Modeling of Volatility with Non-linear Time Series Model
MModeling of Volatility with Non-linear TimeSeries Model a Kim Song Yon, Kim Mun Chol
Faculty of Mathematics,
Kim Il Sung
University, Pyongyang, D. P. R. Korea a Corresponding author. e-mail address: [email protected]
Abstract
In this paper, non-linear time series models are used to describe volatilityin financial time series data. To describe volatility, two of the non-linear timeseries are combined into form TAR (Threshold Auto-Regressive Model)with AARCH (Asymmetric Auto-Regressive Conditional Heteroskedastic-ity) error term and its parameter estimation is studied.
Keywords:
Volatility, Non-linear time series model, ARCH, AARCH (Asymmet-ric ARCH), TAR, QMLE (Quasi Maximum Likelihood Estimation)
In financial practice, the volatility of asset price is an important variable and itsmodeling has a great significance in investment, monetary policy making, finan-cial risk management and etc.A good forecast of the volatility of asset prices over the investment holdingperiod is a good starting point for assessing investment risk. The volatility ofasset is the most important variable in the pricing of derivative securities. To pricean option we need to know the volatility of the underlying asset from now untilthe option expires. In fact, the market convention is to list option prices in termsof volatility units. Nowadays, the definition and measurement of volatility may beclearly specified in the derivative contracts. In these new contracts, the volatilitynow becomes the underlying “asset”.Volatility model has become one of the major tasks in analysis of financialtime series models and interested many scientists.1 a r X i v : . [ q -f i n . S T ] J u l In 1973, the mathematical finance entered a new phase by Black and Scholes[1]. They used a stochastic analysis theory to estimate the price of an option, theholder of which has the right to buy a good (or underlying asset ) with a predeter-mined price K at a predetermined future time T + τ rather than the present time T . (This kind of option is called an European option.)Assume that the dynamics of the price S t of the underlying asset follows dS t = rS t dt + σS t dω t under the risk neutral measure. Here r (short rate) and σ are constants and ω t is a standard Wiener process. Then the time t -price of the option is provided asfollows [1]: C t = S t Φ( d ) − K exp( − rt )Φ( d − σ √ τ ) . Here τ = T − t and the parameter σ is only unknown.This σ is called the volatility and as shown in the above formula, accurate es-timation of σ has become a very important issue in option pricing and estimating.Furthermore, other issues such as estimation of volatility σt related to the time t have been raised.In 1982, Robert Engle suggested a new model which can estimate the volatilityin a more accurate way [3]. He paid attention to the error term in ARCH, whichwas mostly ignored in linear time series models such as AR, ARMA and ARIMA,and proposed a new nonlinear model by adding, instead of simple white noise,the error-term characteristic heteroskedasticity in which the conditional deviationchanges auto-regressively. He proved that if the stock price X t is to be changedinto the following ε t = 100 log( x t /x t − or ε t = x t − x t − x t − , then, it is possible to be modeled as follows: ε t = z t h t ,z t : i.i.d, N (0 , ,h t = α + α ε t − + · · · + α q ε t − q . This is the famous ARCH (q) model. ( h t being the volatility.) This model over-whelmed the statisticians and (in particular) econometricians and became instantlypopular. Engle was awarded the Nobel Prize for Economics in 2003. Thesetypes of non-linear models have been recognized as far superior estimation toolsand meanwhile researches have been made to improve the model for better ap-plicability. As a result, many modifications have appeared to form an ARCHfamily. Since then, other researches have been actively carried out to estimatevolatility with ARCH (q) model. Engle also proposed the method of testing theARCH effect, the effect of conditional deviation and proved the existence of Auto-Regressive Conditional Heteroskedasticity, therefore raising questions towardsI.I.D (Independent Identical Distribution) used for the error-term previously.In 1986, Bollerslev modified Engle’s ARCH (q) model into GARCH (p, q)model [2]. (cid:40) ε t = z t h t , z t : i.i.d,h t = α + (cid:80) qi =1 α i ε t − i + (cid:80) pi =1 β i h t − i . In his paper, he proposed existence, stationary condition and MLE (maximumlikelihood estimation) of the GARCH (1, 1) process. After this, many ARCHmodels such ARCH-M, IGARCH and LogGARCH have been developed. Through-out the researches, the volatility has been proved to be more influenced by ‘badnews’ rather than ‘good news’, that is, to be asymmetric, which resulted in re-searches on asymmetric models.In 1991, Nelson [8] proposed an exponential GARCH (EGARCH) modelwhich shows the asymmetric shock. h t = γ + γ h t − g ( ε t − ,g ( x ) = ωx + λ ( | x | − E | x | ) . But in many research papers, the effective parameter estimation and stationaryconditions was not being clearly explained and this difficulty was thought to behard to overcome [5]. But in 1993, Glosten tried to model asymmetric volatil-ity using Threshold ARCH (TARCH) model and later, many asymmetric modelswere proposed [4].Especially in 2003, Liu Ji Chun developed the asymmetric ARCH (q) model[10].Until now, the constant researches are being made to come up with bettervolatility models which can show effects of various ARCH models.In this paper, the modeling of volatility with the non-linear time series modelis observed based on the analysis of previous researches.It is already well known that volatility and other financial time-series data arewell described by ARCH models. At the same time, these data have systematicchanges after certain time points. For example, financial time-series data changedabruptly after Asian financial crisis and US housing crisis.The most typical model reflecting this type of systematic change is the TARmodel. The concept of this model can be said to be first conceived when in 1953, P.A. P. Moran raised the problem unsolved by linear-type models while he modeledthe ecosystem data of Canadian lynx. As a solution to this problem in 1983, Tongmentioned the limit of the previous research methods which analyzed the time-series data in one frame and proved that it was better to see the time-series data asthe combination of various linear models with different ranges. He came up withthe following model for the Canadian Lynx Ecosystem. x t = (cid:26) .
62 + 1 . x t − − . x t − + ε t , x t − ≤ . , .
25 + 1 . x t − − . x t − + ε (cid:48) t , x t − > . ,ε t ∼ N (0 , . ) , ε (cid:48) t ∼ N (0 , . ) . At the same time, he also showed that if the original data of number of sunspots ( ω t ) from 1,749 to 1,924 is transformed using Box-Cox transformation, or x t =2( ω / t − and then it can be described by the following model: x t = . . x t − + 0 . x t − − . x t − +0 . x t − − . x t − − . x t − + 0 . x t − − . x t − + 0 . x t − + 0 . x t − +0 . x t − + ε t , x t − ≤ . . . x t − − . x t − +0 . x t − + ε t , x t − > . This is a great contribution to analysis of data accompanied by systematic changes.As shown in the model, Threshold-AR model has been changed to be basedon threshold values (11.9824 in the above model) with certain time delays (9 inthe above model) and became the origin of non-linear time series models by com-pletely removing previous linearity. Therefore, we thought that if we are to modelthe volatility data with completely different behaviors after certain events such asfinancial crisis, it was better to combine TAR and AARCH attribute within thesame structure.In this paper, we propose TAR-AARCH (Threshold Autoregressive-AsymmetricAutoregressive Conditional Heteroskedasticity) Model (1)-(3) to describe volatil-ity. x t = t (cid:88) j =1 (cid:32) ϕ j + p (cid:88) k =1 ϕ jk x t − k (cid:33) x t − d ∈ R j ) + ε t (1) ε t = z t h t , z t : i, i, d, N (0 , (2) h t = α + q (cid:88) i =1 ( α i | ε t − i | + β i ε t − i ) (3) α > , q > , p, q : knownHere for t , t , · · · , t l , the sequence of intervals R j , j = 1 , , · · · are as follows: R = ( −∞ , t ] , R = ( t , t ] , R = ( t , t ] , · · · , R l = ( t l − , + ∞ ) , x ∈ A ) = (cid:26) , x ∈ A, , x / ∈ A. As shown in the model (1)-(3), (1) is the TAR model and its error-terms (2)and (3) are AARCH models. In other words, model (1)-(3) form the TAR modelis inclusive of Asymmetric ARCH effect.Also given the consideration that complete form of likelihood function is im-possible in the model, the method of appropriate parameter estimation based onQMLE (Quasi Maximum Likelihood Estimation) has been established and asymp-totic normality of estimators have been proved for this TAR-AARCH model.Then, it is possible to estimate all the parameters of the combined TAR-AARCHmodel after estimation of time delay and threshold parameter through wavelet inthe TAR model [7].
To use QMLE for the model, we should first find ∂h t ∂α t , ∂h t ∂ϕ jk , ∂ε t ∂α i , ∂ε t ∂ϕ jk . But as the(3) shows, it is impossible to estimate ∂h t ( θ ) ∂ϕ jk as the parameter ϕ jk contains abso-lute value term. Therefore usual QMLE method will be ineffective in finding outQMLE for parameter.But if concentrated QMLE is to be used for (1)-(3), this problem can be solved.That is, ¯ α ∆ = ( α , · · · , α q , β , · · · , β q ) can be fixed and QMLE concentrated on ¯ θ = ( ϕ , · · · , ϕ lp ) can be obtained for (1) and its asymptotic normality can beobserved.If we assume that ¯ θ = ( ϕ , · · · , ϕ lp ) is known, obtain QMLE concentratedon ¯ α for x t and demonstrate its asymptotic normality, then we will be able toobtain estimators for parameters ¯ θ and ¯ α through the two steps above. But weshould be able to ascertain if such estimators can be assumed to be the estima-tors for parameters, and if so, how much difference they have when compared toQMLE. For this, the parameters of the TAR-ARCH model with QMLE and itsasymptotic normality already obtained have been divided as based of the abovementioned method to obtain concentrated QMLE and asymptotic normality foreach sub-parameter and the results have been compared with QMLE obtainedfrom [6], to prove efficiency of this method. Let α ∆ = ( α , α , · · · , α q ) ,θ = ( ϕ , ϕ , · · · , ϕ p , ϕ , ϕ , · · · , ϕ p , · · · , ϕ l , ϕ l , · · · , ϕ lp ) . If for TAR-ARCH model, α is known, then QML equation concentrated on θ isformulated as: n (cid:88) t =1 h t ( α ) ε t ( θ ) ( x t − d ∈ R j ) = 0 , n (cid:88) t =1 ε t ( θ ) h t ( α ) x t − k ( x t − d ∈ R j ) = 0 ,j = 1 , · · · , l, k = 1 , · · · , p Theorem 1.
In the model (1) - (3) , let ˆ θ ,n be QML estimator concentrated on θ when α is known, and also that it satisfies the strong stationary condition θ ∈ Θ , α ∈ Θ , ˆ θ ,n ∈ Θ .θ , ∈ Θ is a true parameter of model (1) - (3) when α is known, then ◦ ˆ θ ,n p → θ , , ◦ √ n (cid:16) ˆ θ ,n − θ , (cid:17) ∼ N (0 , I − ( θ , )) . Likewise, QMLE concentrated on and its asymptotic normality can be provenin the same way as Theorem 1.We can see how much difference the QMLE and the concentrated QMLE haveas follows.For this, parameter obtained with QMLE (TAR-ARCH) may be divided into (cid:16) ˜ θ , ˜ α (cid:17) and its Fisher’s Information matrix I − ( θ ) = I − (cid:16) ˜ θ , ˜ α (cid:17) can be com-pared to the Fisher’s Information matrix of the above concentrated QMLE, I − ( θ )= I − ( α ) to prove the following relation between them: (cid:104) var (cid:16) √ n (cid:16) ˜ θ ,n − θ , (cid:17)(cid:17)(cid:105) − ≥ (cid:104) var (cid:16) √ n (cid:16) ˆ θ ,n − θ , (cid:17)(cid:17)(cid:105) − , (cid:2) var (cid:0) √ n ( ˆ α n − α ) (cid:1)(cid:3) − = (cid:2) var (cid:0) √ n ( ˜ α n − α ) (cid:1)(cid:3) − . (˜ θ ,n , ˜ α n : sub-parameters of QMLE ) Therefore it is concluded that if the above method for sub-parameters θ , α is usedto obtain concentrated QMLE for the TAR-AARCH models which are difficult toapply QMLE, the obtained estimators will be acceptable although the efficiencyis a little diminished. As easily seen, in the model (1)-(3), if ¯ α is known, the QML equation concentratedon ¯ θ is as follows: n (cid:88) t =1 h t ( ¯ α ) ε t (¯ θ ) ( x t − d ∈ R j ) = 0 , n (cid:88) t =1 ε t (¯ θ ) h t ( ¯ α ) x t − k ( x t − d ∈ R j ) = 0 ,j = 1 , · · · , l, k = 1 , · · · , p Then, it is possible to prove the following theorem.
Theorem 2.
In the model (1) - (3) , ¯ α is known and ¯ θ ,n is the QML estimatorconcentrated on ¯ θ . Also, it satisfies the strong stationary condition for ¯ θ ∈ Θ , ¯ θ ,n , ¯ α ∈ Θ . If ¯ θ , is the true parameter for model (1) - (3) when ¯ α is known,then the following is true: ◦ ¯ θ ,n p → ¯ θ , , ◦ √ n (cid:0) ¯ θ ,n − ¯ θ , (cid:1) ∼ N (cid:0) , I − (cid:0) ¯ θ , (cid:1)(cid:1) .The QMLE concentrated on ¯ α can be obtained and its asymptotic normality provenin the same way. In this paper, first, two types of non-linear models, TAR and Asymmetric ARCHwere combined into TAR-AARCH model to describe volatility in a more effectiveway. Second, appropriate parameter estimation method was proposed to countervarious difficulties generated by asymmetry of the combined model.
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