1 | P a g e
IMPLIED VOLATILITY SPREADS
Do implied volatility spreads contain
information that can be exploited?
J.J.M. MELMAN
1University of Groningen,
Master’s Thesis Finance
Supervision:
Dr. P.P.M. Smid
2June 2013
ABSTRACT
This paper examines whether a portfolio strategy based on the implied volatility spread is of practical relevance for investors and other parties interested in informed trading. By using the implied volatility spread as a measure of price pressure for stocks, abnormal returns are earned of 54 basis points per month and are robust when controlling for illiquidity. Using the conditions of the sequential trade model of Easley, O’Hara, and Srinivas (1998) there is no evidence found that the implied volatility spread captures informed trading. When including transaction costs the implied volatility spread strategy is unprofitable.
Keywords: Implied volatility spread, abnormal returns, Informed trading. JEL Classification: G11, G14
2 | P a g e
1. Introduction
A subject in the finance literature that has attracted much attention is the forming of portfolios that realize returns higher than expected. In the search for these positive abnormal returns, many different variables have been studied. Some of the variables that have been extensively examined in this context relate to the past stock returns (see, e.g., Hestona and Sadka, 2008; and Novy-Marx, 2012). While these variables come from the stock market, this paper focuses on the implied volatility spread, which comes from the options market and is calculated by subtracting the implied volatility3 of a put from that of a call with an identical strike price, maturity and underlying.
The implied volatility spread belongs to a group of implied volatility measures that all of have their own theoretical background. I have chosen to focus on the implied volatility spread because Cremers and Weinbaum (2010) argue that it captures informed trading activities that are not directly reflected in the underlying market. Hence, if this is true, the construction of portfolios based on implied volatility spreads should result in portfolios that obtain abnormal returns by exploiting public information about price directions from informed trading. For the other implied volatility measures, this relation with informed trading is not found (see, e.g., Xing, Zhang, and Zhao, 2010; Bali and Hovakimian, 2009).
Because the implied volatility spread is argued to capture informed trading activities, the justification for the study goes beyond the facilitation of hedge funds and other individuals/parties interested in realizing abnormal returns. The results, for example, can also facilitate market regulators in their task to ensure the fairness of trading. Since they need to prevent the circumstances that would allow some investors to use their information advantage in an illegal way, the implied volatility spread can be used as a monitoring variable to detect possibly illegal insider trading activities. Moreover, for the same reason, the study might be relevant for dealers and specialists. As noticed by Black (1975), they make profits on investors who do not have an information advantage and lose money on those investors who do have an information advantage. Hence, profits made by the difference between the bid-ask spread for the trades of the uninformed investors have to be sufficient to offset the losses made on the informed investors. Knowing more about where informed trading occurs can help them to facilitate liquidity investors better and manage their adverse selection risks.
The first contribution of this paper to the existing literature is that it addresses the practical relevance of the implied volatility spread for investors. The second is that it examines whether the implied volatility spread (still) captures informative trading or not. To deliver these contributions the paper
3
3 | P a g e
addresses two questions: (1) Do implied volatility spreads provide an abnormal return that can be realized by investors, and (2) do they capture informed trading activity? To answer these questions I use a European sample with options from a range of exchanges and recent data from the period January 2008 until December 2012. In contrast, previous literature that studied the implied volatility spread in depth used pre-crisis, U.S. data.
To answer the questions, stocks are sorted into five portfolios based on their implied volatility spread. From these five portfolios a hedge portfolio is formed that has a short position in portfolio one and a long position in portfolio five to exploit both the price pressure from negative and positive implied volatility spreads. To determine whether the implied volatility spread captures informed trading, the sequential trade model of Easley, O’Hara, and Srinivas (1998) is used. This model provides conditions
under which informed investors trade in the options market. Finally transaction costs associated with the rebalancing of the portfolios are incorporated. The results provided in this study show there are abnormal returns earned by a strategy based on the implied volatility spread. Hence, consistent with previous literature, I find that the implied volatility spread is not just a result of mere noise. There is, however, no indication that the implied volatility spread captures informed trading activities. When transaction costs are incorporated, abnormal returns are insufficient to compensate for these costs and there is no indication that they could become positive by adjusting the strategy. Hence, either for investors or for other parties there is no practical relevance found.
4 | P a g e
2. Literature overview
Section 2.1 of the literature overview introduces the theoretical background of the implied volatility spread and explains why it might capture informed trading activities. Section 2.2 includes an overview of the related empirical literature and provides a table of the results of studies that used the implied volatility spread.
2.1. Theory
An important variable in the field of finance is volatility, which measures the uncertainty of an investment. When investors are considered to prefer certainty, it helps to quantify the risk and hence the expected return. To measure the volatility, one can look back at the past performance of an investment or at the current pricing on the options market. When the second approach is taken, it is necessary to have an options pricing model. One of these models, probably the most widely known, is the options pricing model of Black and Scholes (1973). When using the model, for example, for a European call option on a non-dividend paying stock, one needs to know the strike price, current stock price, volatility of the stock, the risk-free rate, and the time to maturity. Of these parameters, only the volatility is not directly observable and therefore needs to be estimated. Because the other parameters can be observed, it is actually the only unknown and its value can be solved. This particular value for the volatility is called the implied volatility. Hence, it is the volatility that is implied by the prices and values of the other parameters that are observed in the market and the options contract (see, e.g., Hull, 2012).
There is a wide range of options available for each stock. When considering options on the same stock with equal time to maturity, one would expect that the implied volatilities by option prices are the same. However, Rubenstein (1994) finds that since the crash of October 1987 there is a marked difference between options on the same stock, with the same maturity but with different strike prices. For stock options the implied volatility is typically higher for low strike prices compared to high strike prices (Dennis and Mayhew, 2002). This observed pattern, referred to as a smile,4 arises primarily because of the assumption made by Black and Scholes (1973) about the distribution of returns does not hold. In reality the distribution is not the assumed lognormal one, as mentioned by Ederington and Guan (2002), but has fat-tails towards the left, which reflect higher probabilities of extreme downward movements.5
Assuming all the other assumptions of the Black and Scholes (1973) model hold, European call and put options with the same strike price and maturity should still have the same implied volatility (see,
4
In the finance literature the terms for stock options that are also used, instead of smile, are skew (see, e.g., Hull, 2012) and smirk (see, e.g., Zhang, Zhao, and Xing, 2010) because for stock options the curve does not look like a smile.
5
5 | P a g e
e.g., Hull, 2012). This can be explained by considering the no-arbitrage relation called the put-call parity, which is derived by Stoll (1969). In Eq. (1) this relation is stated.
( ) ( ) ( ) (1)
where and respectively are the European option call and put prices with a strike price of on the underlying non-dividend paying stock . Both options have equal expiration dates indicated by the . The relation does not require assumptions about the probability distribution because both sides in Eq. (1) provide exactly the same payoff at . The put-call parity also holds for the options pricing model of Black and Scholes (1973) when the values for the volatility for the underlying in both the call and the put option are the same (see, e.g., Cremers and Weinbaum, 2010). When this is not the case the put-call parity is violated. Therefore, when Eq. (2) is violated, the put-call parity violated.
(2)
where represents the implied volatility of the put and call with the same maturity and strike price. Now, when using a call and a put with the same maturity, strike price, and underlying, the implied volatility spread is obtained by subtracting the implied volatility of the put from that of the call. If this implied volatility spread is positive, the call is relatively expensive. Using Eq. (1), the arbitrage trade consists of a short position in the call, a long position in the put, a long position in the stock and borrowing the present value of the strike price. The part of the arbitrage trade especially relevant for this study is that the stock is purchased.
The intuition behind the implied volatility spread has been made clear by taking a perspective without frictions and using European options for which the put-call parity is a no-arbitrage relation. In reality, however, there are frictions that must be considered. As noted by Barberis and Thaler (2003), there should be a condition that enables investment opportunities that have the same payoffs to be priced differently, which is exactly what frictions do. Consider, for example, transaction costs; the larger they are, the larger the implied volatility spread needs to be for a profitable arbitrage opportunity. In addition to the frictions, exchange traded options are American and therefore have an additional right of exercising before the expiration date. Stoll (1969) incorrectly claims that the put-call parity also holds for dividend-protected American options because there is no rationality for exercising them prematurely. However, Merton (1973) shows it can sometimes be rational to exercise the put options early and provides proof of this. Hence, the put-call parity not a no-arbitrage relation with American options.
6 | P a g e
put and, most importantly for this study, a possibly underpriced stock. There is a relevant question in this context that will be first addressed in the empirical evidence section: What is the relation between the stock and options market, is the options market leading or lagging?
Cremer and Weinbaum (2010) argue that the implied volatility spread captures informed trading and that therefore an investment strategy based on this measure results in abnormal returns. The reasoning why implied volatility spread captures informed trading is that the trading activity of informed investors affects the implied volatility (i.e. prices) of the options, and hence the implied volatility spread.6 In the existing literature there are several arguments documented why informed investors choose to trade in the options market instead of the stock market. Black (1975) argues that informed investors with limited funds are able to increase their exposure towards a certain stock by investing in the options market. This higher exposure comes from the ability to obtain a highly leveraged position in the options market. This argument is supported by the results of Lee and Yi (2001), which show that informed investors are attracted by options with greater financial leverage. Two other arguments are the lack of short sale constraints in the options market (
Afef and Olfa,
2009) and the possibility to bet on volatility (Back, 1993). Important to note is that when the lack of short sale constraints is the main reason why informed investors choose to trade in the options market, one would expect that only the negative implied volatility spreads captures informed trading.The sequential trade model of Easley et al. (1998) offers conditions under which informed investors trade in the options market. When these conditions are satisfied, it is attractive for informed investors to trade in the options market because this enables them to realize a larger profit. For example, when there are many informed investors trading in the stock market and the stock market is relatively illiquid, then it becomes more attractive for informed investors to trade in the options market.
2.2. Empirical literature
The implied volatility spread is a measure from the options market and in this study is used as a measure of price pressure for stocks. For this measure to be an indicator for price directions of the underlying stock, it is important to know whether there is a certain order of information processing between the option and stock markets. While semi-strong market efficiency implies that information that becomes public is immediately incorporated into prices, there is empirical evidence, related to the question whether options markets lead or lag stock markets, that contradicts its efficiency. Manaster and Rendleman (1982) examined the question whether option prices lead stock prices or vice versa by using option prices observed in the market. From these option prices they calculate implied stock prices using the Black and Scholes (1973) formula. By comparing their implied stock prices with the observed stock
6
7 | P a g e
prices they conclude that options prices contain information about stock equilibrium prices that is not fully incorporated in the stock prices. However, they did not account for non-synchronicity, which could explain their results. This however was taken into account by Bhattacharya (1987), but did not change the results of Manaster and Rendleman (1982). Since these studies a lot of empirical research has been performed to determine whether the options market leads the stock market (or the other way around). The results, however, are inconclusive. For example, Stephan and Whaley (1990) find an inverse relation and Chan, Chung, and Johnson (1993) find no leading or lagging relation using high frequency data. It should be noted that these studies do not distinguish between the permanent and transitory price changes. According to Chakravarty, Guien, and Mayhew (2004), only the permanent price change is important to make a judgment about where price discovery occurs. They conclude that in the options market there is significant price discovery and it is bigger when the underlying is relatively illiquid and the bid-ask spread of this underlying is large compared to that of the option. However, for the investment strategy based on the implied volatility spread to realize abnormal returns it is not necessary that there is one order of information from the option market to the stock market. If this order of information processing only counts for the stocks that have options with a relatively large implied volatility spread it is sufficient. At least from existing literature there is not unequivocally evidence that options markets lag stock markets.
The implied volatility spread from the options market is not the only variable that uses implied volatilities. Xing, Zhang and Zhao (2010) study the volatility smirks, which are calculated by subtracting the implied volatility of an at-the-money call from an out-of-the-money put. Xing, Zhang, and Zhao (2010) argue that the out-of-the-money put is attractive when investors have negative information, and hence when this investors group grows the slope of the volatility smirks increases. Following this reasoning, the slope can be used as a measure of jump risk, which is wherefore Yan (2011) uses it. This is logical when considering that a volatility smirk with a steeper slope indicates a relatively fat left side tail of the distribution. Xing, Zhang, and Zhao (2010) show that a weekly rebalanced (hedge) portfolio with a short position in the stocks with the smallest slope and a long position in the stocks with the largest slope results in a significant positive weekly alpha7. To examine whether their measure captures informed trading activity, Xing, Zhang, and Zhao (2010) use the model of Easley et al. (1998) but find statistical support for this.
The implied volatility spread studied in this paper is also used by Cremers and Weinbaum (2010) and is calculated as the difference between the implied volatility of call options and put options with the same characteristics as in the put-call parity. Based on earlier research in the area of put-call parity deviations, Cremers and Weinbaum (2010) examine three possible interpretations of the implied volatility
7
8 | P a g e spread. The first interpretation is that the implied volatility spread is the result of noise. By using the implied volatility spread to construct portfolios, one would not expect to find any significant alphas if this interpretation is correct. However, they do find significant alphas, as stated in Table 1, for the monthly and weekly rebalanced hedge portfolio that has a long position in the stocks with the highest implied volatility spread and a short position in the stocks with the lowest implied volatility spreads. The second interpretation is that the deviations reflect short-sale constraints. This interpretation comes from the study of Ofek, Richardson, and Whitelaw (2004). Using rebate rates as a proxy for shorting difficulties, Cremers and Weinbaum (2010) conclude that the alphas are not driven by shorting difficulties.The third interpretation is that the implied volatility spread captures the trading activity of informed investors. This interpretation is examined using the conditions from the model of Easley et al. (1998). As is consistent with the model, Cremers and Weinbaum (2010) find that the alphas of the hedge portfolio are larger when the stock market is relatively illiquid compared to the options market and smaller when the opposite is true. In addition to this they show that the implied volatility spreads are larger when the probability of informed trading (PIN) in the stock market is high, which is again consistent with the model of Easley et al. (1998). While they mention that their results seem to contradict the efficiency of markets, they test whether these results remain stable over time by separating their sample from January 1996 until December 2005 into two periods. From this separate analysis they conclude that for the most recent sample period, ranging from January 2001 until December 2005, the alphas are lower and argue that this reflects a reduction of mispricing overtime.
9 | P a g e
a jump return. Another volatility variable Bali and Hovakimian (2009) use is the realized volatility. It is standard deviation of the returns of the month before the rebalancing moment. Beside that they use it without any variable from the option market, they subtract the average implied volatility from it resulting in a proxy for volatility risk.
10 | P a g e
Table 1
Results of empirical studies that used the implied volatility spread for portfolio construction
Study Universe1 Period Portfolios2 Measure3 Frequency Weighing Alphas4
Bali and Hovakimian (2009)
NYSE, AMEX, NASDAQ
1996:01-2004:12
Quintile Level Monthly Equally 1.49%
Value 1.14%
Cremer and Weimbaum (2010)
U.S. stocks with exchange listed options
1996:01-2005:12
Quintile Level 4 weeks Value 0.55%
Weekly Value 0.91%
Both 4 weeks Equally 1.60%
Value 1.07%
Weekly Value 2.18%
1996:01-2000:12
Quintile Level 4 weeks Value 0.66%
Weekly Value 1.26%
Both 4 weeks Value 1.77%
Weekly Value 3.07%
2001:01-2005:12
Quintile Level 4 weeks Value 0.42%
Weekly Value 0.65%
Both Weekly Value 1.44%
Baltussen et al. (2012) S&P/Citigroup U.S. Broad Market Index 1250 largest stocks
1996:01-2009:10
Quintile Level Weekly Equally 0.64%
Change Weekly Equally 0.45%
1
Baltussen et al. (2012) and Bali and Hovakimian (2009) use at-the-money options, and Cremers and Weinbaum (2010) use a broader range of options to calculate the implied volatility spreads.
2The term “Quintile portfolios” in the column means that the stocks at the rebalancing moment are allocated across five portfolios
based on their implied volatility spread.
3
The term “Level” in this column means that the portfolios are formed based on the difference between the implied volatilities between the puts and calls on the same stock. The term “Change” means that the portfolios are formed based on the change between implied volatility spreads across time. The term “Both” means that they are combined.
4
11 | P a g e
3. Methodology
To answer the question whether an active portfolio strategy based on the implied volatility spread earns abnormal returns, it is necessary to have an expected return and compare this with the return realized. Similar to the studies stated in Table 1, the realized returns are obtained from five portfolios that are created by using the implied volatility spreads of the stocks in the sample. These implied volatility spreads are used to sort the stocks and allocate them, each month in descending order, to the five portfolios. The stocks with the highest implied volatility spread are allocated to portfolio one and the stocks with the lowest implied volatility spread are allocated to portfolio five. The hedge portfolio has a short position in portfolio five and a long position in portfolio one. Furthermore, there is a one day delay between the obtained implied volatility measure and the portfolio construction. This delay ensures that the results are not coming from spurious predictability and it also makes them relevant for both large and small investors, as noted by Baltussen et al. (2012). In addition to the implied volatility spread, this study will use the change in the implied volatility spreads to sort the stocks. The change is calculated by subtracting the implied volatility spread, two days before the month starts, from the implied volatility spread on the last day of the month. For measuring the performance of the portfolios the returns are calculated as follows. The arithmetic returns of the stocks are calculated by using Eq. (3):
(3)
where is the arithmetic return for stock j over month t, and is the total return index for stock j
over month t. Next, the arithmetic return of the portfolio in excess of the risk-free rate is calculated by using Eq. (4):
∑ ( )
(4)
where is the arithmetic excess return of portfolio over month , is the number of stocks in
portfolio over month , is the weight of stock over month , and is the risk-free rate of stock
over month . The stock weight for equally weighted portfolios is and for the weighted portfolios
the stock weight is its market value divided by the market value of the portfolio. The portfolio’s arithmetic excess mean return is calculated by summing the monthly returns of the portfolio and dividing them by the number of months, as stated in Eq. (5):
∑
(5)
12 | P a g e
three factors of Fama and French (1993) and the momentum factor of Carhart (1997). The Fama and French (1993) three factor model is basically an extension of the Capital Asset Pricing model (CAPM) of Sharpe (1964) and Lintner (1965). However, Fama and French (1993) conclude that value stocks (i.e. stocks with a high book-to-market ratio) and stocks from smaller companies (where size is measured by market capitalization) perform better than growth stocks and larger companies. Therefore they include the small-cap minus big-cap and high-value minus low-value factors. Carhart (1997) includes another factor called momentum, which is based on the results of a trading strategy researched by Jegadeesh and Titman (1993). This trading strategy buys stocks that have recently performed well and sells those that have recently performed poorly. Including this momentum factor, a four factor model is obtained, which is better able to explain returns than both the Fama and French (1993) three factor model and the CAPM model. This four factor model is stated below in Eq. (6).
(6)
where represents the alpha of portfolio , is the arithmetic excess portfolio return of portfolio over month , is the arithmetic excess market return over month , is the arithmetic return on the Fama and French small-cap minus the big-cap portfolio over month , is the arithmetic return on the Fama and French high-value minus low-value portfolio over month , and is the arithmetic
13 | P a g e
4. Data
The dataset contains all the stocks with both call and put continuous option series, as explained in Appendix 1, available within the economic and monetary union (EMU) from January 2008 until December 2012 in Datastream. When options on stock funds are traded on multiple exchanges, the preference is given to the home country of the company. This provides a more diversified sample over different exchanges since many stocks have both options listing on the Eurex and in their home country. In total this resulted in a sample size of 231 stocks. When necessary data are missing of stocks because of data gaps in DataStream, they are not allocated to the portfolios and therefore, in general, the total sample used will be smaller than 231.
14 | P a g e
emphasis is on the ATM-implied volatility and, to save space, the descriptive statistics associated with the weighted implied volatility are included in Appendix 2. Fig. 1 shows how many stocks are incorporated into the samples when using the different continuous data series. It can be observed that the sample used fluctuates more when the change in the weighted/ATM implied volatility spread is used but that overall the sample size used fluctuates between 321 and 316 stocks.
The returns for the four factor model are obtained from the Fama-French website. Since this paper’s focus is on the economic and monetary union (EMU), the European factors are used. These returns need to be adjusted since Fama-French state the returns on the European stocks in dollars. In addition to this they subtracted the U.S. one month T-bill rate to obtain the market return in excess of the risk-free rate. To obtain the excess market return in euros, the following adjustments are made. First the U.S. one month T-bill rate that they subtracted is added back. Next, this monthly return is adjusted by the return on a euro/dollar investment over that period. Finally, the German one month T-bill rate is subtracted. The other returns in Eq. (6) are also adjusted by the return on a euro/dollar investment. However, since the last three factors are all return differences between portfolios, the adjustments have a small impact.
Figure 1: Number of stocks used from the total sample of stocks across time for the different portfolios.
This figure plots the number of stocks included in the portfolios created by using the ATM/Weighted implied volatility spread and the change in this for the January 2008 until December 2012 sample period. The ATM/Weighted implied volatility spread is calculated by subtracting the ATM/weighted implied volatility of the put from that of the call on the last day of the previous month. The change in the ATM/Weighted implied volatility spread is calculated by subtracting from the ATM/Weighted implied volatility spread on the last day of the month the ATM/Weighted implied volatility spread on the day before.
205 210 215 220 225 230 235 N u m b e r o f sto cks u sed
15 | P a g e
Table 3 Panel A provides the descriptive statistics of the implied volatility spreads. Instead of reporting only means, eight percentiles are reported to examine the distribution of the implied volatility spreads. The ATM implied volatility spread percentiles and means show that the implied volatility spreads are overall negative and that there is no specific year in which this is different. This means that, according to the theory provided, there is an overall downward pressure on the stock prices in the sample period. In Table 3 Panel B the percentiles are provided for the weighted implied volatility spread variable. These implied volatility spreads are again on average negative, but the difference between the 10th percentile and 90th percentile is considerably larger. For example, for the whole sample period the difference between the 10th percentile and 90th percentile of the ATM implied volatility spreads is 4.12% and for the weighted implied volatility spread this is 20.03%. Hence, it matters how the implied volatility spread is calculated. The difference between the two implied volatility spread distributions can be explained by the options that are used to calculate them. For the calculation of the weighted implied volatility spreads, options that are more illiquid are also incorporated. This may cause the implied volatility spread to be larger because, for example, transaction costs for these stocks are typically higher. In addition, the distribution of calls and puts across time and across strike prices can deviate and this can have an impact on the resulting
Table 3
Descriptive statistics of implied volatility spreads
Panel A: ATM Implied volatility spreads
2008-2012 2008 2009 2010 2011 2012 P er ce n til e 10.00% -2.93% -2.38% -3.72% -3.01% -2.30% -2.86% 20.00% -0.99% -0.88% -1.30% -1.25% -0.85% -0.89% 30.00% -0.45% -0.44% -0.56% -0.45% -0.44% -0.40% 40.00% -0.26% -0.23% -0.30% -0.25% -0.29% -0.21% 60.00% -0.04% 0.03% -0.01% -0.06% -0.09% -0.02% 70.00% 0.09% 0.15% 0.12% 0.04% 0.02% 0.14% 80.00% 0.30% 0.35% 0.40% 0.24% 0.15% 0.38% 90.00% 1.28% 0.97% 2.20% 1.11% 0.90% 1.52% Mean -0.39% -0.66% -0.12% -0.56% -0.31% -0.32%
Panel B: ATM Implied volatility spreads
2008-2012 2008 2009 2010 2011 2012 P er ce n til e 10.00% -12.60% -12.60% -13.65% -12.20% -11.01% -13.01% 20.00% -6.67% -6.67% -6.82% -6.34% -5.72% -6.49% 30.00% -3.77% -3.77% -4.06% -3.87% -3.58% -3.67% 40.00% -2.30% -2.30% -2.20% -2.23% -2.29% -2.12% 60.00% -0.49% -0.49% 0.04% -0.40% -0.50% -0.20% 70.00% 0.50% 0.50% 1.50% 0.50% 0.33% 0.97% 80.00% 2.72% 2.72% 4.91% 2.14% 2.71% 3.60% 90.00% 7.43% 0.97% 13.22% 7.62% 8.37% 10.23% Mean -1.91% -0.53% -0.39% -1.95% -1.38% -0.22%
16 | P a g e
distribution of the weighted implied volatility spreads (See Appendix 1 for an explanation).
In Table 4 Panel A the market value percentiles show the wide diversity in size of the companies in the sample. When calculating the portfolio returns by weighting the stock returns equally, each implied volatility spread will be considered equally relevant. However, when calculating the portfolio returns by weighting the stock returns within them based on their value, only a small number of implied volatility spreads are relevant (those of the large firms). Therefore, I will use equally weighted portfolios. In Table 4 Panel B the book-to-market ratios of the stocks incorporated into the sample have a less extreme distribution. However, there are some extremes above the 90th percentile, which explains the relative high mean. Table 4 Panel C shows how many of the stock options are traded on each of the six different exchanges incorporated into the sample. The largest number of the stock options are traded on the Eurex. Together with the Euronext.liffe Paris more than half of the total sample (51.9%) of stock options are traded here. Next to the different exchanges, Table 3 also shows to which industry group the different stocks belong. Using the business industry classification (BIC) codes, it is clear that the largest industry group is that of the financials (14.6%).
Table 4
Descriptive statistics of sample
Panel A: market value (in € mln)
17 | P a g e
Table 4 continued Panel C: Industries and exchanges
Industries Exchanges
Financials 47 14.60% Eurex 66 28.57%
Industrials 38 11.80% Euronext.liffe Paris 54 23.38%
Consumer Goods 29 9.00% Euronext.liffe Amsterdam 34 14.72%
Consumer Services 29 9.00% Borsa Italiana 31 13.42%
Technology 20 6.20% MEFF Renta Variable 28 12.12%
Utilities 19 5.90% Vienna Stock Exchange 18 7.79%
Oil & Gas 15 4.70%
Telecommunication
s 11 3.40%
Health Care 8 2.50%
18 | P a g e
5. Results
The results section is divided into three sections. In section 5.1 it is determined whether the implied volatility spread portfolios earn abnormal returns. Section 5.2 investigates whether the implied volatility spread is associated with informed trading activity. Finally, section 5.3 examines if the abnormal returns earned by the implied volatility spread portfolios remain relevant given the incorporation of transaction costs.
5.1 Abnormal Returns
The abnormal returns section is divided into two parts. In section 5.1.1 the main results of the implied volatility spread portfolios are stated and discussed. Here also the change in the implied volatility spread is used. Section 5.1.2 investigates whether the abnormal returns that are found are robust when controlling for illiquidity and the sample is split into two to see how the strategy performs across different exchanges.
5.1.1 Main results
Table 5 Panel A shows the performance of the ATM implied volatility spread portfolios. Considering the alphas of the hedge portfolios, the statistical evidence is not overwhelming for the realization of abnormal returns using a long-short strategy based on the ATM implied volatility spread. The hedge portfolio alpha of 0.54% estimated by the FF4 model is statistically significant and this is actually the only statistical significant evidence in the table. Taking a closer look at the results of the individual portfolios in Table 5 Panel A, it is hard to draw any conclusions because there is no statistically significant evidence found. In contrast, the studies discussed in the literature review find a pattern of monotonically declining alphas and returns from the portfolio that incorporates the stocks with the highest implied volatility spreads towards the portfolio with the stocks that have the lowest implied volatility spreads. The returns in Table 5 Panel A do not show any indication of this pattern across the portfolios. Portfolio 1 seems not to be invested in undervalued stocks at all, since its returns are similar to those of portfolio 2 and 3.
19 | P a g e
four factor model are omitted. Comparing the results of the portfolios that are constructed by using the ATM implied volatility spreads to those constructed using the weighted implied volatility spreads shows that the conclusions are similar. Hence, while using the two weighted and ATM implied volatilities resulted in clearly different spreads, as noted earlier in Table 3, both lead to portfolios that earn similar abnormal returns.
Table 5 Panel C and Table 5 Panel D provide the results of the portfolios that are created by using the change in the ATM/Weighted implied volatility spreads. The portfolios in the tables are the result of calculating the change in the implied volatility spread by subtracting the implied volatility spread two days before the month starts from the implied volatility spread on the next day. The results can be briefly summarized since there is no evidence that using the change in the implied volatility spread results in portfolios that earn abnormal returns.
Table 5
Main results of portfolios formed using the ATM/weighted implied volatility spreads and the change in these spreads
Panel A: ATM implied volatility spread Portfolios
1 2 3 4 5 Hedge
Arithmetic excess mean returns -0.21% -0.27% 0.11% -0.45% -0.69% 0.48%
Alpha FF4 0.19% 0.00% 0.38% -0.13% -0.35% 0.54% (t-value) (0.74) (0.00) (1.24) (-0.41) (-0.93) (2.32)** Alpha FF3 0.14% -0.03% 0.34% -0.16% -0.36% 0.50% (t-value) (0.33) (-0.09) (0.80) (-0.45) (-0.94) (1.47) Alpha CAPM -0.15% -0.22% 0.16% -0.40% -0.64% 0.48% (t-value) (-0.31) (-0.80) (0.42) (-1.18) (-1.50) (1.43)
Panel B: Weighted implied volatility spread Portfolios
1 2 3 4 5 Hedge
Arithmetic excess mean return -0.20% -0.39% -0.16% 0.10% -0.73% 0.53%
Alpha FF4 0.18% -0.10% 0.15% 0.20% -0.34% 0.53% (t-value) (0.45) (-0.36) (0.51) (0.74) (-0.89) (2.37)** Alpha FF3 0.13% 0.00% 0.12% 0.17% -0.38% 0.51% (t-value) (0.25) (-0.01) (0.37) (0.56) (-0.87) (2.04)** Alpha CAPM -0.13% -0.31% -0.08% -0.06% -0.65% 0.52% (t-value) (-0.25) (-1.17) (-0.25) (-0.22) (-1.53) (1.91)*
Panel C: Change in ATM implied volatility spread Portfolios
1 2 3 4 5 Hedge
Arithmetic excess mean return -0.27% -0.52% -0.32% -0.11% -0.19% -0.08%
20 | P a g e
Table 5 continued
Panel D: Change in Weighted implied volatility spread Portfolios
1 2 3 4 5 Hedge
Arithmetic excess mean return -0.33% -0.31% -0.28% -0.33% -0.22% -0.11%
Alpha FF4 0.05% 0.00% -0.00% 0.00% 0.09% -0.04% (t-value) (0.15) (0.01) (-0.02) (-0.02) (0.26) (-0.15) Alpha FF3 0.00% -0.03% 0.03% -0.02% 0.05% -0.05% (t-value) (0.02) (-0.07) (-0.09) (-0.07) (0.14) (-0.18) Alpha CAPM -0.28% -0.25% -0.23% -0.28% -0.16% -0.11% (t-value) (-0.66) (-0.72) (-0.70) (-0.86) (-0.38) (-0.45)
For the January 2008 through December 2012 sample period, all stocks are sorted at the beginning of the month into quintile portfolios. These quintile portfolios are sorted in descending order based on their implied volatility spread measures on the last day of the previous month. For the portfolios in Panel A and B, the ATM/Weighted implied volatility spread is used to rank the stocks. This measure is calculated by subtracting the ATM/weighted implied volatility of the put from that of the call on the last day of the previous month. For the portfolios in Panel C and D, the change in the ATM/Weighted implied volatility spread is used to rank the stocks. This measure is calculated by subtracting from the ATM/Weighted implied volatility spread on the last day of the previous month the ATM/Weighted implied volatility spread on the day before. The hedge portfolio has a short position in portfolio five and a long position in portfolio one. The returns are calculated monthly, equally weighted, and in excess of the German one month T-bill rate (i.e. the risk-free rate). The FF4 alpha is estimated by a four factor model (stated below) that includes the three factors of Fama and French (1993) and the momentum factor of Carhart (1997). The FF3 alpha is estimated by omitting the momentum ( ) factor, and the CAPM alpha is estimated by also omitting the small-cap minus big-cap ( ) and high-value minus low-value factors ( ).
The Newey-West adjusted t-statistics are reported between brackets and statistical significance at the 10%, 5% and 1% level is indicated by *, ** and *** respectively.
5.1.2 Controlling for illiquidity and sub-sample test
One might expect that stocks with a large negative or positive implied volatility spread are at the same time the stocks that are relatively illiquid. The reason for this expectation is that a higher degree of illiquidity prevents arbitrage, due to their higher transaction costs, and hence their higher illiquidity ensures that the implied volatility spread can be larger. Vice versa, when the stock is relatively liquid arbitrage might occur more easily and this results in a smaller implied volatility spread. In addition to this, the period analyzed in this study is characterized by the financial and euro crisis. As Amihud (2002) mentions, a crisis is typically a period in which the illiquidity of relatively illiquid stocks is amplified because there is a ‘flight to liquidity’.
21 | P a g e
five portfolios are merged into one portfolio that represents the portfolio that is invested in the stocks with the highest implied volatility spreads. This is also done for the other implied volatility spread portfolios in the different illiquidity sub-groups and this process results in five portfolios that are controlled for illiquidity.
To measure the stocks illiquidity the Amihud (2002) illiquidity ratio is used with an adjustment. This illiquidity measure considers a stock illiquid when a large price change can be realized with a low volume. Hence, the measure is closely related to volume but it makes a distinction between high volatility stocks and low volatility stocks with similar volumes. The adjustment consists of taking the square root of the daily return-volume ratio, which reduces the impact of extreme ratios and is in line with Hasbrouck (2005). The calculation is stated in Eq. (7):
∑ √| | (7)
where is the illiquidity of stock over month , is the number of days in month for
which both the volume and returns of stock are available, is the return of stock on day , and
is the daily trading volume of stock on day . The daily volume represents the stock value traded on that day.
Table 6 Panel A provides an overview of illiquidity percentiles for the stocks within the portfolios constructed by using the ATM implied volatility spread. Although they seem to be a bit higher, the percentiles do not indicate that there is a clear concentration of illiquid stocks in portfolios one and five. In Appendix 2 Table 2.1 the same overview is provided for the portfolios constructed by using the weighted implied volatility spread and do not show a different pattern. Table 6 Panel B shows the returns of the ATM/weighted implied volatility spread portfolios when controlled for illiquidity. The pattern does not change; portfolios one to four all have similar returns and portfolio five has a relatively large negative return. Table 6 Panel C reports the alphas of the ATM/weighted implied volatility spread hedge portfolios when controlling for illiquidity through the double sorting procedure of Ang. et al. (2006). While the statistical evidence is lower for the realization of abnormal returns, the alphas do not disappear. Overall it can be concluded that the implied volatility spread portfolios’ abnormal returns are robust when controlling for illiquidity.
22 | P a g e
Table 6
Results of implied volatility spread strategy when controlling for illiquidity
Panel A: Illiquidity ratios (÷10,000) of the stocks within the portfolios constructed by using the ATM implied volatility spreads
1 2 3 4 5 P er ce n ti le 10.00% 0.07 0.06 0.06 0.06 0.07 20.00% 0.1 0.09 0.09 0.1 0.11 30.00% 0.14 0.12 0.12 0.13 0.13 40.00% 0.22 0.15 0.16 0.18 0.18 60.00% 0.37 0.27 0.28 0.34 0.38 70.00% 0.55 0.37 0.41 0.46 0.52 80.00% 0.86 0.53 0.69 0.72 0.83 90.00% 1.86 1.2 2.44 1.61 1.78
Panel B: Portfolio returns when controlled for illiquidity
ATM implied volatility spread portfolios Weighted implied volatility spread portfolios
1 2 3 4 5 Hedge 1 2 3 4 5 Hedge
AR -0.27% -0.14% 0.01% -0.37% -0.65% 0.43% -0.23% -0.17% -0.20% -0.18% -0.66% 0.46%
Panel C: Hedge portfolio alphas
CAPM alpha (t-value) FF alpha (t-value) 4-factor alpha (t-value)
ATM hedge 0.38% (1.26) 0.39% (1.23) 0.43% (1.67) *
Weighted hedge 0.43% (1.53) 0.44% (1.57) 0.46% (1.89) *
The portfolios shown in Panel A are created as follows: For the January 2008 until December 2012 sample period all stocks are sorted at the beginning of the month into quintile portfolios in descending order based on the ATM implied volatility spread. The ATM implied volatility spread is calculated by subtracting the ATM volatility of the put from that of the call on the last day of the previous month.
The portfolios shown in Panel B and C are created as follows: For the January 2008 until December 2012 sample period all stocks are sorted at the beginning of the month into quintile portfolios in descending order based on their illiquidity. Their illiquidity is measured with the Amihud (2002) illiquidity ratio using the square root adjustment purposed by Hansbrouck (2005) and stated in Eq. (7). The stocks within the illiquidity quintile portfolios are again sorted into quintile portfolios based on the ATM/weighted implied volatility spread in descending order. Finally, the five portfolios with the highest ATM/weighted implied volatility spreads, and which have different degrees of illiquidity, are merged to create portfolio one. This is also done for the other implied volatility spread portfolios in the different illiquidity sub-groups. This process results in five portfolios that are controlled for illiquidity. The weighted implied volatility spread is calculated by subtracting the weighted implied volatility of the put from that of the call on the last day of the previous month.
The arithmetic excess mean returns (AR) are monthly, equally weighted, and in excess of the German one month T-bill rate (i.e. the risk-free rate). The FF4 alpha is estimated by a four factor model (repeated below from Eq. (6)) that includes the three factors of Fama and French (1993) and the momentum factor of Carhart (1997). The FF3 alpha is estimated by omitting the momentum ( ) factor, and the CAPM alpha is estimated by also omitting the small-cap minus big-cap ( ) and high-value minus low-value factors ( ).
The Newey-West adjusted t-statistics are reported between brackets and statistical significance at the 10%, 5% and 1% level is indicated by *, ** and *** respectively.
23 | P a g e
diversified distributed. Therefore, to see whether the performance of the strategy with the implied volatility spread depends on the sample of exchanges used, it makes sense to split the sample into two samples, one with only stocks traded on these four exchanges (sample 1) and another with only stocks traded on Euronext.liffe Paris and Eurex option exchanges (sample 2). This results in two samples with similar sizes (120 and 111). Table 7 Panel B shows the performance of the portfolios constructed by using the ATM/Weighted implied volatility spreads for the two sub-samples. The hedge portfolio returns of sample 1 are a bit higher and the pattern across portfolios is similar. In Table 7 Panel C the alphas of the hedge portfolio are reported and seem to make the difference smaller. Hence, I can conclude that the strategy works using both sub-samples.
Table 7
Results of implied volatility spread strategy using exchange sub-samples
Panel A: Exchanges where the options of the stocks within the portfolios constructed by using the ATM implied volatility spreads are traded 1 2 3 4 5 Sample Eurex 7.93 10.33 20.7 16.87 9.73 66 Euronext.liffe Paris 10.95 19.45 12.98 6.18 3.77 54 Euronext.liffe Amsterdam 7.33 6.1 3.1 8.2 8.72 34 Borsa Italiana 6.6 6.78 5.7 4.33 7.35 31
MEFF Renta Variable 5.3 2.68 2.93 7.9 9 28
Vienna Stock Exchange 7.65 0.65 0.58 2.28 6.52 18
Panel B: Portfolio returns of the of the exchange groups
ATM implied volatility spread portfolios Weighted implied volatility spread portfolios
1 2 3 4 5 Hedge 1 2 3 4 5 Hedge
AR S1 -0.20% -0.07% -0.16% -0.16% -0.74% 0.53% -0.15% -0.16% -0.21% -0.10% -0.72% 0.57% AR S2 -0.18 % -0.20% -0.01% -0.43% -0.67% 0.49% -0.23% -0.19% -0.34% -0.05% -0.73% 0.51%
Panel C: Hedge portfolio alphas of the exchange groups
CAPM alpha (t-value) FF alpha (t-value) 4-factor alpha (t-value)
ATM hedge S1 0.50% (1.36) 0.54% (1.38) 0.53% (1.72) *
ATM hedge S2 0.46% (1.18) 0.53% (1.43) 0.50% (1.69)*
Weighted hedge S1 0.56% (1.53) 0.54% (1.90)* 0.55% (1.95)*
Weighted hedge S2 0.50% (1.41) 0.52% (1.40) 0.54% (1.72)*
For the January 2008 until December 2012 sample period all stocks are sorted at the beginning of the month into quintile portfolios in descending order based on the ATM/weighted implied volatility spread. The ATM implied volatility spread is calculated by subtracting the ATM/weighted implied volatility of the put from that of the call on the last day of the previous month. For the portfolios in Panel A, only the ATM implied volatility spread is used and the whole sample. This panel reports where the options of the stocks incorporated in the portfolios are traded. For example, the 7.15 in the left corner means that during the January 2008 until December 2012 sample period there are on average 7.15 stocks with options traded on the Eurex in portfolio one. For the portfolios in Panel B and C, the sample is split into two. Sample one (S1) only includes stocks with options traded on the MEFF Renta Variable, Vienna Stock Exchange, Euronext.liffe Amsterdam, and Borsa Italiana exchanges. Sample two (S2) only includes stocks with options traded on the Euronext.liffe Paris and Eurex exchanges.
The arithmetic excess mean returns (AR) are monthly, equally weighted, and in excess of the German one month T-bill rate (i.e. the risk-free rate). The FF4 alpha is estimated by a four factor model (repeated below from Eq. (6)) that includes the three factors of Fama and French (1993) and the momentum factor of Carhart (1997). The FF3 alpha is estimated by omitting the momentum ( ) factor, and the CAPM alpha is estimated by also omitting the small-cap minus big-cap ( ) and high-value minus low-value factors ( ).
24 | P a g e
5.2 Informed trading
To test whether the previously found abnormal returns are the result because the implied volatility spread captures informed trading, I follow Cremers and Weinbaum (2010) and Zhang, Zhao, and Xing (2010) in using the Easley et al. (1998) model. This model provides two usable conditions under which the options market is more attractive for informed investors. These conditions are (1) that the stock market is relatively illiquid compared to the options market and (2) there are many informed investors in the stock market. Similar to Cremers and Weinbaum (2010), I creating two groups; one contains stocks for which the conditions of the Easley et al. (1998) model are satisfied (to the greatest extent) and the other group contains stocks for which they are not satisfied. If the implied volatility spread captures informed trading, one would expect the alphas to be larger for the first group, for which the conditions of the Easley et al. (1998) model are satisfied.
To measure whether the stock market is relatively illiquid compared to the options market, the stocks’ adjusted illiquidity ratio, as defined in Eq. (7) is used. To make it a relative ratio, the trading value of all options (i.e. put and call options for all maturities and strike prices) traded on the stock is used. This relative ratio is defined in Eq. (8).
∑ √| | (8)
where is the relative illiquidity of stock over month , and is the trading value of all options traded (regardless of their maturity, strike price, or whether it is a call or put) on stock on day . Each month the relative illiquidity ratio of the previous month is used to rank the stocks and divide them into two groups of 80 stocks (approximately 33,3% of the sample). One group contains stocks for which the relative illiquidity ratio is the highest and the other group contains stocks for which this ratio is the lowest. Within these groups the stocks are sorted again in quintile portfolios based on their ATM/weighted implied volatility spread.
The other condition is that the concentration of informed investors in the stock market is high. To test for this condition, Cremers and Weinbaum (2010) and Xing, Zhang, and Zhao (2010) both use the probability of informed trading (PIN). This variable, from Easley, Hvidkjaer, and O’Hara (2002), measures the risk of information-based trading and it requires tick-by-tick trade data for its estimation. However, as Aslan, Easley, Hvidkjaer, and O’Hara (2011) have recently noted, when this estimation is done with recent data8 it compromises the maximum likelihood estimation techniques used to obtain the
8
25 | P a g e PINs. To get around this issue, they examine how accounting and market characteristics are related to PIN and they show that a stock’s standard deviation is positively related to its PIN. The explanation, provided by Aslan et al. (2011), for the positive relation is that for stocks with higher standard deviations information is more valuable. Hence, similar to how I formed groups based on the relative illiquidity ratio, I also rank the stocks each month on their standard deviation and divide them into two groups of 80 stocks. The standard deviation is calculated by following Lim (2001), which means that weekly return data from Wednesday to Wednesday closing prices is used to mitigate the bid-ask bounce effect and 52 weeks of past stock returns are used.
Table 8 shows the results for when the relative illiquidity ratio is used to form the groups. In Panel A of the table the relative illiquidity ratio percentiles of the ATM implied volatility spread portfolios are stated. From this panel it can be observed that all portfolios have stocks with large and small relative illiquidity ratios and thus the size of the implied volatility spread seems not to be related to the relative illiquidity ratio. In Table 8 Panel B the returns of the ATM/Weighted implied volatility spread portfolios are stated. The pattern of returns across the portfolios in the different groups is similar to that observed before and the return of the hedge portfolio for the relatively high illiquidity group is higher than that of the relatively low illiquidity group. This is what one would expect but the difference is almost negligible and does not change when estimating the alphas as stated in Panel C of Table 8. Therefore, there is no indication that the implied volatility spread strategy captures informed trading.
26 | P a g e
Table: 8
Testing for informative trading using the relative illiquidity ratio
Panel A: Relative illiquidity ratios (÷10) of the stocks within the portfolios constructed by using the ATM implied volatility spreads
1 2 3 4 5 P er ce n ti le 10.00% 0.11 0.11 0.08 0.09 0.11 20.00% 0.21 0.20 0.15 0.20 0.20 30.00% 0.36 0.33 0.26 0.35 0.35 40.00% 0.45 0.43 0.33 0.45 0.43 60.00% 0.97 0.94 0.78 1.02 0.95 70.00% 1.42 1.35 1.20 1.54 1.41 80.00% 1.91 1.72 1.59 2.01 1.95 90.00% 3.71 2.73 2.72 3.43 3.45
Panel B: Portfolio returns of the relative illiquidity subsamples (S1 high, S2 low)
ATM implied volatility spread portfolios Weighted implied volatility spread portfolios
1 2 3 4 5 Hedge 1 2 3 4 5 Hedge
AR S1 -0.09% -0.32% 0.09% -0.02% -0.70% 0.60% -0.08% -0.16% -0.11% 0.00% -0.64% 0.55% AR S2 -0.38% -0.03% 0.02% -0.22% -0.90% 0.53% -0.27% -0.33% -0.41% -0.23% -0.80% 0.54%
Panel C: Hedge portfolio alphas of the relative illiquidity subsamples (S1 high, S2 low)
CAPM alpha (t-value) FF alpha (t-value) 4-factor alpha (t-value)
ATM hedge S1 0.61% (1.22) 0.72% (1.47) 0.66% (2.61)** ATM hedge S2 0.54% (1.46) 0.52% (1.47) 0.50% (1.47) Difference 0.07% (0.14) 0.20% (0.41) 0.16% (0.41) Weighted hedge S1 0.53% (1.64) 0.55% (1.59) 0.57% (1.70)* Weighted hedge S2 0.56% (1.41) 0.56% (1.36) 0.52% (1.43) Difference -0.02% (-0.04) -0.01% (-0.02) 0.04% (0.04)
The portfolios stated in panel A are created as follows: For the January 2008 until December 2012 sample period all stocks are sorted at the beginning of the month into quintile portfolios in descending order based on the ATM implied volatility spread. The ATM implied volatility spread is calculated by subtracting the ATM implied volatility of the put from that of the call on the last day of the previous month.
The portfolios stated in panel B and C are created as follows: For the January 2008 until December 2012 sample period all stocks are sorted at the beginning of the month into quintile portfolios in descending order based on their relative illiquidity ratio. Their relative illiquidity is measured with Eq. (9). The 80 stocks with the highest relative illiquidity ratios form group S1 and the 80 stocks with the lowest relative illiquidity ratios form group S2. The stocks within the relative illiquidity quintile portfolios once are again sorted into quintile portfolios based on the ATM/weighted implied volatility spread in descending order. The weighted implied volatility spread is calculated by subtracting the weighted implied volatility of the put from that of the call on the last day of the previous month.
The arithmetic excess mean returns (AR) are monthly, equally weighted and in excess of the German one month T-bill rate (i.e. the risk-free rate). The FF4 alpha is estimated by a four factor model (stated below) that includes the three factors of Fama and French (1993) and the momentum factor of Carhard (1997). The FF3 alpha is estimated by omitting the momentum ( ) factor and the CAPM alpha by omitting also the small-cap minus big cap ( ) and high-value minus low-value factors ( ).
27 | P a g e
Table: 9
Testing for informative trading using the standard deviation
Panel A: Annualized standard deviation of the stocks within the portfolios constructed by using the ATM implied volatility spreads
1 2 3 4 5 P er ce n ti le 10.00% 22.02% 19.57% 20.71% 20.89% 23.24% 20.00% 26.36% 23.23% 24.66% 25.17% 27.56% 30.00% 30.41% 26.29% 27.84% 28.86% 31.39% 40.00% 33.83% 29.53% 30.72% 32.28% 35.25% 60.00% 42.69% 37.18% 38.70% 40.42% 44.59% 70.00% 48.71% 41.94% 43.61% 45.19% 48.84% 80.00% 57.23% 47.56% 48.63% 51.65% 56.27% 90.00% 67.31% 56.40% 57.91% 60.99% 66.73%
Panel B: Portfolio returns of the standard deviation groups (S1 large, S2 small)
ATM implied volatility spread portfolios Weighted implied volatility spread portfolios
1 2 3 4 5 Hedge 1 2 3 4 5 Hedge
AR S1 -0.19% -0.15% -0.26% -0.36% -0.80% 0.60% -0.28% -0.34% -0.30% 0.05% -0.86% 0.57% AR S2 -0.10% -0.29% -0.13% -0.15% -0.58% 0.49% -0.24% -0.25% -0.18% 0.08% -0.69% 0.45%
Panel C: Hedge portfolio alphas of the standard deviation groups (S1 large, S2 small)
CAPM alpha (t-value) FF alpha (t-value) 4-factor alpha (t-value)
ATM hedge S1 0.62% (1.36) 0.69% (1.55) 0.65% (1.87)* ATM hedge S2 0.47% (1.64) 0.48% (1.59) 0.48% (1.58) Difference 0.15% (0.28) 0.21% (0.38) 0.17% (0.33) Weighted hedge S1 0.57% (1.25) 0.64% (1.44) 0.60% (1.72)* Weighted hedge S2 0.47% (1.91)* 0.47% (1.95)* 0.46% (2.14)** Difference 0.10% (0.22) 0.17% (0.36) 0.13% (0.31)
The portfolios stated in panel A are created as follows: For the January 2008 until December 2012 sample period all stocks are sorted at the beginning of the month into quintile portfolios in descending order based on the ATM implied volatility spread. The ATM implied volatility spread is calculated by subtracting the ATM implied volatility of the put from that of the call on the last day of the previous month.
The portfolios stated in panel B and C are created as follows: For the January 2008 until December 2012 sample period all stocks are sorted at the beginning of the month into quintile portfolios in descending order based on their relative illiquidity ratio. Their standard deviation is calculated by following Lim (2001) which implies that weekly return data from Wednesday to Wednesday closing prices are used and 52 weeks of past stock returns. It is annualized by multiplying the weekly standard deviation with the square root of 52. The 80 stocks with the highest standard deviation form group S1 and the 80 stocks with the standard deviation form group S2. The stocks within the groups are then sorted into quintile portfolios based on the ATM/weighted implied volatility spread in descending order. The weighted implied volatility spread is calculated by subtracting the weighted implied volatility of the put from that of the call on the last day of the previous month.
The arithmetic excess mean returns (AR) are monthly, equally weighted and in excess of the German one month T-bill rate (i.e. the risk-free rate). The FF4 alpha is estimated by a four factor model (repeated below from Eq. (6)) that includes the three factors of Fama and French (1993) and the momentum factor of Carhard (1997). The FF3 alpha is estimated by omitting the momentum ( ) factor and the CAPM alpha by omitting also the small-cap minus big cap ( ) and high-value minus low-value factors ( ).
28 | P a g e
5.3 Transaction costs
There is extensive literature available concerning investment strategies that have significant abnormal returns but cannot be realized in an economically feasible way due to transaction costs (see, e.g., Lesmond, Schill, and Zhou, 2004; Avramov, Chordia, and Goyal 2006). Hence, to determine whether the abnormal return previously found can be realized by investors, transaction costs need to be considered. The total transaction costs not only depend on the instantaneous costs, such as the bid-ask spread, but also on, for example, the price impact of the order in the long and short run. Even when making a short sidestep to the U.S.-based transaction costs, there is a variety of percentages is used. Some researchers assume a fixed percentage [Jegadeesh and Titman (1993) 0.5%] and some use a range of percentages to account for transaction costs [Chan and Lakonishok (1997) 3.31% for small to 0.9% for large capitalization stocks]. Using a fixed percentage, however, seems highly inappropriate because, as can be observed from Table 6, there are both differences between the illiquidity of the stocks within the portfolios and differences between the portfolios. Hence, to realistically include transaction costs in the analysis, it would be desirable to let them vary.
De Groot, Huij, and Zhou (2012) provide in their paper accurate estimates of transaction costs for sizable strategies from 1995 until 2009 with the 1000 largest European stocks. They use the estimates from Nomura Securities (one of the largest brokers) which incorporates the trades’ bid-ask spread costs, the impact on the equilibrium price of the trade, the short-term price implications and a 3 basis point (bp) commission fee. Using their estimates the transaction costs stated in Table 10 are calculated for several volume groups, into which the stocks of the sample can be allocated. On average, 31 stocks (13.5%) of the sample do not belong to any of the volume groups. For those stocks that do not fit into one of the groups, the transaction costs of the lowest volume decline are used.
Table 10
Overview of transaction costs
Volume ($ mln) <1.1 <2.3 <3.8 <6.2 <8.9 <12.3 <25.7 <32 <117.4 <137.1 137.1>
Basis points 88.0 77.0 72.8 62.5 51.0 37.8 34.0 24.0 22.5 21.0 20.0
Transaction costs are based on the European estimates of Nomura Securities, which are stated in the paper of De Groot et al. (2012). Their table states the distribution of transaction costs for volume deciles of the 1000 largest European stocks. In addition to this, De Groot provided me with a table with the volumes in dollars ($) of these largest companies which enabled me to make the estimated in this table. Since it is not clear which exchange rate should be used to translate the volume deciles, I stated the amounts in dollars and use trading volume denoted in dollars to allocate the stocks within the groups. To match the volumes stated in the table provided by De Groot with the transaction costs stated in the paper of De Groot et al. (2012), interpolation techniques must be used since the reported volume percentiles do not exactly match with the transaction cost deciles.
The distribution for the period 2008 until 2013 requires the assumption that transaction costs are proportional to volume across time and across stocks, and that transaction costs for shorting a stock are equal to those incurred when taking a long position in the stock.
