 y2 (c) The autocorrelation of yt2 is given by ρk = [E(Φ)]2k , and for an AR(1) process it turns out to be ρk = φ 2k . Proof. E(yt ) = E E[yt |yt−1 ] = 0, Var(yt ) = EVar(yt |yt−1 ) + VarE(yt |yt−1 ) = σe2 + E(Φ 2 )Var(yt−1 ), and Var(yt ) = γ y (0) =
σe2 (1−E(Φ 2 ))
E[yt yt−1 ] = E[Φ yt−1 yt−1 ] + E[et yt−1 ] γ y (1) = E(Φ)γ y (0). Similarly, the kth lag autocovariance is γ y (k) = [E(Φ)]k γ y (0). For the proof of part (c), we have E(yt4 ) = E (Φ yt−1 + et )4 4 2 = E(Φ 4 )E(yt−1 ) + 6E(Φ 2 )E(yt−1 )σe2 + 3σe4 3σe4 (1 + E(Φ 2 )) = (1 − E(Φ 2 ))(1 − E(Φ 4 ))
(2.3)
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and hence, the kurtosis is given by K
(y)
=
=
" 2 # 1 − E(Φ 2 ) 3σ4 (1 + E(Φ 2 )) E(yt4 ) = (1 − E(Φ 2 ))(1 − E(Φ 4 )) σ2 E(yt2 )2
3[1 − (E(Φ 2 ))2 ] . 1 − E(Φ 4 )
(2.4)
We now derive the autocorrelation of yt2 of the FCA(1) process 2 2 2 2 2 2 2 ) + E(et2 yt−k ) = E(Φ 2 yt−1 yt−k E(yt2 yt−k ) = E(Φ 2 )E(yt−1 yt−k ) + σe2 E(yt−k ) 2 2 E(yt2 yt−1 ) = E(Φ 2 )E(yt4 ) + σe2 E(yt−1 ) 2 ) E(yt2 yt−1 σ y4
E(Φ 2 ) = y2
ρ1 =
−1
K (y) − 1
2 ] − σ4 E[yt2 yt−1 y
= [E(Φ)]2 .
E(yt4 ) − σ y4
(2.5) y2
Similarly, the kth lag autocorrelation of yt2 , ρk = [E(Φ)]2k .
Example 1. Consider the following FCA(1) model of the form: yt = Φ yt−1 + et
t ∈ T.
If we assume that Φ is a trapezoidal fuzzy number, then its γ -level set is written in this form Φ(γ ) = [Φ1 (γ ), Φ2 (γ )] = [a − (1 − γ )α, b + (1 − γ )β] where α > 0, β > 0 and 0 ≤ γ ≤ 1. The possibilistic kurtosis is given by, K
(y)
1 − (E(Φ 2 ))2 =3 1 − E(Φ 4 )
2 90 144 − 6(b2 + a 2 ) + 4(bβ − aα) + (α 2 + β 2 ) . = 144 30 − α 4 + β 4 + 6 β 3 b − α 3 a + 15 β 2 b2 + α 2 a 2 + a 4 + b4 + 20 βb3 − αa 3 (2.6) Moreover, the kth lag autocorrelation function of yt2 is given by y2 ρk
a+b β −α = + 2 6
2k
.
2.2. RC GARCH and FC GARCH models
Consider the following general class of GARCH( p, q) model for the time series yt : yt =
p
ht Zt ,
ht = ω +
p X i=1
(2.7) 2 αi yt−i +
q X j=1
β j h t− j ,
(2.8)
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783
where Z t is a sequence of independent, identically distributed random variables with zero mean and unit variance. Let u t = yt2 − h t be the martingale difference and let σu2 be the variance of u t . Then, Eqs. (2.7) and (2.8) can be written as yt2 − u t = ω +
p X
2 αi yt−i +
i=1
q X
β j h t− j ,
(2.9)
j=1
φ(B)yt2
= ω + β(B)u t , P Pq where φ(B) = 1 − ri=1 φi B i , φi = (αi + βi ), β(B) = 1 − j=1 β j B j and r = max( p, q). We shall make the following stationarity assumptions for yt2 which has an ARMA(r, q) representation. (A.1) All the zeroes of the polynomial φ(B) lie outside of the unit circle. P∞ 2 (A.2) ψ < ∞ where the ψi0 s are obtained from the relation ψ(B)φ(B) = β(B) with ψ(B) = P∞ i=0i i 1 + i=1 ψi B . For any stationary process the ψ-weight’s are given in Thavaneswaran et al. [10]. The assumptions ensure that the u 0t s are uncorrelated with zero mean and finite variance, and the yt2 process is weakly stationary. In this case, the autocorrelation function of yt2 will be exactly the same as that for a stationary ARMA(r , q) model. The GARCH kurtosis in terms of the ψ weights were given in Thavaneswaran et al. [10]. In this section we propose the corresponding fuzzy volatility models (FCV) and derive the kurtosis in terms of possibilistic mean and possibilistic variance of the fuzzy coefficient. First we give a lemma for the kurtosis of RC GARCH model. Consider the following random coefficient model: p 2 yt = h t Z t , h t = ω0 + (α1 + bt−1 )yt−1 + β1 h t−1 , Z t ∼ N (0, σ Z2 ) and bt ∼ N (0, σb2 ). Lemma 2.1. Under suitable stationary conditions, the kurtosis of yt is given by K (y) =
3[1 − (α12 σ Z2 + β1 )2 ] [1 − (2α1 β1 σ Z2 − 3α12 σ Z4 − 3σb2 σ Z4 − β12 )]
.
The proof is given in Appadoo et al. [3]. 2.2.1. Random coefficient ARCH(1) model In analogy with the RCA models we introduce a class of RCA versions of GARCH models. Consider the following general class of ARCH (1) model for the time series yt : p 2 yt = h t Z t , h t = ω + (α1 + at−1 )yt−1 , (2.10) where Z t is a sequence of independent, identically distributed random variables with zero mean and unit variance. Let u t = yt2 − h t be the martingale difference and let σu2 be the variance of u t . Then, 2 yt2 = ω + (α1 + at−1 )yt−1 + ut .
(2.11)
The following lemma follows from Appadoo et al. [3]. Lemma 2.2. For the model FCA-ARCH(1) p 2 , yt = h t Z t , h t = ω0 + (α1 + at−1 )yt−1 The kurtosis is given by K (y) =
where Z t ∼ N (0, σ Z2 ) and at ∼ N (0, σa2 ).
3[1−α12 σ Z4 ] . [1−3σ Z2 (α12 +σa2 )]σ Z2
In analogy with the RCA models, we introduce a class of FCA versions of GARCH(1, 1) models. Consider the following general class of GARCH(1, 1) models for the time series yt : p 2 yt = h t Z t , h t = ω0 + Φ yt−1 + β1 h t−1 , where Z t ∼ N (0, σ Z2 ). (2.12) The fuzzy coefficient Φ = [Φ1 (γ ), Φ2 (γ )] is the γ -level set, and we assume
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(i) et ∼ N (0, σe2 ) R1 (ii) 0 (Φ12 (γ ) + Φ22 (γ ))γ dγ < 1. Theorem 2.3. Consider the following FCA-GARCH(1, 1) models: p 2 yt = h t Z t , h t = ω0 + Φ yt−1 + β1 h t−1 , where Z t ∼ N (0, σ Z2 ). R1 Under the stationary conditions, 0 (Φ12 (γ ) + Φ22 (γ ))γ dγ + β1 < 1, the possibilistic kurtosis of yt is given by " # 2 1 − E[Φ]σz2 + β1 (y) . K =3 (2.13) 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 Proof. i h h i E yt2 = E h t Z t2 = E[h t ]σz2 .
(2.14)
The unconditional variance is given by E [h t ] = ω0 + E[Φ]E[h t ]σz2 + β1 E[h t−1 ]. Thus, ω0 E [h t ] = 1 − E[Φ]σz2 − β1
and 2 2 h 2t = ω0 + Φ yt−1 + β1 h t−1 . Therefore, E[h 2t ]
=
ω02 + 2ω0 E[h t−1 ] σz2 E[Φ] + β1 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12
.
(2.15)
The possibilistic kurtosis is " 2 #" # 2 + 2ω E[h 2−β 2 ω ] σ E[Φ] + β 1 − E[Φ]σ 0 t−1 1 1 z z K (y) = 3 0 ω02 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 h i " # 0 2−β 2 σz2 E[Φ] + β1 ω02 + 2ω0 1−E[Φω]σ 2 −β 1 − E[Φ]σ 1 1 z z = 3 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 ω02 #" 2 # ω02 1 − E[Φ]σz2 − β1 + 2ω02 σz2 E[Φ] + β1 1 − E[Φ]σz2 − β1 =3 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 1 − E[Φ]σz2 − β1 ω02 " # 2 1 − E[Φ]σz2 + β1 . =3 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 "
(2.16)
If we assume that Φ is a trapezoidal fuzzy number then E[Φ] =
3 (a + b) + (β − α) 6
(2.17)
and 2 (a 2 + b2 ) (bβ − aα) (α + β 2 ) + + . E(Φ ) = 2 3 12 2
(2.18)
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785
Table 1 Monthly log returns for IBM stock γ
z ∗γ
E[Φ]
E[Φ 2 ]
K (y)
0.0999 0.0999 0.0999 0.0999 0.0999 0.0999 0.0999 0.0999 0.0999 0.0999
0.01019 0.01011 0.01006 0.01003 0.01001 0.01000 0.00999 0.00998 0.00998 0.00998
3.4553 3.4491 3.4458 3.4437 3.4422 3.4412 3.4405 3.4400 3.4397 3.4396
2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.645 1.282 1.036 0.842 0.674 0.524 0.385 0.253 0.126 0.000
The possibilistic kurtosis can be obtained by substituting Eqs. (2.17) and (2.18) into Eq. (2.16). With β1 = 0, the possibilistic kurtosis for the FC-ARCH(1) process can be obtained. If we assume that Φ is a triangular fuzzy number then E[Φ] =
6a + (β − α) 6
(2.19)
and E(Φ 2 ) = a 2 +
2 a(β − α) (α + β 2 ) + . 3 12
(2.20)
The possibilistic kurtosis can be obtained by substituting Eqs. (2.19) and (2.20) into (2.16). With β1 = 0, the possibilistic kurtosis for the FC-ARCH(1) process can be obtained. 3. Example This example illustrates how the possibilistic kurtosis can be obtained for a GARCH process. We first apply the modeling procedure to build a GARCH model for the monthly log returns of IBM stock returns ([13]), and then obtained a GARCH(1, 1) model as appropriate. Assuming Z t is an i.i.d standard normal, we obtained the following fitted model with the parameter estimates: p rt = µ + yt , yt = h t Z t 2 h t = ω + Φ yt−1 + β1 h t−1
b µ = 0.0130(0.0023) b ω = 0.00035(0.00011) b = 0.0999(0.0214) Φ βb1 = 0.8187(0.0422). For convenience, we assume that Φ is a symmetric triangular fuzzy number. Then, we obtain i h Φ = [Φ1 (γ ), Φ2 (γ )] = 0.0999 − z ∗γ (0.0214), 0.0999 + z ∗γ (0.0214) , 2
(3.1)
2
where z ∗γ is the 100(1 − γ2 )th percentile of a standard normal distribution. Using Eqs. (2.16) and (3.1), we can build 2 the possibilistic kurtosis (see Table 1). 4. Conclusions Granger [11], a Nobel prize winner (2003), had cited the first author’s work (Abraham and Thavaneswaran, [12]) in his Berkeley Symposium presentation. In this paper, some results in Abraham and Thavaneswaran [12] are extended
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for volatility models with fuzzy coefficient. A numerical example for real data with fuzzy kurtosis calculation is also discussed in some detail. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001) 315–326. R. Fuller, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363–374. S.S. Appadoo, M. Ghahramani, A. Thavaneswaran, Moment properties of some time series models, Math. Sci. 30 (1) (2005) 50–63. C. Carlsson, R. Fuller, Fuzzy Reasoning in Decision Making and Optimization, Physica-Verlag, 2002. L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. C. Carlsson, R. Fuller, A fuzzy approach to real option valuation, Fuzzy Sets and Systems 139 (2003) 297–312. R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18 (1986) 31–43. D.F. Nicholls, B.G. Quinn, Random Coefficient Autoregressive Models: An Introduction, in: Lecture Notes in Statistics, vol. 11, Springer, New York, 1982. A. Thavaneswaran, S.S. Appadoo, M. Samanta, Random coefficient GARCH models, Math. Comput. Modelling 41 (6–7) (2005) 723–733. C.W.J. Granger, Overview of non-linear time series specification in economics, Berkeley NSF-Symposia, 1998. B. Abraham, A. Thavaneswaran, A nonlinear time series model and estimation of missing observations, Ann. Inst. Stat. Math. 43 (1991) 493–504. R.S. Tsay, Analysis of Financial Time Series, John Wiley & Sons, 2002.
Fuzzy coefficient volatility (FCV) models with applications A. Thavaneswaran a,∗ , K. Thiagarajah b , S.S. Appadoo c a Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada b Department of Mathematics, Illinois State University, Normal, IL, USA c Department of Supply Chain Management, University of Manitoba, Winnipeg, Manitoba, Canada
Received 30 November 2005; received in revised form 14 July 2006; accepted 28 July 2006
Abstract Recently, Carlsson and Fuller [C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001) 315–326] have introduced possibilistic mean, variance and covariance of fuzzy numbers and Fuller and Majlender [R. Fuller, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363–374] have introduced the notion of crisp weighted possibilistic moments of fuzzy numbers. In this paper, we propose a class of FCV (Fuzzy Coefficient Volatility) models and study the moment properties. The method used here is very similar to the method used in Appadoo et al. [S.S. Appadoo, M. Ghahramani, A. Thavaneswaran, Moment properties of some time series models, Math. Sci. 30 (1) (2005) 50–63]. The proposed models incorporate fuzziness, subjectivity, arbitrariness and uncertainty observed in most financial time series. The usual forecasting method does not incorporate parameter variability. Fuzzy numbers are used to model the parameters to incorporate parameter variability. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Possibilistic mean; Possibilistic variance; Fuzzy coefficient autoregressive model
1. Introduction The sophisticated analysis used by the financial industry has lent increasing importance to time series modeling. Recently, there has been growing interest in using fuzzy numbers in finance and economics. Many financial series, such as returns on stocks and foreign exchange rates, exhibit leptokurtosis and volatility varying in time. Many decision making problems exhibit some level of arbitrariness, vagueness and fuzziness (Carlsson and Fuller [4] and Fuller and Majlender [2]). These features have been the subject of extensive studies ever since Zadeh [5] has reported them. Decision-making problems, in general, involved uncertainty as their model parameters are not precisely known. As a result there has been growing interest in using fuzzy algebra in such models. Historically, probability theory is presented as forming theoretical foundations for reasoning and decision making in situations involving uncertainty. However, often one is faced with the situations in which decisions are required to be made on the basis of ill-defined variables and imprecise (vague) data. Fuzzy algebra is a simple but potentially a useful way to study impreciseness
∗ Corresponding author. Tel.: +1 204 474 8984; fax: +1 204 474 7621.
E-mail address: [email protected] (A. Thavaneswaran). c 2006 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2006.07.019
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through a cascade of calculations. As an alternative, fuzzy models have been used to model systems that are hard to define precisely, see for example Dubois and Prade [6] for details. The use of fuzzy set theory as a methodology for modeling and analyzing certain financial problems is of particular interest to a number of researchers due to fuzzy set theory’s ability to quantitatively and qualitatively model those problems which involve vagueness and imprecision. Recent studies has shown that a fuzzy random variable can be considered as a measurable mapping from a probability space to a set of fuzzy variables. Fuzzy time series models provide a new avenue to deal subjectivity observed in most financial time series models. We summarize the preliminaries in Section 1. In Section 2, RCA and GARCH models are given and the corresponding FCV models are introduced. Following Carlsson and Fuller [4,7], higher order moments of fuzzy numbers are defined. We also derive the moments and possibilistic kurtosis of the proposed FCV models. Section 3 concludes with an illustrative numerical example. 1.1. Preliminaries and notations Let a set X be a subset of a universal set U , and let R ⊂ X be the set of real numbers. Definition 1.1. Fuzzy set A in X is a set of ordered pairs A = {(x, µ(x)) : x ∈ X }, where µ(x) maps x ∈ X on the real interval [0, 1] and is known as the membership function of x ∈ X . Definition 1.2. A fuzzy set A in R n is said to be a convex fuzzy set if its α-cuts Aα are (crisp) convex sets for all α ∈ [0, 1]. Alternatively, a fuzzy set A in R n is a convex fuzzy set if and only if for all x1 , x2 ∈ R n and 0 ≤ λ ≤ 1, µ A (λx1 + (1 − λ)x2 ) ≥ Min(µ A (x1 ), µ A (x2 )). Definition 1.3. A fuzzy number A˜ ∈ F, is called a trapezoidal fuzzy number (Tr.F.N.) with core [a, b] left width α and right width β if its membership function has the following form a−t 1 − if a − α ≤ t ≤ a α 1 if a ≤ t ≤ b µ(x) = (1.1) t −b 1− if b ≤ t ≤ b + β β 0 otherwise and we use the notation A˜ = (a, b, α, β). It can easily be shown that Aγ = [a1 (γ ), a2 (γ )] = [a − (1 − γ )α, b + (1 − γ )β]
∀γ ∈ [0, 1].
(1.2)
The support of A˜ is (a − α, b − β). Moreover, for any fuzzy number A˜ and a positive real number C, the following relationship holds A˜ ≤ C ⇐⇒
1
Z
(a1 (γ ) + a2 (γ ))γ dγ ≤ C.
(1.3)
0
1.2. Moments Following Carlsson and Fuller [1], crisp possibilistic mean and possibilistic variance of continuous possibility distributions are given below. 1
Z M(A) = 0
(a1 (γ ) + a2 (γ ))γ dγ ,
(1.4)
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where M(A) is the level-weighted average of the arithmetic means of all γ -level sets. 2 ! Z 1 a1 (γ ) + a2 (γ ) Pos[A ≤ a1 ] Var(A) = − a1 (γ ) dγ 2 0 2 ! Z 1 a1 (γ ) + a2 (γ ) − a2 (γ ) dγ Pos[A ≥ a2 ] + 2 0 Z 1 1 (a2 (γ ) − a1 (γ ))2 γ dγ . = 0 2
779
(1.5)
Carlsson and Fuller [1] defined the variance of a fuzzy number A as the expected value of the squared deviations between the arithmetic mean and the endpoints of its level sets. Let A and B ∈ F be fuzzy numbers with A(γ ) = [a1 (γ ), a2 (γ )] and B(γ ) = [b1 (γ ), b2 (γ )], γ ∈ [0, 1]. Goetschel and Voxman [8] introduced a method for ranking fuzzy numbers as Z 1 Z 1 A ≤ B ⇐⇒ (a1 (γ ) + a2 (γ ))γ dγ ≤ (b1 (γ ) + b2 (γ ))γ dγ . (1.6) 0
0
As pointed out by Goetschel and Voxman [8] the definition given in (1.6) for ordering fuzzy numbers was motivated by the desire to give less importance to the lower levels of fuzzy numbers. For a Tr.F.N A the possibilistic mean value is given by Z 1 Z 1 E(A) = γ (a1 (γ ) + a2 (γ ))dγ = γ [a − (1 − γ )α + b + (1 − γ )β]dγ 0
0 1
Z =
γ [a − (1 − γ )α + b + (1 − γ )β]dγ =
0
=
Z
1
[(a − α + b + β)γ + (α − β)γ 2 ]dγ
0
a+b β −α + . 2 6
(1.7)
The possibilistic variance is given by Z Z 1 1 1 1 Var(A) = γ (a2 (γ ) − a1 (γ ))2 dγ = γ [(b + (1 − γ )β) − (a − (1 − γ )α)]2 dγ 2 0 2 0 (b − a)2 (b − a)(α + β) (α + β)2 = + + . 4 6 24 On the other hand if A˜ is a triangular fuzzy number then the possibilistic mean and possibilistic variance are given by (α + β)2 (β − α) and Var(A) = . E(A) = a + 6 24 The definition of the nth moment of a fuzzy number is partly motivated to give a meaningful definition of possibilistic kurtosis for fuzzy numbers. The kurtosis measures the peakedness of unimodal distributions. Following Carlsson and Fuller [1], we define the possibilistic nth moment of a fuzzy random variable A, E(An ), as Z 1 n n E(A ) = a2 (γ ) + a1n (γ ) γ dγ . (1.8) 0
Now, we compute different moments of Tr.F.N’s. Z 1h Z 1 i E(A2 ) = a22 (γ ) + a12 (γ ) γ dγ = [(a − (1 − γ )α)2 + (b + (1 − γ )β)2 ]γ dγ 0
=
(a 2
0
+ b2 ) 2
2 (bβ − aα) (α + β 2 ) + + 3 12
(1.9)
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and E(A ) = 3
1
Z 0
=
[a23 (γ ) + a13 (γ )]γ dγ
Z
1
=
[(a − (1 − γ )α)3 + (b + (1 − γ )β)3 ]γ dγ
0
2 2 β 3 − α3 α a + β 2b βb − αa 2 + a 3 + b3 + + . 20 4 2
(1.10)
Similarly, E(A4 ) =
1
Z 0
=
[a24 (γ ) + a14 (γ )]γ dγ =
1
Z
((a − (1 − γ )α)4 + (b + (1 − γ )β)4 )γ dγ
0
3 2 2 α4 + β 4 β b − α3a β b + α 2 a 2 + a 4 + b4 2βb3 − 2αa 3 + + + . 30 5 2 3
(1.11)
The kurtosis is an important statistical measure of peakness. For any random variable X with finite fourth moments, −µ)4 . Analogously, we define the possibilistic kurtosis of a fuzzy number A the possibilistic kurtosis is given by E(X [Var(X )]2 as K (A) =
E[A − E(A)]4 . [Var(A)]2
(1.12)
2. Moments of FCV models 2.1. RCA and FCA models Random coefficient autoregressive time series were introduced by Nicholls and Quinn [9] and some of their properties have been studied recently in Appadoo et al. [3]. A sequence of random variables {yt } is called an RCA(1) time series if it satisfies the equations yt = (φ + bt )yt−1 + et
(2.1)
t ∈ T,
where T denotes the set of integers and 2 (i) bett ∼ 00 , σ0b σ02 , e
(ii) φ 2 + σb2 < 1. The sequences {bt } and {et } respectively, are the errors in the model. According to Nicholls and Quinn [9], (ii) is a necessary and sufficient condition for the second order stationarity of {yt }. For the RCA model in (2.1), the following theorem and proof are given in Appadoo et al. [3]. Theorem 2.1. Let {yt } be an RCA(1) time series as in (2.2) satisfying as in conditions (i) and (ii), and let γ y (k) be its covariance function. Then, (a) E(yt ) = 0, E(yt2 ) =
σe2 φ k σe2 , the kth lag autocovariance for yt is given by γ y (k) = and the 2 2 1−φ −σb 1−φ 2 −σb2 ρk = φ k for all k ∈ T . That means the usual AR(1) process has same autocorrelation as
autocorrelation for yt is the RCA(1). (b) If {bt } and {et } are normally distributed random variables then the kurtosis K (y) of the RCA process {yt } is given by K (y) =
3[1 − (σb2 + φ 2 )2 ] [1 − (φ 4 + 6φ 2 σb2 + 3σb4 )]
.
For an AR(1) process, K (y) reduces to 3. y2
y2
(c) The autocorrelation of yt2 is given by ρk = (φ 2 + σb2 )k . For an AR(1) process, it turns out to be ρk = φ 2k .
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Corollary 2.1. Let {yt } be an RCA(1) time series of the form yt = θt yt−1 + et satisfying the stationarity conditions, y and let ρk denote its correlation function. Then, (a) when θt = φ + bt , E(yt ) = 0, and E(yt2 ) = y ρk
)}k
φk .
σe2 , 1−φ 2 −σb2
the kth lag autocorrelation for yt is given by
= {E(φ + bt = y (b) when θt = sgn(bt ), bt ∼ N (0, σb2 ) then ρk = [1 − 2F(0)]k , where F is the cumulative distribution function of bt . (i.e.) when the coefficient θt is driven by a binary random variable {bt }, it takes the values −1 and +1. (c) when θt = (φ + |bt |α ), bt ∼ N (µ, σb2 ), the autocorrelation for yt is given by α k (2σ 2 ) 2 y (d) ρk = φ + √bπ 0 α+1 , 2 where 0(.) is the Gamma function. In analogy with RCA models, a sequence of random variables {yt } is called FCA(1) time series if it satisfies the equation yt = Φ yt−1 + et
t ∈ T,
(2.2)
where T denotes the set of integers and Φ(γ ) = [Φ1 (γ ), Φ2 (γ )] is the γ -level set if a fuzzy number. We also assume that (i) et ∼ N (0, σe2 ) R1 (ii) 0 (Φ12 (γ ) + Φ22 (γ ))γ dγ < 1. Note that the expression given in (2.2) for a FCA(1) model is a fuzzified version of the classical AR(1) model and hence subjectivity, arbitrariness, randomness and fuzziness have been incorporated in the model. The sequence {et } is white noise and Φ is the fuzzy coefficient. For an FCA(1) model, (ii) is a necessary and sufficient condition for the second order stationarity of {yt }. So, together with (i), it also ensures strict stationarity. We prove the following theorem for the FCA(1) model given in (2.2). Theorem 2.2. Let {yt } be an FCA(1) time series satisfying conditions (i) and (ii), and let γ y denote its autocovariance function. Then, i h [E(Φ )]k σ 2 σ2 (a) If E(yt ) = 0, E(yt2 ) = 1−E(eΦ 2 ) then the kth lag autocovariance for yt is given by γ y (k) = 1−E(Φ 2 )e , and the autocorrelation for yt is ρk = [E(Φ)]k for all k ∈ T . h i Φ 2 ))2 (b) If {et } is a normally distributed random variable then the kurtosis K (y) of {yt } is given by K (y) = 3 1−(E( , 4 1−E(Φ ) and for an AR(1) process it reduces to 3. y2 y2 (c) The autocorrelation of yt2 is given by ρk = [E(Φ)]2k , and for an AR(1) process it turns out to be ρk = φ 2k . Proof. E(yt ) = E E[yt |yt−1 ] = 0, Var(yt ) = EVar(yt |yt−1 ) + VarE(yt |yt−1 ) = σe2 + E(Φ 2 )Var(yt−1 ), and Var(yt ) = γ y (0) =
σe2 (1−E(Φ 2 ))
E[yt yt−1 ] = E[Φ yt−1 yt−1 ] + E[et yt−1 ] γ y (1) = E(Φ)γ y (0). Similarly, the kth lag autocovariance is γ y (k) = [E(Φ)]k γ y (0). For the proof of part (c), we have E(yt4 ) = E (Φ yt−1 + et )4 4 2 = E(Φ 4 )E(yt−1 ) + 6E(Φ 2 )E(yt−1 )σe2 + 3σe4 3σe4 (1 + E(Φ 2 )) = (1 − E(Φ 2 ))(1 − E(Φ 4 ))
(2.3)
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and hence, the kurtosis is given by K
(y)
=
=
" 2 # 1 − E(Φ 2 ) 3σ4 (1 + E(Φ 2 )) E(yt4 ) = (1 − E(Φ 2 ))(1 − E(Φ 4 )) σ2 E(yt2 )2
3[1 − (E(Φ 2 ))2 ] . 1 − E(Φ 4 )
(2.4)
We now derive the autocorrelation of yt2 of the FCA(1) process 2 2 2 2 2 2 2 ) + E(et2 yt−k ) = E(Φ 2 yt−1 yt−k E(yt2 yt−k ) = E(Φ 2 )E(yt−1 yt−k ) + σe2 E(yt−k ) 2 2 E(yt2 yt−1 ) = E(Φ 2 )E(yt4 ) + σe2 E(yt−1 ) 2 ) E(yt2 yt−1 σ y4
E(Φ 2 ) = y2
ρ1 =
−1
K (y) − 1
2 ] − σ4 E[yt2 yt−1 y
= [E(Φ)]2 .
E(yt4 ) − σ y4
(2.5) y2
Similarly, the kth lag autocorrelation of yt2 , ρk = [E(Φ)]2k .
Example 1. Consider the following FCA(1) model of the form: yt = Φ yt−1 + et
t ∈ T.
If we assume that Φ is a trapezoidal fuzzy number, then its γ -level set is written in this form Φ(γ ) = [Φ1 (γ ), Φ2 (γ )] = [a − (1 − γ )α, b + (1 − γ )β] where α > 0, β > 0 and 0 ≤ γ ≤ 1. The possibilistic kurtosis is given by, K
(y)
1 − (E(Φ 2 ))2 =3 1 − E(Φ 4 )
2 90 144 − 6(b2 + a 2 ) + 4(bβ − aα) + (α 2 + β 2 ) . = 144 30 − α 4 + β 4 + 6 β 3 b − α 3 a + 15 β 2 b2 + α 2 a 2 + a 4 + b4 + 20 βb3 − αa 3 (2.6) Moreover, the kth lag autocorrelation function of yt2 is given by y2 ρk
a+b β −α = + 2 6
2k
.
2.2. RC GARCH and FC GARCH models
Consider the following general class of GARCH( p, q) model for the time series yt : yt =
p
ht Zt ,
ht = ω +
p X i=1
(2.7) 2 αi yt−i +
q X j=1
β j h t− j ,
(2.8)
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783
where Z t is a sequence of independent, identically distributed random variables with zero mean and unit variance. Let u t = yt2 − h t be the martingale difference and let σu2 be the variance of u t . Then, Eqs. (2.7) and (2.8) can be written as yt2 − u t = ω +
p X
2 αi yt−i +
i=1
q X
β j h t− j ,
(2.9)
j=1
φ(B)yt2
= ω + β(B)u t , P Pq where φ(B) = 1 − ri=1 φi B i , φi = (αi + βi ), β(B) = 1 − j=1 β j B j and r = max( p, q). We shall make the following stationarity assumptions for yt2 which has an ARMA(r, q) representation. (A.1) All the zeroes of the polynomial φ(B) lie outside of the unit circle. P∞ 2 (A.2) ψ < ∞ where the ψi0 s are obtained from the relation ψ(B)φ(B) = β(B) with ψ(B) = P∞ i=0i i 1 + i=1 ψi B . For any stationary process the ψ-weight’s are given in Thavaneswaran et al. [10]. The assumptions ensure that the u 0t s are uncorrelated with zero mean and finite variance, and the yt2 process is weakly stationary. In this case, the autocorrelation function of yt2 will be exactly the same as that for a stationary ARMA(r , q) model. The GARCH kurtosis in terms of the ψ weights were given in Thavaneswaran et al. [10]. In this section we propose the corresponding fuzzy volatility models (FCV) and derive the kurtosis in terms of possibilistic mean and possibilistic variance of the fuzzy coefficient. First we give a lemma for the kurtosis of RC GARCH model. Consider the following random coefficient model: p 2 yt = h t Z t , h t = ω0 + (α1 + bt−1 )yt−1 + β1 h t−1 , Z t ∼ N (0, σ Z2 ) and bt ∼ N (0, σb2 ). Lemma 2.1. Under suitable stationary conditions, the kurtosis of yt is given by K (y) =
3[1 − (α12 σ Z2 + β1 )2 ] [1 − (2α1 β1 σ Z2 − 3α12 σ Z4 − 3σb2 σ Z4 − β12 )]
.
The proof is given in Appadoo et al. [3]. 2.2.1. Random coefficient ARCH(1) model In analogy with the RCA models we introduce a class of RCA versions of GARCH models. Consider the following general class of ARCH (1) model for the time series yt : p 2 yt = h t Z t , h t = ω + (α1 + at−1 )yt−1 , (2.10) where Z t is a sequence of independent, identically distributed random variables with zero mean and unit variance. Let u t = yt2 − h t be the martingale difference and let σu2 be the variance of u t . Then, 2 yt2 = ω + (α1 + at−1 )yt−1 + ut .
(2.11)
The following lemma follows from Appadoo et al. [3]. Lemma 2.2. For the model FCA-ARCH(1) p 2 , yt = h t Z t , h t = ω0 + (α1 + at−1 )yt−1 The kurtosis is given by K (y) =
where Z t ∼ N (0, σ Z2 ) and at ∼ N (0, σa2 ).
3[1−α12 σ Z4 ] . [1−3σ Z2 (α12 +σa2 )]σ Z2
In analogy with the RCA models, we introduce a class of FCA versions of GARCH(1, 1) models. Consider the following general class of GARCH(1, 1) models for the time series yt : p 2 yt = h t Z t , h t = ω0 + Φ yt−1 + β1 h t−1 , where Z t ∼ N (0, σ Z2 ). (2.12) The fuzzy coefficient Φ = [Φ1 (γ ), Φ2 (γ )] is the γ -level set, and we assume
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(i) et ∼ N (0, σe2 ) R1 (ii) 0 (Φ12 (γ ) + Φ22 (γ ))γ dγ < 1. Theorem 2.3. Consider the following FCA-GARCH(1, 1) models: p 2 yt = h t Z t , h t = ω0 + Φ yt−1 + β1 h t−1 , where Z t ∼ N (0, σ Z2 ). R1 Under the stationary conditions, 0 (Φ12 (γ ) + Φ22 (γ ))γ dγ + β1 < 1, the possibilistic kurtosis of yt is given by " # 2 1 − E[Φ]σz2 + β1 (y) . K =3 (2.13) 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 Proof. i h h i E yt2 = E h t Z t2 = E[h t ]σz2 .
(2.14)
The unconditional variance is given by E [h t ] = ω0 + E[Φ]E[h t ]σz2 + β1 E[h t−1 ]. Thus, ω0 E [h t ] = 1 − E[Φ]σz2 − β1
and 2 2 h 2t = ω0 + Φ yt−1 + β1 h t−1 . Therefore, E[h 2t ]
=
ω02 + 2ω0 E[h t−1 ] σz2 E[Φ] + β1 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12
.
(2.15)
The possibilistic kurtosis is " 2 #" # 2 + 2ω E[h 2−β 2 ω ] σ E[Φ] + β 1 − E[Φ]σ 0 t−1 1 1 z z K (y) = 3 0 ω02 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 h i " # 0 2−β 2 σz2 E[Φ] + β1 ω02 + 2ω0 1−E[Φω]σ 2 −β 1 − E[Φ]σ 1 1 z z = 3 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 ω02 #" 2 # ω02 1 − E[Φ]σz2 − β1 + 2ω02 σz2 E[Φ] + β1 1 − E[Φ]σz2 − β1 =3 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 1 − E[Φ]σz2 − β1 ω02 " # 2 1 − E[Φ]σz2 + β1 . =3 1 − 3σz4 E[Φ 2 ] + 2σz2 β1 E[Φ] + β12 "
(2.16)
If we assume that Φ is a trapezoidal fuzzy number then E[Φ] =
3 (a + b) + (β − α) 6
(2.17)
and 2 (a 2 + b2 ) (bβ − aα) (α + β 2 ) + + . E(Φ ) = 2 3 12 2
(2.18)
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785
Table 1 Monthly log returns for IBM stock γ
z ∗γ
E[Φ]
E[Φ 2 ]
K (y)
0.0999 0.0999 0.0999 0.0999 0.0999 0.0999 0.0999 0.0999 0.0999 0.0999
0.01019 0.01011 0.01006 0.01003 0.01001 0.01000 0.00999 0.00998 0.00998 0.00998
3.4553 3.4491 3.4458 3.4437 3.4422 3.4412 3.4405 3.4400 3.4397 3.4396
2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.645 1.282 1.036 0.842 0.674 0.524 0.385 0.253 0.126 0.000
The possibilistic kurtosis can be obtained by substituting Eqs. (2.17) and (2.18) into Eq. (2.16). With β1 = 0, the possibilistic kurtosis for the FC-ARCH(1) process can be obtained. If we assume that Φ is a triangular fuzzy number then E[Φ] =
6a + (β − α) 6
(2.19)
and E(Φ 2 ) = a 2 +
2 a(β − α) (α + β 2 ) + . 3 12
(2.20)
The possibilistic kurtosis can be obtained by substituting Eqs. (2.19) and (2.20) into (2.16). With β1 = 0, the possibilistic kurtosis for the FC-ARCH(1) process can be obtained. 3. Example This example illustrates how the possibilistic kurtosis can be obtained for a GARCH process. We first apply the modeling procedure to build a GARCH model for the monthly log returns of IBM stock returns ([13]), and then obtained a GARCH(1, 1) model as appropriate. Assuming Z t is an i.i.d standard normal, we obtained the following fitted model with the parameter estimates: p rt = µ + yt , yt = h t Z t 2 h t = ω + Φ yt−1 + β1 h t−1
b µ = 0.0130(0.0023) b ω = 0.00035(0.00011) b = 0.0999(0.0214) Φ βb1 = 0.8187(0.0422). For convenience, we assume that Φ is a symmetric triangular fuzzy number. Then, we obtain i h Φ = [Φ1 (γ ), Φ2 (γ )] = 0.0999 − z ∗γ (0.0214), 0.0999 + z ∗γ (0.0214) , 2
(3.1)
2
where z ∗γ is the 100(1 − γ2 )th percentile of a standard normal distribution. Using Eqs. (2.16) and (3.1), we can build 2 the possibilistic kurtosis (see Table 1). 4. Conclusions Granger [11], a Nobel prize winner (2003), had cited the first author’s work (Abraham and Thavaneswaran, [12]) in his Berkeley Symposium presentation. In this paper, some results in Abraham and Thavaneswaran [12] are extended
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for volatility models with fuzzy coefficient. A numerical example for real data with fuzzy kurtosis calculation is also discussed in some detail. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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