Test relationship between index’s implied volatility and its options trading volume after the subprime-crisis

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Amsterdam Business School

Executive Master

Program: International Finance 2014 – 2016

Master Thesis

Student Name: Xu, Lei Student Number: 10839518

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Test Relationship Between Index’s Implied Volatility

and Its Options Trading Volume After the subprime-crisis

ABSTRACTS

Volatility is an unobservable factor in the option market. It would be ideal if the volatility is monitored by watchdogging trading volume. The aim of this research is to quantify and model the phenomena that trading volume and volatility proliferated together for years in the option market after the Subprime crisis. We find no correlation between the trading volume in the option market and the implied volatility of the underlying S&P 500 index.This study explores the relationship between trading volume of S&P 500 index options and implied volatility of the underlying by beginning with a multiple linear model. The findings conclude that the trading volume of the S&P 500 index options have explanatory power on its implied volatility derived by BMS formula in a multiple linear model. To deploy AR(1) model to further study the relationship between them, there is a dynamic relationship inferred by statistical analysis. Moreover, independent variables in AR(1) model are able to explain the variation of the dependent variable better than in the multiple linear model.

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1. INTRODUCTION

Financial options and the option market have developed extremely fast in the past 30 years in view of the versatile option’s characteristics, which can be employed to hedge exposures, speculate and arbitrage, etc. The recent subprime crisis caused American and even the whole world’s economy to plunge into a great recession. Its occurrence was from July 2007 to the end of 2008 and afterwards financial markets worldwide became more fluctuating. In addition, it led to a significant change in the behavior of traders in financial markets. Hence, after the subprime crisis, financial options on all asset classes are recognized and increased its popularity as effective tools in risk management. Apparent nervousness is prevailing and persistent in the options market. In the meantime, the trading volume of option contracts keeps making new records. This phenomenon caught the attentions of financial market observers, who raised the question if two variables have a connection.

An option is a derivative security, a contract giving the buyer the right to buy or sell a

defined quantity of a defined underlying asset. Through options, the buyer is able to transfer risk to their counterparties, and in return, their counterparties gain a fee. Options can be used to limit downside risk, while maximizing the upside profit. In some studies, the findings are that the subprime crisis changes the behaviors of investors, that they become much more risk-averse than pre – subprime- crisis. Due to the option’s hedging property, options meet investor’s demand as a risk management tool. Moreover, risk management is required by regulators. One of the primary reasons causing the subprime crisis is that practitioners in the financial sector recklessly invested in risky assets. As a take-away from this failure, there has been a drive by central banks globally for greater transparency of asset risks in financial institutions. So most of financial institutions are heavily regulated, carefully monitor total risks and strategically utilize financial options to absorb risks in exceed of the required levels (Hull, 2012). In more detail relative to neutralizing exposures, financial institutions and non-financial corporations rebalance their portfolios frequently, in order to maintain close to zero values of Greek letters. In this sense, more volatile the market seems, the more option contracts are traded. As a result of risk - aversion, options are recognized and enter the phase of high demand in global financial markets.

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- 4 - | P a g e In the option exchange market, the listed option product’s quotes are displayed on the dashboard. The quotes include basic information, such as ticker, strike price, expiry price, bid / ask price, and trading volume. Trading volume is a fundamental indicator about its listed instrument. Hence, a number of empirical work has been focused on the trading volume in financial markets. Intuitively, trading volume quantifies the extent of investors’ activities in the market. In Clark’s (1973) mixture model, trading volume is a proxy for the speed of information flow. Dufee (1992) used trading volume as a proxy for noise trading behavior. Trading volume literally mean the demand in the market. Overall, it measures the degree of trading activities in reaction to new information inflows.

In addition to trading volume, the last bid and ask prices are also observable variables in the option market. The bid-ask spread is the difference between the bid price for an option contract and its ask price. Market makers and traders prepare to supply at the ask prices and sell at the bid prices. They earn bid-ask spreads from trading in the option market and are compensated for the risk of loss caught in the wrong side of the trade. Sometimes when market is volatile, market makers and trader widen the spread of bid-ask in order to prevent investors from trading during such time; moreover, investors are very sensitive to the bid-ask spread since it is a hidden cost incurring when trading option contracts in the market (Mitchell, 2016). Therefore, the study suggests that the value of implied volatility is likely influenced by the variable of bid-ask spread percentage in option market. The bid-ask spread percentage is calculated by dividing the bid-spread spread by two times the price of option without bid-ask spread. (See equation 4.1)

In respect of pricing options, the Black-Scholes-Merton Model (BSM) is one of most

important theories in the contemporary finance. The economists Black, Scholes, and Merton developed an option pricing model in 1973 (MarKenzie, 2006). Before that, there was no mathematical formula to calculate the option’s theoretical value, even though traders had an intuitive sense about what matters for the value of options. Comparable to the binomial tree formulae, BSM is built on the assumption that without an arbitrage opportunity, a riskless portfolio earns the return of the riskless rate in theory.

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- 5 - | P a g e Superior to other pricing models, the BSM necessitates five observable variables in the financial markets to obtain an option’s value. These variables are: underlying asset time-zero price (S), option strike price (K), underlying stock volatility (𝝈𝝈), the time to maturity (τ), and compounded risk-free rate (r). The BSM calculates the intrinsic value of a call option (C) or a put option (P) as follows:

C(S,K,𝝈𝝈,τ,r) = S0*N(d1) - K𝒆−𝒓𝝉 N(d2) (1.1)

And P (S,K,𝝈𝝈,τ,r) = K𝒆−𝒓𝝉 N(-d2)- S0N(-d1) (1.2)

Where d1 = (In(S0/K) +(r-σ2/2) T)/(σ√τ)

And d2 = d1- σ√τ

The BSM assumes that the underlying asset prices have a lognormal distribution and that their returns distribute normally, which are theoretically acceptable. So we are able to approximately calculate option values based on the assumption. By contract, in the reality, most financial variables are more likely to experience big move than the normal distribution would suggest. The heavy tailed distribution depicts the change of market prices more appropriate than normal distribution.

Abdullah and Hanani (2014) found that implied volatility rises significantly during the subprime crisis: it is three times the rate before the crisis. Distinct from historical volatility, implied volatility is derived from the market price of options, based on the BSM. Implied volatility is the expectation of the volatility for the underlying asset for the remaining life of the option, an ex-ante forecast that is often claimed to be informatively superior to

predictions based solely on past realized volatility (Knight, Satchell, 2007). Hence, implied volatility is far more functional than historical volatility and thereby investors can apply corresponding strategies to code with the risk if they foresee uncertainty. Related to implied volatility, another renounced empirical finding is volatility smile, also called volatility skew. Illustrated by volatility smile, volatility decreases as the strike price increases for equity options. Besides the strike price influence, there also seems to be a term structure pattern in volatility smiles: the smiles are strongest in short-term options and flatten out monotonically with increasing time-to-maturity (Sadayuki Ono, 2004).

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- 6 - | P a g e As mentioned above, trading volume is the proxy for new information inflow to investors and implied volatility is the forward outlook of market fluctuation according to the market view. Implied volatility indicates the risk degree of underlying asset from the viewpoint of the market. In common, both variables imply views of the underlying asset. Moreover, the spread of bid-ask spread is sensitive to its instrument’s volatility owing to the fact that market makers protect themselves from risk with the bid-ask spread. Hence, current studies explore their relationship by starting with multiple linear regression. This study expects their relationship tends to be more complicate than linear.

2. LITERATURE REVIEW

The theory, supporting the consistent relationship between trading volume and volatility is Mixture Distribution Hypothesis (MDH). The MDH is introduced by Clark on the quantitative grounds. It conjectured the dynamic of both quantities were dependent on latent events leading to a joint distribution where the volatility and the trading volume are both described by log-normal distribution (Queirós, 2016). Financial instruments price volatility and

activities of trading on the instrument are both driven by the arrival of information in the financial markets. The information includes unexpected good news and bad news. And then investors and the market respond to the news, as the chain reaction is set into motion. The MDH posits that price volatility and trading volume are determined by the same information arrival rate (Martens and Luu, 2002). Namely, volatility and trading volume are shared the same underlying. In the sense that participants in either the stock exchange or the option exchange are exposed to identical and equivalent news, the theory applies to the option market as well.

In opposition to the MDH, there is the scenario of Sequential Arrival of Information

Hypothesis (SAIH), introduced by Copeland. The SAIH conjectures that information arrives to agents at different times so that the final steady states in the market is led by a sequent of local steady states (Martens and Luu, 2002). And the financial markets are somehow analogous to the liquidity in the see. Inspired by the scenario, even though trading volume and volatility are triggered by the same cause, distinct agents are reacted to it in their special way and at their particular time. Nevertheless, the data is regressed and analyzed by

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- 7 - | P a g e using the time serial data so that the result is expect to be disagreeable with the theory of consistent correlation between the trading volume and the volatility. In essence, it is of no doubt that they are related in practice.

There are a number of studies focusing on trading volume predicting stock return volatility in stock market. Studies showed that trading volume does not linearly Granger-cause return volatility but may nonlinearly Granger-cause return volatility (Brooks 1998; Heimstra and Jones 1994). In other words, the form of the relationship between trading volume and volatility is ambiguous.

However, in future markets, the study (Bessembinder and Seguin, 1992) examines the relationship between futures trading activity and equity volatility. Their result is consistent with the idea that increased trading in future markets leads to greater volatility and

destabilization in the share market. Similarly, both Wang (2002) and Pan, Liu and Roth (2003) provide with their studies a much more precise identification of trading activities. Their findings concluded that the trading demand in the futures market was correlated with price volatility. It can be easily imagined in practice as traders foresee a more fluctuating price in coming future, they would like to fix their position with future contacts. Meanwhile, the hedging pressure could make the less informed traders panic and nervous in the spot market. On the other hand, the findings are not always compatible with one another. For instance, there is the empirical evidence that the effects of future-trading on spot volatility in the USA is inconclusive.

Further reviewing literatures researching the subject in the option market, some studies investigate the relation between market volatility and trade activities. Chatrath,

Ramchander and Song (1995b) found that the cash market volatility increased due to an increase in the level of option trading. Besides, Hagelin (2000) considered the relationship between cash market volatility and option trading under different market conditions. Park, Switzer and Bedrossian (1999) used the most actively traded equity options to investigate the relationship between trading activity of stock options and the volatility of the underlying stocks. The authors found that the unexpected trading activity in option markets indeed holds strong explanatory power over the volatility of the underlying stock returns. Ho, Zheng

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- 8 - | P a g e and Zhang (2012) research is one of the latest to our knowledge and is done after the subprime crisis. In their work, a sample of the 15 stocks with the highest option trade volume in the New York Stock Exchange from 2002 until 2006 is analyzed. Their ﬁndings are consistent with the theory that a higher level of trading activity in the option market leads to a higher degree of volatility in the underlying stock.

Previous research examined the relation between trading volume and stock volatility in the stock market, future market and option market. Besides, they research the topic on

particular stocks, actively traded stocks, stocks in a particular industry or indices. Intuitively, the research using index data generates more unbiased results than when using particular stock data. The index represents the overall capital market and the US national economy, which is a deep market, whereas a particular stock has a thin market. Nevertheless, the findings of previous studies on the relationship between trading volume and volatility is never consistent and uniform so far until now.

3. METHODOLOGY

The current study involves collecting S&P 500 index option (SPX) trading data in option marketing, deriving implied volatility according to the modified BSM, and then analysing the relation between its option trading volume and its implied volatility. The data is collected from the Chicago Board Options Exchange (CBOE), which is the exclusive home for S&P 500 index options and US’s largest option exchange. The sample data is selected from the population and prepared for regression analysis by the using statistics software - Eviews. Hypothesis testing determines whether the relation between option trading volume and implied volatility expected from financial theory will be upheld or not with typical 95% confidence interval.

A good reason to select the S&P 500 index option in the study, is that the S&P 500 Index is widely regarded as the leading benchmark of the overall U.S. stock market, comprising 500 leading companies. In addition, its contracts are the most-actively traded index options in the U.S. So using its option data avoids irregular or pre-event option trading by traders in

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- 9 - | P a g e possession of important or private information. In total seven and a half years (January 2009 – June 2016) daily data of SPX options is retrieved from CBOE’s historical options data, in line with researching period after end of 2008. The daily data comprises the trade date,

expiration date, strike price, trade volume, bid / ask price, underlying price, and more.

In the CBOE, listed SPX options have a large variety of moneyness, maturity, and strike price due to SPX’s popularity. Inevitably, the whole data set has a huge size. As matter of factor, part of the huge size is not actively traded and illiquid, especially deep out of money options. Hence, population sampling is conducted to scrutinize if observations are representatives. In addition, the purpose of the data selection is to evade the bias of option pricing models, under-pricing the values of deep-out-of-the-money ‘s S&P 500 index options, as well as the impact of option maturity and moneyness on implied volatility. In theory, the lognormal distribution assumption of Black -Scholes-Merton is reasonable and accepted. But in reality, lognormal distribution underestimates the extreme movements in the underlying stock price. The result of using a lognormal model is that deep-out-money options would be under-priced compared to its intrinsic value. Meanwhile, the value of implied volatility depends also on the moneyness of the option. In order to by-pass the deficiency of the Black Scholes Model and minimise the impacts of moneyness on implied volatility, the study selects the data with the narrow moneyness range of 99% -101%, close to at-the-Money. Similar to moneyness, maturity of options also has impact on the variable of implied volatility. So the option data with no more than 60 days’ time to maturity, is selected for the test.

Implied volatility is calculated from the modified BSM, which takes dividends into account. Assume companies pay continuous dividends, which is yield of the dividend (q). In addition, the average US dollar’s 3-month LIBOR rate is taken as a proxy for the risk free rate (r). As shown by the equation (2.1) and (2.2), the value of call option (C) and put option (P) are derived by substituting the underlying security price (S), strike price (K), time to expire (T) and risk-free rate (r). Also observed market values of the call option (Cmarket) and put option

(Pmarket)’s equations are as below (3.1) and (3.2).

C = S𝒆−𝒒𝑻_{*N(d1) – K𝒆}−𝒓𝑻_{N(d2) (2.1)}

And

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- 10 - | P a g e Where d1 = ln� 𝑆 𝐾�+�𝑟−𝑞+12𝜎2�∗𝑇 𝜎√𝑇 And d2 = d1 – 𝜎√𝑇

Implied volatility is obtained by using Visual Basic for Application in which the Goal Seek function is employed. Regardless of bid / ask spread, the option’s observed market value should be equal to its theoretical value.

Cmarket = Ctheory (S,K,T,r,𝝈𝝈,q)

And

Pmarket = Ptheory (S,K,T,r,𝝈𝝈,q)

Subsequently, bid-ask spread percentage is obtained by the calculation as followings (4.1). The formula is dividing the bid-spread spread by two and the price of option regardless of bid-ask spread, respectively.

Bid-ask spread percentage = (bid-ask spread) / (2*option price) (4.1)

Daily series of trading volume, directly observed in the option market and derived daily series of implied volatility and bid-ask spread percentage, are imported into the statistic software Eviews. Prior to data importation, trading volume series has large scale numbers relative to the other two series, it is transformed by taking a natural logarithm. Eviews runs multiple linear regression where the dependent variable (y) is a time series of underlying S&P 500 index’s implied volatility and the independent variable (x2) is a time series of trading

volume and the independent variable (x3) is a time series of SPX option’s bid-ask spread

percentage. The estimation method is OLS and parameter coefficients in the model yt

=β+β2x2t+ β3x3t +μt are derived in Eviews. Meanwhile, the other statistics are also presented

in the regression output.

After all, based on the regression output, we can judge how significant and precise the parameter coefficients of the model are at typical 5% significance level. Of course, other

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- 11 - | P a g e statistics determine how fit the model for the relationship among variables. In the case the proposed multiple regression model is unable to explain the relationship as the plan, the ARMA model is superior to linear model. The implied volatility illustrates the property of serial correlation and ARMA model is ideal so far for autoregressive series. But the challenge for serial correlation is that the data is ordered (over time), which cause error term turn out to be correlated. One of solutions is to put residual laps into the model as independent variables if we could not introduce any explanatory independent variables into the model.

4. DATA AND DESCRIPTIVE STATISTICS

For the purpose of the analysis, initially we obtained daily data from January 2009 – June 2016, which is the post subprime crisis period. The daily data comprise all S&P 500 index options (call and put) available to trade in CBOE. The exchange market is able to provide a huge amount of data, which consist of all sort of S&P index options with a wide range of time to maturity and a large variety of strike prices. The data in 2009 taken as example, in order to suit all diverse needs of traders, CBOE tailor-customize 41 classes of options based on S&P 500 index. Moreover, the 41 classes are categorized into three styles, namely standard, LEAP, and weekly. Considering the options with particular characteristics can complicate the calculation, the study only selects the standard SPX options, excluding the variants of SPX.

With regard to the population sampling, even though we only select the standard SPX options as our study object, the data set has more than one million daily observations in total and it is problematic to run an iterative calculation deriving implied volatility, inserting 6 variable in the BSM. Therefore, the study uses population sampling to lead to accurate result and shorten time consumption of the calculation. Firstly, most of deep out of the money options and options with long-term maturities are latent. They are not

representative options, which are likely to mislead the result. As an opposition, close to the money options and short-term option close to money are most popular amongst options. Secondly, volatilities implied by the BSM is not constant over the moneyness. Options deep in the money and deep out of the money have higher implied volatility than those at the money, the so-called volatility skew. Meanwhile, the BSM formula tents to underestimate

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- 12 - | P a g e deep out of money option volatilities, failing to capture the risk of extreme events. Hence, we filter data of the standard style S&P 500 index options (SPX) with a range (99% - 101%) of moneyness. Re-emphasize that using this selection we are able to effectively evade the bias of the BSM and the impact of moneyness on implied volatility. Likewise, the volatility smile is more skewed in short term options than long-term options. To mitigate the influence of time to maturity, we pick the data with no more than 2-month expiration. In the end, the quantity of 40,095 daily observations are qualified as sample data.

Prior to plugging the variables (S, K, r, q, τ, c/p) into the pricing model, the study shows individual variable’s descriptive statistics, graphs or values. First, Figure 1 shows the underlying S&P 500 index price (January 2009 – June 2016) histogram and statistics.

The view displays the frequency distribution of underlying S&P 500 index price (2009 – 2016) in the histogram. During this period, S&P 500 index prices mostly stay and linger at the prices around 1,300 and 2,050. And standard deviation in the past 7 years is 355.40, the dispersion of the series. Further, the skewness is -0.438 and it is a negative value, which implies that the distribution of the series has a long left tail. Since Kurtosis is less than 3, the distribution is flat relative to normal distribution. Finally, Jarque-Bera is a test statistic for testing whether the series is normally distributed, while the p-value is 0. Therefore, the null hypothesis that the series of S&P index price is normally distributed at the 5% significance level can be rejected.

Figure 1:

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- 13 - | P a g e Figure 2 shows the movement of strike prices and underlying prices at time elapse. It is expected to see the high correlation (0.99) and overlap between the two series because the study focuses on approximate at-the-money options.

Table 1 shows the values of the risk free rate and yearly dividend yield in the past 8 years respectively. Risk-free rate is not observable from the market, but the study uses USD 3-month Libor rate as a proxy for the risk-free rate. The 3-3-month LIBOR is a representation of 3-month maturity of the London Interbank offered rate (LIBOR). Although LIBOR is not theoretically risk free, LIBOR is considered as a good proxy against other short-term instruments’ return rate. Additionally, for options on the S&P 500 index, it is reasonable to assume that dividends are paid continuously, equivalent of the index’s yearly average dividend yield.

Figure 2:

Strike Prices Corrected to underlying S&P 500 Index Prices

Table 1:

Proxy for Risk-free Rate (r) and S&P 500 index Dividend Yield (q)

US Dollar LIBOR

3 Month interest rates yearly dividend yield S&P 500 index Year Risk-free Rate (r)* Dividend Yield Rate (q)**

2009 0.675% 2.02% 2010 0.34% 1.83% 2011 0.38% 2.13% 2012 0.43% 2.20% 2013 0.27% 1.94% 2014 0.23% 1.92% 2015 0.32% 2.11% 2016 0.66% 2.04% *http://www.global-rates.com/interest-rates/libor/american-dollar/2009.aspx **http://www.multpl.com/s-p-500-dividend-yield/table

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- 14 - | P a g e To examine series of call / put at the money option market prices underlying the S&P index, its histogram is given as Figure 3 below. In comparison with the histogram and stats of underlying S&P index prices, the distribution of at the money options disperses and deviates less. Meanwhile its distribution frequency looks more like normal distribution than Figure 1. On the contrary to underlying S&P index prices, at the money options have a right long tail. The reason behind the right skewness is that at-the-money options are near the money and have an increasing opportunity to make the profit and of course its price is relatively high. Even though it distributes close to normality, the null hypothesis of normal distribution is rejected due to an insignificant P-value.

Finally, using all the observations in the sample and substituting them in the BSM formula, the calculations imply series of volatility correspondingly. Meanwhile, option trading volume is observable data in the CBOE. As we spot from the data sheet, the numbers of trading volume are large integers relative to the values of implied volatility, which are decimals. In order to rescale the data of trading volume, trading volume series is taken natural

logarithms. Hence, taking a logarithm on the data is likely to improve the linear relationship among the three. Further, the series of bid-ask spread percentage is obtained according the formula (3.1). Hence, implied volatility, bid-ask spread and trading volume series are ready for regression in Eviews. Before that, let the review the three series’ distribution histograms as Figure 4.

Figure 3:

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- 15 - | P a g e Three series look noticeable different and are far from the normal distribution. To our knowledge about volatility, implied volatility should bound around the mean and eventually reverts to the mean. Moreover, implied volatility would exhibit clustering and the auto-correlation property. On the other hand, trading volume spikes toward one direction, no bounce relative to implied volatility. Subsequently, the study tests the correlation and linear regression model in Eviews and the results appear in the following section, Empirical Results and Discussion.

5. EMPIRICAL RESULTS AND DISCUSSION

Barbosa and Saldas (2013) commented that one of the most consensual of these determinants of volatility in the literature is stock trading volume. Several papers have demonstrated that there is a positive relationship between the two. Before further testing the regression model of the relationship of trading volume, implied volatility and bid-ask spread, the examination on the correlation of the series is performed in Eviews.

Figure 4:

Distribution Histograms of Trading Volume Series, Implied Volatility Series and Bid / Ask Spread Series

Table 2:

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- 16 - | P a g e Table 2 shows the output of the correlation coefficient, t-statistic and p-value from Eviews. The result is considered surprisingly low after reviewing that several papers concluded there is a positive association between trading volatility and implied volatility. As a matter of fact, our test result shows the correlation coefficients between trading volume and implied volatility and between implied volatility and askbid spread percentage are 0.0084 and -0.0500, respectively. From the statistical point of view, the p-value of correlation between the trading volume and the implied volatility is 0.0969, which is significant, so that the null hypothesis cannot be rejected at the significance level of 5% that the two serious are

unrelated. Whereas the p-value of correlation between the implied volatility and the bid-ask spread percentage is 0.00, the null hypothesis can be rejected. Caught us by surprise, there is a positive correlation between the trading volume and the bid-ask spread percentage. So the positive correlation can be interpreted either investors in the options market are not influenced by the widen bid-ask spread or the trading demand presses the fees up charged by market makers and traders. Conclusively, it is evident that the result does not justify previous papers’ conclusion of consistent correlation between trading volume in option markets and the implied volatility.

Overall, the result tables are obtained from Trading Volume multiple linear regression, which are Table 3. We examine the results and the parameter estimates subsequently. Without exception, we analyze the standard errors, the t-ratios and the p-values associated with two-sided to test the current study’s hypothesis at a 5% marginal significance level. Inevitably, the models are revisited and its goodness of fit is verified by checking R-squared and the residual plot.

The general aim of the research is to quantify and model the phenomena that trading volume and volatility proliferated together for years in the option market after the Subprime crisis. Due to the Subprime crisis, investors adapted their behaviors and risk-toleration and central banks started to impose rigid regulations to financial institutions. As the situation in financial sector changes, it is valuable and meaningful to test if the theory, a higher level of trading activity in the option market leads to a higher degree of volatility in the underlying stock, can also be applied to implied volatility and trading volume in the option market. Table 3 shows the Implied Volatility as dependent ‘s linear regression result from Eviews, statistic software by processing sample data under OLS method. A constant (β1), and slopes

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- 17 - | P a g e estimated in a multiple linear regression where the dependent variable is a time series of implied volatility and the independent variable is a time series of trading volume and bid-ask spread percentage. The true relation is estimated as follows:

yt = 0.125 + 0.00038x2t -0.06043x3t+ μt

Next, we revisit the study’s hypothesis that there exists a linear relationship between implied volatility and trading volume in the option market. In other words, using both the test of significance and confidence interval approaches, we test the null hypotheses that β2

≠ 0 or β3 ≠ 0 against a two -sided alternative. Under the test of 5% significance approach,

the first null hypothesis that β2 ≠ 0 will not be rejected if the test statistic lies within the

non-rejection region i.e. if the following condition holds: ‘ - tcrit ≤ 𝛽2−𝛽2∗_{𝑆𝐸(𝛽2)} ≤ +𝑡crit

Rearranging, the null hypothesis would not be rejected if ‘ -tcrit *SE(β2) ≤ β2 – β2* ≤ +𝑡crit*SE(β2)

Find 𝑡crit = tꝏ;5% = ±1.96, SE(β2) = 0.0001 and β2*= 0.00038

So the condition is upheld and the first hypothesis would not be rejected.

Subsequently, the second null hypothesis that 𝛽3 ≠ 0, will not be rejected if the test statistic lies within the non-rejection region i.e. if the following condition holds: ‘ - tcrit ≤ 𝛽3−𝛽3∗_{𝑆𝐸(𝛽3)} ≤ +𝑡crit

Table 3

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- 18 - | P a g e Again rearranging, the null hypothesis would not be rejected if

‘ -tcrit *SE(β3) ≤ β – β* ≤ +𝑡crit*SE(β3)

Find 𝑡crit = tꝏ;5% = ±1.96, SE(α) =0.0058 and β3*=-0.0604

Again, so the condition is upheld and the first hypothesis would not be rejected.

Under the hypothesis tests at 5% significance level, the estimate model cannot be rejected and linear parameter coefficients of repressors are all significant. Next, another question that can be raised is that if it is a good model for the true relationship between them from a statistical point of view. However, from the result table, the R-squared value is 0.286%, which can be interpreted as the dependent variable of implied volatility has 0.286%

variation coming from the two independent variables, which is a relatively small proportion compared to the whole 100%. In other words, trading volume can influence implied volatility by a marginal 0.286% and the rest of 99.7% is influenced by other factors, which is failed to specified in the regressed model.

Continuously to evaluate the goodness of fit of the model, the error term should be examined by various regression diagnostic, heteroscedastic test, autocorrelation test and normality test. If they are detected in the disturbance, the assumptions under OLS method are violated and the coefficient estimates are no longer the best. Unfortunately, from the mentioned test statistical results, the errors are conclusively heteroscedastic, lagged and abnormal distributed (see Figure 9 -11). The similar conclusion as adjusted R-squared is reluctantly derived from diagnostic tests, that the multiple linear model is of

misspecification with two explanatory variables.

Further to the study to model the relation between the two, the autocorrelation, one of dominant characteristics from implied volatility is considered. Relative to autocorrelation, AR(1) model is ideally to handle the data with lags. Due to the serial correlation, the one period lag has the explanatory power on the dependent variable. So one period lag of the implied volatility series is added in to the model as another dependent variable. Table 4 shows the result output generated by Eviews. Based on the parameter estimates, the model containing lag of the explanatory variable is as follows:

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- 19 - | P a g e y = .0050 - .0005*X1t - .0066*X2t + .9752*y(t-1)

According to the statistical p-values, four parameter coefficients are sufficiently significant so that the null hypothesis, all independent variable have effects in the model, cannot rejected. Furthermore, the R-squared and adjusted R-squared are both more than 95%. So the model of AR (1) almost explains most of the variability in dependent variable, which is a good sign, but not sufficient to conclude the goodness of fit yet until the residual series is tested. Nevertheless, the errors of this model is non-normality, heteroscedastic and correlated (see Table 5 in Appendix). As see from the plot of residual, all combined

dependent variables are infeasible to explain the spiking volatility during the subprime crisis, extreme events. Most extreme observations occurred in that period, which are spotted in the error term and cause residual non-normality and heteroscedasticity. Those facts cannot be solved by adding the lagged dependent variables into the model. Therefore, the model is reasonably fit with variables, evaluated with high adjusted R-squared.

6. ROBUSTNESS CHECKS

The linear regression models are estimated by using the Ordinary Least Square (OLS). OLS estimators are very sensitive to the presence of an observation that lies outside the norm for the regression model of interest. It is very likely that within our sample data of 40.095 observations, there are some extreme data and outliers, which can result in coefficient estimates that do not accurately reflect the underlying statistical relationship. On the other hand, Robust Least Square is designed to be robust and less sensitive to outliers. Meanwhile Table 4:

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- 20 - | P a g e the residual diagnostics shows that the model is not completely satisfactory because no non-normality, heteroscedasticity and correlation are found in the residual series. however, the Newey-West method is able to estimate the regression with consistent presence of both heteroscedasticity and autocorrelation of unknown form.

The result (Table 8 in Appendix) derives that the values of parameter coefficients in the tests are not significant at the 5% because of both having a zero p-value. Therefore, the

hypothesis of no effect on the dependent variable, can be rejected, which is the same conclusion derived from the OLS method previously. In the end, the conclusion from robust regression verify and confirm our previous findings.

7. CONCLUSION

Prior empirical research has documented findings that option trading volume has explanatory power over returns’ volatility and it is robust after controlling the conditions obtained analyzing the data from 16 European and US banks. Another similar study’s findings are consistent with the theory that a higher level of trading activity in the option market leads to a higher degree of volatility in the underlying stock. However, these studies either have used particular stock data for analysis or focus on the historical volatility, instead of implied volatility. In this study, we tested relationship between trading volume of S&P 500 index option and implied volatility of the underlying.

The study explored the relationship between trading volume of S&P 500 index option and implied volatility of the underlying by beginning with a multiple linear model and continuing with AR(1) model. We find no correlation between trading volume in option market and the implied volatility of the underlying S&P 500 index. Regressed in the multiple linear model, we find that trading volume of S&P 500 index option have explanatory power on its implied volatility derived by the BMS formula. Compared to the linear model, dynamic model AR(1) is capable to capture the movement of implied volatility better by adding one more

dependent variable, one period lag. In addition, the improvement made in our study was controlling the moneyness impact on the values of implied volatility. Most notably, it was

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- 21 - | P a g e the first study to my knowledge to investigate the phenomena of propagating trading volume accompanied by a highly volatile in option market since the subprime crisis, while the findings were contradictory to the phenomenon from the correlation point of view. Even though the two are related in both the multiple linear and AR(1) model, the persistent heteroscedasticity and correlation are remained in the residuals, respectively. Finally, the study laid a foundation for researchers with an interest to further explore the relationship in a more complicate model.

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APPENDIX

Table 5: AR(1) Model’s Errors Normality Test

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- 24 - | P a g e Table 9: Result of Heteroscedasticity Test on Line Model’s Residual (p-value=0) Table 7: AR(1) Model’s Errors Serial Correlation Test with Zero P-Value

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- 25 - | P a g e Table 10: Result of Normality Test on linear Model’s Residual (p-value=0)