Clarifying the dynamics of the relationship between option and stock markets using the threshold vector error correction model

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Mathematics and Computers in Simulation 79 (2008) 511–520

Clarifying the dynamics of the relationship between option and stock markets using the threshold vector error correction model Ming-Yuan Leon Li ∗ Department of Accountancy and Graduate Institute of Finance and Banking, National Cheng Kung University, No. 1, Ta-Hsueh Road, Tainan 701, Taiwan Received 12 October 2007; received in revised form 12 February 2008; accepted 12 February 2008 Available online 19 February 2008

Abstract This work examines how the option and stock markets are related when using the threshold vector error correction model (hereinafter referred to as threshold VECM). Moreover, compared to previous studies in the literature of application of threshold models, this study not only investigates the impacts of price transmission mechanisms on stock return means but also the volatilities of returns. The model is tested using the U.S. S&P 500 stock market. The empirical findings of this investigation are consistent with the following notions. First, the equilibrium re-establishment process depends primarily on the option market and is triggered only when price deviations exceed a critical threshold. Second, arbitrage behaviors between the option and stock markets increase volatility in these two markets and reduce their correlation. © 2008 IMACS. Published by Elsevier B.V. All rights reserved. JEL classification: G12; G13; G14; C53 Keywords: Option; Threshold model; Implied stock prices; BS model

1. Introduction There has been a significant body of research on the interrelation between option and stock markets, with a particular focus on the following areas: Panton [45] deals with the issue by using call option prices as an explanatory variable for underlying stock prices; Anthony [2] uses option market trading volumes as a proxy for information flow; Bakshi et al. [3] investigate whether call prices and underlying stock prices move in the same direction. Furthermore, Manaster and Rendleman [36], Bhattacharya [7] and Stephan and Whaley [47], Chan et al. [15] adopt the differences between implied and observed stock prices as the information content, and assumed that information value exists in implied stock prices, and that observed stock prices move in the direction of the implied process. Like the studies of Manaster and Rendleman [36], Bhattacharya [7] and Stephan and Whaley [47], Chan et al. [15], this work employs the implied stock prices approach.1 However, unlike previous studies, this investigation employs a

∗

Tel.: +886 6 2757575x53421; fax: +886 6 2744104. E-mail address: [email protected]. 1 According to the Black–Scholes [10] model (hereafter, the BS model), option price is a function of the current values of the underlying stock, the instantaneous variances of rate of return, the time to maturity of the option, risk-rate rate of interest, and the exercise price of the option. The implied stock price is determined by an observed option price and known values of all parameter inputs in the BS model except stock price. 0378-4754/$32.00 © 2008 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2008.02.023

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new framework for examining questions regarding the dynamic relationships between option and stock markets. The ideas of this study are presented as follows. Theoretically, the implied stock price calculated using the BS model and the observed stock price should converge and arbitrage opportunities should not exist.2 However, owing to possible misspecifications of the BS model and market imperfections, this study establishes a non-linear adjustment mechanism in which the speed of convergence toward the equilibrium relationship increases with absolute price discrepancies. Building upon the above, this work investigates non-linear relationships between option and stock markets using a regime-varying framework. Briefly, this study uses the threshold VECM (vector error correction model) to define two different market regimes based on magnitude of price deviation: (1) when the deviation between the implied and the observed stock prices exceeds a critical threshold the benefits of adjustment exceed the costs, and thus economic agents take significant actions to restore system equilibrium; (2) when the price discrepancy falls below the critical threshold, the convergence towards the equilibrium is unremarkable. The rest of this paper is organized as follows. Section 2 details the method for calculating the implied stock price. Section 3 then examines the interrelation dynamics between option and stock markets using a vector error correction model (hereafter VECM), namely a no-threshold system. To provide a contrast, Section 4 then describes a method of examining the information content of price discrepancies of implied and observed stock prices using threshold VECM. Conclusions are finally drawn in Section 5, along with recommendations for future research. 2. Calculation of implied stock prices This study adopts two prices to examine the dynamic relationship between the option and stock markets: (1) the daily S&P 500 stock index for the observed price (OBS hereafter) and (2) the corresponding implied stock price (IMP hereafter) derived from the BS model. Furthermore, this study uses the price deviation from equilibrium between the OBS and IMP prices as a proxy of information content. The BS formula for call option is presented as follows: Ct = St × N(d1 ) − E × e−rt ·T N(d2 )

(1)

where √ ln(St /Et · e−rt ·T ) √ + 0.5σt T σ T √ d2 = d1 − σt T

d1 =

(2) (3)

In Eqs. (1)–(3), Ct denotes the call option price, St represents the OBS price, σ t is the standard error of the return rate on the St , Et denotes the exercise price and rt represents the risk-free interest rate at time t, respectively. Additionally, T is maturity by annually and N(·) is the cumulative normal density function. This study adopts a numerical technique for inverting the BS model (see Eqs. (1)–(3)) to calculate the IMP price and denotes it as St∗ . Notably, σ t in the BS model (see Eqs. (1)–(3)), that is, the standard error of returns on the St cannot be obtained from the database. Engle [18] propose the autoregressive conditional heteroskedasticity (ARCH) model to picture timevarying volatility, and subsequent studies, such as Bollerslev et al. [12], Li et al. [29], and McAleer [38], forming the ARCH family of models. Of these developments, the most commonly used method to characterize variance dynamics of stock returns is GARCH (generalized ARCH) models developed by Bollerslev [11]. Subsequently, Nelson [41] develops the Exponential-GARCH (EGARCH) to capture the leverage effect stated by Black [9] and Schwert [46]. Glosten et al. [23] establish the GJR-GARCH model to accommodate asymmetric behavior between negative and positive shocks. Engle and Ng [21] proposed non-linear asymmetric GARCH (NGARCH); and Zakoian [51] generated the threshold GARCH (TGARCH) model. Furthermore, Chou [16] devised the GARCH-M (GARCH in the mean) model, in which the heteroskedastic variance term is incorporated into the return mean equation. To permit the conditional correlation to vary over time in the bivariate GARCH system, Engle and Kroner [19] designs the BEKK-GARCH (Baba-Engle-Kraft-Kroner-GARCH) model; Engle [20] proposes the DCC-GARCH (the dynamic conditional correlation GARCH) model; and Tse and Tsui [49] designs TVC-GARCH (the time-varying correlation GARCH) by allowing the conditional correlation to vary over time. 2

Please refer to Black [8], Boyle and Vorst [14] and Leland [28] for the related discussions.

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Fig. 1. Natural log values of the observed and implied prices.

The target of this work is not to find the best GARCH model for the variance. Therefore, for convenience, this study employs the stationary AR(1)-GARCH(1,1) model for variance estimation. Given that Rt stands for the rates of return for the St price, the AR(1)-GARCH(1,1) model used in this work is as follows3 : Rt = π0 + π1 Rt−1 + εt ,

|π1 | < 1

(4)

where the shocks are given by εt = σt · vt ,

iid

vt ∼N(0, 1)

2 σt2 = ω + κ · ε2t−1 + τ · σt−1

(5)

and ω > 0, κ ≥ 0, τ ≥ 0 are sufficient conditions to ensure that the conditional variance σt2 > 0. This study maximizes the log likelihood function with respect to π0 , π1 , ω, κ and τ using the E-view program. Certain studies dealing with structural properties of the GARCH-based models are briefly summarized as follows.4 The necessary and sufficient conditions for the existence of the second and fourth moments are studied by Ling and Li [31] and Ling and McAleer [32,33], Nelson [42] and He and Ter¨asvirta [24]. Nelson [42] and Bougerol and Picard [13] derive the log-moment condition for strict stationarity and ergodicity. Lee and Hansen [27] and Jeantheau [25] study the consistency properties. Last but not the least, the log-moment condition of the GJR model is examined by McAleer et al. [39]. Last, Ling and McAleer [34] study the moment conditions for consistency and asymptotic normality for the multivariate GARCH models and indicate that the second and fourth moment conditions are far more straightforward to check in practice. This study adopts a rolling estimation process to include recent market information and continuously repeat the work of model estimation on stock variances. In particular, at time t, 84 historical data (3-month daily data) are incorporated into the estimation of the GARCH model parameters, and after which the variance at time t, σ t , is estimated. The sample period for the GARCH model comprises 774 trading days (from 1 September 2002 to 2 September 2005). The present tests involving an 84 prior-trading-days estimation window yield 690 out-of-sample variance estimates (from 2 December 2002 to 1 September 2005). Last, this work collects certain necessary variables for the IMP price, except for σ t , from the DataStream database and adopts a numerical technique to invert the BS model (see Eqs. (1)–(3)) and then calculates the IMP price, denoted as St∗ . Fig. 1 describes the natural log value of the OBS price in addition to the IMP price, that is, st = ln(St ) and st∗ = ln(St∗ ). Next, the study develops the following regression to examine whether the OBS and IMP prices for the S&P 500 stock 3 4

When q = 0, it becomes the ARCH (q) specification of Engle [18]. The author would like to thank an anonymous referee for this suggestion.

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Table 1 Unit root and cointegration tests for the case of U.S. S&P 500 stock market S&P 500 stock market

Log levels % returns Error correction term

OBS price

IMP price

−1.123257 −12.61968* −6.010830*

−1.169417 −12.38666*

Note: 1. This paper adopts a 3-month prior-trading-days estimation window (84 daily observations) and 690 calculated implied stock prices from 2 December 2002 to 2 September 2005 are obtained. 2. The unit test for the log levels and return rates of the observed and implied stock prices is augmented Dickey–Fuller tests for unit roots (ADF). Cointegration tests are based on conventional procedure. 3. * denotes significance at 1%.

market are cointegrated: st = λ0 + λ1 · st∗ + zt

(6)

st∗

where st and denote the natural log value of the OBS and IMP stock prices at time t, respectively. The zt variable provides a clear measure of the deviation term from the equilibrium relationship of the OBS price and the corresponding IMP price derived from the BS model, namely the error correction (EC) term. If the EC, zt is a stationary I(0) variable, then the price series st and st∗ are cointegrated. Table 1 lists the unit root and cointegration test for the IMP and OBS prices for the S&P 500 stock market. The present empirical results reveal that the series of both the IMP and OBS prices are non-stationary. However, the logarithmic first difference of stock price, including the IMP and OBS stock prices, is stationary. Additionally, the cointegration test indicates that the EC term, zt , is stationary. 3. Examining the dynamics of the interrelation between option and stock markets via a no-threshold system The empirical results presented here indicate that the two price series, the IMP and OBS prices, are nonstationary but cointegrated. As is well known, if two time series data are cointegrated, the model should include the EC term, in which case the VECM specification is given by: st = α + β · zt−1 +

p 

γi · st−i +

i=1

st∗

=

α∗

+ β∗

· zt−1 +

p  i=1

q 

∗ ηj · st−j + et

j=1

γi∗

· st−i +

q 

(7) η∗j

∗ · st−j

+ e∗t

j=1

where denotes the difference operator (such as, st = st − st − 1 ), while st and st∗ are the natural log prices of the OBS ∗ ), and IMP prices at time t, respectively. Notably, this investigation sets the EC term, zt − 1 to (st−1 − λ0 − λ1 · st−1 ∗ which represents the last period disequilibrium between st − 1 and st−1 prices. The covariance matrix of two error terms, et and e∗t in Eq. (7), is presented as follows:          et σ 0 1 ρ σ 0 ∼ iid 0, · · (8) 0 σ∗ ρ 1 0 σ∗ e∗t where the σ and σ* denote the standard error of et and e∗t , respectively and the ρ represents the correlation coefficient between them. The VECM suffers two limitations. First, the VECM assumes that a tendency towards a long-run equilibrium is present during every time period. Nonetheless, movement towards the OBS–IMP equilibrium may not be necessary during every period. The existence of potential misspecifications of the option pricing model and other market imperfections may prevent the making of continuous adjustments. Second, the VECM adopts the unrealistic framework of constant variances and correlation.5 5

The characteristic is consistent with the notion that arbitrage threshold behavior causes the relationship between time series of the implied and observed stock prices to differ by regime, and thus so too do the variance and correlation of option and stock markets.

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Table 2 Parameter estimates of VECM S&P 500 stock market Observed stock price equation ( st ) α (constant term) ∗ ) β (zt − 1 = st − 1 − λ0 − λ1 ·st−1 γ 1 ( st − 1 ) ∗ ) η1 ( st−1 γ 2 ( st − 2 ) ∗ ) η2 ( st−2 γ 3 ( st − 3 ) ∗ ) η3 ( st−3 γ 4 ( st − 4 ) ∗ ) η4 ( st−4

0.0004 (0.0003) −0.0222 (0.0503) −0.1312 (0.0825) 0.0281 (0.0774) 0.1006 (0.0823) −0.1152 (0.0766) 0.0752 (0.0805) −0.0593 (0.0746) −0.0055 (0.0735) 0.0674 (0.0678)

Implied stock price equation ( st∗ ) α* (constant term) ∗ ) β* (zt − 1 = st − 1 − λ0 − λ1 ·st−1 ∗ γ1 ( st−1 ) ∗ ) η∗1 ( st−1 ∗ γ2 ( st−2 ) ∗ ) η∗2 ( st−2 ∗ γ3 ( st−3 ) ∗ ) η∗3 ( st−3 ∗ γ4 ( st−4 ) ∗ ) η∗4 ( st−4

0.0004 (0.0004) 0.1375 (0.0536)*** 0.1075 (0.0878) −0.1737 (0.0824)*** 0.1357 (0.0876) −0.1530 (0.0815)** 0.1736 (0.0856)*** −0.1109 (0.0793) 0.0522 (0.0782) 0.0443 (0.0721)

Covariance matrix σ σ* ρ

0.0085 0.0091 0.8649

Note: 1. Please refer to this study’s Eq. (7) for the model specification of the equation of the OBS and IMP prices. This study sets the lag number order in the VECM to four, namely p = 4 and q = 4. 2. Please refer to this study’s Eq. (8) for the specification of covariance matrix. The value in the parenthesis denotes the standard error of parameter estimate. 3. The ***, ** and * denote the significance in 1%, 2.5% and 5%, respectively. 4. The β and β* estimates are insignificant and significant in 1%, respectively.

Table 2 lists the parameter estimates of the VECM. To concentrate on the process of disequilibrium in the optionstock markets, this work discusses two parameters of the EC term, namely β and β*. The β and β* estimates, as shown in Table 2, are insignificant and significant in 1%, respectively.6 This finding is consistent with the following notion: once the OBS–IMP price relationship deviates from the long-term cointegrated equilibrium, price adjustment in the option market is generally required to re-establish equilibrium during the next period. This finding provides one explanation for the conclusions of Manaster and Redlman [36], Bhattacharya [7] and Stephan and Whaley [47], Chan et al. [15]: the informational content in the price differences between the IMP and OBS prices cannot significantly predict future OBS price changes. The option market is generally considered a superior vehicle compared to the stock market owing to lower trading costs and limitations. This investigation thus posits that the OBS–IMP price discrepancies from the equilibrium condition are mainly adjusted in the option market, rather than the stock market. 4. Examining the dynamics of the interrelation between option and stock markets via a threshold system This study hypothesizes that the possibility of misspecification of the option pricing model and other market imperfections may create a band within which the IMP and OBS stock prices are free to diverge. To capture this discrete adjustment process in the option-stock market, this work establishes a system in which the conventional 6

Both of the β and β* estimates have right sign. Specifically, the signs of the β and β* estimates are negative and positive, respectively.

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VECM is combined with threshold techniques to capture the non-linear interrelationship between the option and stock markets. As with the VECM, this study first regresses st on st∗ : st = λ0 + λ1 · st∗ + zt

(9)

The next step is to address and examine the behavior of threshold cointegration. The k is defined as an observable discrete regime variable, the threshold VECM is then established below7 : st = αk=1 + βk=1 · zt−1 +

p 

γik=1 · st−i +

=

α∗,k=1

+ β∗,k=1

· zt−1 +

∗ ηk=1 · st−j + ek=1 j t

j=1

i=1

st∗

q 

p 

γi∗,k=1

· st−i +

i=1

st =

αk=2

+ βk=2

· zt−1 +

p 

, η∗,k=1 j

∗ · st−j

γik=2

· st−i +

q 

ηk=2 j

(10) ∗ · st−j

+ ek=2 t

j=1 p  i=1

if |zt−1 | ≤ θ

+ e∗,k=1 t

j=1

i=1

st∗ = α∗,k=2 + β∗,k=2 · zt−1 +

q 

γi∗,k=2

· st−i +

q 

, η∗,k=2 j

∗ · st−j

if |zt−1 | > θ

+ e∗,k=2 t

j=1

where the θ denotes the threshold parameter. Two market regimes are defined in the above setting: (1) regime I or the central regime (namely k = 1), where |Zt − 1 | ≤ θ, while (2) regime II or the outer regime (namely k = 2), where |Zt − 1 | > θ. The dynamics of the mispricing term, zt , depend on the market regime where that term is located. Particularly, in the central regime: –θ < Zt − 1 < θ (namely, k = 1), the mispricing term, zt − 1 is too small to trigger arbitrage trading; however, in the outer regime: zt − 1 < −θ or zt − 1 > θ (namely, k = 2), the mispricing term, Zt − 1 is sufficiently large to initiate arbitrage trading and thus the OBS–IMP deviation from the equilibrium condition, zt − 1 , helping to significantly affect the next period values of the IMP and OBS prices. Consequently, the estimates of βk = 2 and β*,k = 2 based on regime II are expected to be significant, while the estimates of βk = 1 and β*,k = 1 for regime I are expected to be insignificant. Generally, one of the difficulties in operating threshold models is estimating the threshold parameter, θ, in Eq. (10). This investigation designs grid-search procedures for threshold parameter estimation. The grid-search procedures are as follows: (1) st is regressed on st∗ and then the observations of the EC term, zt are obtained; (2) a series of arranged EC terms is established that orders the observations of zt according to the value of zt − 1 , rather than time; (3) by assigning two small numbers to serve as the initial values of θ and −θ, such as 0.005 and −0.005, the series of arranged EC terms can be split into two different regime areas: inside/outside the thresholds; (4) the regressions of Eq. (10) are estimated for each regime area and the residual sum of square RSS is calculated and saved; (5) the values of θ and −θ are increased using a single grid with small values of 0.0001 and −0.0001, and procedure (4) is then repeated for the new values of θ and −θ; (6) procedures 4 and 5 are then repeated, the RSS value is derived for each value of θ, and the value of θ for which the RSS is minimized is chosen. Notably, to avoid the problem of a small number of observations existing for any particular regime, particularly the outer regime, it is necessary to restrict the value of the threshold parameters. This investigation uses the 4% and 40% percentiles of the EC term, zt − 1 , to bound the threshold parameter. That is, the observation percentage of the outer regime is allowed to range from 4% to 40%.

7 Balke and Formby [4] represent one of the first studies to introduce the threshold cointegration model for capturing the non-linear adjustment behaviors in spot-futures markets. The concept of combining non-linearity and cointegration has generated considerable applied interest, including the following applications: Yadv et al. [50], Martens et al. [37] and Lin et al. [30] dealing with the spot-futures relationship; Anderson [1] that focus on the yields of T-Bills; Michael et al. [40] and O’Connell [43] that focus on the exchange rates; Balke and Wohar [5] examining interest rate parity; Obstfeld and Taylor [44], Baum et al. [6], Enders and Falk [17], Lo and Zivot [35] and Taylor [48] focusing on purchasing power parity.

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Table 3 Parameter estimates of threshold VECM S&P 500 stock market Outer regime (k = 2), |Zt − 1 | > 0.0152, Obs % = 5.84%

Central regime (k = 1), |Zt − 1 | ≤ 0.0152, Obs % = 94.16%

Observed stock price equation ( st ) αk (constant term) ∗ ) βk (zt − 1 = st − 1 − λ0 − λ1 ·st−1 γ1k ( st−1 ) ∗ ) ηk1 ( st−1 γ2k ( st−2 ) ∗ ) ηk2 ( st−2 γ3k ( st−3 ) ∗ ) ηk3 ( st−3 k γ4 ( st−4 ) ∗ ) ηk4 ( st−4

−0.0020 (0.0021) −0.0924 (0.1287) −0.0532 (0.2551) −0.0353 (0.2182) −0.3929 (0.2822) 0.0457 (0.1960) −0.2385 (0.2852) 0.0556 (0.2079) 0.2356 (0.2757) −0.1448 (0.1996)

0.0005 (0.0003) 0.0831 (0.0659) −0.2108 (0.0937)*** 0.0983 (0.0912) 0.1453 (0.0943) −0.1312 (0.0922) 0.1216 (0.0898) −0.1022 (0.0864) −0.0643 (0.0797) 0.1136 (0.0761)

Implied stock price equation ( st∗ ) α*,k (constant term) ∗ ) β*,k (zt − 1 = st − 1 − λ0 − λ1 ·st−1 ∗,k γ1 ( st−1 ) ∗ η∗,k 1 ( st−1 ) γ2∗,k ( st−2 ) ∗ η∗,k 2 ( st−2 ) ∗,k γ3 ( st−3 ) ∗ η∗,k 3 ( st−3 ) γ4∗,k ( st−4 ) ∗ η∗,k 4 ( st−4 )

0.0024 (0.0023) 0.3850 (0.1431)*** 0.0074 (0.2835) −0.0093 (0.2426) −0.6543 (0.3137)*** 0.1408 (0.2178) −0.3695 (0.3170) 0.1727 (0.2311) 0.3167 (0.3064) −0.2370 (0.2218)

0.0002 (0.0003) −0.0079 (0.0685) 0.0572 (0.0974) −0.1474 (0.0948) 0.2513 (0.0980)*** −0.2125 (0.0959)*** 0.2615 (0.0933)*** −0.1935 (0.0898)*** 0.0074 (0.0829) 0.0815 (0.0791)

Covariance matrix σk σ*,k ρk

0.0103# 0.0114# 0.6603

0.0083 0.0086 0.8937#

Notes: 1. Please refer to this study’s Eq. (10) for the model specification of the equation of the observed and implied stock prices. Notably, this study sets the lag number order in the threshold VECM to four, namely p = 4 and q = 4. 2. Please refer to this study’s Eq. (11) for the specification of covariance matrix. 3. The value in the parenthesis denotes the standard error of parameter estimate. 4. The ***, ** and * denote the significance in 1%, 2.5% and 5%, respectively. 5. The # denotes the relatively greater estimate under the outer/central (namely, k = 2 or 1) regime. 6. For the case of U.S. S&P 500 stock market, both of the estimates of βk = 2 and βk = 1 are insignificant. Nonetheless, the β*,k = 2 and β*,k = 1 estimates are significant in 1% and insignificant, respectively. This finding reveals that the tendency to return equilibrium situation mainly occurs in the option market and it is significant (insignificant) in the outer (central) regime for the case of U.S. S&P 500 stock market.

Next, by considering two pairs of error terms, namely (ek=1 and e∗,k=1 ) and (ek=2 and e∗,k=2 ), the two sets of t t t t covariance matrix are established as follows:          ek=1 σ k=1 1 ρk=1 0 0 σ k=1 t ∼ iid 0, · , if |zt−1 | ≤ θ · 0 σ ∗,k=1 0 σ ∗,k=1 1 ρk=1 e∗,k=1 t          (11) ek=2 σ k=2 1 ρk=2 0 0 σ k=2 t ∼ iid 0, · , if |zt−1 | > θ · 0 σ ∗,k=2 0 σ ∗,k=2 ρk=2 1 e∗,k=2 t The threshold VECM has two key advantages compared to the conventional VECM. First, the discrete adjustments in the threshold system release the unrealistic assumption that the tendency of the option-stock market moving towards equilibrium exists during every time period. Second, the threshold VECM considers the concept of arbitrage threshold and applies it to identify the outer/central regime at each time point, and then calculates the variance and correlation parameters for each regime. The threshold VECM can thus overcome the limitations of constant variance and correlation.

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Fig. 2. Time series of price deviation (zt − 1 ) and the threshold parameter estimate (θ).

Table 3 lists the parameter estimates of the threshold VECM. The empirical findings are presented as follows. First, the estimate of the threshold parameter, θ, is 0.0152 and the observation percentage for the outer regime, namely the situation of |Zt − 1 | > 0.0152, is 5.84%.8 Fig. 2 illustrates the time series of the price deviation, zt − 1 , and the θ estimate for the U.S. S&P 500 stock market. Second, both the βk = 2 and βk = 1 estimates for the equation of OBS price are insignificant.9 However, for the equation of IMP price, the β*,k = 2 and β*,k = 1 estimates are positively significant in the 1% level and insignificant, respectively. This finding reveals that the tendency of the OBS–IMP prices to return to equilibrium involved in the option market is significant during the outer market regime, but insignificant under the central regime condition. This finding is consistent with the notion that the disequilibrium revision process in the option-stock market is limited because of misspecifications of the option pricing models and certain market imperfections. Third, examining the estimates of the covariance matrix (see Eq. (11)), the estimates of σ k = 2 and σ*,k = 2 for the outer regime are 0.0103 and 0.0114, and exceed the estimates of σ k = 1 (0.0083) and σ*,k = 1 (0.0086) for the central regime, respectively. By contrast, the estimate of ρk = 2 (0.6603) for regime II is smaller than ρk = 1 (0.8937) for regime I. Furthermore, the estimates of σ and σ* obtained with the VECM (see Table 2) are 0.0085 and 0.0091 which lie within the zones of [σ k = 2 , σ k = 1 ] and [σ*,k = 2 , σ*,k = 1 ], respectively. Furthermore, the estimate of ρ (0.8649) generated by the VECM (see Table 2) is inside the region of [ρk = 2 , ρk = 1 ]. This finding clearly indicates that the conventional VECM, which ignores the non-linear adjustment in the OBS–IMP prices across various market regimes, underestimates the degree of co-movement between the option and stock markets during central regime periods and overestimates it during outer regime periods. The above phenomena are detailed below. First, based on the findings presented here, the outer market regime is associated with more distinct arbitrage behavior: simultaneous short selling in the spot market and purchasing in the call option market when the mispricing term, Zt − 1 , is negative, and vice versa when Zt − 1 is positive. This arbitrage behavior clearly causes the OBS price in the stock market and the IMP price in the option market to tend to move in opposite directions and thus reduces the scale of co-movement between them. Moreover, this arbitrage behavior in the option-stock markets increases volatility in both markets. Furthermore, the rate of increase is examined from the central regime (k = 1) to the outer regime (k = 2). For the option market, σ*,k = 2 and σ*,k = 1 are 0.0114 and 0.0086, respectively and the rate of increase is roughly 32.6%. Correspondingly, the σ k = 2 and σ k = 1 for the stock market are 0.0103 and 0.0083, respectively, and the increase rate is 24%. The finding is consistent with the notion that arbitrage trading increases volatility in both option and stock 8

Notably, the calculated observation percentage of the outer regime does not reach the restriction boundary, namely 4% and 40%. The sign of the βk = 2 and βk = 1 estimate is negative and positive, respectively. However, in theoretical, the parameters of the EC term in the OBS price equation, that is the βk = 2 and βk = 1 , should be negative. 9

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markets; however, the increasing-variance effect is larger for the option market than the stock market. This phenomenon is consistent with the finding that the disequilibrium correction is greater for the option market than for the stock market, particular during the outer regime periods. 5. Conclusions and extensions This investigation is one of the first studies on the application of threshold systems to the dynamics of the interrelation between the option and stock markets. This work examines the effects of arbitrage threshold on mean returns and volatility. This study obtains the following findings. First, the disequilibrium adjustment process in the option-stock market occurs mainly in the options market and is triggered only when the OBS–IMP price deviation exceeds a critical threshold. Second, arbitraging between the option and stock markets increases volatility in both markets and reduces their correlation. Several caveats should be noted. First, this investigation adopts a basic form of the BS model. Future researches could consider the use of put-call-parity rather than use a single option pricing model.10 Second, price discrepancy may also result from the use of a rolling estimation process to yield volatility inputs. Future studies may use alternative volatility measurements. Third, this work only considers a two market system, including option and stock markets. Future studies could incorporate futures markets and design a tripe threshold VECM.11 Acknowledgements I would like to make an acknowledgement to the anonymous reviewer for his very helpful comments. I also would like to thank Cindy Huang for providing some of the data used in this study. I gratefully acknowledge funding from the National Science Council of Taiwan (NSC96-2416-H-006-023-MY3). References [1] H.M. Anderson, Transaction cost and non-linear adjustment towards equilibrium in the US treasury-bill market, Oxford Bull. Econ. Stat. 59 (1997) 465–484. [2] J.H. Anthony, The interrelation of stock and options market trading-volume data, J. Finance 43 (1988) 949–964. [3] G. Bakshi, C. Cao, Z. Chen, Do call prices and the underlying stock always move in the same direction? Rev. Financ. Stud. 13 (2000) 549–584. [4] N.S. Balke, T.B. Fomby, Threshold cointegration, Int. Econ. Rev. 38 (1997) 627–645. [5] N.S. Balke, M.E. Wohar, Nonlinear dynamics and covered interest rate parity, Empirical Econ. 23 (1998) 535–559. [6] C.F. Baum, J.T. Barkoulas, M. Caglayan, Nonlinear adjustment to purchasing power parity in the post-Bretton Woods era, J. Int. Money Finance 20 (2001) 379–399. [7] M. Bhattacharya, Price changes of related securities: the case of call options and stocks, J. Financ. Quant. Anal. 22 (1987) 1–16. [8] F. Black, Fact and fantasy in the use of options, Financ. Anal. J. 31 (1975) 61–72. [9] F. Black, Studies of stock price volatility changes, in: Proceedings of the 1976 Meeting of the Business and Economics Statistics Section, American Statistical Association, 1976, pp. 177–181. [10] F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Pol. Econ. 81 (1973) 637–659. [11] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, J. Econometrics 31 (1986) 307–327. [12] T. Bollerslev, R.Y. Chou, K.F. Kroner, ARCH modelling in finance: a review of the theory and empirical evidence, J. Econometrics 52 (1992) 5–59. [13] P. Bougerol, N. Picard, Stationarity of GARCH processes and of some non-negative time series, J. Econometrics 52 (1992) 115–127. [14] P.P. Boyle, T. Vorst, Option replication in discrete time with transaction costs, J. Finance 48 (1992) 271–293. [15] K. Chan, Y.P. Chung, H. Johnson, Why option prices lag stock prices: a trading-based explanation, J. Finance 48 (1993) 1957–1967. [16] R.F. Chou, Volatility persistence and stock valuations: some empirical evidence using GARCH, J. Appl. Econometrics 3 (1988) 279–294. [17] W. Enders, B. Falk, Threshold-autoregressive, median-unbiased, and cointegration tests of purchasing power parity, Int. J. Forecasting 14 (1998) 171–186. [18] R.F. Engle, Autoregressive conditional heteroscedasticity with estimates of variance of United Kingdom inflation, Econometrica 50 (1982) 987–1007. [19] R.F. Engle, K.F. Kroner, Multivariate simultaneous generalized ARCH, Econometric Theory 11 (1995) 122–150. [20] R.F. Engle, Dynamic conditional correlation—a simple class of multivariate GARCH models, J. Bus. Econ. Stat. (2002) 339–350. [21] R.F. Engle, V.K. Ng, Measuring and testing the impact of news on volatility, J. Finance 48 (1993) 1749–1778.

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See Kamara and Miller [26] for the related discussions. See Fleming et al. [22].

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