The effect of increasing trading volume in index futures on equity correlations.

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The Effect of Increasing Trading Volume in

Index Futures on Equity Correlations

Supervisor: DR. Simon Broda

Author: Tjandra Kentjana

Amsterdam Business School, University of Amsterdam

Master in International Finance

Thesis

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Abstract

Index futures have become more popular, indicated by the increase of the trading volume of index futures year over year. The previous research done by Gulen, H. and Mayhew, S. found that there was an increase in conditional covariance as an effect of index futures trading volume during its introduction and listing. This paper further researched the effect of trading volume of futures markets on equity correlations by observing implied correlations and correlation measures on selected 50 S&P 500 constituents. We found that the time series of correlations and implied correlations are stationary. Further to this, we discovered that the trading volume has an effect on equity correlation, which improves its forecasting power, with the exception of the implied correlation (2-year index). The statistic result concerning the implied correlation (2-year index) failed to reject the null hypothesis, which states that there is no improved forecast power.

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1 Introduction

The activity of the stock market started some time ago, notably in the year 1531 in Belgium, 1773 in London and 1792 in New York. The trading activity had then evolved to include index futures in addition to the regular stocks. Index futures instrument is a standardized agreement to sell or buy a stock index at a future date at an agreed price. The instrument was introduced in 1982 in the United States, and can be used to hedge market exposure and to measure market expectation on future economic outlook. For a specific index, namely S&P 500, the index futures contract was firstly launched on 21 April 1982. Since then, the trading volume on index futures has increased significantly and today, they are traded in most markets around the world. The biggest option exchange is Chicago Board Options Exchange (CBOE), located in Chicago, USA; CBOE was founded in 1973 and went public in 2010, listed on the NASDAQ.

The market enthusiasm on options has increased year over year, highlighted by the 44% increase on trading volume in 2006 compared to prior year. The CBOE annual market statistics show some key facts of trading volume, denominated in number of contracts. In 2007, the average daily volume was 3,8 million contracts and the annual trading volume was 945 million contracts. The trading volume continued to grow and has reached an average daily trading volume of 4,7 million contracts and an annual trading volume of 1,2 billion contracts in 2013.

Following the trend of trading volume on index futures, some research which mainly focused on volatility has been published. Volatility measures how a security or index return deviates or disperses and is used to determine the riskiness of an asset. High volatility means that a security or index can deviate in a large range in price or return and low volatility describes less fluctuation in price or return.

One of the papers which focuses on volatility and covariance was written by Gulen and Mayhew (2000) and concluded that the increase in conditional volatility in the United States and Japan is related to futures trading, both during the introduction period and after the

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index futures had been traded. They observed 25 countries spread across America, Africa, Europe and Asia region, during which they used various models to derive the conclusion on the increase of conditional volatility in the United States and Japan. In nearly every other selected country, they detected no significant increase.

Their paper also noted an increase on conditional covariance with the world market after introduction of futures contracts, which means that the local market would become more integrated with the world market. Based on this finding, it would be especially interesting to research further especially the effect of trading volume of index futures on equity correlations. This thesis will focus on several correlation and implied measures: historical correlation, the implied correlations which are calculated by CBOE, and correlation which is calculated from the Dynamic Conditional Correlation (DCC) Multivariate GARCH model. Correlation is a measure of the linear relationship between two variables, identifying if they have strong correspondences. Correlation is needed to identify the benefit of diversification when composing a portfolio, or performing an asset allocation and risk assessment. In addition, correlations are key measures for several finance activities, namely hedging and pricing decision for some structured products. Hedging needs the review of correlation between its underlying assets and adjust the strategy accordingly. Correlation can, as well, serve as a trading strategy, namely dispersion trading strategy, which observes the trend of implied correlation. When the implied correlation index is considered high, it means that the premium of an index option is relatively higher than a single-stock option. This situation may create an arbitrage opportunity by taking a long position on the single-stock option and taking a short position on the index option.

Implied correlation is a measure of the market’s expectation of the future correlation of the index component. Currently, a publicly available implied correlation index is traded by CBOE, which states that the market expectation is impliedly expressed in SPX index option and single stock prices of the S&P 500 constituents. The instrument was introduced in July 2009 and CBOE constructed a historical record dated back to January 3, 2007.

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The goal of this paper is to observe the effect of the increase of trading volume in index futures on equity correlations. The observation on equity correlations will focus on the S&P 500 index by taking the 50 biggest market capitalizations of the S&P 500 constituents, in order to observe their correlation and implied correlation measures. The correlation measures will focus on the historical correlation and correlation calculated with the DCC Multivariate GARCH model. The implied correlation will be taking two implied correlation indexes listed on CBOE. S&P 500 is the best single index that represents the large-cap U.S equities with total constituent assets approaching USD 1,6 trillion. Reviewing the factsheet of S&P 500 at Standard and Poor’s, the 50% of its sector breakdown comprises of industrial sector as the following: information technology (19,4%), financials (16,1%), health care (13,7%), consumer discretionary (11,9%) and energy (10,4).

This paper will serve statistical evidence of the trading volume effect on equity correlation in the United States. The previous research did by Gulen, H. and Mayhew, S. found that conditional covariance with the world market increases in 21 out of 25 countries with relatively high statistical significance. Along with the finding, we should expect that over time countries will be more integrated with the world even without index futures. Kearney, C. and Poti, V. also confirmed the increase on equity correlations in their paper, “Correlation Dynamics in European Equity Markets”, along with the structural break during the monetary integration in the Euro-zone. After the structural break, the correlation continued to increase again.

In general, the explained previous two papers observed the increase of correlations in relation to index futures trading volume and the pattern of correlation in relation to the Euro-zone monetary integration. This paper will look at equity correlations from a different angle, specifically the effect of the increase of trading volume in index futures.

The statistical test will be performed using the Diebold-Mariano (DM) statistical test to conclude whether there is an improved forecast on the correlation and implied correlation measures. The DM test will take the forecast error from each forecast model of each correlation and implied correlation. Therefore each measure will have two forecast models, one with trading volume and one without trading volume. The absolute value of the DM

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statistic result should be higher than 1,96 to reject the null hypothesis that the first model and the second model have the same forecasting powers. By observing the DM statistic result, the effect of increasing trading volume of index futures on equity correlation can be concluded.

2 Literature Review

2.1 Correlation

Correlation coefficient between two variables in a time series data is defined as:

,

This correlation coefficient is not a time varying correlation because of the assumption that conditional variances and conditional correlations are constant over time. Correlation coefficient ranges between -1 and 1. Correlation above zero indicates positive association amongst the two variables, meaning the increase or decrease of one variable and the other variable are in the same direction. Correlation below zero indicates that each variable moves in the opposite direction and zero correlation indicates no correspondence between the variables. Many researches have been done to find the best calculation to model time-varying correlation, starting with the easiest way, which is historical correlation, Exponentially Weighted Moving Average (EWMA) continuing on to a more complex model, the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. One of the GARCH models is taken as one of the correlation methods in this thesis. Sub section 3.3.2 provides further explanation regarding Dynamic Conditional Correlation Multivariate GARCH.

The DCC model is linear but provides univariate estimation very simply with two-step methods based on the likelihood function (Sheppard, 2001). The DCC model is reliable in different conditions and provides sensible empirical results (Sheppard, 2001). Within the same paper, Sheppard compared 8 different methods in estimating time-varying correlation. Based on the VAR on the long short portfolio test and autocorrelation test in the first squared standardized residual, the DCC model was the best.

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In a simple situation, correlations are driven by covariance and variance of assets. In a broader context, macro economy can drive correlations. A research that was conducted by Poti V. and Kearne, C. and also leveraged the DCC model provided evidence that there was structural break in market index correlations during the beginning of the monetary integration process Euro zone (KEARNEY, 2006). Similar results previously found by Longin and Solnik also indicated that correlations tend to rise with the degree of international equity market integration. (Erb et al., 1994; Longin and Solnik, 1995).

2.2 Implied correlation

Implied correlation is another way to measure correlation, deriving correlation by involving options and underlying assets of the options. Publicly available implied correlation can be found at CBOE and they are traded as an index. Sub section 3.3.1 provides further explanation regarding CBOE implied correlation. A research in examining the forecasting power of one of the implied correlation indexes, the ICJ index, relative to future S&P 500 returns was conducted by Zhou. Zhou found that historical movement and the current weekly change of ICJ was strongly linked to the S&P 500 index in the future. In other words, ICJ can be used to forecast the S&P 500 returns in the next seven thru then months (Zhou). Another similar paper that discussed the forecast performance of implied correlation index was written by Skintzi and Apostolos, where they compared implied correlation and historical correlation to realized correlation. Implied correlation has a biased forecast of realized correlation, but it defeated the historical correlation as a forecast of the realized correlation (SKINTZI, 2005).

Moreover, it is an important conclusion that implied correlation index has a strong dynamic dependence and fluctuates substantially over time (SKINTZI, 2005). Hence due to the strong dynamic dependence, the implied correlation index is not suggested for diversification purpose.

A paper that observed conditional volatility and conditional covariance was written by Gulen, H. and Mayhew, S. Their paper identified that there was an increase of conditional volatility in the United States and Japan, as well as an increase in conditional covariance in 21 countries out of the total 25 selected countries. The increase was identified before and after the introduction of equity-index futures trading in twenty-five countries (GULEN, 2000). As previously discussed in the introduction, the trading volume continued to increase. In 2007,

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the annual trading volume was 945 million contracts, and it reached 1,2 billion contracts in 2013, or in other words, has increased by 21,25% in 7 years. Based on the finding that there was an increase on conditional volatility and conditional covariance in the United States and the fact that the trading volume increases year over year, we are interested to further analyzing the effect between the trading volume and equity correlations in the United States.

3 Methodology

3.1 Data descriptive statistics

Table 1 discloses some descriptive statistics for variables that were used in this thesis. The returns are taken daily and are denominated in USD. The returns were then converted to log returns. The implied correlations are expressed in percentage. Trading volume is denominated in number of contracts and the period is from 2007 thru 2013, involving 1762 observations.

3.2 Research questions

The purpose of this paper is to observe the effect of the increase of trading volume on equity correlation in the United States, by mainly obtaining statistical evidence on the existence of an improved forecast on equity correlations. Below is the research question that will be covered:

 Does trading volume of index futures improve the forecasts of correlation and implied correlation measures?

3.3 Data analysis

This research took sample of 50 single-stock of S&P500 constituents and have been selected based on market capitalization. The historical prices information was obtained from Datastream. The implied correlation index prices and the trading volume were obtained from the Chicago Board Options Exchange (CBOE). The time frame covered 2007 to 2013. The calculation of the DCC Multivariate GARCH model and the Diebold Mariano test were

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performed in Matlab, which used the UCSD GARCH toolbox developed by Kevin Sheppard. The time series analysis and the OLS regression were performed in E-views. In summary, below are the steps that were performed to answer the research questions:

 Obtaining the CBOE implied correlation index from 2007 thru 2013  Obtaining index futures volume trading

 Calculating average historical correlation and it is measured at time t over a 120-days window.

 Calculating average correlation using the DCC multivariate GARCH model

 Performing time-series analysis to conclude stationarity of the correlations and implied correlation time series

 Building forecasting models for both time series correlations and implied correlations on trading volume and on a constant only and perform Ordinary Least Square (OLS) regression to obtain the forecast error for each forecast model.

 Performing Diebold Mariano test on forecast error of each model to detect improved forecasting power from the forecast models, with the following hypothesis:

| |

The null hypothesis notation states that the loss function that is a direct function of forecast error of model 1 has equal accuracy forecast with model 2 (Diebold, 1995). The alternative hypothesis states that both models have different accuracy forecast. This thesis used 5% significance level, which means that the absolute value of the statistic must be higher than 1,96 to reject the null hypothesis that the first and second forecast models have the same forecasting power.

The following subsections will discuss further calculation methodologies that are used in this thesis.

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3.3.1 Implied correlation

The CBOE S&P 500 Implied Correlation Index White Paper (Chicago Board Options Exchange, 2009) stated that the index reflects the weighted average correlation of the S&P 500 index. It measures daily expected average correlation of the S&P 500 index using SPX option implied volatilities and a weighted portfolio of the implied volatilities of options on stocks in an SPX tracking basket. SPX tracking basket is a subset of the S&P 500 which consists of the 50 largest components, defined by market capitalization. The implied correlation is calculated as follows:

∑

2 ∑

∑

The index price is updated every 15 seconds during the trading day and is calculated in several steps:

1. CBOE maintains a SPX tracking basket which contain 50 stocks selected based on the biggest market capitalization. The tracking basket then facilitates weight calculation that is attached to the implied volatilities on stocks in the implied correlation equation. The weight of a stock is calculated from market capitalization of that stock divided by the total market capitalization of the tracking basket.

2. Determine the SPX options implied volatility and implied volatilities for options on stocks of all the selected 50 stocks.

3. Derive the implied correlation or

The underlying thought of implied correlation index started when random mismatch was observed between the movement of implied volatility of an index option and its constituents. The increase of implied volatility of an index was not always followed by the increase of implied volatility of the constituent. This mismatch was attributed to the market’s changing views on correlation.

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The market’s view on correlation is implicitly measured from the relationship of implied volatilities of options on an index and its constituent. The implied correlation index may be used as trading signals for dispersion strategy as it shows the premium between the index options and its constituent options.

A long position on volatility dispersion strategy means buying at-the-money straddles in options on index components and selling at-the-money index option straddles (Chicago Board Options Exchange, 2009). Straddle is defined as taking a position in put and call together for an index or single stock with the same strike price and expiration date. One interpretation of this strategy, when implied correlation index is low, is that index option premiums are relatively high compared to equity options. Therefore, it may be profitable to sell the index options and buy equity options.

3.3.2 Dynamic Conditional Correlation Multivariate GARCH

The enormous journey to find the best estimate for correlations has been published in various academic journals since 1900. One of those, in 1982, Robert F. Engel made an important breakthrough by finding time-varying volatility (ARCH). In 1986, Bollerslev introduced the GARCH model which solved the length of ARCH lags (q) problem. The correlation estimate method that will be used in this thesis is the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) and will focus on the Dynamic Conditional Correlation (DCC) model. The DCC model was proposed by Robert Engel in 2002, which assumes that conditional correlations changes overtime, and is a contrast to the Constant Conditional Correlation (CCC) model. The CCC model assumes that the conditional correlation is constant over time, , . The DCC model assumes that the

conditional correlation adopts a GARCH pattern. On their paper, Engel and Sheppard (2001) showed that this model solved the problem of multivariate conditional variance estimation by estimating univariate GARCH models for each asset, and then, using transformed residuals resulting from the first step, estimating a conditional correlation estimator through loglikelihood function. This model maintains the Bollerslev’s model as well as producing time varying correlations. On the conditionally multivariate GARCH model, the assets returns have zero expected value and conditional covariance Ht. The asset return at (n x 1) vector can be described as follows:

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| ~ 0,

and represent the time varying standard deviations and correlation matrices respectively from the univariate GARCH model. The second step, the loglikelihood function in defining Qt is described as follow:

∑ log 2 2 | | log | | ;

where ~ 0, or is the standardized residual as is defined as:

/

q, the unconditional covariance of the standardized residual resulted from the multivariate GARCH estimation, is defined as:

,

̅

,

̅

Finally, the correlation between two variables will be calculated as follows:

, ,

,

∗

,

The DCC multivariate GARCH has many advantages; one of them being that the DCC model is easy to be extended by including exogenous variables (Sheppard, 2001). Furthermore, the number of parameters that need to be estimated during the DCC calculation is independent from the number of series to be correlated. This means the DCC model can afford to estimate a large number of correlations matrices.

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3.3.3 The Diebold-Mariano Test

This thesis used the Diebold-Mariano (DM) statistic to test the forecasting power or the predictive accuracy, which is derived from ratio of the mean of the squared forecast errors over the weighted sum of the covariance of the squared forecast errors. The first step was to populate the forecast errors from the models that are compared.

The DM statistic is calculated as follow:

̅

_∑

, m1 repesents model 1 and m2 repesent model 2

2 ∑

,

The null hypothesis states that both forecast models have equal accuracy level, and the alternative hypothesis states that the the two forecast models have different accuracy levels. The interpretation will be derived by comparing the DM statistic result to the range of

/2 to /2 . The used test significance level is 5% and therefore the lower and upper boundaries will be 1,96 and 1,96 respectively. If the DM result falls outside of the range of -1,96 and -1,96; then there is a difference between the two forecast models, which rejects the null hypothesis. The second model is more accurate than the first model, when the t-statistic is positive.

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3.4 Expected results

The result of this thesis should show the effect of trading volume on equity correlation, mainly if the trading volume would improve the forecasting power of both implied correlations and correlations. The conclusion whether there is an improved forecast for all types of time series correlations and implied correlations will be indicated by the Diebold-Mariano statistic test. We used a 5% significance level, which means that the absolute value of the DM statistic test should be higher than 1.96.

4 Analysis

4.1 Historical correlation

The historical correlation is calculated averagely from the correlation matrix. The correlation matrix contains correlation from each pair of stocks of the selected 50 assets at time t over a 120-days window. The calculation process required that each pair of stock has at least 120 return data at time t and that the historical correlation takes the average of the correlation matrix at time t. The historical correlation utilizes the rolling-window method to obtain time-varying correlations, in which the length of the rolling window itself, influences the limited changing and the smoothness of the resulted time-varying correlation. Overall, a shorter rolling window tends to produce an abnormal historical correlation time series, but on the other hand, it can better represent a simultaneous correlation series. 120-day window was therefore used in this paper to obtain time-varying historical correlations. The limitation of using the method is that the correlation time series resulting from the sequential overlapping samples of size t will tend to exhibit so-called “ghost features” (Alexander, 1996), because the drift of the biggest asset movement will be still reflected in the following correlation series until the rolling window that contains the big asset move is completely switched. This condition leads to the tendency that the time series is auto correlated and consequently

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causes limitations for further time series analysis. Further to this, the historical correlation does not take into account the dynamic properties of time series, namely the conditional variance, which becomes the main focus in the DCC multivariate GARCH model. Beyond all the downsides that are previously mentioned, the historical correlation is the simplest approach to produce time-varying correlation and provides preliminary analysis of correlation and its connection with trading volume, which is the main purpose of this paper. As found by Zhou that the forecast power of historical correlations is underperformed compared to implied correlations, we still believe that this method can serve as an opening benchmark and gives introduction on how the time-varying correlation will look.

The next sections will discuss different time series of implied correlation and the DCC multivariate GARCH correlation.

The plot of the average historical correlation can be viewed in the below graph:

The statistics of time-series of correlation and implied correlation is presented in the first part of Table 2. It shows that the average of implied correlations exceeds the historical correlation. The average implied correlation scores 59,13% and 61,51%.

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4.2 Implied correlation

The time series of implied correlation includes 1 year index and 2-year index that are available at CBOE. The 2-year index of implied correlation is higher in almost all periods compared to the 1 year index. Additionally this 2-year implied correlation index has the highest mean amongst the four correlations and implied correlation measures. The highest implied correlation was reached at 1,0593 on 20 November 2008. The CBOE noted that on that day, the implied volatility index was also noted to have reached a record high at 0,8086. The implied correlation was greater than one due to low moneyness or the option position was out of the money compared to its underlying security. This unacceptable level marked that the implied correlation measure is not suggested at low moneyness level. Looking at the 7-year history, the implied correlation surpassed one only three times. This fluctuation supports the finding of Skintzi that implied correlation fluctuates substantially over time.

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4.3 The DCC multivariate GARCH model

The process started with the d-mean process of the selected assets returns and then continued with the univariate GARCH estimation to obtain the standardized residuals ( ). The standardized residuals from the first step were then used in the likelihood function to obtain with the following equation:

,

̅

,

̅

Finally correlation at time t is calculated averagely from the correlation matrix at time t, which contains correlation from each pair of stocks of the selected 50 assets at time t.

The result below discloses paramaters and standard error from DCCmv GARCH: Standard error

DCCmv GARCH 0.0038 0.9527 0.0046

Table 2 shows that DCC GARCH correlation has the lowest average at 13,97% compared to the other time series correlation and implied correlations. In addition to the lowest average correlation, the DCC GARCH also has the lowest standard deviation at 1,10% which makes the trend on the graph deviate between 37,5% and 43,5%.

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4.4 The stationarity of the implied correlations and correlations time series The Augmented Dickey Fuller test, first lag, was applied on all time series correlations and implied correlations. The results showed that all time series correlations and implied correlations followed stationarity process or mean reverting, which means that all time series will fit in regression process.

The regression equations during the stationarity test are as follows: Historical correlation

∆ Δ Implied correlation: 1 year index

∆ Δ 1 1

Implied correlation: 2-year index

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DCC multivariate GARCH correlation ∆ Δ

Below is the detail result from the Augmented Dickey Fuller test:

Historical correlation has a computed ADF test statistic of -13.57397, which is lower than the critical value at 1% significant level. Conclusion: the historical correlation time series is stationary.

Implied correlation: 1 year index has a computed ADF test statistic of -29.13582, which is lower than the critical value at 1% significant level. Conclusion: the Implied correlation: 1 year index time series is stationary.

Implied correlation: 2-year index has a computed ADF test statistic of -27.81349, which is lower than the critical value at 1% significant level. Conclusion: the Implied correlation: 2-year index time series is stationary.

DCC multivariate GARCH correlation has a computed ADF test statistic of -26.92678, which is lower than the critical value at 1% significant level. Conclusion: DCC multivariate GARCH correlation time series is stationary.

4.5 The Diebold-Mariano test

The Diebold Mariano (DM) test is calculated from forecast errors. The forecast error is calculated from rolling one step ahead from two models. Therefore, the first step is then building forecast models for each correlations and implied correlation time series, one model is with trading volume and another is without trading volume. This paper includes, in total, four-time series correlation:

 Historical correlation

 Implied correlation: 1 year index  Implied correlation: 2-year index  DCC multivariate GARCH correlation

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Below are the two structural forecast models that are applied to each correlation and implied correlation time series:

Historical correlation (M1) With trading volume

Without trading volume

Implied correlation: 1 year index (M2) With trading volume

1 Without trading volume

1 Implied correlation: 2-year index (M3) With trading volume

2

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DCC multivariate GARCH correlation (M4) With trading volume

The DM statistic results from the four forecast models are as follow:

M1 M2 M3 M4

Diebold

Mariano statistic -2.2918 -2.1402 -0.6175 -2.3799 Bellow result summary from the Diebold Mariano statistic test shows a mixed result as the following:

Historical correlation (M1)

The DM test was performed based on forecast error from both M1 forecast models. The absolute value of DM of model 1 is 2.2918, which is higher than 1.96, therefore rejecting the null hypothesis at the 5% level of significance. It concludes that the forecasting power of the first model (with trading volume) is better than the second forecast model (without trading volume) or in other words, is not the same as the second forecast model.

Implied correlation: 1 year index (M2)

The DM test was performed based on forecast error from both M2 forecast models. The absolute value of DM of model 2 is 2.1402, that is higher than 1.96, therefore rejecting the null hypothesis at the 5% level of significance. It concludes that the forecasting power of the

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first model (with trading volume) is better than the second forecast model (without trading volume) or in other words, is not the same as the second forecast model.

Implied correlation: 2-year index (M3)

The DM test was performed based on forecast error from both M3 forecast models. The absolute value of DM of model 3 is 0.6175, that is lower than 1.96, thus failing to disprove the null hypothesis at the 5% level of significance. It concludes that there is no significance difference between the forecasting power of the first model (with trading volume) and the second forecast model (without trading volume). Although the ending result has failed to reject the null hypothesis, we observed that the original DM statistic test is negative which means that the first forecast model (with trading volume) is better than the second forecast model (without trading volume).

DCC multivariate GARCH correlation (M4)

The DM test was performed based on forecast error from both M4 forecast models. The absolute value of DM of model 4 is 2.3799, that is higher than 1.96, therefore rejecting the null hypothesis at the 5% level of significance. It concludes that the forecasting power of the first model (with trading volume) is better than the second forecast model (without trading volume) or in other words, is not the same as the second forecast model.

5 Conclusion

The purpose of this paper is to research the effect of trading volume of index futures on equity correlation in the United States, which is observed on selected S&P 500 constituents. The effect of trading volume is measured by identifying if the increase of the trading volume will improve the forecasting power of correlations and implied correlations. This paper found a mixed result, where the trading volume increased the forecasting power of historical correlation, implied correlation (1 year index) and correlation DCC multivariate GARCH.

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The opposite result was found in implied correlation (2-year index) whereby the trading volume did not improve its forecast.

The result of this paper supports the findings from previous research done by Gulen, H. A. and Mayhew, S., that there is an increase in conditional covariance in some countries and world returns when the trading futures were listed. In this case, this thesis supports the effect of trading volume specifically in the United States. The negative result that appeared on the implied correlation year index) still shows that the first model of implied correlation year index) with trading volume was better than the second model of implied correlation (2-year index) without trading volume, despite the conclusion of an improved forecast was rejected at 5% significance level.

The observed implied correlations were taken from CBOE and the historical data is available from the beginning of 2007. Observing implied correlation with a longer time frame would lead to more robust conclusions.

Based on the above findings that the trading volume will increase the forecast power of both the correlations and implied correlation with lower significance level to the implied correlation (2-year index), the correlations and implied correlation time series might be reliable indicators to detect if there is an arbitrage opportunity and therefore they can serve as information to construct a trading strategy. Some identified areas for future research include: testing if the trading volume will improve correlations forecast for hedgers and speculators by providing better trading signal for dispersion strategy. Another possible research area could be comparing the effect of trading volume outside the United States to see if geography could be a distinct variable.

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6 References

Alexander, C. (1996). Volatility and correlation forecasting. In The Handbook of Risk

Management and Analysis. John Wiley and Sons, New York.

Chicago Board Options Exchange, I. (2009). CBOE. Retrieved June 27, 2014, from The CBOE S&P 500® Implied Correlation Indexes: http://www.cboe.com/micro/impliedcorrelation/ImpliedCorrelationIndicator.pdf

Diebold, F. a. (1995). Comparing predictive accuracy. Journal of Business and Economic Statistics,

13, 253-265.

Engle, R. (2012). Dynamic Conditional Correlation: a simple class of multivariate Generalized Autoregressive Conditional Heteroskedasticity models. Journal of Business

and Economic Statistics, 339-350.

Erb, C. H. (1994). Forecasting international equity correlations. Financial Analysts Journal, 32-45.

GULEN, H. A. (2000). Stock index futures trading and volatility in international equity markets. The Journal of Future Markets, 20, 661-685.

KEARNEY, C. A. (2006). Correlation dynamics in European equity markets. Research in

International Business and Finance, 20, 305-321.

LINDER, D. A. (2014). A framework for robust measurement of implied correlation. The

Journal of Computational and Applied Mathematics, 271, 39-52.

Sheppard, R. F. (2001). Theoretical and Empirical Properties of Dynamic Conditional Multivariate GARCH. National Bureau of Economic Research, 1-43.

SKINTZI, V. D.-P. (2005). Implied correlation index: a new measure of diversification. The

Journal of Futures Markets, Vol. 25, 171–197.

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7 Appendix

Table 1: Descriptive statistics

This table discloses the descriptive statistics for variables that were used in this thesis. Returns are daily and are dominated in USD. Implied correlations are expressed in percentage. Trading volume is dominated in number of contracts. The sample period is from 2007 thru 2013, involving 1762 observations.

Variables Mean Min. Max. Std. Dev. Skewness Kurtosis AAPL 0.11% -19.75% 13.02% 2.28% -0.47 6.27 ABT -0.01% -71.48% 9.19% 2.15% -20.84 692.25 AMGN 0.03% -9.90% 13.03% 1.77% 0.55 7.75 AMZN 0.13% -13.68% 23.86% 2.77% 1.09 11.39 AXP 0.02% -19.35% 18.77% 2.86% 0.06 8.28 BA 0.02% -8.24% 14.38% 1.99% 0.09 4.34 BAC -0.07% -34.21% 30.21% 4.28% -0.24 14.47 BMY 0.04% -8.96% 8.91% 1.57% 0.06 3.96 C 0.00% -49.47% 227.93% 7.07% 18.93 617.25 CMCSA 0.01% -41.79% 21.93% 2.40% -2.93 59.15 COP 0.00% -23.71% 15.36% 2.14% -1.17 15.46 CSCO -0.01% -17.69% 14.80% 2.12% -0.51 10.07 CVS 0.05% -22.49% 13.04% 1.79% -1.33 21.51 CVX 0.03% -13.34% 18.94% 1.86% 0.13 14.21 DIS 0.05% -10.23% 14.82% 1.93% 0.23 7.04 GE -0.02% -13.68% 17.98% 2.25% -0.02 8.62 GOOG 0.05% -12.34% 18.23% 2.01% 0.45 10.39 GS -0.01% -21.02% 23.48% 2.88% 0.32 11.98 HD 0.04% -8.58% 13.16% 1.91% 0.39 4.44 HPQ -0.02% -22.35% 15.78% 2.25% -0.54 11.43 IBM 0.04% -8.64% 10.90% 1.49% -0.04 5.18 INTC 0.01% -13.22% 11.20% 2.01% -0.15 4.44 JNJ 0.02% -7.97% 11.54% 1.07% 0.56 14.29 JPM 0.01% -23.23% 22.39% 3.18% 0.31 10.75 KO -0.01% -69.56% 13.00% 2.10% -20.63 690.58 LLY 0.00% -13.18% 13.41% 1.53% -0.24 10.28 MCD 0.04% -8.32% 8.97% 1.28% -0.03 5.19 MDT 0.00% -14.20% 9.80% 1.64% -0.77 9.34 MMM 0.03% -8.95% 9.42% 1.55% -0.21 5.09 MO -0.05% -120.17% 15.17% 3.21% -30.12 1120.01 MRK 0.01% -15.94% 11.91% 1.80% -0.47 10.64 MS -0.05% -29.97% 62.59% 4.20% 1.43 37.40 MSFT 0.01% -12.46% 17.06% 1.89% 0.19 8.98 ORCL 0.05% -12.39% 12.28% 2.00% -0.17 5.12 OXY 0.04% -20.45% 16.64% 2.56% -0.27 9.00

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Variables Mean Min. Max. Std.

Dev. Skewness Kurtosis

PEP 0.02% -12.71% 8.20% 1.20% -0.41 11.85 PFE 0.01% -10.89% 9.69% 1.55% -0.13 5.90 PG 0.01% -8.23% 9.73% 1.21% -0.27 7.77 QCOM 0.04% -15.36% 15.68% 2.07% -0.06 7.78 SLB 0.02% -20.34% 13.90% 2.60% -0.61 7.35 T 0.00% -8.04% 15.08% 1.56% 0.53 10.41 TXN 0.02% -15.81% 9.81% 1.95% -0.48 4.74 UNH 0.02% -20.62% 29.83% 2.43% 0.48 22.45 UPS 0.02% -8.88% 8.94% 1.56% -0.01 4.94 USB 0.01% -20.05% 20.57% 2.80% -0.01 11.52 UTX 0.03% -9.17% 12.79% 1.69% 0.24 6.32 VZ 0.02% -8.41% 13.66% 1.53% 0.26 7.62 WFC 0.01% -27.21% 28.34% 3.48% 0.75 15.14 WMT 0.03% -8.41% 10.50% 1.29% 0.12 7.74 XOM 0.02% -15.03% 15.86% 1.73% 0.07 14.48 IC1 _59.14% _19.92% _105.93% _11.18% _0.14 _0.05 IC2 _61.51% _20.39% _87.10% _10.70% _-0.32 _-0.43 TrdVol _1,975,897 _314,971 _7,314,048 _821,620 _1.59 _5.05

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Table 2: Implied correlations and correlations: summary statistics

The first table shows statistics summary: mean and standard deviation for correlation and implied correlation time series. The historical correlation is calculated averagely from each pair of stocks of the selected 50 S&P 500 constituents at time t over a 120-days window. The implied correlation takes the implied correlation indexes, which are issued (1 year and 2-year maturity) by CBOE. The last correlation is calculated averagely using the DCC multivariate GARCH model. The last table provides some results (standard error of regression and the Diebold-Mariano statistics) from the OLS regressions of the forecast model. The regression involved two forecast models for each correlations and implied correlations measure, one with trading volume and without trading volume. The trading volume is obtained from CBOE.

Time series Mean Std. Dev.

Historical correlation 0.42678 0.11781

Implied correlation: 1 year index 0.59137 0.11185 Implied correlation: 2-year index 0.61512 0.10700

DCC multivariate GARCH 0.39165 0.01109

Time series Standard error of regression DM statistics

Historical correlation: -2.2918

Forecast with trading volume 0.117169 Forecast without trading volume 0.117807

Implied correlation: 1 year index: -2.1402

Implied correlation: 2-year index: -0.6175

DCC multivariate GARCH: -2.3799

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8 Acknowledgments

I would like to express my gratitude and thank to DR. Simon Broda for his supervision and guidance on this thesis.

Special thanks to Patricia, and to my fellow classmates: Carla, Hanimario and Paulrich who always give moral support.

The effect of increasing trading volume in index futures on equity correlations.