Stock Returns and Implied Volatility: A New VAR Approach

Abstract

The authors re-examine the return-volatility relationship and its dynamics under a new vector autoregression (VAR) identification framework. By analyzing two model-free impliedvolatility indices - the well-established VIX (in the United States) and the recently published VKOSPI (in Korea) - and their stock market indices, the authors find an asymmetric volatility phenomenon in both the developed and emerging markets. However, the VKOSPI shows impulse response dynamics that are quite different from those of the VIX. This finding can be attributed to the unique characteristics of the KOSPI200 options market, which determine the dynamics of the VKOSPI. [PUBLICATION ABSTRACT]

Headnote

Abstract

The authors re-examine the return-volatility relationship and its dynamics under a new vector autoregression (VAR) identification framework. By analyzing two model-free impliedvolatility indices - the well-established VIX (in the United States) and the recently published VKOSPI (in Korea) - and their stock market indices, the authors find an asymmetric volatility phenomenon in both the developed and emerging markets. However, the VKOSPI shows impulse response dynamics that are quite different from those of the VIX. This finding can be attributed to the unique characteristics of the KOSPI200 options market, which determine the dynamics of the VKOSPI.

JEL G10 G15

Keywords Asymmetric volatility; vector autoregression; VIX; VKOSPI

(ProQuest: ... denotes formulae omitted.)

1 Introduction

Financial economists have documented asymmetric return-volatility relationships in global stock markets. That is, stock market returns and volatility are inversely related, and this relationship is more noticeable for negative returns than for positive returns. Thus, in the time-series framework, negative return shocks have the greater impact on volatility. This asymmetric volatility phenomenon has been traditionally explained by two hypotheses: the leverage hypothesis and the volatility feedback hypothesis.

The leverage hypothesis explains the asymmetric relationship between individual stocks' returns and their volatilities based on the observation that a decrease in the stock price of a company results in its equity portion of the firm value becoming smaller while the debt value relative to the total firm value increases. As a result, the stock becomes more risky. Many early studies, including Black (1976), Christie (1982), Schwert (1990), and Duffee (1995), have adopted this hypothesis to explain the asymmetric volatility phenomenon detected in global stock markets.

The volatility feedback hypothesis is more general and is based on the positive inter-temporal relationship between expected return and conditional volatility. Whether positive or negative, a shock initially increases both current and future volatilities. However, since the increased volatility levels up the expected return and decreases the current stock price, a negative shock results in a stronger increase in volatility, whereas a positive shock restrains the volatility increase. The most representative studies that adopt this volatility feedback hypothesis are those by French, Schwert, and Stambaugh (1987) and Campbell and Hentschel (1992).

While these two competing hypotheses have been developed separately, recent studies tend to examine the two hypotheses simultaneously. For example, Bekaert and Wu (2000) propose a unified framework in which these two hypotheses can be investigated and tested together. Wu (2001), examining the hypotheses simultaneously, finds that both the leverage and volatility feedback effects generate asymmetric volatility.

More recently, related studies suggest that the two traditional hypotheses are not adequate to fully account for the asymmetric volatility in daily or higherfrequency levels, and they investigate volatility within the frameworks of tradingbased or behavioral explanations. Avramov, Chordia, and Goyal (2006) argue that trades by uninformed individual investors generate asymmetric volatility. Hibbert, Daigler, and Dupoyet (2008), and Han, Guo, Ryu, and Webb (2012) claim that the psychological bias of market participants is the main cause of the asymmetric volatility phenomenon observed in daily and intraday data.

Although numerous studies have examined asymmetric volatility at the firm and market levels, and many possible explanations for the phenomenon have been suggested and discussed, it remains unclear whether previous studies' methods1 can provide findings that fully explain the dynamic return-volatility relationship. This is because such methods primarily rely on simple regression analyses or GARCH models, which possibly fail to describe the mechanism of the volatility response to shocks. In contrast, in this paper, we re-examine the asymmetric relationship between returns and volatility by employing a new vector autoregression (VAR) approach recently suggested in Lee's (2010) seminal paper. By adopting this VAR identification framework to analyze the dynamic impulse response behavior of volatilities to positive and negative return shocks, we can better understand the dynamics of the volatility response that has been overlooked in previous studies.

Another contribution of the current study is to examine the asymmetric behavior of the VKOSPI (Volatility Index of KOSPI200), a volatility index implied by the KOSPI200 options product, which represents the most liquid options contract in the world. In addition, we compare the asymmetric behavior of the VKOSPI with that of the VIX, on which most previous studies have focused. Given that little research has examined the VKOSPI, which provides the valuable information on market sentiment and investor attitude toward risk in the Korean market that represents a leading emerging economy, and given that substantial differences may exist between emerging and developed markets, this study has an additional academic value of providing new insights into these issues.

Our empirical results show that the asymmetric volatility phenomenon occurs in both developed (the United States) and emerging (Korea) markets. However, the dynamic volatility responses show quite different patterns across the markets. We explain the difference based on the KOSPI200 options market's unique characteristics and immaturity, which determine the dynamics of the VKOSPI.

This paper is organized as follows. In Section 2, we introduce the KOSPI200 options market and the VKOSPI. In Section 3, we discuss the new VAR framework employed in this study. Empirical results and discussions are provided in Section 4. We give our conclusions based on the findings of this study in Section 5.

2 The KOSPI200 Options Market and the VKOSPI

For the identification of the dynamic and asymmetric return-volatility relationship within this study's framework, which will be described in Section 3, implied volatility is a more appropriate construct than realized or historical volatility. This is because implied volatility can gauge the expectations and sentiments of market participants, whereas realized or historical volatilities contain little such information. Therefore, basing our analysis on each market's implied volatilities will provide rich implications for that market.

Among the implied volatility candidates, model-free implied volatility is known to have more explanatory power than the others that are dependent on option pricing models, such as the Black-Scholes or Heston models.2 The most widely used model-free implied volatility indicator is the VIX, which represents the volatility index implied by the S&P500 option prices. The S&P500 options market and the VIX of the U.S. market have been discussed in numerous academic papers, and their characteristics are fully analyzed and well-known to academics and market practitioners.3 In contrast, only a handful of studies have analyzed the KOSPI200 options, the most actively traded options in the world.4 Further, to the best of our knowledge, only two published articles (Ryu, 2012; Han et al., 2012) examine the VKOSPI. Given that the KOSPI200 index options are top-tier options products due to their high trading volume and investor interest, there is good reason to conduct research efforts based on the model-free implied volatility of the emerging market.

Since the Korea Exchange (KRX) introduced the KOSPI200 index options in 1997, the trading volume of the KOSPI200 options has sharply increased. Now, the KOSPI200 options market is the most liquid derivatives market in the world. From its earliest stage, it was dominated by highly speculative investors. Although the trading volume of professional and experienced investors has steadily increased, and their trading activity now accounts for a significant portion of the total trading volume, speculative traders and domestic individual investors are still major market players in the KOSPI200 options market.

Inspired by the great success of the KOSPI200 options market, the KRX published the VKOSPI, the official volatility index for the KOSPI200 stock index, on April 13, 2009. The VKOSPI is calculated from the KOSPI200 index and options prices based on the model-free method.5 Therefore, the VKSOPI reflects the market sentiment and investor expectation embedded in the market prices of KOSPI200 options. Further, the VKOSPI can be regarded as a representative market indicator of the Korean market in that the transactions of the stocks underlying the KOSPI200 index and its options account for a dominant portion of the total transactions in the Korean financial market.

Ryu (2012) and Han et al. (2012) also find that the VKOSPI has desirable qualities as a stock market indicator, containing significant and meaningful information on the Korean financial market. They additionally report that the VKOSPI captures the major shocks to the global economy and shows movements similar to the VIX. In addition, the elaborate logic represented in its construction of equations (see Ryu, 2012) makes the VKOSPI a low-noise indicator and an accurate fear-gauge for the Korean market.

3 Empirical Framework

Lee (2010) has developed his new VAR identification framework to examine the asymmetric effects on stock market returns of positive and negative inflation shocks that have the same magnitude but opposite signs. Further, with a slight modification, the new VAR framework can be used for investigating the asymmetric and dynamic relationship between any two economic variables. In this section, we briefly discuss Lee's framework, which we employ in this study to investigate the asymmetric volatility phenomenon.

We consider the following bivariate models:

... (1)

... (2)

where ... and is the lag operator. In addition, bij, Bij(L), Aij(L), and óij are the elements of a 2-by-2 matrix of B0, B(L), A(L), and Ù, respectively (i, j=1, 2). The elements of A(L) and Ù are obtained through the least squares estimation of Equation (1). By comparing Equations (1) and (2), we can obtain all elements of B(L) if each element of B0 is identified.6

We can identify and estimate four components of B0 by using the relationship B0(B0)T=Ù, which is obtained by taking the variance of each side of the relationship ...

... (3)

... (4)

... (5)

... (6)

Equation (3) illustrates how we can estimate the elements of B0 from the estimated Ù matrix. Equations (4), (5), and (6) represent the three restrictions implied by Equation (3).

To identify the four components of B0, we need one additional restriction. We use the following additional restriction for the identification:

... (7)

Equation (7) reflects the requirement that positive and negative shocks on the first variable Y1t are the same size but with opposite signs. This restriction helps us to identify positive and negative return shocks and allows us to examine the dynamic effects of each type of return shock on the volatility. Therefore, Equations (4)-(7) yield the estimate of each element of the B0 matrix.7 We employ this VAR framework to examine the asymmetric return-volatility relationship. If we set the first variable as the stock market return and the second variable as the stock market volatility, we can then analyze the dynamic responses of the volatility to positive and negative return innovations of the same magnitude.

4 Empirical Results and Discussions

We compare daily VIX and VKOSPI data for the period from April 13, 2009, the date VKOSPI was announced, to September 9, 2011. To analyze the stationary process, we use the first-differenced volatilities and the corresponding stock market index (that is, S&P500 and KOSPI200) returns.8

We estimate the VAR model by using U.S. market data (S&P500 index returns and VIX) and Korean market data (KOSPI200 index returns and VKOSPI), separately. The first variable (Y1t) in the BVAR is the log return on the stock index and the second variable (Y2t) is the differenced implied volatility index in each market. Thus, the first error term, e1t, indicates a positive return shock, and the second error term, e2t, indicates a negative return shock. During the estimation procedure, we determine the lag-order of the VAR model by conducting a sequential likelihood ratio test, as in Rapach (2001). The test shows that the most appropriate number of lags for the VAR model is 6 and 8, respectively, for the U.S. and Korean market datasets. The estimated elements of the B0 matrix for the U.S. market data are somewhat different from those for the Korean market data: the coefficients estimates b11, b21, and b22 are 0.0085 (0.0086), -0.3627 (0.0251), and 1.9059 (1.3794), respectively, for the U.S. (Korean) market data.9

Figures 1 and 2 show the dynamic impulse responses of stock returns and implied volatility to positive and negative return shocks for the U.S. and Korean market data, respectively. The figures illustrate the asymmetric effects of positive (Panel A) and negative (Panel B) return innovations constructed to have the same magnitude (i.e., b11+b12=0).10 To obtain the standard error bands of the impulse responses, we generate 1,000 bootstrap replications, as in Runkle (1987) and Rapach (2001). Figure 3 shows the upper and lower standard error bands for each impulse response.

Panel B of Figure 1 shows the response of changes in VIX to the S&P500 return shocks. Positive stock returns induce a decrease in volatility, and negative stock returns induce an increase in volatility. However, the magnitudes of the effects are quite different: negative return shocks have much stronger impacts on volatility changes than positive return shocks. The patterns of impulse responses shown in Figure 1 are consistent with the traditional hypotheses that explain the asymmetric volatility phenomenon.

The return-volatility relationship in the Korean market is somewhat different from that in the U.S. market. As shown in Panel B of Figure 2, both positive and negative return shocks initially influence volatility in the same direction, inducing an increase in volatility. However, a positive return shock induces only a slight initial increase in volatility, whereas a negative return shock induces a strong initial increase in volatility. That is, the negative return-volatility relationship induced by negative return shocks dominates the positive relationship induced by positive return shocks. As a result, consistent with previous studies, we observe asymmetric return-volatility relationships in the Korean market within KOSPI200 index return and VKOSPI data.

However, previous studies in this area employing the asymmetric GARCH model and simple regression approach have overlooked some unique features of the dynamic patterns of the volatility responses that exist in the emerging market, although they have reported the asymmetric volatility phenomenon. While the magnitude of the initial increase of the implied volatility in response to a positive return shock is small, this is in stark contrast with the result in the U.S. market in which the implied volatility decreases in response to a positive return shock. We attribute this unique pattern observed in the Korean market to the characteristics of the KOSPI200 options market and the trading behavior in the Korean financial market.

One explanation for this pattern in the Korean market is that some options buyers overreact to the market signal. Uninformed or less informed traders tend to overly buy call options in response to a positive return shock, and this causes the additional increase of the call prices, leading to an increase of the implied volatility. If the magnitude of this increase is larger than that of the decrease caused by the positive return shock (as asymmetric volatility theories suggest), then the net effect we observe is an initial increase of the volatility in response to the positive return shock, as shown in Panel B of Figure 2. In contrast, traders might overly buy put options when they face a negative return shock, leading to an additional increase of the put prices and increased implied volatility. In this case, the direction of the volatility change is consistent with asymmetric volatility theories (i.e., sharply increased volatility in response to a negative shock.).

Among market practitioners, it is widely believed that, in the KOSPI200 options market, domestic individual investors tend to regularly and overly buy options and overreact in response to positive news of the underlying market (Kim and Ryu, 2012). Further, the existence of special options accounts makes buying KOSPI200 options easier to implement than writing them. Since early in the history of the KOSPI200 options market, the KRX has promoted options trading by inducing individual investors to open special accounts that prohibit them from writing options, instead of requiring relatively lower levels of margin accounts. Given that noisy individuals with little wealth and trading experience prefer using the special accounts, they are even more likely to overreact and be affected by the behavioral biases. These tendencies seem to result in somewhat different patterns in the VKOPSI responses, compared to the U.S. market responses.

Table 1 reports the forecast error variance decomposition of stock returns and volatility and shows the proportion of returns and volatility that can be explained by positive and negative return shocks. In the United States as well as in Korea, more than 90% of the volatility forecast error variance is explained by negative return shocks. Further, for three trading days following the arrival of a negative return shock, it explains more than 99% of the forecast error variance of volatility in the U.S. market. In the Korean market, it explains a lower portion of about 96%. This suggests that the asymmetric volatility phenomenon is somewhat stronger in the U.S. market. In the Korean market, although the negative return shock plays a dominant role, compared to the U.S. market, the positive return shock shows a relatively significant influence on the dynamic stock-volatility relationship.

5 Conclusions

Employing the new VAR identification framework proposed by Lee (2010), this study clearly demonstrates the negative and asymmetric relationship between stock market returns and implied volatility in the U.S. and Korean markets. Additionally, we find differences in impulse response dynamics between the United States (a developed market) and Korea (an emerging market), which can be attributed to the unique characteristics and the market behavior of the KOSPI200 options market.

This paper demonstrates that the new VAR framework can be employed to investigate the asymmetric and dynamic relationship between two economic variables. A number of studies have examined various properties of the VIX, including asymmetric volatility, but few have focused on the VKOSPI. In this regard, this paper contributes to the literature by being the first to analyze the asymmetric and dynamic responses of the VKOSPI by employing the new VAR framework.

This study is expected to be a stepping-stone for further empirical research on the VKOSPI and other implied volatility indices of global financial markets. Some possible extensions are as follows. To describe the volatility dynamics, investigators might decompose the VKOSPI into observed and unobserved components based on state-space models (e.g., Kalman filtering). Researchers could apply signal noise filtering techniques to eliminate the noise that might be embedded in the VKOSPI.11 Recently, the KRX has been preparing to launch some new derivatives underlying the VKOSPI (e.g., VKOSPI futures and VKOSPI options). The asymmetric volatility may show different patterns after these VKOSPI-related derivatives are actively traded.

Sidebar

Citation Bong Soo Lee and Doojin Ryu (2013). Stock Returns and Implied Volatility: A New VAR Approach. Economics: The Open-Access, Open-Assessment E-Journal, Vol. 7, 2013-3. http://dx.doi.org/10.5018/economicsejournal. ja.2013-3

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The Editor

Footnote

1 Bekaert and Wu (2000) provide a good review of previous research.

2 If we derive implied volatility from option pricing models, it contains some model bias, of which representative examples are volatility smiles or smirks of the Black-Scholes model.

3 See the recent studies of Giot (2005a, 2005b), Banerjee, Doran, and Peterson (2007), Becker, Clements, and McCelland (2009), and Duan and Yeh (2010).

4 Some recent studies, such as Ahn, Kang, and Ryu (2008, 2010), Ryu (2011), and Kim and Ryu (2012), have begun to address the market microstructure issues of the KOSPI200 options market

5 Refer to Ryu (2012) for further details about the VKOSPI.

6 Equations (1) and (2) imply that B(L) is equal to [I-A(L)L]-1B0, where I is a 2-by-2 identity matrix.

7 As in Lee (2010), we assume that b11 is greater than zero. We also take the higher value of the two possible solutions of b22.

8 The quoting unit for the stock indices is a point, and the volatilities are represented as percentages.

9 For brevity, we present only the estimated element of the B0 matrix and the level of estimated coefficients related to the impulse responses (see Figures 1, 2, and 3). The estimated coefficients of A(L) can be provided by the authors upon request.

10 Note that the two structural shocks, e1t and e2t, are normalized (i.e., Var(et)=I). Therefore, the figures present impulse responses of each variable to "unit" positive and negative shocks.

11 However, considering the desirable properties and elaborate nature of the VKOSPI, explained in Section 2, their probability is relatively low.

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