Efficiency in the German index option market: cross market and internal market efficiency tests

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Efficiency in the German index option market: cross market

and internal market efficiency tests

Dirk W. Snoek

a, *

Supervisor: Dr. Sibrand Drijver

b

a_{MSc
Finance,
Faculty
of
Economics
and
Business,
University
of
Groningen.
Student
number:
1544284} b _{University
of
Groningen
and
TKP
Investments,
the
Netherlands}

THESIS INFO Date: Juli 2 2014 Word count: 12,783 JEL Classification: G13 G14 G15 Keywords: Market efficiency Index option market Pricing relations Exchange Traded Funds ABSTRACT

This study covers several aspects of the efficiency of the German index option market. This is done by testing several no-‐arbitrage conditions that are based on relative pricing relations and that should not be violated in a perfectly efficient market. The frequencies and sizes of violations of these various conditions are reported for three consecutive annual subsamples to determine if the overall level of market efficiency improves over time. Moreover, this level of market efficiency is compared to levels reported in similar studies on the German and other index option markets. Aside from that, the conditions tested are categorized into two types of efficiency: the cross market efficiency which is the joint efficiency of the index option market and the index itself, and the internal index option market efficiency which does not relate to the underlying instrument. The distinction between these two types of efficiency is made to test whether the increased use of Exchange Traded Funds (ETFs) contributes to the level of DAX index option market efficiency. This will be the case if the level of cross market efficiency improves more than the level of internal market efficiency. The effects of transaction costs and short selling constraints are shown by considering different types of traders. The results show that the German market’s efficiency is similar to that of other European markets. There is only little evidence that the efficiency of the German index option market improves over time and there is no evidence found that suggests that this improvement is caused by the increased volume of ETFs. I would like to thank Optiver and BinckBank for providing me with information on current transaction costs for different types of traders. I would also like to thank my supervisor dr. Sibrand Drijver for all his efforts, ideas and very useful feedback.

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1. Introduction

The Chicago Board Options Exchange (CBOE) was the first exchange to start trading index options in 1983. Since then, index options have become a widely used tool for investors around the world. An index option is similar to any other option contract, except that the underlying instrument is the index itself, instead of for instance a single stock or currency. This feature of the index option provides the investor with a high level of diversification, since the value of the index is based on the performance of a large number of securities. Index options are therefore often used for managing the systematic risk of investment portfolios. Another reason for their success is that index options are typically cash settled, making this tool simple and inexpensive to implement in several investment strategies.

Index options have also become a frequently used financial instrument in Europe. In August 1991, the German DAX index started trading in index options on the German derivatives exchange, which was at the time still called the Deutsche Terminbörse (DTB) and is since 1998 known as the Eurex Exchange.1

This study aims to test the efficiency of the German index option market, where efficiency is defined as the degree to which the prices of this index and its derivatives reflect all available market information.

Testing option-market efficiency can be done in several ways. One way to do this is to compare observed transaction prices of the index options with the prices implied by a theoretical model, where the Black and Scholes (1973) is the best known and most used model for calculating option prices. As Mittnik and Rieken (2000) point out, this method has a major drawback since it bases the pricing relations between put and call options on the assumption that the theoretical pricing model is valid. To avoid having to use theoretical models, the present study uses market efficiency tests that are designed to be solely based on the no-arbitrage principle. This principle is based on the simple assumption that investors prefer more to less and that if a riskless profit opportunity exists, an arbitrageur will enter the market and take advantage of this mispricing. In an efficient market, the arbitrage opportunities are immediately eliminated and all asset prices reflect their fundamental values.

A major issue in this theory is that arbitrageurs are sometimes limited by market frictions to fully exploit the riskless profit opportunities. The strategies to benefit from these arbitrage opportunities require trades in multiple assets and thus require transaction costs. If the underlying instrument of the option is a single stock or currency, these strategies are simple and relatively cheap. In the case of index options, however, the underlying instrument is a stock portfolio of typically many assets, namely the index itself. This makes the execution of the arbitrage strategies a lot more complex and costly,

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considering that sometimes an investor may need to replicate this index to eliminate an arbitrage opportunity. For the better part of the 20th century, this meant that an investor would need to take a position in all stocks traded on an index, which sometimes consists of thousands of different assets. To tackle this obstacle for investors, Exchange Traded Funds (ETFs) were introduced in the North American stock markets in the early 1990s. These funds are designed to track the performance of an index and trade just like a regular stock. For instance, the S&P 500 Depositary Receipt (SPDR, or “Spider”) mimics the performance of the S&P500 index. ETFs are initially specified as a constant multiple of the underlying index, usually 1/10th or 1/100th, making these instruments much easier to trade than the traditional ways of replicating an index.

In the first years after their introduction, ETFs represented only a small fraction of the total trades in index funds. However, their share has grown spectacularly since and the biggest ETF, the SPDR, now has a trading volume of nearly $400 billion a month.2 Since 2000, ETFs are also listed on European stock exchanges, where the Deutsche Börse’s XTF Exchange Traded Funds segment is Europe’s leading market.3

It can be argued that since ETFs make replicating an index less complex and costly, investors possess a very useful tool to execute arbitrage strategies, making the markets more efficient. Some earlier studies have tested the impact of ETFs on market efficiency. A good example is a study by Ackert and Tian (2001), who use five tests to measure the influence of the introduction of SPDRs on the S&P500 index options. All of these tests are independent of an option pricing model and are therefore true tests of market efficiency instead of joint tests of market efficiency and model validity and specification, as was also pointed out by Mittnik and Rieken (2000). The tests examine both violations in pricing relations across option and stock markets (boundary conditions test and put-call parity test) as violations in pricing relations on the option market that are independent of the stock market (box spread test, call and put spread test and option convexity test).

2

ETFs with most volume, ETF Channel

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While Ackert and Tian (2001) compare the results of two subsamples, one before and one after the introduction of SPDRs, the present study uses three consecutive, annual subsamples and therefore focuses more on the development of the level of efficiency over time. This ‘trend’ is examined using the same model independent efficiency tests, which all depend on a set of conditions regarding the (relative) pricing of options. Formally, the research question for this study is formulated as follows:

Does the DAX index option market become more efficient over time?

The hypothesis tested to answer this question is the following:

H0: The level of pricing efficiency of DAX index options improves over time.

By using these tests, not only can be determined whether the index option market has become more efficient in relation to the stock market (cross market efficiency) and independent of the stock market (internal market efficiency), but also whether the level of efficiency of the former relation has increased more than the latter. If this is the case, one could argue that the increase in efficiency can be (partly) attributed to the increased use of ETFs in the financial markets. To see if this can be an explanation for increased efficiency over time, the results of the cross market efficiency tests are compared to the internal market efficiency tests, forming the following hypothesis:

H0: The cross market efficiency of the DAX index options market improves more than the internal

market efficiency of this market over time.

Ackert and Tian (2001) report only little evidence that supports this hypothesis. Acknowledging the fact that their study was based on data from a market where ETFs only had a small share and that this share has increased substantially since, this study also aims to examine the more recent influence of ETFs on index option market efficiency.

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the impact of ETFs can be even better examined than in the previous studies, since the market share of ETFs has increased rapidly in recent years.

The remainder of this study is organized as follows. In section 2 the literature concerning efficiency of index option markets and the relation between ETFs and market efficiency is reviewed. Section 3 discusses the German DAX index market. Section 4 provides the theoretical background for the tests used in this paper. In section 5 the data and methodology of this study are discussed. Section 6 discusses the results of the used tests and compares these to similar studies. The final section concludes this study.

2. Literature review

2.1. Index option market efficiency

The efficiency of the index option market is a widely studied topic. As stated in the introduction, a distinction between two types of studies can be made: those that use a theoretical pricing model, like the Black and Scholes Option Pricing Model (1973) (e.g. Evnine and Rudd, 19854; Cavallo and Mammola, 2000) and those that use model independent tests (e.g. Kamara and Miller, 1995; Ackert and Tian, 2000, 2001; Mittnik and Rieken, 2000a, b). Given the fact that the former type involves a joint test of the market efficiency and the validity of the option pricing model and its specifications, most studies base their definition of an efficient market on the no-arbitrage principle.

Another important distinction that can be made between option market efficiency studies is the type of efficiency that is tested. The first type is the cross market efficiency, which tests the joint efficiency of the index option market and the underlying instrument. The second type of efficiency tests focuses on the internal efficiency of the index option market, leaving the underlying instrument out of the test. The former type is typically performed using the boundary conditions test and the put-call parity test, as was made famous by Stoll in 1969. The latter type uses tests that concern only the relative pricing of the options in the very same option market. Examples of these tests are the box spread, call and put spreads and option convexity, which is tested by constructing butterfly spreads.

While most studies focus on the US market (Evnine and Rudd, 1985; Kamara and Miller, 1995; Ackert and Tian, 2000, 2001), some studies test the efficiency of European index option markets: Mittnik

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and Rieken (2000a, b) on the German DAX index, Capelle-Blancard and Chaudhury (2001) on the French CAC40 index, and Cavallo and Mammola (2000) and Brunetti and Torricelli (2007) on the Italian MIB30 index. Most of these studies focus only on the cross market efficiency (Evnine and Rudd, 1985; Kamara and Miller, 1995; Mittnik and Rieken, 2000a, b; Cavallo and Mammola, 2000), while others only take internal index option market efficiency into account (Ackert and Tian, 2000; Brunetti and Torricelli, 2007). The other two mentioned model independent studies (Ackert and Tian, 2001; Capelle-Blancard and Chaudhury, 2001) combine both cross market and internal market efficiency. The findings of all mentioned studies are summarized in table 1 and show a tendency of improved index option market efficiency over the years.

2.2. Market efficiency and ETFs

To provide investors with more simple and cost-efficient ways to mimic an index, major US brokerage firms started developing program trading facilities in the late 1970s, which allowed investors to trade a whole stock basket in a single transaction. More and more investors became interested in these programs and the exchanges began developing mutual funds that replicate their indices. This development in the financial markets led to the introduction of Exchange Traded Funds (ETFs) in North America. The first equity-like index fund was introduced on the Toronto Stock Exchange on March 9, 1990, tracking the Toronto 35. The units of this fund were called the Toronto Index Participations (TIPs) and they were traded on the stock exchange. Characterized by extremely low management fees, they became a huge success. Even though they became too expensive for the exchange and were terminated in 2000, they inspired other exchanges to develop a similar fund. In 1993, the American Stock Exchange (AMEX) began trading the SPDR, which is now known as the world’s first ETF.5 As stated in the introduction, the market share of ETFs has been increasing rapidly since their introduction and ETFs are now a substantial part of the modern financial world. This is not only true for the US market, but also for the European markets, where the German XTF trading platform is the biggest.

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Table 1

Findings of previous studies on market efficiency.

* Conditions tested if listed: 1: Boundary; 2: Put-call parity; 3: Box spread; 4: Call- and put spreads; 5: Convexity.

Study Market Data sample Data Tests used* Conclusions

period type

Evnine and S&P100 / June – Aug. 1984 Intra-daily 1, 2 Significant violations, inefficient

Rudd, 1985 MMI market

Kamara and S&P500 Jan. – March 1989 Daily 1, 2 Substantially less frequent

Miller, 19956 May 1989 Intra-daily violations compared to earlier studies

Ackert and S&P500 Jan. 1986 – Daily 3, 4, 5 No evidence for improved efficiency

Tian, 2000 Dec. 1996 over time

Ackert and S&P500 Feb. 1992- Daily 1 – 5 Option market efficiency improved over

Tian, 2001 Jan. 1994 time7

Mittnik and DAX Feb. 1992 – Daily 1 Decreasing violations over the years,

Rieken, 2000a Sept. 1995 market becomes more efficient

Mittnik and DAX Feb. 1992 – Daily 2 PCP is rejected, but market efficiency is

Rieken, 2000b Sept. 1995 not8

Cavallo and MIB30 Dec. 1996 – Daily 2 Results support market efficiency

Mammola, 2000 Sept. 1997

Capelle-Blancard CAC40 Jan. 1997 – Intra-daily 1-5 Results support market efficiency and Chaudhury, 2001 Dec. 1999

Brunetti and MIB30 Sept. – Dec 2002 Intra-daily 3, 4, 5 Arbitrage only possible in frictionless

Torricelli, 2007 markets

Since ETFs provide investors with a cheap and simple tool to replicate the performances of an index, the efficiency of the market should be increased if the volume of ETFs traded increases. A way to measure the influence of ETFs on market efficiency is to compare the degree of improvement over time between the cross market efficiency and the internal index option market efficiency. If the former improves to a higher degree than the latter, the influence of ETFs on market efficiency is positive. To my

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Kamara and Miller point out that, before their paper in 1995, all other put-call parity tests were performed on American-style options, including Evnine and Rudd (1985). Because these types of options have a possibility of earlier exercise, the put-call parity may not hold for the American-style options. They base their results on European-style options only and find significantly less violations than earlier put-call parity studies.

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Ackert and Tian (2001) come to the conclusion that the improved option market efficiency cannot be directly linked to the introduction of SPDRs in the US index option market.

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best knowledge, the only study that has examined the link between ETFs and index market efficiency was done by Ackert and Tian in 2001. They based their study on the S&P500 and compared the efficiency of two data samples: one before and one after the introduction of the ETFs on this index, the SPDRs. Their results support the hypothesis that option market efficiency improved over time. They state that even though the replication of the index with actual stock trading may not have been costless prior to the introduction of the SPDRs, there is only little evidence that suggests that the cross market efficiency has been improved since the introduction of ETFs.

3. The German Market

The DAX index, created in 1988, is the most important index in Germany and tracks the performance of the 30 largest companies on the German equities market. It is capitalization-weighted and its base value at the time of introduction was 1,000 points. The index has reasonably large contracts, where 1 index point equals € 25.9 The companies included in the index are trading on the Frankfurt Stock Exchange and represent about 80% of the free-float market cap authorized in Germany.10 The DAX is calculated as a performance index and is therefore adjusted for dividends, which are assumed to be reinvested into the shares, stock splits and changes in capital stocks. The value of the index, therefore, is not affected by any of these events. The prices are taken from Xetra, the electronic trading system operated by Deutsche Börse. Options on this index started trading in August 199111 and like most other index options, these are European-style options. The exercise prices of these options have fixed increments of 50 (100) index points for the options that expire quarterly (monthly). The expiration day is the third Friday of the expiration month and all options are cash settled. Like in most European countries, there are several short selling constraints and restrictions in Germany, which also apply to stocks (ETFs in this study) and are therefore also taken into account for the efficiency tests.12

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What is the DAX Exchange?, Trade the Markets

10_{Factsheet DAX} 11

25 years of the DAX – the facts, Deutsche Börse

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ETFs were introduced in Germany in the year 2000 and about 98% of these funds are traded on Xetra.13 Their volume has increased significantly since their introduction, resulting in an average monthly volume of up to 3.3 million for their most successful fund (iShares MSCI Germany Index Fund).14 There are seven different ETFs on the DAX index and they have a total of more than € 23 billion in assets under management.15 These ETFs typically retain their profits, meaning that similarly to the DAX index, they reinvest all dividends of the stocks in their portfolios. This way, the funds can mimic the performance of the DAX more closely, as is argued by Schlusche (2009). Since the DAX is a relatively expensive index, most of the ETFs have a base denomination of 1/100th of the index.16

4. Tests on market efficiency

The present study tests several pricing relationships of German index options for arbitrage opportunities. These arbitrage opportunities can be detected if the pricing relationships are violated by the actual prices of the transactions. As stated in the introduction, this study tests both the cross market efficiency and the internal option market efficiency, following Ackert and Tian (2001). The cross market efficiency is tested by two pricing relationships (lower boundary conditions for put and call index options and the put-call parity conditions) that include a position in the underlying asset. To test this efficiency, only the most liquid options are taken into account, so that the violations found could have led to actual realized profits. These are the options with a strike price that is closest to the index level at the time of the transaction and are called at-the-money (ATM) options. The internal index option market efficiency is tested by pricing relationships that do not require a position in the underlying asset (box spreads, call and put spreads and butterfly spreads). To make a fair comparison to the violations found in the cross market efficiency, the options chosen for the tests that include a spread are the ones that are both liquid and at the same time generate a relatively high percentage of violations. As Vipul (2009) states, these violations tend to occur relatively more often as the options diverge from the 100% moneyness level. To match both requirements, the options chosen for the box spreads and the call and put spreads are the ones that are slightly in- or out-of-the-money (ITM or OTM). The option price convexity test uses all three levels of moneyness of the options by constructing butterfly spreads.

13

ETF exchange-traded funds, Xetra

14_{Germany ETF List, ETF Database} 15

25 years of the DAX – the facts, Deutsche Börse

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The parameters for these tests are defined as:

bid price of European ITM , ATM or OTM call option; ask price of European ITM, ATM or OTM call option;

bid price of European OTM , ATM or ITM put option; ask price of European OTM, ATM or ITM put option;

bid price of the index; ask price of the index;

strike price, where ; time to maturity of the option; risk-free interest rate;

percent of proceeds available to short sellers, where ;

transaction costs of buying or selling calls, puts, ETF shares or German government bonds (Bunds), where or .

The first test performed in this study is the lower boundary condition, which was developed by Merton (1973). It is the most stringent test used in this study and therefore only detects extreme outliers in the option pricings. Because of this feature, the lower boundary test has a double function in this study: not only does it contribute to testing the cross market efficiency of DAX index options, it also clears the dataset of outliers for the rest of the tests used, since the detected violations of this test will not be taken into account for the other pricing relations. For this condition to be valid, it is assumed that the options used are European-style and that there are no dividends paid on the stocks which are represented by the index. Since the DAX is a performance index and the options traded on it are European-style, both assumptions apply to this study. The lower boundary conditions are given by the following inequalities:

( ) _(1a) ( ₎ _(1b)

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option, shorting the index (using ETFs) and lending an amount at the risk free rate (using Bunds). This will lead to a riskless profit of [( ) ] at time .

Since taking advantage of this type of mispricing requires a short position in the index and since there are short selling constraints in Germany, the profits from using this arbitrage strategy should be adjusted. This is done by including a profit reducing factor ) in the equation. Since the strategy of profiting from underpriced index put options does not require a short position in the index, this factor is not needed in equation (1b).

The second pricing relationship used to examine the cross market efficiency is the put-call parity. This relationship was developed by Stoll (1969) and is based on the fact that there are two ways to purchase options: a call option can be purchased directly, but can also be created by a combination of a long position in the underlying asset and a long position in an otherwise equal put option on the underlying asset. Both holdings will yield the same return and the latter method is therefore also referred to as a synthetic call option. The same goes for the two ways of purchasing a put option, either buying it directly or taking a short position in the underlying asset in combination with a long position in an otherwise equal call option on this underlying asset. Because of these different methods of creating a call or put position, the call and put options should be priced correctly relative to each other. Accounting for bid-ask spreads, interest rates over the time to maturity and transaction costs, the conditions that must hold for an efficient market are as follows:

( ) ₍ ₎ _(2a) ( ( ) ₎ _(2b)

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To test the internal efficiency of the index option market, three pricing relationships that do not require a position in the index are used. The first relationship is tested using a box spread, which is constructed with two European-style calls and two European-style puts and is a combination of a bull call spread and its corresponding bear put spread. These options all have the same underlying asset and the same time to maturity, but the two strike prices involved differ: one pair of put and call has a lower strike price and the other pair has a higher strike price . The bull call spread involves buying the call option with the lower strike price and selling the call with the higher strike (Fig. 1). For the bear put spread, the composition is reversed: the put with the higher strike is purchased and the put with the lower strike is sold (Fig. 2). This yields the same price relationships as expressed in the put-call parity equations, except that in this case the underlying asset is left out.

Fig. 1 Fig. 2

Call Spread Payoff. Put Spread Payoff.

When combining the payoffs of both these spreads, the payoff will always be the difference between the selected strike prices , regardless of the price of the underlying at time . This structure is known as the box spread and is graphically displayed in Fig. 3.

The box spread is expressed by the following conditions:

( ) _(3a) ( ) _(3b)

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Fig. 3 Fig. 4

Box Spread Payoff. Call Convexity (Butterfly Spread Payoff).

These call and put spreads are also tested individually, using the following conditions:

_(4a)

_(4b)

Both conditions are violated if the price of the ITM option is relatively high compared to its OTM equivalent.

The final test examines the convexity of the option prices as a function of the strike price over their different levels of moneyness. Three call (put) options are combined to create a riskless position using a call (put) butterfly spread. The call butterfly spread is constructed by buying a call option with a strike of

, selling short 2 call options with strike , and finally buying a call option with strike , where . The payoff of this structure is shown in Fig. 4. The put butterfly spread is constructed in a similar fashion, taking long positions in the highest and lowest strike price puts and selling short two put options with the middle strike level. The payoff of the put butterfly is exactly the same as its call butterfly equivalent. The conditions to test these spreads are as follows:

(5a)

(5b)

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relative to its ITM and OTM equivalents. Assuming , arbitrageurs can profit from this mispricing by taking a long position in a portfolio that consists of 50% ITM options and 50% OTM options. This position can be financed by selling the number of options held in this position of the ATM equivalent options. Since this strategy requires the purchase of two different options, the minimum volume needed to exercise it is twice as high as that of the other arbitrage strategies considered in this study. Table 2 provides an overview of the conditions used in this study and the types of mispricing that could cause their violations.

Table 2

Overpricing (O) and Underpricing (U) of in-the-money (ITM), at-the-money (ATM) and out-of-the-money (OTM) call and put options and their violations of the conditions.

Call Put

ITM ATM OTM ITM ATM OTM

Boundary (1a) U

(1b) U

Put-call parity (2a) U O

(2b) O U

Box spread (3a) U O U O

(3b) O U O U

Call spread (4a) O U

Put spread (4b) O U

Call convexity (5a) U O U

Put convexity (5b) U O U

5. Methodology and data

The present study not only tests the cross- and internal market efficiency of DAX index options, it also examines the degrees of improvement of these efficiencies over time. To do this, subsamples are taken over three consecutive annual periods, starting on January 1st, 2010. The trend regarding the level of improvement in efficiency is examined and is the primary tool for the analysis. This trend is tested by looking at both the relative frequency of the violations and the average values of these violations.

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January 1st, 2010 – December 31st, 2012 are taken from Yahoo Finance. Since the DAX is a performance index, dividends paid in the period until expiration of the options need not be considered.

The data are filtered so that the maximum maturity of the options is 3 months. For each day, all index options of which the strike price was closest to the index level were selected and labeled ATM options. To match the options for the spread tests, the moneyness of the options is an important selection criterion. As stated in the previous section, these options have to divert from the 100% moneyness to effectively discover pricing violations, but should not be too far in- or out-of-the-money to avoid illiquidity. An increment of 100 index points is therefore chosen, making the ITM (OTM) call (put) option 100 index points lower and the OTM (ITM) call (put) option 100 points higher than its ATM equivalent. As a proxy for the risk-free rate, the Euribor17 rates of 1, 2 and 3 months are used. For options with a time to maturity less than 30 days, the number of remaining days is taken as a weighted average over the 1-month rates. For instance: on the 22nd of October 2010, a call option expiring on the 21st of November 2010 has 28 days till expiration. Its assumed interest rate is therefore (28/30)*(0.7%) = 0.657%, which is based on the 1-month Euribor rate for October, 0.7%. For a call option with that same maturity date, the interest rate on the 16th of November 2010 will be (3/30)*(0.85%) = 0.085%. This results in significantly lower interest rates for shorter times to maturity, even if the 1-month Euribor increases, like in this example.

There are four types of transaction costs considered in this study: the bid-ask spreads on the index and its options, the costs of buying or selling an ETF to mimic a position in the index, index option transaction costs and ), which are assumed to be equal per contract of 100 options and the costs of buying a German government bond . The sizes of these transaction costs are hard to determine. As Capelle-Blancard and Chaudhury (2001) point out, the transaction costs vary over time and may also be dependent on the arbitrage strategy used and the size of the transaction. They are also dependent on the type of trader that executes the strategy: professional traders pay significantly less than retail traders. To cope with this, two types of traders are considered in this study: the professionals, who work at high frequency trading (HFT) firms and non-professional traders, who are active traders but are not considered professionals and therefore have significantly higher transaction fees.

Another factor that can also be viewed as a type of transaction costs is the profit reducing factor that applies to short sellers and which is a direct effect of the short selling constraints that apply on the German stock market. Since the level of transaction costs have a significant influence on the number

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of pricing relationship violations detected, this study considers five different scenarios concerning these costs to determine the magnitude of its influence:

Scenario A assumes frictionless markets, where all mentioned transaction costs are zero and is set at 1.0, implying no extra costs for short sellers. As the transaction costs are zero, it does not matter whether the arbitrageur is a professional or not. Since bid-ask spreads are also a form of transaction costs, these are also not taken into account for this scenario.

Scenario B assumes the arbitrageur is a professional trader, working at a HFT firm and for that reason has extremely low transaction costs. The short selling constraints are still left out , but the bid-ask spread is included and is assumed constant. To construct bid and ask prices for the index and its options, an approach similar to Kamara and Miller (1995) is used. This method deducts 0.5 index points from the index to produce a bid price and adds this same amount to yield an ask price, making the bid-ask spread of the index 1 index point, which is a typical spread for the DAX index.18 For the index options, the typical spread is € 0,1419 and is constructed using the same method.

Following Cavallo and Mammola (2000) and Capelle-Blancard and Chaudhury (2001), this study includes a fixed fee for replicating the index. In these studies, this fee was assumed 5 index points. Considering the fact that these studies were performed on data up to 1999, before ETFs were introduced in Europe, these costs are nowadays a fraction of what they used to be. This is also due to the fact that the options on the DAX are scaled to 1/5th of the index. This means that for every point increase of the DAX, a call option (with a delta of 1.0) will yield a cash value increase of € 5.00, where the cash value of the index point itself is € 25.00.20 For the professionals, these costs are assumed to be € 0.50 for every replicated position. The transaction costs for call and put options are assumed € 0.005 per contract and the costs of buying a government bond are set at € 0.09.21

In Scenario C the same assumptions of Scenario B are used, except that in this scenario the short selling constraints are included. Following Kamara and Miller (1995), Ackert and Tian (2000, 2001) and Capelle-Blancard and Chaudhury (2001), 99% of the proceeds are available to the short seller

18

According to a DAX derivative trader at Optiver

19

According to a DAX derivative trader at Optiver

20_{Contracts Specifications for Futures Contracts and Options Contracts at Eurex Germany and Eurex Zürich,}

Eurex

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. Since the only pricing relations that require a short position on the index are (1a) and (2b), Scenario C is equal to the previous scenario with respect to the other conditions.

Scenario D assumes the arbitrageur is not a professional, but is still an active trader. According to BinckBank, this type of trader typically pays € 1,95 per option contract. Replication of the index is assumed to cost this type of trader € 20,00 and the transaction costs for the government bonds are taken from the website of the Boerse Stuttgart22 and are assumed a constant € 33,55 per transaction.23

Scenario E assumes the same transaction costs as under D, but also acknowledges the German short selling constraints and sets to 0.99. For the same reason mentioned in Scenario C, the differences with the previous scenario are only seen in conditions (1a) and (2a).

The five different scenarios are summarized in table 3.

Table 3

An overview of the different scenarios.

Scenario A: no transaction costs; B and C: professional traders; D and E: non-professional traders.

Sce- Bid-ask Bid-ask Index option Index replication Transaction Short nario spread index spread options transaction costs costs costs Bund sale

in index points in € in € in € in € proceeds % = ) A 0.0 0.0 0.0 0.0 0.0 1.0 B 1.0 0.14 0.005 0.5 0.09 1.0 C 1.0 0.14 0.005 0.5 0.09 0.99 D 1.0 0.14 1.95 20.0 33.55 1.0 E 1.0 0.14 1.95 20.0 33.55 0.99

For each trading day during the test period, the five pricing relationships are tested. The boundary conditions (1a) and (1b) are tested for each option individually. For the put-call parity, all ATM call and put options with the same maturity are matched. The box spread and call and put spreads are constructed by matching OTM and ITM call and put options with the same maturity. For the box spread this includes two pairs of put and call options, where the call (put) spread is composed of ITM and OTM call (put) options with the same maturity. Finally, the call (put) option convexity test is performed on a

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Transaction fees, Börse Stuttgart

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combination of three call (put) options with the same maturity, but different strike prices. The data can be found in the appendix.

As stated earlier, this study focuses mostly on the level of improvement in efficiency over time. To compare the level of efficiency between the annual subsamples, the difference in frequency of the violations between the subsamples is expressed as a Z-statistic, following Capelle-Blancard and Chaudhury (2001):

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where and are the number of violations as a percentage over their corresponding number of observations, and . The Z-statistic is provided three times to compare the difference of the first year with the second (2010-2011), the first with the third (2010-2012) and the second with the third year (2011-2012). This statistic provides a result for the consistency of the trend of improved efficiency over time. If all three Z-statistics show the same sign, i.e. they are all negative or all positive, a consistent trend is implied. If there is a sign change among these values, the trend is not considered consistent.

To see if ETFs have had a positive influence on the market efficiency, the development in cross market efficiency is compared to the development of internal market efficiency over time. This is done by comparing the average relative results of the efficiency tests for all three annual subsamples. If the cross market efficiency level shows a higher degree of improvement than the internal market efficiency over time, it could be argued that the increased use of ETFs has contributed to the level of efficiency of the German index option market.

6. Empirical results

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support for the hypothesis that the increased use of ETFs has led to a higher degree of efficiency in the German index option market in recent years.

For every section, the results of the efficiency tests are shown for all three annual subsamples. The most important statistics used are the frequency of the violations, expressed as a percentage of the number of observed transactions, and the average size (in €) of these violations. The relative change in terms of frequency between the subsamples is expressed as a Z-statistic, where negative values imply that the frequency of violations has increased between two subsamples and positive values imply the reverse development.

6.1. Lower boundary conditions

Table 4 shows the results of the lower boundary test of the call options. Scenarios C-E are left out, since there are no violations detected for the professional traders when short selling constraints are taken into account and for both scenarios concerning the non-professional traders.

Table 4

Results of the lower boundary call options test.

* significant at 10% level; ** significant at 5% level; ***significant at 1% level.

Lower boundary call options 2010 2011 2012

Number of observations 1,343,235 1,020,347 923,647

Scenario A

Violations 15,402 11,666 12,514

Violations % 1.15 1.14 1.35

Mean violation size 12.70 20.11 0.11

2010-2011 2010-2012 2011-2012

Z-statistic 0.24 -13.76*** -13.23***

Scenario B

Violations 8,058 11,666 0

Violations % 0.60 1.14 0

Mean violation size 19.47 17.53 0

2010-2011 2010-2012 2011-2012

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Out of the 1,343,235 observations, scenario A detects 15,402 (1.15%) violations over the year 2010. This is approximately the same as the 1.14% of violations in 2011, resulting in a low, non-significant and positive Z-statistic of 0.24. In 2012 the percentage of violations is higher than in the previous years with 1.35%. This yields a significant negative Z-statistics of 13.76 and 13.23 for the comparison of 2010-2012 and 2011-2012, respectively. For this scenario, the level of efficiency based on condition (1a) therefore seems to be decreasing over the years in terms of frequency. The average values of the violations, however, tell another story: they increase at first, but are only a fraction in 2012 of what they are in the years before that, indicating that the DAX index call options are more efficiently priced in 2012 compared to 2010 and 2011.

Concerning the average size of the violations, it should be noted that this statistic should have a smaller share in the determination of the development of the level of market efficiency than the frequency of the violations. This is because, as Brunetti and Torricelli (2007) argue, the average values of the arbitrage opportunities are likely to be reduced when transaction costs are taken into account. This is especially true for scenarios A-C in this study, since the transaction costs for these scenarios are significantly lower than for the other two scenarios. This makes the average violation size a less reliable indicator of market efficiency for all conditions tested in this study. For this condition under scenario A, the extremely low average violation size of 2012 should therefore only have a small positive influence on the verdict of market efficiency concerning the index call options.

For scenario B, the results are quite different. The percentage of violations in 2010 is much lower than under the first scenario, while for 2011 this percentage is the same as under scenario A. For 2012 there are no violations detected. The level of efficiency seems to decrease at first, but increases again in 2012. This observation also explains the sign change of the Z-statistics, implying an inconsistent trend. The average size of the violation shows a slight decrease over the years in which violations are detected. This indicates that, on average, arbitrageurs can profit less form executing arbitrage strategies concerning the call boundary.

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Table 5

Results of the lower boundary put options test.

Lower boundary put options 2010 2011 2012

Number of observations 1,154,225 937,612 800.826

Scenario A

Violations 5,082 14,037 18,133

Violations % 0.44 1.50 2.26

2010-2011 2010-2012 2011-2012

Z-statistic -75.63*** -102.88*** -36.84***

Scenario B

Violations 5,082 14,037 18,133

Violations % 0.44 1.50 2.26

2010-2011 2010-2012 2011-2012

Z-statistic -75.63*** -102.88*** -36.84***

When comparing these results with the results of the call boundary condition, a number of differences can be noted: first of all, the difference in relative frequency is much bigger over the years for this condition. This can be seen by looking at the sizes of the Z-statistics, which are all bigger and statistically significant. Second, there is no sign change in the Z-statistics; they are all negative. This indicates that the level of pricing efficiency of the put options decreases continuously over the years. Third, there is no difference in the frequency of the violations between the two shown scenarios. This means that the low transaction costs of the professional trader do not influence the number of violations. Only the average sizes of the violations decrease under scenario B relative to A, which is a logical result of increased transaction costs. For both these scenarios, the average size of the violations is highest in 2010, which gives a conflicting indication of the development of the level of market efficiency over the years, compared to the frequency statistic. Again, the latter statistic is more reliable than the former.

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frequent violations than this study does. The results on the CAC 40 index, as provided by Cavallo and Mammola (2000), are more similar to the scores found for the DAX.

There is only one scenario (B of the call boundary condition) that supports the hypothesis that the market has become more efficient over recent years. When frictionless markets are assumed, the condition regarding the call options gets violated more often than the put condition. This is in line with the findings of Mittnik and Rieken (2000a), who report a similar overall frequency of the violations on the DAX index option between 1992 and 1995. They also conclude that there is only little evidence that the market becomes more efficient over time, but that these arbitrage profits quickly diminish when transaction costs and short selling constraints are taken into account. Ackert and Tian (2001) draw a similar conclusion on the US market when studying the effects of the introduction of the SPDRs.

6.2. The put-call parity (PCP) conditions

Table 6 shows the results of the put-call parity conditions (2a and 2b). When comparing the frequency of the violations of these conditions with the boundary conditions, the results look surprising. This can, however, be explained by looking at the equations used to determine the pricing relationship: the left hand side of equation (2a) is the negative of the left hand side of equation (2b). When outliers are not removed from the dataset and transaction costs are not taken into account, the percentage of the violations of conditions (2a) and (2b) should sum to 100. Since the boundary conditions have identified a few violations and these outliers are removed from the dataset, the percentages of scenario A should sum somewhat close to 100. Taking this feature of the relational conditions into account, the frequency of the violations should be considered both an absolute and a relative statistic: it shows not only the absolute frequency of violations when transaction costs are included, but also the relative degree of efficiency between call and put options.

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Table 6

Results of the put-call parity test.

Put-call parity 2010 2011 2012 2a 2b 2a 2b 2a 2b Number of observations 796,813 626,017 596,892 Scenario A Violations 690,993 93,281 510,740 89,574 446,378 128,777 Violations % 86.72 11.71 81.59 14.31 74.78 21.57

Mean violation size 25.70 2.86 38.49 10.65 18.51 7.43

2010-2011 2010-2012 2011-2012

Z-statistic 82.79*** -45.60*** 175.90*** -153.51*** 91.23*** -104.95***

Scenario B

Violations 661,657 93,281 505,382 89,574 434,140 128,769

Violations % 83.04 11.71 80.73 14.31 72.73 21.57

2010-2011 2010-2012 2011-2012

Z-statistic 35.39*** -45.60*** 144.43*** -153.50*** 104.93*** -104.93***

Scenario C

Violations 17,166 93,281 85,523 89,574 5,717 128,769

Violations % 2.15 11.71 13.66 14.31 0.96 21.57

2010-2011 2010-2012 2011-2012

Z-statistic -248.24*** -45.60*** 58.15*** -153.50*** 281.05*** -104.93***

Scenario D

Violations 23,350 250,468 134,466 161,973 12,376 305,507

Violations % 2.93 31.43 21.48 25.87 2.07 51.18

2010-2011 2010-2012 2011-2012

Z-statistic -335.81*** 73.21*** 32.46*** -237.91*** 352.30*** -297.25***

Scenario E

Violations 4,129 250,468 313 161,973 0 305,507

Violations % 0.52 31.43 0.05 25.87 0 51.18

Mean violation size 38.66 8.01 34.13 13.75 0 10.26

2010-2011 2010-2012 2011-2012

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but become more skewed in the following subsamples. The average values of the violations peak in 2011 for both sides of the put-call parity, but this is also the year that the sum of violation frequencies of both conditions is the lowest. This indicates that 2011 is the most efficient year in terms of frequency of the violations, even though the differences are very small.

When looking at scenario B, the most striking finding is that the results for condition (2b) do not change in terms of frequency. Apparently, the low transaction costs of professional traders do not affect the number of violations of this condition. Condition (2a) is only slightly influenced by the transaction costs, making the total sum of violations lower than under scenario A. The trend of more balanced relative pricing of call and put options over the years does not change: the Z-statistics are all significant and do not show a sign change for both conditions. The average values of the violations are slightly lower than under scenario A for all cases. The total frequency of the violations is the highest in 2011 under this scenario, which is the exact opposite of the results under A. The differences are, again, small though.

Scenario C clearly shows the impact of short selling constraints in Germany. While the results of condition (2b), which does not require a short position in the index, remain the same, the frequency of violations is reduced significantly for condition (2a). This serves as evidence that transaction costs are not the only obstacles for arbitrageurs; governmental policies also belong in that category. The short selling constraints also affect the trend that is found under the earlier scenarios. Condition (2a) shows a sign change of the Z-statistics, which indicates that 2011 is a less efficient year than the other two subsamples. This is also supported by the fact that the sum of the frequencies of the violations is the highest in 2011.

Scenarios D and E take the higher transaction costs of non-professional traders into account. Under D, the same trend for condition (2a) is shown as under C, with the lowest degree of efficiency in 2011. This, however, changes for condition (2b), where the frequency of violations more than doubles in 2012 with respect to the previous scenario. This results in a peak of the total frequencies of violations in 2012 under both scenarios involving non-professional traders.

Based on the PCP conditions, there are definitely results that suggest that the market is not efficient. These arbitrage opportunities, however, tend to disappear when transaction costs are included. The same results are reported by Mittnik and Rieken (2000b) and Ackert and Tian (2001), who also find some signs of option mispricing, but not enough to reject market efficiency.

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in terms of frequency of the violations. This is not true for the average values of these violations, but this statistic is less reliable. Therefore, based on the put-call parity, no strong evidence is found for an improving degree of efficiency over recent years.

6.3. The box spread conditions

The first set of conditions used to test the internal index option market efficiency consists of the box spread conditions (3a) and (3b). For this set of conditions, as well as those used for the call and put spreads and butterfly spreads, no short position in the index is required. Scenarios C and E will therefore yield the same results as scenarios B and D, respectively. For that reason, scenarios C and E are not mentioned in any of the internal market efficiency test results.

The box spread has a similar pricing relationship to the put-call parity, except that in this case the underlying instrument is left out of the equations and two pairs of matched call and put options are used. The left hand side of equation (3a) is therefore, like in the PCP equations, the negative of the left hand side of equation (3b). Again, the sum of the frequencies of the violations should be somewhat close to 100 for each year under scenario A.

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no violations for the first condition of the box spread and only 0.34% for the second condition. The average size of the violations found by them is, similar to these results, much higher than the results of Ackert and Tian (2001) and Capelle-Blancard and Chaudhury (2001).

Table 7

Results of the box spread test.

Box spread 2010 2011 2012 3a 3b 3a 3b 3a 3b Number of observations 47,068 72,249 40,457 Scenario A Violations 2,276 42,939 1,695 63,305 5,379 34,750 Violations % 4.84 91.23 2.35 87.62 13.30 85.89

2010-2011 2010-2012 2011-2012

Z-statistic 21.88*** 20.16*** -43.25*** 24.62*** -61.53*** 8.14***

Scenario B

Violations 1,871 1,858 372 12,740 849 1,896

Violations % 3.98 3.95 0.51 17.63 2.10 4.69

2010-2011 2010-2012 2011-2012

Z-statistic 36.85*** -81.56*** 16.34*** -5.35*** -20.82*** 73.36***

Scenario D

Violations 1,024 190 372 211 849 24

Violations % 2.18 0.40 0.51 0.29 2.10 0.06

2010-2011 2010-2012 2011-2012

Z-statistic 22.96*** 3.15*** 0.79 10.89*** -20.82*** 9.93***

A big difference with the PCP results is the influence of the transaction costs. Under scenario B the frequencies of the violations drop significantly, which is not the case for the PCP conditions. Scenario D shows an even lower percentage of violations, which indicates that transaction costs make the market seem more efficient, while this does not have to be the case. The average values of the violations are, however, much higher than under the PCP conditions. They also seem to increase as the transaction costs get higher.

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6.4. The call and put spread conditions

Table 8 shows the results of the call spread condition (4a). Scenario C is not included because no violations are detected when transaction costs of non-professionals are taken into account. The results of scenarios A and B are simple and straightforward: the only year in which violations are found is 2010. The frequency is the same for both scenarios and the average size of the violations decreases as the transaction costs are included. These results indicate that 2010 is the least efficient year and thus supports the hypothesis that efficiency has increased over time.

Table 8

Results of the call spread test.

Call spread 2010 2011 2012

Number of observations 176,465 267,470 211,102

Scenario A

Violations 41 0 0

Violations % 0.02 0 0

Mean violation size 5.31 0 0

2010-2011 2010-2012 2011-2012

Z-statistic 6.40*** 6.40*** 0

Scenario B

Violations 41 0 0

Violations % 0.02 0 0

Mean violation size 3.23 0 0

2010-2011 2010-2012 2011-2012

Z-statistic 6.40*** 6.40*** 0

Table 9

Results of the put spread test.

Put spread 2010 2011 2012

Scenario A

Violations 0 0 0

Violations % 0 0 0

Mean violation size 0 0 0

2010-2011 2010-2012 2011-2012

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The put spread condition, shown in table 9, is not violated in any of the scenarios. This indicates that based on the put spread condition, the market seems perfectly efficient. It does, however, not provide any information on the development of the level of efficiency over recent years.

Previous studies generally also find few violations of the call and put spread conditions. Capelle-Blancard and Chaudhury (2001) report violations in only 0.01% of the time for both conditions. Brunetti and Torricelli (2007) report 0.02% for the call spread and 0.08% for the put spread. The DAX index option market therefore seems to be in line with the French and Italian markets.

6.5. The put and call convexity conditions (butterfly spreads)

Table 10

Results of the call butterly spread test.

Call convexity 2010 2011 2012

Scenario A

Violations 2,475 321 237

Violations % 1.50 0.15 0.13

2010-2011 2010-2012 2011-2012

Z-statistic 43.24*** 43.85*** 1.66*

Scenario B

Violations % 1.31 0.15 0.13

2010-2011 2010-2012 2011-2012

Z-statistic 39.52*** 40.17*** 1.66*

Scenario D

Violations % 0.92 0.05 0.13

2010-2011 2010-2012 2011-2012

Z-statistic 36.52*** 31.66*** -8.55*

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consistent improvement in the level of efficiency. For the comparison of 2010 with the following two years, the Z-statistics are positive and significant at the 1% level. The comparison of 2011 and 2012 also shows an increase in efficiency, albeit only significant at the 10% level. The average values of the violations remain somewhat stable at first, but increase in 2012.

The results for the put convexity are very similar under these scenarios. The Z-statistics are all significant and show a consistent trend. The increase in terms of efficiency is much lower in 2011, compared to the call convexity condition. This lower rate of improvement is, however, fully compensated by the results of 2012, which is even a more efficient period for the put convexity than for the call convexity.

Table 11

Results of the put butterly spread test.

Put convexity 2010 2011 2012

Scenario A

Violations % 2.10 1.85 0.06

2010-2011 2010-2012 2011-2012

Z-statistic 5.32*** 56.43*** 57.45***

Scenario B

Violations % 2.10 1.80 0.06

2010-2011 2010-2012 2011-2012

Z-statistic 6.40*** 56.34*** 56.59***

Scenario D

Violations % 0.64 0.08 0.05

2010-2011 2010-2012 2011-2012

Z-statistic 27.14*** 28.28*** 2.76***

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year. The results of this scenario are, in comparison with previous studies, not surprising. Capelle-Blancard and Chaudhury (0.26% and 0.40%) and Brunetti and Torricelli (0.00% and 0.01%) find similar results.

The results of both convexity conditions do support the hypothesis that the option market has become more efficient in recent years. In all cases, 2010 is the least efficient year, while 2012 has the best overall efficiency score.

6.6. The influence of ETFs

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Table 12

Average relative pricing performances for all conditions.

Negative average percentages indicate a higher level of efficiency compared to the average for that condition. The average values of the violations are shown in brackets and are constructed in a similar way.

Deviations from average 2010 2011 2012

Cross market efficiency

(1a) -0.02% 0.16% -0.15% (-2.95) (-4.78) (7.73) (1b) -0.64% 0.06% 0.58% (-2.49) (3.03) (-0.55) (2a) 7.24% 10.95% -18.19% (-4.99) (-6.85) (11.84) (2b) -4.35% -4.47% 8.81% (3.63) (-3.47) (-0.16)

Internal market efficiency

(3a) 0.12% -2.41% 2.29% (6.72) (-27.49) (20.77) (3b) -0.56% 2.76% -2.21% (-2.83) (0.45) (2.38) (4a) 0.01% -0.005% -0.005% (-1.90) (0.95) (0.95) (5a) 0.75% -0.38% -0.37% (13.10) (5.97) (-19.08) (5b) 0.64% 0.27% -0.91% (14.84) (12.67) (-27.51)

As the table shows, the cross market efficiency tests demonstrate a more than average efficiency in the market for 2010 in three out of the four conditions when looking at the frequency of the violations. Only in one case, (2a), the level of efficiency is higher for 2012 than it is for 2010. And even in that case, efficiency decreases in 2011 with respect to the level of 2010. The average values of the violations show results that support the improved efficiency hypothesis: in three out of four cases, the average value is lower in 2010 than it is in the other years. Yet, this results in lower than average violation sizes in 2012 for only two conditions.

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All in all, neither types of market efficiency show results of consistent improvement over time. The scores of the cross market efficiency are worse than those of the internal market efficiency. The hypothesis that the increased volume of ETFs has contributed to the level of market efficiency is therefore not supported by these results.

7. Conclusion

This study examines the efficiency of the German index options market by using model independent pricing relation tests, which rely on the no-arbitrage principle. The lower boundary conditions show a fairly high level of efficiency, as compared to the US market. The scores on this test are very similar to the results of previous studies on the European index option markets. This test provides only little evidence that the efficiency of the DAX index options improved over time. The put-call parity test demonstrates many violations of the pricing relationship. These violations, however, tend to diminish when transaction costs are included. Since there are many inconsistencies found in the level of improved efficiency, this test does not support the hypothesis that German index options are more efficiently priced over time. The box spread test shows a high frequency of violations when frictionless markets are assumed. The frequency in the German market exceeds the one in the American market for this scenario. When transaction costs are taken into account, the reverse results are found. Again, the results for this scenario are very much in line with those of the other studied European markets. They, however, also do not suggest improved efficiency over time. The call and put spread conditions suggest that the German market is very efficient: there are only violations found for the year 2010, which also implies improvement of efficiency. The convexity conditions results show a low frequency of inefficiencies and are therefore similar to other studies of European index option markets. The level of improvement in efficiency is the most consistent in this section and it therefore implies that the German index options are more efficiently priced over time. Considering the overall results of the tests, there is, however, not much evidence that supports the first hypothesis stated in this study.

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References

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Capelle-Blancard, G., Chaudhury, M., 2001. Efficiency Tests of the French Index (CAC40) Options Market, Unpublished working paper. University of Paris, France.

Cavallo, L., Mammola, P., 2000. Empirical tests of efficiency of the Italian index options market, Journal of Empirical Finance 7, 173-193.

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Mittnik, S., Rieken, S., 2000a. Lower-boundary violations and market efficiency: Evidence from the German DAX index options market, Journal of Futures Markets 20, 405-424.

Mittnik, S., Rieken, S., 2000b, Put-call parity and the informational efficiency of the German DAX-index options market, International Review of Financial Analysis 9, 259-280.

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Online

Information on the DAX:

25 years of the DAX – the facts, from Deutsche Börse, available at http://www.deutsche-boerse.com

Contracts Specifications for Futures Contracts and Options Contracts at Eurex Germany and Eurex

Zürich, available at https://www.eurexchange.com/blob/exchange-en/3138-36774/114082/3/data/cs_

history_24072006_en.pdf

Deutsche Börse AG History, from Funding Universe, available at

http://www.fundinguniverse.com/company-histories/deutsche-b%C3%B6rse-ag-history/