Option Pricing Models with HF Data: An

Journal of CENTRUM Cathedra ● Volume 5, Issue 1, 2012 ● 70-90

JCC Journal of CENTRUM Cathedra

Option Pricing Models with HF Data: An Application of the Black Model to the WIG20 Index Ryszard Kokoszczyński

University of Warsaw, Warsaw, Poland

Paweł Sakowski


Robert Ślepaczuk *


Abstract In this paper, we compared several Black option pricing models by applying different measures of volatility and examined the Black model with historical (BHV), implied (BIV), and several different types of realized (BRV) volatility. The main objective of the study was to find the best model; that is, the model that predicts the actual market price with the minimum error. The high frequency (HF) data and bid-ask quotes (instead of transactional data) for the Warsaw Stock Exchange (WSE) were used to omit the problem of nonsynchronous trading and to increase the number of observations. Several error statistics and the percentage of price overpredictions (OP) showed the results that confirmed the initial intuition that the BIV model is the best model, the BHV model is the second best, and the BRV is the least efficient among the models studied. Keywords: Option pricing models, financial market volatility, high frequency financial data, realized volatility, implied volatility, microstructure bias, emerging markets, Warsaw Stock Exchange JEL Classification codes: G14, G15, C61, C22

Option trading dates back to the 17th century, when options were part of, and one of the reasons for, the South Sea bubble and the Amsterdam tulip mania. However, rapid growth in the options market came in the 1970s only. First, two seminal papers by Black and Scholes (1973) and Merton (1973) introduced the BlackScholes-Merton (BSM) model, a formula for valuing European options. In 1973, the Chicago Board of Options Exchange (CBOE) was founded, which heralded the beginning of trading on standardized listed options; the CBOE adopted the BSM model for option pricing in 1975. The rapid growth of option markets, due to the combination of a seemingly reliable pricing formula and a good exchange mechanism, brought a considerable amount of data and stimulated the intensive development of the option pricing research. Soon, empirical studies showed clearly that some theoretical assumptions of the BSM model are not fully supported by these data (Bates, 2003) and that the BS formula exhibits substantial pricing biases across both moneyness and maturity (Bakshi, Cao, & Chen, 1997). A number of new models were then proposed, each of them relaxing some of the restrictive assumptions of the BSM model (Broadie & Detemple, 2004; Garcia, Ghysels, & Renault, 2010; Han, 2008; Mitra, 2009). There is also a growing literature devoted to comparisons of their various features although even the best metric for model comparison is a controversial issue (Bams, Lehnert, & Wolff, 2009).

Option Pricing Models with HF Data: An Application of the Black Model to the WIG20 Index

71

Notwithstanding the numerous criticisms, the BSM model is still widely used, both as some kind of benchmark in comparative studies mentioned earlier and among financial practitioners. A detailed analysis of the literature shows that the BIV model calculated on the basis of the last observation performs quite well even when compared with many different pricing models such as standard BSM model, BRV model, Generalized autoregressive conditional heteroskedasticity (GARCH) option pricing models, or various stochastic volatility (SV) models (An & Suo, 2009; Andersen, Frederiksen, & Staal, 2007; Bates, 2003; Brandt & Wu, 2002; Ferreira, Gago, Leon, & Rubio, 2005; Mixon, 2009; Raj & Thurston, 1998). However, practically all these studies used data from the mature capital markets of the United States of America, the United Kingdom, and Japan. Thus, the purpose of this study was to find the best option pricing model, but for a different market such as an emerging market. We used the WSE data for our study not only because the Polish market is best known to us but also because this market is the largest in the region1 and can therefore lead to general conclusions. The choice of the market for the present study also derived from the very limited knowledge available regarding option pricing in the Polish capital market. Most studies exist only in Polish or in the form of unpublished papers, which makes them practically inaccessible to a wider audience. Moreover, these studies are usually limited to GARCH option pricing models (Osiewalski & Pipień, 2003). The purpose of this paper was thus to close this gap, at least partially, as the only study covering similar issues is Fiszeder’s (2010) although it is limited to daily data. There are several ways of measuring and estimating volatility. A number of studies have indicated plausibly that it makes a difference what kind of volatility –historical, implied, or realized– has been applied (Ammann, Skovmand, & Verhofen, 2009; Berkowitz, 2010; Martens & Zein, 2004). That observation has been one of the major factors defining the scope of this study. Thus, in order to verify the initial hypothesis for this study, a few additional questions were as follows: • What kind of volatility process should be used in the black model? • What length of time period (parameter n – responsible for the memory of the process) should be used for averaging volatility in the estimation? • What is the optimal interval (delta) for estimating volatility? • Do errors depend on the option’s time to maturity (TTM) and moneyness ratio (MR)? These questions were the reason HF data (10s data interval, based on tick data) for WIG20 index 2 option quotes (bid and ask) were used in order to increase the observed liquidity of the market and to remove nonsynchronous bias. The remaining sections of this paper are organized as follows. The next section indicates the methodology for option pricing with a special focus on various volatility measures and their estimators. Afterwards, there is a detailed section with the description of the data used and of the volatility processes that were studied. HF data brought a significant number of specific technical issues that constrained to some extent the whole research endeavor. The results section indicates the results of the study in detail. The last section shows the implications these results have for other financial models and for further research on option pricing.

Option Pricing Methodology The main assumption was that one can price a European style option on the WIG20 index applying the Black model for futures option contract, where the WIG20 index futures contract is the basis instrument. This is possible for two reasons 3: • WIG20 index futures mature exactly the same day as WIG20 index options, and the expiration prices are set exactly in the same way; and, • WIG20 index options are European-style options, so there is no need to worry about early expiration as in the case of American options. The use of the Black model instead of the BSM model was motivated by the following facts: • It was not necessary to calculate the dividend yield for the index when using the Black model, so we omited the problem of the BSM assumption that dividends are paid continuously; thus, we did not have to calculate the dividend yield nor set the exact time of dividend payment; • It was possible to use the data from the period between 9:00 a.m. and 9:30 a.m. each day though index quotation starts only at 9:30 a.m., which gives a longer trading day.

6

6

was possible to use the data fromln( / K )between / 2 Ita.m.   2T 9.00 theFperiod wasand possible 9.30 a.m. to use each theday, data from the period between 9.00 a.m. and 9.30 a.m. each day, (3) d1   T 72 Option Pricing Models with HF Data: An Application of the Black Model to the WIG20 Index ough index quotation starts only at 9.30 a.m., which2gives a though longer trading index quotation day. starts only at 9.30 a.m., which gives a longer trading day. ln( F / K )   T / 2  d1   T (4) d2  formulas forT the model (Black, for 1976) are presented below: The formulas for the BlackThe model (Black,1976) areBlack presented The below: formulas the Black model (Black, 1976) are presented below:  rT

e  rT [ FNof (da1 )call  KN  e  rT [ FN (d1 )  KN (d 2 )] and p are respectivelyc valuation and(da2 )]put futures option, T is the(1)cexpiration pe

 rT

[ KN (d 2 )  FN (d1 )] (2)p  e [ KN (d 2 )  FN (d1 )]  rT

is the risk-free rate, F – the futures price, K – the underlying strike, and N(.) is the

:

where:

(1) (2)

where:

tive standard normal distribution.

ln( F / K )   2T / 2 ln( F / K )   2T / 2 (3d )1  (3)  T were examined with three different types of  T Some properties of the Black model ln( F / K )   2T / 2 ln( F / K )   2T / 2 d2   d1   T (4d )2   d1   T (4) y estimators: historical volatility, realized volatility, and implied volatility. We used  T T where c and p are respectively valuation of a call and a put futures option, T is the expiration date, r is the riskc and p aredata respectively valuation callfutures andon a put where option, cthe andunderlying pfor TareisWIG20 respectively the expiration valuation of the a call and a put futures option, T is distribution. the expiration equency (10-s data interval, based thefutures tick index free rate, Fof–athe price, K –data) strike, andoption N(.) is cumulative standard normal d1 

Some properties the Black model were examined with three different types of volatility estimators: r is the risk-free rate, F – the futures price, K –ofthe strike, and N(.) date, restimators is the risk-free rate,isFthe – the futures price, K – the underlying strike, and N(.) is the (bid and ask) to calculate them. Formulas for allunderlying three are presented below.

historical volatility, realized volatility, and implied volatility. We used high frequency data (10s data interval, based on the tick data) for WIG20 index option quotes (bid and ask) to calculate them. Formulas for all three ative standard normal distribution. cumulative standard normal distribution. estimators are presented below. The historical volatility (HV) estimator (standard deviation for log returns based on

1. model The historical volatility deviation logexamined returns based on thedifferent daily interval) Some properties of the Black were examined with(HV) Some threeestimator properties different (standard types of theof Black model for were with three types of

y interval) was directly derivedwas from directly derived from

lity estimators: historical volatility, realized volatility, and volatility implied estimators: volatility.historical We usedvolatility, realized volatility, and implied volatility. We used n N n N 1 1 n 2 n VAR = (ri ,tthe − rtick ) 2 data) for WIG20 index option (5) VAR  ( r  r ) ( 5 ) ∑∑ ∆ option  i ,t frequency  frequency data (10-s data interval, high data) for WIG20 dataindex (10-s data interval, based on ( N * n ) − 1 ( based N  n) on  1 thet 1tick t i = 1 = 1 i 1 ∆ s (bid and ask) to calculate them. Formulas for all three quotes estimators (bid are andpresented ask) to calculate below. them. Formulas for all three estimators are presented below. 7 where: The historical volatility (HV) estimator (standard deviation for log returns based on 1. The historical volatility (HV) estimator (standard deviation for log returns based on n 7 on– the variance of log returns calculated high VAR∆calculated close price for interval day t with thefrequency sampling frequency equal Δ, frequency – variance of the logi-th returns with high data on with the basis of last n data on the basis of last n days; ily interval) was directlyrderived from daily interval) directly derivedfrequency from – log return for iththe interval on daywas t with sampling equal to Δ, which is calculated in the i,t 7 C – close price for the i-th interval on the day t with the sampling frequency equal Δ, i,t number of Δ intervals during the the stock market session, following way: ∆

N  nn N n N  1 1 2 VARnn  (riiN  r ) 22 – number of Δ intervals(during 5VAR ) n the  the stock market (rsession,   i ,t  r ) ,,tt Δ memory of the process measured the calculation respective (on N  the nlogC )day days, 1 ttt11with ( NC * n−) log  1 tC used – close price for the i-thr interval sampling frequencyof equal Δ, r = log ii  11 thein 1 i 1 = logC –*in

(5)

(6)

- log return for i-th interval on day t with sampling frequency equal to Δ, which is n – memory of the process measured in days, used in the calculation of respective number of Δmeasures. intervals during the the stock market session, s:–and average th where: C – close price for the i interval on the day t with the sampling frequency equal Δ; i,t

i,t

i ,t

i–1,t

ed in the following way:i,t

i ,t

i −1,t

estimators and average measures.

– memory the for process measured inondays, used of in last the ncalculation average log of return the i-th interval the days with of therespective sampling n – number of Δbasis intervals N n – variance of log returns n during the stock market session; Δ calculated with high frequency – variance on the basis of logof returns last n calculated with high frequency data on the basis of last n VARdata    – average log return for the i-th (interval on the basis of last n days with the sampling r C r  log C  log 6)the calculation ors and average measures. n – memory of the process measured in days, used in of respective estimators and i ,t i 1,t Δ, which is calculated in the following way:i ,t average measures; days, Δ, which is calculated in the following way: frequency

– average log return for the i-th interval on the basis of last n days with the sampling n N log return for the ith interval on the basis of last n days with the sampling frequency Δ, r 1 – average - log return for i-th rinterval  riis on daywhich t with - logfollowing equal returnto for Δ, way: which i-th interval is(7) 1on dayn tNwith sampling frequency equal to Δ, which is rfrequency  i,t in the ,t sampling calculated cy Δ, which is calculated N in the * nfollowing t 1 i 1 way: r ri ,t (7 )  N  * n t 1 i 1 ated in the following way: calculated in the following way: n N n N∆ 1 1 e of the historical volatility i=1r and NΔ=1 for every ri,t (daily log returns) r  estimator (7) ∑∑ ri ,t r = (7 )  i ,t In the case of the historical volatility i=1 and NΔ=1 for every ri,t (daily log returns) N n t 1 i 1 N * nestimator t =1 i =1



r  log C  log Ci1,t (6) ∆ r  log Ci,t  log Ci1,t (6) i ,t formulas (5), (6) and (7). Moreover, wei,t used the constant value of parameter n=21, i,t In the case of the historical estimator = 1(7). andMoreover, NΔ = 1 for ri,tconstant (daily log returns) and Cn=21, in and volatility Ci,t in formulas (5), (6)i and weevery used the value of parameter i,t case of the historical volatility estimator i=1 and NΔ=1 for every ri,t (daily log returns) formulas (5), (6), and (7). Moreover, we used the constant value of parameter n = 21, because we wanted to we wanted to reflect the historical volatility from the last trading month.reflect the historical volatility from the last trading month. because we the wanted totrading the historical volatility from in formulas (5), (6) andreflect (7). Moreover, we used the constant valuelast of parametermonth. n=21,

2. estimator The realized volatility estimator was based onestimator the following formula: The realized volatility (RV) was based on the(RV) formula: 2.following The realized volatility (RV) was based on the following formula: e we wanted to reflect the historical volatility from the last trading month. N

2 2. The realized volatility estimator was based on the following formula: RV(RV)  ,t   ri ,t i 1

N

RV  ,t   ri ,t

N

(8) ,t   ri ,t RV

2

i 1

(8)

2

(8) estimator was based on the most recent observation, implied (IV) e implied volatility (IV) estimator was based on3.theThemost recentvolatility observation, i 1

σ was derived from the Black formula with the assumption that other parameters σThe wasimplied derivedvolatility from the(IV) Black formulawas withbased thetherefore assumption otherobservation, parameters estimator on the mostthat recent

and the valuation results were given. The implied volatility for the previous observation was re σ wasresults derivedwere fromgiven. the Black formula volatility with the assumption that other parameters aluation The implied for the previous observation was calculated separately for each class of the time to maturity (TTM) and of the moneyness ratio valuation were given. The implied volatility for the previous observation ratio was separatelyresults for each class of the time to maturity (TTM) and of the moneyness

3. The implied volatility (IV) estimator was based on the most recent observation,

nd the valuation results were given. The implied volatility for the previous observation was

therefore σ was derived from the Black formula with the assumption that other parameters

alculated separately for each class of the time to maturity (TTM) and of the moneyness ratio

and the valuation results were given. The implied volatility for the previous observation was


73

MR), that is, for 50 different classes; the details of thiscalculated option classification are class presented in to maturity (TTM) and of the moneyness ratio separately for each 8of the time

3. then Thetreated implied volatility (IV)that estimator was based on the most recent observation, therefore σ was (MR), is, the for 50 different classes; the to details of this option classification are presented in ection 4. This estimator was as an input variable for volatility parameter y to historical volatility which is based from on information from many (n>1), a that other parameters and the valuation results derived the Black formula withperiods the assumption 4. This estimator was then treated as an input variable for the volatility parameter to given.model The IVwith forSection the previous observation wasthe calculated separately for each class of the TTM88 alculate the theoretical value forwere the Black implied volatility (BIV) for next volatility estimator requires information only from one single period (interval Δ). and of the MR, that is,calculate for 50 the different classes; thethe details this with option classification are for presented theoretical value for Blackof model implied volatility (BIV) the next bservation. Contrary to historical volatility which is based on information from many periods (n>1), aa in the results section. This estimator was then treated as an input variable for the volatility parameter Contrary to historical volatility which is based on information from many periods (n>1), re, the procedure of averagingtoand annualizing a realized volatility estimator observation. calculate the theoretical value for the BIV modelisfor the next observation. volatility estimator information In the next step, the historical volatility wasrealized annualized and transformed into standard realized volatility estimator requires requires information only only from from one one single single period period (interval (interval Δ). Δ). In the nextand step,transformed the historical volatility was annualized andbecause transformed standard Inin theformula next step, into 8standard deviation thisinto is the padifferent from that presented (9)5:the HV was annualized 4 4 Therefore, volatility : the used in theinBlack model eviation because this is rameter the parameter used the Black model Therefore, the: procedure procedure of of averaging averaging and and annualizing annualizing aa realized realized volatility estimator estimator is is8 4 : deviation because this is the parameter used in the Black model

1 annual_ stdvolatility n slightly from that formula Contrary to historical which information from periods a (9) slightly different from many that presented presented in)(n>1), formula (9)5:: [ RV ]annual 252isn based [on RV ] ,t different (annual 10in   _ std _ std n n ton historical volatility which is SD based on252 information from many periods (n>1), HV  SD n t 1252Contrary N VAR ( 9 ) HV   * N  * VAR (9)a    n realized volatility estimator requires information only from one single period (interval Δ). n annual_ requires std ninformation 1 realized volatility estimator only one single period (interval(10 Δ). 1 annual_ std [ RV ]  n  252 [[from RV   ,t [ RV ]  252 RVa ]]RV (10))    ,t estimator requires inforese volatility estimators, several typestoofHV option pricing models were examined: n Contrary which is based on information from many periods (n > 1), t  1 n t 1 Therefore, the procedure of averaging and annualizing a the realized volatility estimator isannualizing Therefore, procedure of averaging a realizedand volatility estimator is mation only from one single period (interval Δ). Therefore, the and procedure of averaging annualizing a RV 5 With these volatility estimators, several types of option pricing models were examined: Black model with historical volatility (sigma as standard deviation, n=21) – BHV; 5 from estimator is slightly different presented in formula 5 volatility estimators, several (9) types: of option pricing models were examined: : thesethat slightly different from that presented in formula (9)With n

5



slightly different from that presented in formula (9) :

 the Black with (sigma Black model with realized volatility (realized volatility an model estimate sigma; volatility RV the as Black model withofhistorical historical volatility (sigma as as standard standard deviation, deviation, n=21) n=21) – – BHV; BHV; 1 n 1 n annual_ std n annual_ std n [ RV ]  252 [ RV ] ) [ RV ]   252  the[ Black RV ] model ( 10 ) volatility   ,t as an estimate of sigma;(10 volatility (realized ,t n tthe Black model with realized volatility (realized as an estimate of sigma; RV RV calculated on the basis of observations with a different interval  with and realized a different n t volatility 1 1



was calculated on the of with aa different interval  and was calculated on the basis basis of observations observations with pricing different interval and aa different different With these volatility estimators, several types ofmodels option models were examined: meter in the processestimators, of averaging) – BRV With these volatility several types option pricing were examined: With nthese volatility several types; of estimators, option pricing models wereofexamined: 6

parameter n in the process of averaging) – BRV66;

;as standard ndeviation, parameter n in the process of averaging) –(sigma BRV deviation, • The Black volatility model with historical volatility (sigma as standard = 21) – n=21) BHV;– BHV; the Black model with historical volatility Black model with implied (implied an estimate of sigma; IV– was  the Black model with volatility historical volatility (sigma with asasstandard deviation, n=21) BHV; • The Black model realized volatility (RV as an estimate of sigma; RV was calculated on the  the Black model with implied volatility (implied volatility as an estimate of IV was the Black model with implied volatility (implied volatility as as annan estimate of sigma; sigma; IVaverwas  the Black model with realized volatility (realized volatility estimate of sigma; RV basis of observations with a different interval D and a different parameter in the process of ulated the model previous separately forvolatility each time maturity and  thefor Black with observation, realized volatility (realized as antoestimate of sigma; RV calculated for the previous observation, separately for each time to maturity and aging) – BRV 6; calculated for on the the previous for each time toand maturity and was calculated basis ofobservation, observationsseparately with a different interval a different • The Black model with implied volatility (IV as an estimate of sigma; IV was calculated for the eyness classes 50 different groups) – BIV. was calculated on the basis of observations withmoneyness a different interval  and agroups) different classes --each 50 different ––-BIV. previous observation, separately for TTM and MR 50 different groups) – BIV. moneyness classes 50 different groups) BIV. 6 parameter n in the process of averaging) – BRV ; 6these models in order to verify the the following statistics were calculated–for allFinally, ; parametererror n in the process of averaging) BRV the following error statistics were calculated for all these models in order to verify the Finally, the following error statistics were calculated for all these models in order to verify the

 the Black model with implied volatility (implied volatility as an estimate of sigma; IV was Finally, the following error statistics were calculated for all these models in order to verify the research hypothesis: research hypothesis:  the Black model with implied volatility (implied volatility as an estimate of sigma; IV was research hypothesis: hypothesis: calculated for the previous observation, separately for each time to maturity and

t Mean Squared for Errorthe (RMSE):  Root Error • Root mean squared errorMean (RMSE): calculated previous observation, separately for Squared each time maturity and Root Mean Squared Errorto(RMSE): (RMSE):

moneyness classes - 50 different groups) – BIV.

n 1 moneyness classes - 50 different 2 1 groups) – BIV.Finally, 1 n ( Blackfor statistics calculated all these models in order to verify(11 the RMSE o RMSE  ( Black i  MIDi ) 2 the following error (11were ) o RMSE  n ( Black ii   MID MIDii )) 2 (11)) o   i  1 n i 1 n i 1 Finally, the following error statistics were calculated for allhypothesis: these models in order to verify the research where: where: where: research hypothesis:  Root Mean Squared Error (RMSE): (midquote inprice this (midquote research study); MIDi – means the market MIDi price - means the market in this research study), i - means the market price (midquote in this research study), means theMean market price (midquote in this researchMID study),  Root Squared Error (RMSE): Black i – means the Black model price (BHV, BRV, or BIV).1 n 2

n

 ( Black Black price or BRV i  MIDi ) means the the Black Blacko model modelRMSE price (BHV, (BHV, BRV or BIV), BIV), Blackii -- means

n i 1 means the Black model price (BHV, BRV or nBIV), 1 • Mean absolute percentage error (MAPE): 2  Mean Absolute Percentage RMSE  (11) o Mean Absolute Percentage Error Error (MAPE): (MAPE):  ( Black i  MID i) n i 1 where: n Absolute Percentage Error (MAPE): n 1 n Black i − MIDi 9 n Black i  MIDi (MAPE 12 )  11in MAPE = ∑ MIDi - means the market price (midquote thisBlack research study), i  MID i MAPE  n  n 1i =1n Black MIDi MID where: MID i  1 n i 1 MIDii i i (12)(BHV, BRV or BIV), MAPE   Black means the of Black model price overprediction (OP): i - Percentage overprediction (OP): • (midquote Percentage price overpredictions (OP): n i 1 inof MID i this research study), MID - means the market price

(11)

9 ((12 12))

i

1 n  n i 1



Mean Absolute Percentage Error (MAPE):

1 n  OPi n i 1 n Black i  MIDi 1 MAPE   Mean Absolute Percentage Error (MAPE): n MID

price Blacki - means the Black OPmodel  OPi (BHV, BRV or BIV), 

(13)

where:

1 if OPi   0 if

n Black i MAPE MIDi  1  Black i  MIDi n i 1 MIDi Black i  MIDi

he Description of Volatility Processes

OP 

i 1

(13) (12)

i

1 if Black i  MIDi OPi   (12) 0 if Black i  MIDi

The Data and the Description of Volatility Processes The Data

The empirical analysis was based on high-frequency financial data for WIG20 index cal analysis was based on high-frequency financial data for WIG20 index options and WIG20 futures7, supplied by the Information Products Section of the WSE. 20 futures7, supplied by the Information Products Section of the WSE. These data covered the period from January 2, 2008, to June 20, 2008. Because of welld the period from January 2, 2008, to June 20, 2008. Because of well-

9

Percentage of overprediction (OP):

1 Option Pricing Models with HF Data: An Application of the Black Model to the WIG20 Index

74

OP 

n

 OP n i 1

i

(13)

where:

e:

1, if OPi   0, if

Black i  MIDi Black i  MIDi

Data and the Description of Volatility Processes

Data

The Data and the Description of Volatility Processes Data

The empirical analysis was based on high-frequency financial data for WIG20 index

The empirical analysis was based on HF financial data for WIG20 index options and WIG20 futures7,

7 , suppliedbybythe theInformation Information Products ons and WIG20 futuressupplied ProductsSection SectionofofthetheWSE. WSE. These data covered the period from January 2,

2008, to June 20, 2008. Because of well-known statistical problems, tick data were aggregated to 10s quotes.

e data covered the period from January 2, 2008, to June 20, 2008. Because of well-

The number of 10s bid-ask quotes for a trading day depends on the trading hours for option and futures

wn statistical problems, tick data wereThe aggregated 10-s quotes. contracts. tradingtotakes place from 9:00 a.m. to 4:30 p.m. for the time period under consideration8. Only

those quotes for which both bid and ask quotes were available simultaneously were taken into account, so that it was possible to calculate the mid quotes . These quotes were later treated as the market consensus of opandplace werefrom used fora.m. comparison with prices obtained from the option pricing models. on and futures contracts. tion The investors trading takes 9:00 to 4:30 p.m. for theoretical the time Although these quotes do not represent actual prices at which transactions take place, most researchers who Only those quotes for pricing which both bid and quotes od under consideration8.test alternative option models and ask include thewere Black-Scholes model among models tested use bid-ask quotes (mid-quotes) as these enable them to avoid microstructural noise effects (Dennis & Mayhew, 2009). In able simultaneously were taken into account, so that it was possible to calculate the mid addition, Ait-Sahalia and Mykland (2009) stated explicitly that quotes “contain substantially more informa9 tion treated regarding strategic behaviour ofinvestors market makers” as thethe market consensus of option and were and that they “should be probably used at least for es . These quotes were later comparison purposes whenever possible” (p. 592). The number of 10-s bid-ask quotes for a trading day depends on the trading hours for 9

for comparison with theoretical prices obtained from the option pricing models.

There were no corrections for outliers because of the need to show fully the properties of models tested,

even for options pricestransactions and short take TTM, which ough these quotes do not represent actual with priceslow at which place, mostare usually excluded from similar studies. Additionally, the Warsaw Interbank Offered Rate (WIBOR) interest rate (converted into 10s intervals) was used as the

archers who test alternative optionfree pricing andpricing include models, the Black-Scholes interest rate models in option and TTMmodel was calculated in seconds.

ng models tested use bid-ask quotes (mid-quotes) as these enable to avoid These operations led to complete data forthem 128 index options (65 call and 63 put options expiring in March,

June, and September). Thus, the sample period (118 trading days with 2701 observations for each day) included In addition, Ait-Sahalia and 31 871 810s observations (mid quotes, WIBOR rates, TTM, and strike prices for each option). These data were then used the process of calculation volatility parameters (HV, RV, and IV) and later on for theoretical land (2009) stated explicitly thatin quotes “contain substantiallyofmore information option valuation (BHV, BRV, and BIV models).

ostructural noise effects (Dennis & Mayhew, 2009).

rding the strategic behaviour of market makers” and that they “should be probably used at for comparison purposes whenever possible” (p. 592).

Descriptive statistics for WIG20 futures time series Table 1 summarizes the descriptive statistics for 10s interval data for continuous futures contract (with and without the opening jump effect representing the rate of return for the period between the closing of the market in the evening of the previous day and its opening in the morning of the current day – described respectively as Rf and Rf’) in order to show the distribution for the basis instrument. This distribution is shown in order to check the crucial assumption of option pricing models tested in this study; that is, the normality of returns. 10 The statistics presented below seem to confirm the belief that the distribution of HF data is not exactly normal.11


Table 1 Descriptive Statistics for Index Futures Returns (with and without Opening Jump Effect) Rf a N Mean Median Std Deviation Range Minimum Maximum Kurtosis Skewness Kolmogorov-Smirnov Jarque-Berra

Statistic p-value Statistic p-value

318717 -0.000000787 0 0.0003985 0.07847 -0.047473855 0.030991753 1369.388606 -7.31722 Test for Normality 0.3683