Testing the Monotonicity Property of Option Prices

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Testing the Monotonicity

Property of Option Prices

Master thesis

Msc Business Administration

Specialization Finance, profile Risk & Portfolio Management March 2008 Student S.H.K. Hollink Student number: 1257196 hollink@gmail.com Institution

Faculty of Management and Organization University of Groningen

Supervisor University of Groningen

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Most papers concerning option pricing are based on models of Black and Scholes (1973), Merton (1973), Cox and Ross (1976), Derman and Kani (1994), Rubinstein (1994), Bergman, Grundy and Wiener (1996), Bakshi, Cao and Chen (1997) and Dumas, Fleming and Whaley (1998). All these models share the assumption that the underlying asset follows a one-dimensional diffusion process. These one-dimensional diffusion process models share three properties: the perfect relation property, the option redundancy property and the monotonicity property. The first property is based on the assumption, that all options on an underlying asset are perfectly correlated with each other and with the underlying asset. Moreover the price of the underlying asset is the only source of uncertainty for all the options. The option redundancy property is based on the assumption that all options are redundant securities, because options can be replicated by the underlying asset and a risk-free asset. The monotonicity property is based on the assumption that call option prices increase monotonically and put option prices decrease monotonically with the price of the underlying asset. This paper focuses on the last-mentioned property, and uses current option prices to test this property. This leads to the main research question:

“Do call prices and the underlying stock price always move in the same direction, and do put prices and the underlying stock price always move in the opposite direction”?

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number of violations? Furthermore the influence of the volatility of the underlying asset on the violation occurrence will be determined. Does the volatility of the AEX explain the number of violations? Finally, the relation between the put-call parity and the violation rate will be reviewed. Are there arbitrage opportunies for violated options? The rest of this paper proceeds as follows. Section 2 contains a theoretical explanation of the one-dimensional diffusion option pricing model and empirical findings of previous studies. Section 3 describes the methodology applied. Section 4 presents the AEX options and the four options on AEX listed companies and contains two tables with descriptive statistics. Section 5 describes the empirical results. Section 6 offers a summary and several concluding remarks.

2 Literature

This section presents the theoretical and empirical background used in this research. The first subsection gives a view on the one-dimensional diffusion option pricing model. 2.2 presents standard price movements of call and put options derived from the monotonicity property. 2.3 gives an overview of empirical findings of previous studies.

2.1 A one-dimensional diffusion option pricing model

This model states that the price of an asset follows one-dimension diffusion:

dS(t) = µ [t, S]S(t) dt + σ [t, S] S(t) dW(t), t ≥ 0, (1)

Where µ [t, S] is the drift rate and σ [t, S] is the volatility rate, these are both function of at most t and S(t). In addition, S is the price of the underlying asset and t stands for time. W (t) is a standard Brownian motion. All the papers mentioned in the introduction used equation (1) as a basis for their own model. The only difference between the models used by Black and Scholes (1973), Cox and Ross (1976), Dumas, Fleming and Whaley (1998) and Bergman, Grundy and Wiener (1996) is the proxy for σ [t, S]. Using Ito’s lemma and the fact that options are a function of the underlying asset price and time the following two equations are stated:

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Where the subscripts on C and P refer to the first partial derivative. Bergman, Grundy and Wiener (1996, p1582) use equation 2 and 3 for the following proposition:

Let the underlying price S(t) follow a one dimensional diffusion function as described in equation (1), then the option delta of any European call written on the asset must always be nonnegative and bounded from above by one:

0 ≤ Cs < 1 (4)

The delta of any European put, denoted by Ps must be non positive and bounded below by -1:

-1 ≤ Ps < 0 (5)

Equations 4 and 5 have to be correct for the monotonicity property to hold. In the following subsection there are four standard price movements of European call and put options stated, based on the equations 4 and 5. In this research these equations are also used for the American options. Bergman et al. (1996) en Bergman (1998) proved that the American call option is also increasing and convex in the stock price and that the American put option is also decreasing in the stock price.

2.2 Standard price movements of call and put options

According to the fact that option prices and the price of the underlying asset react on the same market information, which is the basis for the monotonicity property, the following price movements should be standard:

1) If the price of the underlying asset changes, the price of the call option should move in the same direction. Therefore ∆S∆C > 0.

2) If the price of the underlying asset changes, the price of the put option should move in the opposite direction. Therefore ∆S∆P < 0.

3) The price changes of the call and the put option should always be smaller than or equal to the price changes in the underlying asset. Because of that ∆C/∆S < 1 and ∆P/∆S > -1 4) Call and put options on the same underlying asset with the same strike price and the same

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2.3 Violations

The standard price movements of options mentioned in 2.2 can be violated in several ways. This paper differentiates 5 types of violations:

Type 1: For call options: ∆S∆C < 0 caused by either ∆S > 0 but ∆C < 0 or ∆S < 0 but ∆C > 0. For put options: ∆S∆P > 0 caused by either ∆S > 0 but ∆P > 0 or ∆S < 0 but ∆P < 0. Type 2: ∆S ≠ 0 but ∆C = 0 or ∆P = 0

Type 3: ∆S = 0 but ∆C ≠ 0 or ∆P ≠ 0

Type 4: ∆C/∆S ≥ 1 and ∆S ≠ 0 or ∆P/∆S ≤ -1 and ∆S ≠ 0 Type 5: ∆C∆P > 0

For type 1 violations the restriction of ∆S∆C < 0 is used in stead of ∆S∆C ≤ 0 because when ∆S∆C = 0, this paper uses a type 2 or type 3 violation for the option. It is also possible that both ∆S and ∆C are zero, than there is no violation at all. For the same reason this paper uses the restriction ∆S∆P > 0 instead of ∆S∆P ≥ 0. Moreover, the put call combinations are only violated where ∆C∆P > 0 instead of ∆C∆P ≥ 0, to exclude the cases where ∆S, ∆C and ∆P are zero. 2.4 Empirical findings of previous studies

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pricing models should allow more than one state variable to evolve stochastically. They mention the variable volatility, used in the Heston’s (1993) stochastic volatility option pricing model. Because this state variable is imperfectly correlated with the underlying price, options prices can move independently of the underlying price. This could be a reason for violations of the monotonicity property. BCC found that 47% of the type 1 violations can be explained by the volatility of the underlying asset.

Pérignon (2006) tested the monotonicity property for options on the European, British, French, German and Swiss stock indices. Pérignon used a sample period of one year with intraday sample intervals. Pérignon used in contrast to BCC, observed transaction prices, this to control explicitly for the impact of microstructure effects on his findings. Pérignon found an overall violation rate of 6% to 35%. He found a violation rate of 7% to 32% for call options and 6% to 35% for put options. Based on an interday interval he found a violation rate of 13.1% for call options on the European stock index and 11.3% for put options on the European stock index. These are type 1 violation rates, the only type of violations where Pérignon tested the dataset for. Pérignon divides the type 1 violations into two groups: the first group with an increasing value of the call or put option and a second group with a decreasing value of the call or put option. The second group violations occur approximately three times as often compared with the first group. Pérignon shows that the violation rate decreases with the length of the time interval for all the 5 indices. For the call options on the European stock index the violation rate decreases from 31.4%, based on a 30 minute interval, to 13,1% based on an interday interval. For put option these violation rates are respectively 31.1% and 11.3%. Pérignon also found a negative relation between the violation rate and the trading activity, because the true option price is better reflected if there is a high trading activity. Furthermore, Pérignon found no relationship between violation rates and the moneyness or maturity of the option, he found comparable violation rates throughout the different categories. According to Pérignon the violations of the monotonicity property are not limited to a special class option, but are a market wide phenomenon. Similarly to BCC, Pérignon claims that a large part of the violations is due to microstructure related effects. Another reason behind the violations according to Pérignon is rational trading tactics followed by traders in a market with relatively limited liquidity. According to Pérignon another reason behind the violation of the monotonicity property is concurrent changes in volatility.

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dataset. Nordén tested with different binominal models the prices, the implied volatility and the delta and vega of the options. Next he tested if there were significant differences between the predicted option price changes and the actual changes. For both hedging strategies Nordén found significant differences. The hedge performances of the delta-vega-neutral portfolios are better than the delta-neutral strategies but still show differences. Nordén recommendation is to use more complicated stock price dynamics to hedge American equity options.

Table 1 contains the main findings of BCC, Pérignon and Nordén that are useful for this paper. The percentages of BCC are based on the cash index of the S&P 500 because this paper also uses the cash index of the AEX index. Furthermore these percentages are based on interday intervals, because this paper will also calculate the violation rates based on interday intervals. The difference between the total number of violations of BCC and Nordén is mainly due to the type 3 violations. The Swedish stocks show more often no price changes on a daily basis than the S&P 500 index. Table 1 does not contain type 5 violation percentages because type 5 violation percentages are not part of the total violation percentages. Every type 5 violation consists of a type 1 violation and this type 1 violation is already part of the total violation percentage. BCC found a type 5 violation percentage of 11% and Nordén of 13.55%. They both show that call options and put options move more often down together than up together.

Table 1: Summary of findings by BCC, Pérignon and Nordén

Calls Puts

BCC Pérignon Nordén BCC Pérignon Nordén

Type 1 9.1 13.1 8.75 5.4 11.3 9.18

Type 2 3.6 n.a. 4.09 2.8 n.a. 5.34

Type 3 0 n.a. 6.06 0 n.a. 6.86

Type 4 11.5 n.a. 13.54 13.2 n.a. 12.36

Total 24.2 13.1 30.95 21.4 11.3 31.17

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price and the price of the underlying asset. In their opinion the findings of BCC cannot be used to invalidate the correctness of the one-dimensional diffusion option pricing model. Fact is, that BCC corrected their violation rate for micro structural effects and still found a violation rate that was too high to ignore.

3 Methodology

3.1 Reasons behind the violations

BCC, Pérignon, Nordén and Dennis and Mayhew give several causes of violations of the monotonicity property. Standard theory states that the value of an option depends on several underlying variables:

V = V ( S, σ , t , K , q , r)

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σ. The interest rate have changed two times in the sample period, this is too little to test the influence of the interest rate on the violation rate. The dividend is absorbed in the AEX index; therefore it is complicated to test the actual influence of the dividend rate on the violation rate. The influence of the dividend on the violation rate for options on the four companies is also hard to test, because the dividend of the four companies is relatively constant. The dividend is used however, in calculating the implied volatility of the AEX and the prices of the stocks of the four companies.

3.2 Expectations

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always shows some change. The violations of type 2 and 4 are probably due to a tick size minimum or other micro structure related effects (trading volume). Type 1 violations are presumable the most constant type and less related to microstructure factors. Violations of type 5 are also expected, with a bigger chance that the call options and put options move down together than moving up together, based on the research of BCC. Nordén shows the same subdivision of violation types 1,2,4 and 5 as BCC, but shows a higher type 3 violation percentage. Apart from the number of type 3 violations, the expected subdivision of the different types of violations of the options on the four companies are comparable with the expectation of the options on the AEX. 3.3 Hypotheses

Based on the theoretical and empirical review discussed in chapter two, and the expectations discussed in the section 3.2, the following hypothesis will be used in this paper:

H1: The monotonicity property holds for option prices on the AEX index.

H2: The monotonicity property holds for option prices on the four most traded options

on AEX listed companies.

H3: There is no relationship between the violation rate and moneyness or maturity

H4: Microstructure factors have no influence on the occurrence level of violations

H5: There is no relationship between the violation rate and the volatility of the

underlying asset.

H6: Call option prices and put option prices, with the same strike price and time to

maturity, always move in the opposite direction. 3.4 Methods

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are summed up. The total number of type 3 violations is the enumeration of the observations where ∆S = 0 and where ∆C ≠ 0 or ∆P ≠ 0, and for the total number of type 4 violations the observations, where ∆C/∆S ≥ 1 and ∆S ≠ 0 or ∆P/∆S ≤ -1 and ∆S ≠ 0 are summed up. The total number of type 5 violations is determined by counting the observations where ∆C∆P > 0 for call and put options with the same strike price and the same time to maturity. Secondly, the violation rates of type 1,2,3 and 4 are grouped by moneyness and time to maturity. There are three groups of moneyness (in the money, at the money and out of the money, see section 4.1) and three groups of time to maturity (short term, medium term and long term, see section 4.1). Thus the total violation rate for call options and the total violation rate for put options are both split up in 36 specified violation rates1. Next there will be determined if the different violation types, the moneyness and the time to maturity have an influence on the violation rate. Thirdly, the influence of the trading volume and the bid-ask spread on the violation rate will be determined. The violation rates will be subdivided by trading volume and the size of the bid-ask spread. After that the distribution of the violations between the different groups, based on trading volume and the size of the bid-ask spread, will be reviewed. Fourthly, the different type of violations will be checked for a relationship with price changes in the underlying asset. All the observations will be grouped, based on the magnitude of the price change of the underlying asset. Next the distribution of the violation rates between the different groups, based on price changes of the underlying asset, will be reviewed. Fifthly, the implied volatility will be calculated for the AEX index and for the four companies. The used implied volatility is an average of implied volatilities calculated with 9 options with different combinations of time to maturity and different strike prices. After that there will be reviewed if the change in volatility could be the reason behind the violation.Finally, the type 5 violations will be subdivided by call options and put options that move up together and move down together.

4 Data

4.1 Data Description

The data used in this paper are end of day observations of the AEX index and end of day bid-ask quotes for AEX index options. These bid-ask quotes are actual quotes of the market makers, quotes where the market makers were obliged to buy or sell for at that moment. The stock market

1

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closes 11 minutes later than the option market. This time gap will be discussed in section 5.6. This paper also uses data containing end of day bid-ask quotes and the end of day underlying stock prices of options on four AEX listed companies. For these options on companies there is also a time gap of eleven minutes. The choice for the four different companies is based on the number of different option contracts available for trading. The four companies with the highest number of different option contracts are ABN AMRO, Fortis, ING and Royal Dutch Shell. The fact that three of the four companies are financials, should not bias the findings of this paper. The assumption is made that the industry or sector where the company is part of, does not have an influence on the relation between the price changes of the options and price changes of the underlying stocks. The option prices of these four companies are used in this research. The data are collected from the Officiële Prijscourant for the sample period February 1, 2007 to July 31, 2007. The Officiële Prijscourant of the 12th of June was not available and two other issues did not contain all the option data, these days are left out of the research.

Table 2: Description Dataset

AEX ABN Amro Fortis ING Royal Dutch

Shell Series 447 102 108 108 108 Total Observations 22,454 2220 2358 2358 2358 Transaction Observations 18,008 1819 1838 1927 1817 Contracts Traded 11,303,939 2,995,861 579,652 1,188,523 1,130,870 Tick size 0.05 0.05 0.05 0.05 0.05

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transaction observations is approximately 80% of the total number of observations. Thus on 80% of the observation days in the sample period there have been trading in the options. The row “contracts traded” displays the total volume of contract trades for the sample period used in this paper. The total volume of contracts traded for ABN AMRO is considerable higher than the volume of the other three companies. This is probably due to the acquisition rumors around ABN AMRO. For AEX option contracts and the four company option contracts there is a minimum tick size of 0.05 Euro: if an option price moves, it has to move with a minimum of 0.05 Euro.

4.2 Descriptive Statistics

Table 3 contains the descriptive statistics for the call and put options on the AEX. The table displays the average price, the bid ask spread and the total number of observations of call options and put options on the AEX. The table subdivides these statistics by moneyness and time to maturity. There are three degrees of moneyness: in the money (ITM), at the money (ATM) and out of the money (OTM), and it is determined by the value of S/K for call options and K/S for put options. A call option is called:

in the money, if S/K ≥ 1.03;

at the money, if S/K 0,97 – 1.03; out of the money, if S/K ≤ 0.97.

For the put option the opposite is true. The three maturity classifications are short term, medium term and long term. An option is called:

short term, if the maturity ≤ 2 months; medium term, if the maturity 2–6 months; long term, if the maturity ≥ 6 months.

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The average price is also an increasing function of the degree of moneyness, which is logical because a call option price depends on S minus K. The average price of the put options is also an increasing function of the time to maturity and the degree of moneyness. The number of total observations is based on only the transaction observations mentioned in section 4.1. The reason for the lower number of observations with a long time to maturity, compared with the other time to maturity options, is the lower liquidity of the long term options.

Table 3: Descriptive Statistics of AEX options

Calls Puts

Moneyness

Time-to-Expiration Time-to-Expiration Short Medium Long

Sub-total Short Medium Long

Sub-total O T M average price 0.49 3.25 5.52 0.89 4.59 6.24 bid-ask spread 0.08 0.20 0.23 0.09 0.19 0.23 number of observations 816 1201 792 2809 1675 2222 1076 4973 A T M average price 6.56 15.29 23.64 7.27 15.38 24.91 bid-ask spread 0.16 0.25 0.35 0.17 0.26 0.37 number of observations 1425 1124 289 2838 1708 1225 281 3214 I T M average price 34.96 44.02 40.12 28.00 40.02 61.92 bid-ask spread 0.28 0.38 0.39 0.34 0.49 0.63 number of observations 1346 845 137 2328 739 769 338 1846 total observations 3587 3170 1218 7975 4122 4216 1695 10033

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term contracts. As mentioned before, if there is high interest in contracts, there is a high liquidity which results in lower bid-ask spreads.

Table 4: Descriptive statistics of options on the four AEX listed companies

Calls Puts

Moneyness

Time-to-Expiration Time-to-Expiration Short Medium Long

Sub-total Short Medium Long

Sub- total O T M average price 0.211 0.515 1.103 0.206 0.827 1.301 bid-ask spread 0.069 0.078 0.081 0.070 0.083 0.086 number of observations 335 369 350 1054 403 435 478 1316 A T M average price 0.764 1.214 1.798 0.914 1.577 2.091 bid-ask spread 0.083 0.082 0.084 0.084 0.082 0.09 number of observations 503 507 455 1465 465 464 386 1315 I T M average price 2.654 2.940 3.732 2.572 2.831 3.560 bid-ask spread 0.096 0.091 0.097 0.099 0.092 0.092 number of observations 448 434 440 1322 342 314 273 929 subtotal 1286 1310 1245 3841 1210 1213 1137 3560 5 Results 5.1 Violations

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put option the violation rate is 25.93%. These violation percentages are slightly lower than expected.

Table 5: Total violation rates for call and put options

Call options Put Options

AEX 20.31 21.66

4 Companies 27.41 25.93

5.2 Influence of moneyness and time to maturity on the violation rate

Table 6 contains the violation percentages of different type of violations across moneyness and maturity. The right bottom of the columns “calls” and “puts” show the total violation percentages of 20.31% and 21.66%. The largest part of the 20.31% violation rate for call options is due to type 4 violations with 9.13%. Thus in almost 10% of the cases investors overreact according to a price change in the underlying asset. 7.3% of the total violation rate is due to type 1 violations, where the price of the option moves in the opposite direction as the price of the underlying asset. The last part of the total violation rate of call options is caused by type 2 violations. In 3.89% of the cases, the price of the call option did not move while the price of the AEX had changed. Table 6: Violation percentages for call and put options on the AEX

Calls Puts

ITM ATM OTM Tot ITM ATM OTM Tot

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Table 6 contains all the violation percentages grouped by moneyness and maturity. The type 1 violations are split up in violations for short term, medium term and long term options and for ITM, ATM, and OTM options. The type 1 violation percentages are relatively constant across the times to maturity with total percentages of 7.92%, 7.26% and 9.93%. The moneyness seems to have a negative influence on the amount of type 1 violations, for every time to maturity the OTM options have the highest violation percentage. For every time to maturity the ATM options have the second largest violation percentage, and the ITM options the lowest percentage. According to type 2 violations, the OTM options have the highest violation percentage for every time to maturity. Striking is the 24.02% for type 2 violations for short term OTM options. The reason for this high violation percentage is probably the tick size problem. The price of these options is close to zero, namely 0.05 Euro, and there is a small chance that these options become valuable again because of the short time to maturity. The type 2 violation percentage for ITM options is low for every time to maturity. The total type 2 violation percentage for ATM options is 10 times as high compared to the ITM options, this is mainly caused by the short term ATM options with a violation percentage of 1.54%. There is no clear relation noticeable between the time to maturity and type 2 violations. There are no type 3 violations because the price of the AEX changed every day in the sample period. Type 4 violations are mainly due to ITM options with a total violation percentage of 23.84%. Thus in almost 25% of all the cases concerning ITM options, investors overreact on a price change of the AEX. For ATM options this is 4.51% and for OTM options this is 1.6%. The type 4 violation percentage is an increasing function of the degree of moneyness. The type 4 violation rates seems to be a decreasing function of the time to maturity but this is only the case for the total violation percentages, and does not hold throughout the different degrees of moneyness. The part of table 6 with the sum of all the different type of violations shows varied percentages. There is no clear distinction between the violation percentages related to the moneyness and time to maturity.

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maturity the lowest. The type 1 violation percentage is the highest for short term options for every degree of moneyness, followed by medium term and long term options. The time to maturity seems to have a negative influence on the type 1 violation percentage. There is no clear pattern noticeable for type 2 violation percentages. The highest type 2 violation percentage is for short term OTM options, with 14.87%. The reason for this is the same as for call options, the value of the option is fixed at 0.05 Euro. The distribution of the type 4 violation percentages for put options on the AEX is comparable with the percentages for the call options on the AEX. The highest type 4 violation percentages are caused by ITM options followed by the ATM and OTM options, and these higher percentages remain higher across different times to maturity. According to the time to maturity, the short term options generate the highest type 4 violation percentages, followed by medium and long term options. This distribution stays the same through almost every degree of moneyness, only the violation percentage for short term OTM options differs. The total violation percentages show one pattern throughout the different degrees of time to maturity and moneyness. The violation percentages for short term options are the highest, followed by medium term and long term options, this is true for every degree of moneyness, and therefore the time to maturity seems to have an influence on the violation rates of put options on the AEX.

Concluding, for the different degrees of moneyness there are two patterns for violation percentages noticeable, namely, a negative relationship between the number of type 1 and 2 violations and the degree of moneyness and a positive relationship between the type 4 violations and the degree of moneyness. These relationships are true for both call options and put options through all the degrees of maturity. These influences of the degree of moneyness are comparable with the findings of BCC, only the type 1 violation percentages of the S&P 500 remain constant for different degrees of moneyness. The time to maturity seems to have a negative influence on the type 4 violation percentages of call and put options on the AEX. Type 1 and 2 violation percentages remain constant throughout the different times to maturity. These results differ with the findings of BCC, they found a positive relation between type 1 violation percentages and the time to maturity. The type 2 and 4 violation percentages remain constant for different times to maturity according to BCC.

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Table 7: Violation percentages for call and put options on the 4 AEX listed companies

Calls Puts

Type 1 short 6.03 7.75 8.66 7.39 3.80 5.81 7.69 5.87 medium 5.99 6.31 7.32 6.49 2.87 4.96 7.36 5.28 long 6.36 6.37 7.14 6.59 2.93 3.89 6.07 4.57 total 6.13 6.83 7.69 6.82 3.23 4.94 6.99 5.25 Type 2 short 2.90 5.37 23.88 9.33 4.39 6.45 24.32 11.82 medium 2.76 6.51 13.01 7.10 6.05 6.25 12.64 8.49 long 3.18 4.62 9.71 5.54 5.49 6.74 9.62 7.65 total 2.95 5.53 15.37 7.34 5.27 6.46 15.12 9.35 Type 3 short 2.01 1.19 1.19 1.48 1.75 1.51 1.49 1.57 medium 1.38 1.18 1.08 1.22 0.96 1.29 1.61 1.32 long 1.82 1.76 0.86 1.53 1.10 1.55 1.67 1.50 total 1.74 1.37 1.04 1.41 1.29 1.44 1.60 1.46 Type 4 short 29.46 8.55 3.88 14.62 30.41 8.39 2.73 12.73 medium 20.51 8.68 3.79 11.22 19.11 8.19 2.99 9.15 long 17.27 7.25 3.14 9.64 13.92 8.29 3.35 7.56 total 22.47 8.19 3.61 11.85 21.74 8.29 3.04 9.86 All Types short 40.40 22.86 37.61 32.81 40.35 22.15 36.23 31.98 medium 30.65 22.68 25.2 26.03 28.98 20.69 24.60 24.24 long 28.64 20.00 20.86 23.29 23.44 20.47 20.71 21.28 total 33.28 21.91 27.71 27.41 31.54 21.14 26.75 25.93

5.3 Influence of microstructure factors on the violation rate

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have a high violation rate. The bid-ask spread of these options is often the minimum tick size. Type 4 violations show an inverse relation with the bid-ask spread compared with the type 1 and 2 violations. The type 4 violation percentages increase in the first four groups and decrease in the last group.

Table 8 Microstructure effects on call options on the AEX Euro bid-ask spread Violation rate (%) Trading volume Violation rate (%) ≤ 0.1 (12.15) 1-14 (5.42) 11.30 4.51 {5.23} {17.27} 0.1-0.2 (7.52) 14-115 (6.64) 1.61 4.22 {8.40} {11.75} 0.2-0.3 (4.50) 115-350 (8.75) 0.52 4.34 {14.53} {8.22} 0.3-0.4 (3.81) 350-935 (9.29) 0.30 3.84 {18.36} {7.77} > 0.4 (4.46) ≥ 935 (8.45) 0.30 2.05 {15.77} {6.27}

The trading volume has a positive influence on the type 1 violation rate, the violation rate only decreases slightly in the last group. The type 2 and 4 violation percentages appear to be a decreasing function of the trading volume. The over and under reaction of the option quotes seems to occur when the options have a low liquidity level. BCC found the same relation between type 4 violation percentages and the bid-ask spread and the trading volume. BCC found however no relations between type 1 and 2 violation percentages and the bid-ask spread and the trading volume.

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The type 1 violation rate percentage increases with the trading volume, only the second last group shows a decrease in violations. The type 2 and 4 violation percentages are decreasing functions of the trading volume.

Table 9 Microstructure effects on put options on the AEX Euro bid-ask spread Violation rate (%) Trading volume Violation rate (%) ≤ 0.1 (17.31) 0-14 (9.28) 9.55 7.59 {2.29} {9.97} 0.1-0.2 (12.15) 14-115 (10.73) 2.61 5.28 {3.88} {5.89} 0.2-0.3 (6.93) 115-350 (13.85) 1.79 3.85 {8.91} {5.83} 0.3-0.4 (6.37) 350-935 (11.99) 1.2 3.06 {11.78} {3.39} > 0.4 (5.55) ≥ 935 (15.77) 0.57 1.96 {13.21} {2.39}

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Table 10 Microstructure effects on call and put options on the four companies Euro bid-ask

spread

Violations (%)

Call options Put options

≤ 0.05 (11.43) (13.31) 8,57 16.10 {12.86} {3.72} 0.05-0.10 (9.40) (8.51) 7.24 8.92 {16.95} {13.28} > 0.10 (11.54) (9.62) 1.92 13.46 {13.46} {15.38}

Table 11 shows the different violation percentages of the call and put options on the four companies divided by trading volume. The type 1 violation percentages of the call options are constant for the different groups of trading volume. The violation percentages of the put options are also constant but slightly smaller than the percentages of the call options. The last group however, shows a much lower violation percentage. The type 2 violation percentages for call options decrease between the first and the second group and remain constant for the last three groups. This is comparable with the violation percentages for the call options on the AEX. The type 2 violation percentages for the put options decrease in the first three groups and remain constant in the last two groups, this is comparable with the put options on the AEX. The type 4 violation percentages for call options show a clear negative relationship with the trading volume, with a violation percentage of approximately 16% in the first group, this is comparable with the findings of the call options on the AEX. The type 4 violation percentages of the put options show also a negative relationship with the trading volume.

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Table 11 Microstructure effects on call and put options on the four companies

Trading volume Violations (%)

Call options Put options

1-35 (7.41) (4.83) 11.24 13.88 {16.40} {12.10} 35-115 (6.02) (4.62) 7.96 11.46 {12.61} {8.10} 115-245 (6.77) (3.65) 5.11 8.00 {10.23} {7.71} 350-935 (5.92) (5.70) 6.60 5.50 {9.50} {8.70} ≥ 935 (6.64) (1.52) 6.86 5.56 {8.77} {4.55}

5.4 Influence of price changes of the underlying asset on the violation rate

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Table 12 Average price changes of the violated call options and the AEX

Panel A ∆S > 0 and ∆C < 0 ∆S < 0 and ∆C > 0

Time to maturity Time to maturity

Moneyness Short Medium Long Short Medium Long

OTM Average ∆S 2.620 2.314 2.187 -2.194 -1.277 -1.155 Average ∆C -0.059 -0.190 -0.053 0.138 0.163 0.152 ATM Average ∆S 1.574 1.711 1.520 -1.195 -1.152 -1.006 Average ∆C -0.372 -0.285 -0.169 0.598 0.259 0.324 ITM Average ∆S 1.565 1.52 1.52 -1.761 -1.039 -1.183 Average ∆C -0.461 -0.40 -0.25 0.265 0.252 0.346

Panel B ∆S > 0 and ∆C = 0 ∆S < 0 and ∆C = 0

OTM Average ∆S 3.496 3.232 2.454 -3.779 -1.914 -1.322

ATM Average ∆S 4.578 1.040 - -1.736 -1.594 -1.990

ITM Average ∆S - - - -0.510 - -0.510

Table 13 contains the average price changes of the violated put options on the AEX and the price changes of the AEX. For the put options and the underlying asset, the average price changes are also higher than the tick size minimum. Thus the observations are relevant for this paper.

Table 13 Average price changes of the violated put options and the AEX

Panel A ∆S > 0 and ∆P > 0 ∆S < 0 and ∆P < 0

OTM Average ∆S 1.753 1.543 1.487 -1.273 -1.116 -1.129 Average ∆C 0.401 0.368 0.308 -0.132 -0.205 -0.180 ATM Average ∆S 1.103 - 0.939 -1.151 -0.843 -1.367 Average ∆C 0.446 - 0.688 -0.434 -0.355 -0.756 ITM Average ∆S 1.520 1.520 1.520 -0.982 -0.913 -1.063 Average ∆C 0.185 0.20 0.250 -0.688 -0.299 -0.478

Panel B ∆S > 0 and ∆P = 0 ∆S < 0 and ∆P = 0

OTM Average ∆S 3.330 2.485 1.197 -2.810 -0.920 -

ATM Average ∆S 0.807 1.520 0.450 -2.537 -0.724 -0.590

ITM Average ∆S 3.949 - - -2.197 -2.161 -0.991

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minimum. With a small positive increase of the average price of the underlying stock this is probably due to the time decay of the value of the option. Also the medium ATM options (type 1) show average price changes below the tick size minimum. Call options with type 2 violations show also average price changes of the underlying asset close to and below tick size minimum.

Table 14 Average price changes of the violated call options and their underlying stocks

Panel A ∆S > 0 and ∆C < 0 ∆S < 0 and ∆C > 0

OTM Average ∆S 0,125 0,086 0,106 -0,376 -0,226 -0,254 Average ∆C -0,037 -0,039 -0,059 0,038 0,044 0,113 ATM Average ∆S 0,074 0,091 0,099 -0,067 -0,216 -0,277 Average ∆C -0,064 -0,046 -0,073 0,371 0,030 0,057 ITM Average ∆S 0,056 0,069 0,105 -0,127 -0,187 -0,187 Average ∆C -0,065 -0,080 -0,096 0,669 0,269 0,542

Panel B ∆S > 0 and ∆C = 0 ∆S < 0 and ∆C = 0

OTM Average ∆S 0,172 0,112 0,076 -0,181 -0,221 -0,183

ATM Average ∆S 0,102 0,073 0,045 -0,173 -0,048 -0,046

ITM Average ∆S 0,064 0,065 0,055 -0,030 -0,125 -0,088

Table 15 Average price changes of the violated put options and their underlying stocks

Panel A ∆S > 0 and ∆P < 0 ∆S < 0 and ∆P > 0

OTM Average ∆S 0,149 0,148 0,131 -0,174 -0,185 -0,237 Average ∆C 0,058 0,043 0,088 -0,036 -0,096 -0,089 ATM Average ∆S 0,086 0,098 0,086 -0,116 -0,143 -0,218 Average ∆C 0,070 0,127 0,092 -0,053 -0,041 -0,047 ITM Average ∆S 0,110 0,113 0,083 -0,069 -0,098 -0,031 Average ∆C 0,119 0,163 0,075 -0,049 -0,125 -0,032

Panel B ∆S > 0 and ∆P = 0 ∆S < 0 and ∆P = 0

OTM Average ∆S 0,176 0,102 0,119 -0,170 -0,077 -0,055

ATM Average ∆S 0,065 0,059 0,042 -0,068 -0,076 -0,065

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Table 15 shows the average price changes for the violated put option and the underlying stock. The average price change for short and medium OTM options (type 1) are also close or below the tick size minimum. Furthermore, several ATM and ITM options (type 2) are close or below the tick size minimum. To determine how strong the relation is between the tick size minimum and the violation rate of the options on the companies, the type 1 and 2 violation rates in table 16 are adjusted for the influence of the tick size minimum. The violations where ∆S and ∆C or ∆S and ∆P are both between Euro -0,05 and Euro 0,05, are omitted. The difference between violation percentages in table 16 and the violation percentages of table 7 indicates that 30 to 40% of the type 1 and 2 violation percentages could possibly be explained by the tick size minimum. Table 16 Violation percentages for type 1 and 2 violations after adjustment for tick size

Calls Puts

Type 1 short 3.57 4.77 6.57 4,82 2.92 4.09 5.21 4.13 medium 4.38 3.94 4.61 4,27 2.23 3.66 4.83 3.71 long 4.32 4.18 5.14 4,50 0.73 2.07 4.18 2.64 total 4.08 4.30 5.41 4.53 2.05 3.35 4.71 3.51 Type 2 short 1.12 3.98 21.19 7,47 2.05 3.44 18.36 8.02 medium 0.92 2.96 10.03 4,27 3.18 3.45 7.59 4.86 long 1.82 1.98 6.86 3.29 1.47 3.89 4.81 3.69 total 1.29 3.00 12.52 5.02 2.26 3.57 9.88 5.56

5.5 Influence of the volatility of the underlying asset on the violation rate

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Table 17 Volatility changes that could cause violations

Calls Puts Type 1 ∆S > 0 ∆C < 0 ∆V < 0 ∆S > 0 ∆P > 0 ∆V > 0 ∆S < 0 ∆C > 0 ∆V > 0 ∆S < 0 ∆P > 0 ∆V < 0 Type 2 ∆S > 0 ∆C = 0 ∆V < 0 ∆S > 0 ∆P = 0 ∆V > 0 ∆S < 0 ∆C = 0 ∆V > 0 ∆S < 0 ∆P = 0 ∆V < 0 Type 4 ∆S > 0 ∆C > 0 ∆V > 0 ∆S > 0 ∆P > 0 ∆V < 0 ∆S < 0 ∆C > 0 ∆V < 0 ∆S < 0 ∆P > 0 ∆V > 0

Table 18 shows the percentages where the volatility could be a reason behind a violation. For call options on the AEX, after a positive price change, the volatility has decreased 69.73% of the time. For call options after a negative price change of the AEX volatility increased 46.30% of the time. For put options this occurs respectively 57.01% and 51.68% of the time. A large number of violations can be explained by concurrent changes in the volatility of the underlying asset. This is comparable with findings of BCC and Pérignon.

Table 18 Violation percentages that could be caused by a change in volatility of the AEX

Changes in S Calls Puts

Type 1 ∆S > 0 _79.45 _75.40 ∆S < 0 _61.97 _56.03 Type 2 ∆S > 0 _84.08 _37.50 ∆S < 0 _72.03 _18.56 Type 4 ∆S > 0 _57.56 _55.02 ∆S < 0 _29.13 _73.14 Total ∆S > 0 _69.73 _57.01 ∆S < 0 _46.30 _51.68

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Table 19 Violation percentage that could be caused by a change in volatility of the four companies

calls puts

Changes in

S ABN Fortis ING RDS ABN Fortis ING RDS

Type 1 ∆S > 0 _90.91 _95.35 _{90.77 84.21} _94.74 _65.0 _{64.29 26.67} ∆S < 0 _92.5 _66.67 _{71.43 88.24} _94.12 _50.0 _66.67 _62.5 Type 2 ∆S > 0 _41.03 _60.47 _{48.72 66.67} _78.57 _56.52 _{40.35 44.93} ∆S < 0 _84.85 _80.77 _65.22 _62.5 _66.67 ₂₄ _{41.54 39.44} Type 4 ∆S > 0 _82.76 _96.88 _87.3 _65.91 _94.44 _55.1 _44.05 _48.0 ∆S < 0 _86.21 _81.58 _{84.75 67.69} _76.67 _64.71 _25.0 _54.29 Total ∆S > 0 _75.0 _83.05 _{79.64 69.44} _81.60 _51.90 _{46.15 51.75} ∆S < 0 _87.5 _79.45 _{78.13 68.12} _80.38 _55.50 _{36.47 46.34}

5.6 Put-call combinations and their violation rate

In this section the violation rates of put options and call options with the same exercise price and the same time to maturity will be reviewed. This section also discusses the influence of the time gap between the end of day option prices and the end of day prices of the underlying asset. Furthermore it will be determined if there are joint violations of the monotonicity property, and if call options and put options move more often up together than down together. Finally there will be determined if there are arbitrage opportunities for the call and put options with joint violations. The time gap could be a reason behind the violation percentages. If for example the price of the underlying stock or index moves in the last ten minutes, this could cause a violation. When S shows a positive price change 10 minutes before closing time (St-10 ) and the call option shows

also a positive price change and the put option a negative price change there are at that time no violations. When the positive price change of the underlying asset however, change into a negative price change in the last ten minutes there will be two violated options. Both call option and put option will show a wrong type 1 violation. The difference between St and St-10 can also

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combination where both call option and put option show a type 1 violation. In this case the difference between St and St-10 could have caused the type 1 violations. If the underlying asset

shows a negative price change at t but a positive price changes at t-10, this difference could have caused the type 1 violation for both the call option and the put option. Other combinations that could be caused by the time gap are put-call combinations where put options and call options show both type 2 or both 4 violations. If the underlying asset shows a price difference at t but shows no price difference at t-10 thiscould be a reason behind the type 2 violations of the call option and the put option. If the call option and the put option are both type 4 violated it also could be caused by the difference in the price change of the underlying asset at t and t-10.

Table 20 shows the violation percentages for the different put-call combinations on the AEX. The 2.36% where both the call option and the put option show a type 1 violation could be caused by the time gap. After correction for these violations there is still a percentage of 8.66 % of type 1 violations for call options and 4.70% for put options. These corrected violation percentages are without the possible type 1 violations that could occur for the put-call combinations that show no violations in table 20 but are type 1 violated after correcting for the time gap. Another possibility is that, with end of day prices of the underlying asset, the call option shows a type 1 violation and the put option shows no violation. If the delta of S for end of day prices is the opposite of the delta of St-10 , the violations should be the other way around. In this paper the assumption is made

that S has an equal chance to change from a negative delta to a positive delta as from a positive to a negative delta in the closing ten minutes. Hence this has no influence on the total violation percentages. In 0.22% of the cases the put-call combinations show both a type 2 violation. This is a low percentage that has no significant influence on the total type 2 violation percentage.

Table 20 Violation percentages for put-call combination of options on the AEX puts

Type 1 Type 2 Type 4 No

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In 3.56.% of the cases the put-call combinations are type 4 violated. After adjustment for these violations the type 4 violation percentages for call options and put options are respectively 7.60% and 6.18%. There is also a possibility that the option shows no violations for St but a type 4

violation for St-10 . Therefore, the adjusted number of type 4 violations could be higher.

Table 21 shows the violation percentages for the call and put options on the four companies. The adjusted percentages for type 1,2 and 4 violations are for call options respectively 4.19%, 4.40% and 7.60, for put options these percentages are respectively 3.25%, 6.70% and 6.18%. The time gap seems to have an influence on the violation rates but is not the main reason behind all the violations.

Table 21 Violation percentages for put-call combination of options on the companies puts

Type 1 Type 2 Type 4 No

violation Total calls Type 1 1,87% 1,71% 1,28% 1,21% 6,06% Type 2 0,61% 1,32% 1,74% 2,05% 5,72% Type 4 1,31% 2,03% 2,09% 4,25% 9,69% No violation 1,33% 2,96% 3,16% - 7,45% Total 5,12% 8,02% 8,27% 7,51%

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Table 22 Joint violations of put-call combinations on the AEX and the four companies

∆S>0 ∆S<0 Total ∆C>0, ∆P>0 ∆C<0, ∆P<0 ∆C>0, ∆P>0 ∆C<0, ∆P<0 AEX 1.83% 2.92% 1.55% 6.40% 12.71% ABN 1.15% 2.95% 1.54% 5.51% 11.15% Fortis 0% 2.13% 0.83% 1.30% 4.26% ING 0% 0.54% 0% 0.11% 0.65% RDS 0.12% 0.72% 0.72% 2.38% 3.93%

A last factor that is researched in this paper is the put-call parity. Are there arbitrage opportunities for jointly violated options? The following put-call parity model is used:

S0 - D / (1+r) T

+ P0 = X / (1+r) T

+ C0 (6)

Where r is the risk free rate, T the time to maturity and D the dividend. For r the 1-month Euribor is used. For every jointly violated put call combination for increasing and decreasing prices of the underlying asset, the difference between the right hand side minus the left hand side of equation 6 is calculated. Table 23 contains the average differences for the different put-call combinations on the AEX and the four companies. The highest average differences are noticeable for put-call combination where one option shows a type 1 violation and the other a type 4 violation and where the underlying asset shows a negative price change. A reason for this could be a call option price that is too low or a put option price that is too high. It is difficult to draw conclusions from these findings because the time gap also has an influence on the put-call parity. Further research should make clear if there are real arbitrage opportunities for the jointly violated put-call combinations. Table 23 Average arbitrage opportunities for jointly violated options on the AEX and the four companies

Calls Puts AEX Companies

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6 Conclusion and Discussion

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Testing the Monotonicity Property of Option Prices

Testing the Monotonicity

Property of Option Prices

Table of Contents