The PCP was developed by Stoll ( 1969 ) to set up a relationship between the monetary values of put and call options. As with any theoretical account, PCP is besides based on three premises. They are the undermentioned: ( I ) involvement rate does non alter in clip, it is changeless for both adoption and loaning, ( two ) the dividends to be received are known and certain, ( three ) the underlying stock is extremely liquid and no transportation barriers exist.

## 2.1 What is the put-call para?

See two options, a European call and a European put, written on the same stock, and with the same work stoppage monetary value and clip to expiry. Let K be the work stoppage monetary value of the option, I„ be the clip to expiry, the current monetary value of the underlying is St, the hazard free rate is r, c be the market monetary value of the call, and p the market monetary value of the put.

See the following two portfolios:

Portfolio A: one European call option plus an sum of hard currency equal to K*exp ( -r*I„ ) .

Payoff

K

0 St

K

Figure 2.1: The final payment of Portfolio A

Portfolio Bacillus: one European put option plus on portion

Payoff

K

0 K St

Figure 2.2: The final payment of Portfolio B

At termination of the options ( i.e. at clip T=t + I„ ) , both portfolios will be deserving soap ( SN‚ , K ) . Because the options are European, they can non be exercised prior to the termination day of the month. The portfolios must hence hold indistinguishable values today. This means that: degree Celsius + K*exp ( -r*I„ ) = p + St, this relationship is known as put-call para.

If the PCP does non keep, there are arbitrage chances. Arbitrage is the process of come ining a sequence of trades that generate a riskless net income. A tool that is frequently used in arbitrage trades is short merchandising, or shorting. This tool means selling an plus that is non owned which is done by borrowing the plus from person who does have it, selling it, so purchasing it back at a ulterior day of the month, and eventually returning it to the party from whom it was borrowed. Such trade is profitable if the monetary value of the plus has fallen over the period between the sale and the redemption.

## 2.2 Examples

Here are two illustrations on the arbitrage opportunities if the PCP does non keep.

Example 1 ( Hull, 2003, p.208 )

St=31 ; K=30 ; I„=0.25 ; r=0.10 ; c=3.00 p=2.25

In this instance:

Portfolio A: degree Celsius + K*exp ( -r*I„ ) = 3.00+30exp ( -0.10*0.25 ) = 32.26

Portfolio B: P + St = 2.25+31 = 33.25

Portfolio B is overpriced comparative to portfolio A, in order to work this ; investor should buy the securities in A, and short the securities in B. That is: purchase the call option ; short one put option ; and short one portion. If investor does this, he will instantly bring forth an sum of hard currency:

-3+2.25+31 = 30.25

Which investor would put at the riskless rate for 3months, after which clip investor would hold:

30.25*exp ( 0.1*0.25 ) = 31.02

If the monetary value of the underlying at termination is greater than 30, investor would exert the call. If the monetary value of the underlying at termination is less than 3, the put would be exercised. Either manner, investor would be buying one portion for 3 at termination. This portion would so be used to shut out the short place on the portion.

The net net income would be:

31.02-30.00 = 1.02

Example 2

However, if portfolio A is overpriced comparative to portfolio B, in order to work this, investor should short the securities in A, and purchase the securities in B. That is: short the call option ; purchase the put option ; and purchase one portion.

St=31 ; K=30 ; I„=0.25 ; r=0.10 ; c=3.00 p=1.00

In this instance:

Portfolio A: degree Celsius + Kexp ( -rI„ ) = 3.00+30exp ( -0.10*0.25 ) = 32.26

Portfolio B: P + St = 1+31 = 32

Portfolio A is over-priced comparative to Portfolio B.

The initial investing would be:

-3+1+31 = 29

In order to finance this investing, investor would necessitate to borrow 29 at the riskless rate. After 3months, investor would necessitate to refund an sum:

29*exp ( 0.1*0.25 ) = 29.73

At the terminal of the three month period, either the short call or the long put will be exercised ; either manner investor will sell the portion for 30. The net net income is:

30-29.73 = 0.27

## Chapter 3 Literature Review

As mentioned in chapter 2, we know the PCP is an of import fiscal definition. However, since the PCP was developed, big sum of trials had been carried out. Trials of the PCP theory have yielded assorted consequences, but bulk of the trials had a similar consequence: the put-call para was invalid.

## 3.1 The put-call para was valid

As mentioned before, the PCP was developed by Stoll ( 1969 ) to set up a relationship between the monetary values of put and call options. However, subsequently he recognised that in the existent fiscal universe, some factors like minutess costs and revenue enhancements could impact the consequences. By presuming the appropriate involvement rate was the three-month Treasury rate and adding this rate into his informations, Stoll tested two sets of informations which were “ regular companies ” and “ new concern ” . The consequences showed that there were jobs with the coefficient of the involvement rate term. He once more recognised that there were some other factors could impact the involvement rate term, so despite those the jobs, Stoll so concluded that the PCP theory was moderately valid.

By looking through old surveies, Klemkosky and Resnick ( 1979 ) found that these surveies consisted with the PCP, they besides pointed out that in the relationship some inefficiency found to be ( set call para and market efficiency ) . However, the options market in 1979 is different from 10 old ages ago. They recognised that the construction of the options markets has been changed, hence some of the antecedently noted defects needed to be amended. They added a dividend term into the original theoretical account. The empirical consequences showed that about half long hedges trials and short hedges trials were profitable. That means the theoretical accounts tested were consistent with the PCP theory. Subsequently in 1980, Klemkosky and Resnick found that there was “ perfect foresight ” feature in their old work, so they corrected their theoretical account by adding the clip slowdown in. A figure of hedges showed profitableness every bit good. That was once more concluded that PCP and market efficiency obtained.

There were still other surveies which concluded the PCP was valid, nevertheless, harmonizing to the objects of this thesis which was seeking to happen out the annulment of the PCP theory, we are now focus on the surveies which concluded the PCP was invalid.

## 3.2 The put call para was invalid

## 3.2A The put call para on the European options

Loudon ( 1988 ) tested the PCP by supplying Australian grounds. He collected the information of the monetary values of indistinguishable put and call options together with the implicit in portion monetary value from the Sydney Stock Exchange Market during the calendar twelvemonth of 1985. The empirical consequences showed that ascertained misdemeanors of the theory are existed, which are the boundary misdemeanors. He did some analysis of these misdemeanors. He tested the effects of some factors: institutional factors, monetary value non-simultaneity, dealing costs, stock monetary value scope, clip to adulthood and in or out of the money class. He found that these ascertained misdemeanors can non be explained by the presence of non-simultaneous monetary value informations. There is no important relationship was found between the extent of monetary value non-simultaneity and ascertained misdemeanors. However, the being of dealing costs shows to hold the most important influence on the misdemeanors observed.

Taylor ( 1990 ) found the misdemeanor of PCP by supplying grounds of the pricing of options traded on the Australian Options Market ( AOM ) . He collected AOM informations for BHP and Woodside from 1982 to 1985. The derivation of put call relationship assumes frictionless markets, that means dealing costs, differential adoption and loaning rates and revenue enhancements are all assumed non to be. He tested the PCP theory and found several groundss of misdemeanor of the para conditions. But misdemeanors of this type were non that easy to develop after seting possible dealing costs into consideration. Although dealing costs can non explicate the being of these misdemeanors, they do supply a principle for their non-exploitation.

Harmonizing to Guo and Su ( 2006 ) , the original PCP may non use to the to a great extent traded options on dividend-paying securities, because the original put-call para dealingss assumed that the implicit in security does non pay dividends before the termination of the options. However, in the existent economic universe a mass of stocks and about all stock indices pay dividends. Therefore, their trials were based on the world that the implicit in security pays dividends ; they improved the put-call para expression. The trials presented a fluctuation of the dealingss when the underlying securities pay dividends. The consequences provided theoretical Scopess for the options monetary values when the underlying stock wages dividend. Besides the consequences spread out application of put-call para dealingss to all options on currencies and dividend-paying stocks and stock indices.

Brunetti and Torricelli ( 2003 ) tested the PCP by concentrating on a European market, the Italian Mib30 index option market ( MibO ) , instead than on North American markets. They collected the information from the period 1 September 2002 to 31 December 2002. They tested the PCP in the entire absence of clashs in the first measure, and so in the 2nd measure they added the bid-ask spread into the analysis. Finally the committee costs had been included every bit good. The consequence showed that during the analysis period, the Mibo market was efficient. Besides the function of clashs in the trials of the PCP has been stood out.

Ahn, Byoun and Park ( 2003 ) researched arbitrage chances of the KOSPI 200 options in Korea, a comparatively new market but the fastest growth and the most actively traded index option market in the universe. Their trials focus on the PCP conditions, such as: dealing costs. The information they used was composed of KOSPI 200 options from January 1998 to September 1999, but excepting the first several months because of possible unusual activities at the beginning subdivision. They found that overall there were a sensible measure of misdemeanors of the PCP conditions. The frequence of misdemeanor scopes from 25.4 % to 49.9 % for different clip periods.

Li ( 2006 ) focused on the arbitrage efficiency of the Nikkei 225 index options market in the Osaka Securities Exchange ( OSE ) with the intent of supplying grounds on the size and frequence of the arbitrage chances in the PCP model. He investigated the arbitrage efficiency of the Nikkei 225 options market by utilizing both an ex station and ex ante trial. The ex station trial showed that there 2.74 per centum of the sample misdemeanors of the PCP and 22.62 index points arbitrage net income for the OSE houses. The ex ante trial showed that both the two Numberss are decreased because of clip slowdown. Then Li used the arrested development analysis to back up the ex station and ex ante trials consequences. To reason, there was no strong grounds against the efficiency of the Nikkei 225 options market.

Kamara and Miller ( 1995 ) tested the PCP status utilizing European options to avoid the early exercising job. They collected the information from the Standard and Poor ‘s 500 stock index traded on the Chicago Board Options Exchange ( CBOE ) . By utilizing day-to-day and intra-daily monetary values, they found misdemeanors of PCP are much less frequent and smaller than those antecedently surveies which utilizing American options. The consequence showed that the divergences from PCP conditions are related to the liquidness hazard in the stock and options markets. Therefore, the frequence and size of divergences increased with the increasing of liquidness hazard. They pointed out that the command monetary values of call and put options rise related to their PCP-implied command monetary values and the ask monetary values fall related to their PCP-implied ask monetary value. Furthermore, the consequences suggest that the trading schemes underlying PCP are capable to important liquidness hazard.

## 3.2B Put-Call para on the American options

There are few surveies of PCP on the American options. Merton ( 1973 ) showed PCP need non keep for the American options because the possibility of early put exercising can non be wholly eliminated when the portfolio is established.

Evnine and Rudd ( 1985 ) used the S & A ; P100 options and Major Markets Index ( MMI ) to prove both European PCP and American PCP. The consequence showed that there were a important figure of possible net income chances. The call overpricing misdemeanors for both the S & A ; P100 and MMI occurred throughout the period. The options often violate the arbitrage boundary and the PCP ; besides, the options are well mispriced comparative to theoretical values.

Gould and Galai ( 1974 ) tested the PCP on securities by adding the dealing costs in. They found that the minutess costs must be assumed to do the theoretical account comply with the informations. The basic theoretical account was supported after including instead big minutess costs. They besides found that similar divergences from the efficient market hypothesis have shown up in related survey by other research workers ; nevertheless their accounts of these consequences appeared to be wrong on theoretical Scopess.

## 3.3 Deviations from the put-call para

Any divergences from the PCP can be used by investors to do risk-free net incomes.

Martijn and David ( 2007 ) investigated misdemeanor from PCP by utilizing different implied volatility or volatility spread between call and set options with the same implicit in monetary value, the same work stoppage monetary value and the same termination day of the month. They found that divergences from PCP contain information about subsequently coming stock monetary values. There was important grounds of predictability. By commanding for size, they found that divergences from PCP are more possible to look when underlying stocks face more information hazard. Overall, they suggested that the monetary value of an option could be affected by its demands.

Ariful, Meher and Geoffrey ( 2004 ) tested the impact of minutess cost on divergences from PCP belongingss. They used the foreign exchange options which traded on the Philadelphia Exchange ( PHLX ) from 01 August 2005 to 31 July 2006. By utilizing the intra-daily option, they studied the impact of alternate steps of dealing cost on PCP divergences. Three steps of dealing costs were used: foremost, a minimal minutess cost that involve merely initial bid-ask spreads ; 2nd, dealing costs associated with trades closed out prior to termination ; and 3rd, a entire dealing costs step. The consequences indicated how minutess costs can impact the reading of divergences from PCP.

## 3.4 Drumhead

As mentioned antecedently, there were many surveies on the PCP ; most of them had a similar consequence which indicated that the PCP was non hold in the existent fiscal society. The dealing costs are one of the grounds that had been showed in most of the surveies.

Besides, there are divergences from the PCP. Any divergences from the PCP can be used by investors to do risk-free net incomes. Some surveies indicated that dealing costs can impact the reading of divergences from the PCP.

## Chapter 4 Methodology

This chapter describes the methodological analysis in the survey and the processs for informations aggregation and information analysis. Methodology includes the undermentioned constructs as they relate to a peculiar subject or field of enquiry: a aggregation of theories, constructs or thoughts ; comparative survey of different attacks ; and review of the single methods. Methodology refers to more than a simple set of methods ; instead it refers to the principle and the philosophical premises that underlie a peculiar survey relation to the scientific method.

In this thesis, the writer is traveling to prove the put-call para by utilizing the most late options informations. The writer will work out the divergence from the put-call para. By utilizing the divergence as a dependant variable and put into a simple arrested development theoretical account, the writer will happen the factors that can impact the divergence. Besides, the writer will explicate the command and offer monetary value which appear in the information.

## 4.1 The arrested development theoretical account

Arrested development theoretical accounts are used to foretell one variable from one or more other variables. Besides, arrested development theoretical accounts provide the scientist with a powerful tool, leting anticipations about yesteryear, present, or future events to be made with information about yesteryear or present events.A

In order to explicate the divergence from put-call para, the writer uses a arrested development theoretical account. The theoretical account is as the followers:

Diff=I?a‚ˆ +I?a‚?*I„ +I?a‚‚*r +I?a‚?*St +I?a‚„*K +u

In this arrested development theoretical account, diff ( the divergence ) is the dependent variable, I„ ( clip to expiry ) , R ( riskless rate ) , St ( the current monetary value of the underlying ) , and K ( strike monetary value ) are the explanatory variables. The error term is u .

The divergence ( diff ) is the difference between the two portfolios in the put-call para. As what has been mentioned in chapter 2, Portfolio A is the call option + an sum of hard currency equal to K*exp ( -r*I„ ) ; Portfolio B is the put option + one portion. Let diff = Portfolio B – Portfolio A = ( p + St ) – ( hundred + K*exp ( -r*I„ ) ) . If diff is positive, that means portfolio B is over-priced comparative to portfolio A, and frailty versa.

It is obvious that the put-call para does non keep in the existent economic universe, there might be some grounds. By making a simple reasoning backward on the above theoretical account, the writer will happen out which explanatory has important consequence on the dependant variable.

In the option data the writer collected, there are two different option monetary values, the command and offer. So the above theoretical account will alter to the following two theoretical accounts:

Let diff_bid be the difference between the two portfolios when they take command monetary values, diff_bid = ( p_bid + St ) – ( c_bid+K*exp ( -r*I„ ) ) ; allow diff_offer be the difference between the two portfolios when they take offer monetary values, diff_offer = ( p_offer + St ) – ( c_offer + K*exp ( -r*I„ ) ) .

Model 1: diff_bid=I?a‚ˆ +I?a‚?*I„ +I?a‚‚*r +I?a‚?*St +I?a‚„*K +u

Model 2: diff_offer=I?a‚ˆ +I?a‚?*I„ +I?a‚‚*r +I?a‚?*St +I?a‚„*K +u

Subsequently, a new variable tau-squared ( I„A? ) was added into the theoretical accounts, in order to prove the additive relationship between dependant variable and I„ . The theoretical accounts became:

Model 3: diff_bid=I?a‚ˆ +I?a‚?*I„ +I?a‚‚*r +I?a‚?*St +I?a‚„*K +I?a‚…*I„A? +u

Model 4: diff_offer=I?a‚ˆ +I?a‚?*I„ +I?a‚‚*r +I?a‚?*St +I?a‚„*K +I?a‚…*I„A? +u

The utilizations of above theoretical accounts can assist to explicate why there is a divergence from the put-call para. As we know, if the put-call para does non keep, there are arbitrage chances. The above theoretical accounts can besides assist the investors to hold on the arbitrage chances.

## 4.2 The option informations

The writer is traveling to briefly depict the option informations that has been used in the trial of put-call para.

The European option informations that used in the trial is called CBOE RUSSELL 2000 INDEX-RUT. The CBOE Russell 2000 ( RUT ) Index is a taking benchmark for the public presentation of small-capitalization stocks. The CBOE Russell 2000 Index was created in 1984 by Frank Russell Company and was designed to track the public presentation of small-cap companies. RUT options make it simple to take part in the small-cap market. ( COBE website, 2010 ) All the European options started at 1 to 30 April 2009, with assortment I„ ( clip to expiry ) from 2days to 626days.

The American option informations that used in the trial is called AFLAC INC. AFLAC Incorporated ( AFLAC ) is a general concern keeping company and Acts of the Apostless as a direction company, supervising the operations of its subsidiaries.A AFLAC sells auxiliary wellness insurance policies to more than 40 million people worldwide. Because 80 per centum of the company ‘s gross revenues are made in Japan, it has been inquiring the federal authorities to coerce Japan to open its insurance markets to more competition. The company besides lobbies on a assortment of wellness attention issues, including the conflict over intensifying prescription drug monetary values. ( Open Secrets, 2010 ) All the American options started at 1 to 30 April 2009, with assortment I„ ( clip to expiry ) from 1days to 661days.

The riskless rate that is used in the theoretical account is the LIBOR rate. However, merely up to one twelvemonth LIBOR rate can be found, therefore, the writer left the options with more than 365days clip to expiry out of the informations. Then the I„ of all the European options is from 2 to 353, and I„ of all the American options is from 2 to 290.

## 4.3 Drumhead

In this thesis, the theoretical accounts are the chief tool to look into the subject. The stata is the chief package that used in this thesis. The most of import theoretical accounts are merely four, they are:

Model 1: diff_bid=I?a‚ˆ +I?a‚?*I„ +I?a‚‚*r +I?a‚?*St +I?a‚„*K +u

Model 2: diff_offer=I?a‚ˆ +I?a‚?*I„ +I?a‚‚*r +I?a‚?*St +I?a‚„*K +u

Model 3: diff_bid=I?a‚ˆ +I?a‚?*I„ +I?a‚‚*r +I?a‚?*St +I?a‚„*K +I?a‚…*I„A? +u

Model 4: diff_offer=I?a‚ˆ +I?a‚?*I„ +I?a‚‚*r +I?a‚?*St +I?a‚„*K +I?a‚…*I„A? +u

Besides, the writer described the information that used in the thesis, which are the most late informations collected from the yokel finance. The LIBOR rate is besides an of import variable in making the trials. The writer collected the LIBOR rates from the ECOWIN, and used them into the trials.

## Chapter 5 Findingss

After making arrested development on the theoretical accounts old mentioned in last chapter, the writer has following consequences which will be explained and discussed. In this chapter, the writer will prove both the European and American options utilizing the precisely same manner. Besides, the writer will briefly speak about the differences of command and offer monetary values that showed in the consequences and convey the old surveies back to happen out the ground of the differences.

## 5.1 European option

Stock figure: 102434. Name: CBOE RUSSELL 2000 INDEX-RUT

Obviously, from the options informations, the put-call para is non hold can be found out. The difference between two portfolios is from -2.9324405 to 20.691776 when they take command monetary values and from -2.7493894 to 36.068069 when they take offer monetary values.

The distribution of the divergence is as follows:

Graph 5.1: The distribution of diff_bid

Graph 5.2: The distribution of diff_offer

Besides the writer summarises on the two divergences: diff_bid and diff_offer, see the figure below:

Figure 5.1: Compare the diff_bid and diff_offer on European options

Variable

Observations

Mean

Std. Dev.

Minute

Soap

diff_bid

3239

4.735678

4.44433

-2.932441

20.69178

diff_offer

3239

4.070535

3.543508

-2.749389

36.06807

The consequence in table 1 ( see page 39 ) shows that all the explanatory variables have important effects on the dependant variable. The clip to expiry ( tau ) and current monetary value of underlying stock ( St ) are positive important, the LIBOR rate ( liborrate ) and the work stoppage monetary value ( K ) are negative important. The consequence in table 2 ( see page 39 ) besides shows that all the explanatory variables have important effects on the dependant variable and all of them are positive important.

In both table 1 and table 2 ( see page 39 ) , the writer found that the clip to expiry has big positive consequence on the divergence. The coefficients are +20.43712 and +5.747666. That means the divergence decreases when clip is near to expiry. The undermentioned two graphs focal point on the relationship between clip to expiry and divergence which show that there seems an increasing heteroscedasticity relationship between the two variables.

Graph 5.3: The lowess drum sander of diff_bid

Graph 5.4: The lowess drum sander of diff_offer

In order to prove whether there are jobs of heteroscedasticity between the divergence and clip to expiry, we need to make hettest. The consequences show that the qi squared statistics are 70.91 and 244.72. And the p-value ( prob & gt ; chi2 ) are both equal to 0.0000. That means there is strong grounds to demo that we have increasing heteroscedasticity.

Because of the presence of heteroscedasticity, the standard mistakes are wrong ; hence, the t-ratios and assurance intervals are besides wrong. An alternate redress for heteroscedasticity is merely to rectify the standard mistakes, since these are the lone measures that are being estimated falsely. The usage of robust option corrects the criterions mistakes irrespective of the signifier of heteroscedasticity ( see the figure below ) .

Figure 5.2: Compare the non-robust and robust on European options

Dep.var.

Exp.var.

Coef.

Std.Err.

T

p-value

95 % conf.Interval

diff_bid

incorrect

tau

18.04837

.1249502

144.44

0.000

17.80338 18.29336

_cons

0.256618

.0422

6.08

0.000

.1738765.3393595

Correct ( robust )

tau

18.04837

.118097

152.83

0.000

17.81682 18.27992

_cons

0.256618

.0452737

5.67

0.000

.1678499.3453861

diff

_offer

incorrect

tau

10.68668

.1965082

54.38

0.000

10.30138 11.07197

_cons

1.418425

.0663677

21.37

0.000

1.288298 1.548552

Correct ( robust )

tau

10.68668

.2485504

43.00

0.000

10.19934 11.17401

_cons

1.418425

.0642217

22.09

0.000

1.292505 1.544344

As mentioned in last chapter, a new variable will be added into the theoretical account to prove the additive relationship between dependant variable and clip to expiry, which is tau- squared ( tau2 ) . In making the arrested development on new theoretical accounts once more ; the consequences are in table 3 and 4 ( see both tabular arraies in page 40 ) . By concentrating on the adjusted R-squared to see if the new added variable tau2 is relevant, because adjusted R-squared has the desirable belongings that is merely rises if the added variable has some explanatory power and if the added variable has no explanatory power, adjusted R-squared falls.

Figure 5.3: Compare the consequences without tau2 and with tau2 on European options

Dependent variable

Models

Adjusted R-squared

diff_bid

Without tau2

0.9238

With tau2

0.9265

diff_offer

Without tau2

0.6249

With tau2

0.6256

Since both adjusted R-squared are risen, it can be concluded that the added variable tau2 has some explanatory power. The consequence in table 3 ( see page 40 ) shows that there is strong grounds that tau2 is positive important ( coefficient is 7.612881, p-value is 0.000 ) . The consequence in table 4 ( see page 40 ) shows that there is grounds that tau2 is negative important ( coefficient is -3.368799, p-value is 0.0007 ) . The consequences indicate that the relationship between dependant variable ( diff_bid or diff_offer ) and clip to expiry ( tau ) is non additive.

By utilizing a simple multinomial arrested development theoretical account: diff = I?a‚ˆ +I?a‚?*I„ +I?a‚‚*I„A? ( 1 ) , the maximal degree of diff at a certain I„ can be worked out.

Differentia equation ( 1 ) with regard to I„ , it becomes:

( a?‚diff ) /a?‚I„ =I?a‚? +2I?a‚‚*I„ ( 2 )

Let ( 2 ) equal to 0, so there is:

I„= – ( I?a‚?/2I?a‚‚ )

After making arrested developments on the above theoretical account with two different dependent variables ( diff_bid and diff_offer ) , the consequences come to:

diff_bid=0.4822 +15.9548*I„ +2.5776*I„A? ;

diff_offer=0.8669 +15.806*I„ -6.3029*I„A? .

Work out I„ equal to -3.0949 and +1.2539. It can be found that, when taken command monetary value, the I„ is a negative figure which is impossible. However, when taken offer monetary value, the soap diff_offer is at point I„=1.2539, and diff_offer now equal to 10.7762.

See the following two graphs:

diff_bid diff_offer

10.7762

0.8669

0.4822

0 I„ 0 I„

1.2539

Figure 5.4: The relationship between diff and I„ on European options

## 5.2 American option

Stock figure: 100892. Name: AFLAC INC

Again, from the computation, we besides can happen out that the put-call para is non keep for American option either. The difference between two portfolios is from -.6563612 to 1.347102 when they take command monetary values ; from -.7673744 to 35.73 when they take offer monetary values.

The distribution of the divergence is as the followers:

Graph 5.5: The distribution of diff_bid

Graph 5.6: The distribution of diff_offer

When summarise once more on both divergences, the consequences are:

Figure 5.5: Compare the diff_bid and diff_offer on American options

Variable

Observations

Mean

Std. Dev.

Minute

Soap

diff_bid

1327

.4663752

.3537124

-.6563612

1.347102

diff_offer

1327

.3953503

.4122343

-.7673744

1.752388

The consequence in table 5 ( see page 41 ) shows that there is strong grounds that all the explanatory variables have important consequence on the dependant variable, furthermore all of the four explanatory variables are positive important. The consequence in table 6 ( see page 41 ) shows that the clip to expiry ( tau ) , LIBOR rate ( liborrate ) and strike monetary value ( K ) are positive important consequence on the dependant variable, but there is no grounds that the current monetary value of underlying stock ( St ) has consequence on dependant variable. Both tabular arraies showed that the clip to expiry ( tau ) is significantly greater than others. The coefficients are 1.130066 and 0.9446083.

Again after concentrating on the relationship between clip to expiry and divergence, the following two graphs show that there seems an increasing heteroscedasticity relationship between the two variables.

Graph 5.7: The lowess drum sander of diff_bid

Graph 5.8: The lowess drum sander of diff_bid

To prove the heteroscedasticity of the American options, the informations must be capable to the same trial performed in chapter 5.1 in the European options. The consequences show that the qi squared statistics are 25.00 and 188.89. And the p-value ( prob & gt ; chi2 ) are both equal to 0.0000. That means there is a strong grounds to demo that we have increasing heteroscedasticity.

Because of the presence of heteroscedasticity, the standard mistakes are wrong, therefore, the t-ratios and assurance intervals are besides wrong. An alternate redress for heteroscedasticity is merely to rectify the standard mistakes, since these are the lone measures that are being estimated falsely. The usage of robust option corrects the criterions mistakes irrespective of the signifier of heteroscedasticity.

Add I„A? into the theoretical account to prove the additive relationship between dependant variable and clip to expiry. The consequences in table 7 and 8 ( see both tabular arraies in page 42 ) shows, foremost, the new added variable I„A? has explanatory power ( adjusted R-squared increased from 0.8687 to 0.8748 and from 0.8997 to 0.9034 ) ; secondly, both I„A?s have negative important consequence on the dependant variable ( coefficients are -1.26426 and -1.151113, both p-value equal to 0.000 ) .

As antecedently did to work out the maximal degree of diff at a certain I„ , the consequences are:

diff_bid=0.0085 +1.7206*I„ -0.6165*I„A? ;

diff_offer=-0.0801 +1.7819*I„ -0.6312*I„A? .

The I„ has been worked out equal to 1.3955 and 1.4115. Then, diff_bid equal to 1.2090 when I„ = 1.3955 ; diff_offer equal to 1.1775 when I„ = 1.4115.

The undermentioned two diagrams show the relationship between diff and I„ :

diff_bid diff_offer

1.2090 1.1775

0.0085 0 I„

-0.0801 1.4115

0 1.3955 I„

Figure 5.6: The relationship between diff and I„ on American options

## 5.3 Bid and Offer monetary value

The command is the highest monetary value that a prospective purchaser is willing to pay for a specific security. The offer, besides called the request monetary value, is the lowest monetary value acceptable to a prospective marketer of the same security. The highest command and lowest offer are quoted on most major exchanges, and the difference between the two monetary values is called the bid-ask spread. The bid-offer spread is the sum by which the ask monetary value exceeds the command. This is basically the difference in monetary value between the highest monetary value thatA a purchaser is willing to pay for an plus and the lowest monetary value for whichA a marketer is willing to sell it. The size of the bid-offer spread in a security is one step of the liquidness of the market and of the size of theA dealing cost.

In the trial of the European options, some differences had been showed between these two monetary values which are unexpected. Look back Figure 5.4, when command monetary values are taken into the trial, it was found that the divergence from PCP rises monotonically with addition in I„ . However, when offer monetary values had been used in the trial, things changed. The divergence from PCP rises at a diminishing rate in I„ , and there is a peak point that the differences would get down diminishing from that point if the clip to expiry ( I„ ) keeps on increasing.

The old surveies had looked at similar difference in unexpected differences found in the trial show here. Kalman and Steven ( 1981 ) found dealing costs causes bid-offer spreads. Besides, Thomas and Francis ( 1993 ) examined the cross-sectional distribution of bid-ask spreads in the S & A ; P 100 index options market, in which they found cross-sectional differences in bid-ask spreads are straight related to differences in market-making costs. Harmonizing to Harold ( 1968 ) , the bid-offer spread is an recognized step of liquidness costs in exchange traded securities and trade goods. On any standardised exchange two elements comprise about all of the dealing costA – securities firm fees and bid-ask spreads. Under competitory conditions the bid-ask spread measures the cost of doing minutess without hold. The difference in monetary value paid by an pressing purchaser and received by an pressing marketer is the liquidness cost. Since securities firm committees do non change with the clip taken to finish a dealing, differences in bid-ask spread indicate differences in the liquidness cost.

Therefore, after the above discussing, it can be concluded that the unexpected difference between command and offer monetary values appeared in the old trials is because of the dealing costs. The dealing costs can be measured by the bid-offer spreads.

## 5.4 Drumhead

The writer did the trials in order to happen out the divergences from the PCP and discuss and explicate these divergences.

In concentrating on the relationship between clip to expiry and differences, it is showed that the longer the clip to expiry, the larger the divergences, ceteris paribus. Therefore, the divergences from put-call para are more likely to happen in options with implicit in stocks that face more information hazard because the longer the clip to expiry, the more hazard demand faced.

At the terminal, the writer explained the command and offer monetary values, which caused the unexpected consequences during the trials.

## Chapter 6 Decision

By looking at what has been antecedently discussed and all the consequences, the writer can pull following decision:

The PCP does non keep for both European and American options used in the research. The differences ( diff_bid and diff_offer ) between the two portfolios, A and B, are positive or negative, ne’er zero.

There is ever divergence from the PCP. From the above research, when use the divergence from the PCP as a dependant variable in a theoretical account, there is grounds that all the explanatory variables in the theoretical account have important effects on the dependant variable.

European option:

diff_bid = -18.5974** + 20.4371**I„ – 0.9803**r + 0.0477**St – 0.0060**K

diff_offer = -22.1961** + 5.7477**I„ + 2.0043**r + 0.0402**St + 0.0085**K

American option:

diff_bid = -0.4627** + 1.1301**I„ + 0.0643**r + 0.0173**St + 0.0013**K

diff_offer = -0.5666** + 0.9446**I„ + 0.1312**r + 0.0020**St + 0.0175**K

** indicates strong significance ( p-value & lt ; 0.01 )

This thesis is interested in the consequence of the clip to expiry ( I„ ) The consequences show there is grounds that clip to expiry is positive significance, which the longer the clip to expiry, the larger the divergences, ceteris paribus. There is an increasing heteroscedasticity relation between divergences and the clip to expiry ( I„ ) , which would do the incorrect of standard mistakes, t-rations and assurance intervals. Through utilizing an option called “ robust ” to rectify the criterions mistakes irrespective of the signifier of the heteroscedasticity.

Besides, the consequences showed a non-linear relation between divergences and the clip to expiry ( I„ ) when the squared of clip to expiry ( I„A? ) was added into theoretical accounts. There is grounds that I„A? is besides important, but the effects are different depends on the option monetary values. There is grounds that I„A? has negative consequence on the divergences from the chosen American option whatever the monetary values are. However, grounds showed that I„A? has positive effects on the divergences from the chosen European option when the command monetary values had been used and the effects are negative when the offer monetary values have been used.

Subsequently, after utilizing a individual multinomial arrested development theoretical account to work out the peak point of the difference at a certain clip to expiry ( I„ ) , the consequences showed that, for the European option, when taken command monetary value, the I„ is a negative figure which is impossible ; when taken offer monetary value, the peak point is equal to 10.7762 at I„=1.2539. For the American option, peak point is 1.2090 at I„ = 1.3955 when its command monetary value ; peak point is 1.1775 at I„ = 1.4115 when its offer monetary value. At last, the writer used the old literature to explicate the unexpected differences showed in the consequences.

The failure of the PCP relationship to depict the relationship between option and stock monetary values has deductions to plus pricing theoretical accounts like the Black/Scholes option pricing theoretical account. When PCP is a theoretical underpinning of the pricing theoretical account, so the pricing theoretical account may necessitate to be revisited to find whether or non it can be improved.