The quest for the best option pricing model is at least 40 years old, but going back into the past, we could find its traces even few centuries earlier (e.g., the speculation during tulipomania or the South Sea bubble).

The futures option pricing model (Black 1976) began a new era of futures option valuation theory. The rapid growth of option markets in the 1970s

The Chicago Board of Options Exchange was founded in 1973 and it adopted the Black-Scholes model for option pricing in 1975.

brought rapidly a lot of data and stimulated an impressive development of research in this area. Quite soon, numerous empirical studies put in doubt basic assumptions of the Black model: they strongly suggest that the geometric Brownian motion is not a realistic assumption. Many underlying return series display negative skewness and excess kurtosis (see Bates 1995, Bates 2003). Moreover, the implied volatility calculated from the Black-Scholes model often vary with the time to maturity of the option and the strike price (cf. Rubinstein 1985, Tsiaras 2009). These observations drove many researchers to propose new models that each relaxes some of those restrictive assumptions of the Black-Scholes model (Broadie and Detemple 2004, GarciaDetailed analysis of the literature (An and Suo 2009, Andersen

Our motivation for this paper is to check the results of Kokoszczyński

The WIG20 is the index of twenty largest companies listed on the Warsaw Stock Exchange (further detailed information may be found at

The complex comparison of Black model with different volatility assumptions presented only for an emerging market is definitely not enough to formulate conclusions of a more general nature. Therefore, we have decided to compare the results for the Polish emerging market with a similar research for the developed Japanese market. For this purpose, we choose the Nikkei 225 index option market (European style), which can be regarded as one of the most important option markets in the world, especially when we consider the level of its innovation and complexity. As a result, we hope we will be able to suggest some more general conclusions.

After a thorough analysis, we can say that the literature regarding the Japanese capital market and especially European style index options, is not so rich and this is our second motivation to write this paper. The reason for this can be that Nikkei 225 index is the basis instrument for many different derivatives that are quoted on many different exchanges and the literature is widely dispersed. We can easily find some papers focusing on pricing American style options or options quoted in different currency than Yen. On the other hand, the papers in English focusing on the European style Nikkei 255 index options are not so numerous.

Unfortunately, because of language barrier we were not able to extend our literature review to papers written in Japanese.

The literature on American style options shows quite good results for the Black model (Raj and Thurston 1998), sometimes better than for various GARCH models (Iaquinta 2007). When we consider the second case (options in other currencies), we actually model not only option prices but the exchange rate fluctuations as well (Wei 1995); thus, the comparison of their results with ours could be regarded as not valid.Therefore, we are left with the very limited number of studies that focus on the European style options or otherwise touch this subject, sometimes only in an indirect way. Li (2006) shows that Nikkei 225 is rather an efficient market (in the sense of lack of arbitrage possibilities analysed through the existence of put-call parity). Yao

This review, covering those Nikkei 225 index options studies that are comparable with our approach, justifies quite strongly the positive assessment of the BSM model. We are going to check this by using the high-frequency data from 2008.

The structure of this paper has been planned in such a way as to answer the following detailed questions:

Which model from among those we test can be treated as the best one?

Can we observe any distinctive patterns in option pricing taking into account moneyness ratio (MR) and time to maturity (TTM)?

Can we distinguish any patterns of liquidity behaviour in a developed market using transactional data?

What is the proper measure of average pricing error? Is it robust to large errors that are likely to emerge when analysing HF data?

Is there any substantial difference between the results for a developed (this paper) and an emerging market (Kokoszczyński

The remainder of this paper is organized as follows. The second section describes some methodological issues. Next section presents data and the fluctuations of volatility processes derived from transactional data. The fourth section discusses the liquidity issues. Results are presented in section five and the last section concludes.

The basic pricing model we choose is the Black-Scholes model for futures prices, that is, the Black model (Black 1976). We call it further in the text the BHV model – the Black model with historical volatility. Below are formulas for this model:

where:

where _{f}

There are two reasons why we decided to use the Black model instead of the standard Black-Scholes model. First, we are able to relax the assumption about continuous dividend pay-outs.

In this way, we are able to eliminate two possible source of pricing error: the necessity to estimate the dividend yield and the assumption about continuous payouts.

Second, we can use additional data because usually derivatives (options, futures, etc.) are quoted much longer than the basis instruments (e.g., Nikkei 225 index).To further justify such an approach, we assume that we can price a European style option on Nikkei 225 index applying the Black model for futures contract (with historical, realized and implied volatility), where Nikkei 225 index futures contract is the basis instrument. This is possible due to the following facts:

Nikkei 225 index futures expire exactly on the same day as Nikkei 225 index options do, the expiration prices

are set exactly in the same way, we study only European-style options; hence, early expiration – like in the case of American options – is impossible.

Early expiration of American-style option could result in the significant error in the case of such a pricing, because of the difference in prices of index futures and of Nikkei 225 index before the expiration date (the basis risk).

One of the most important issues about option pricing is the nature of an assumption concerning the specific type of volatility process. Therefore, we check the properties of the Black model with three different types of volatility estimators: historical volatility, realized volatility and implied volatility, and additionally, we use the Heston model and the GARCH option pricing model. Below we provide a brief description of each of these volatility estimators and models.

The historical volatility (HV) estimator is based on the formula:

where:

_{i,t}

_{i,t}

_{Δ} – number of Δ intervals during the stock market session

In this research, we use N_{Δ} = 1, and hence, the HV estimator is simply standard deviation for log returns based on the daily interval. This approach is commonly used by the wide range of market practitioners.

The second approach is the _{Δ}.

The

We divide 320 options (160 call and 160 put options) into 5 money-ness ratio classes and 5 time-to-maturity classes. The details of this classification are presented in Section 3.

Hence, for each observation, we have 50 different IV values (5 × 5 × 2). These values are then treated as an input variable for volatility parameter in calculations of the theoretical options price for the Black model with the implied volatility (BIV) for the next observation.Before entering into the formula for the Black model, the HV and RV estimators have to be annualized and transformed into standard deviation. The formula for the annualization of the HV estimator is as follows:

Contrary to the HV estimator, which is based on information from many periods (

Having all these volatility estimators and additionally the Heston and GARCH (1,1) option pricing models we present below, we study several types of option pricing models, which will be described in details in section 2.5.

Many classical option pricing models (e.g., the Black model) assume the constant level of volatility of log-returns of basis instruments. However, in reality, many financial time series are characterized by time varying volatility. GARCH models are one possible way to relax this initial assumption. They were proposed by Engle (1982) and Bollerslev (1986). GARCH model describes the dynamic of returns of the basis instruments with following equations:

_{t}

Finally, the GARCH models are also used in the option pricing models. Duan (1995) presents the methodology of European style call option pricing with the assumption that returns of the basis instrument can be described with the GARCH process. In order to become risk neutral in this approach, we differentiate between physical and martingale (risk free) probability measure. Garcia and Renault (1998) describe theoretical aspects of using GARCH models in risk hedging strategies, while Ritchken and Trevor (1999) use GARCH models in the American style option pricing applyingtrinomial trees. Duan

Option pricing based on GARCH model has been done here according to Duan (1995). This approach assumes that log returns undergo GARCH-M(

where parameters are denoted in the same way as in earlier formulas, and additionally,

The pricing of options are conducted assuming local risk-neutral valuation. It requires modification of log returns processes in such a way that the conditional variance one step ahead remains unchanged and simultaneously conditional expected return equals risk-free rate (Fiszeder 2008). Introduction of risk-neutral probabilistic measure

The dynamic of basis instrument log returns with respect to measure

The formula describing the dependence between the price of basis instrument on maturity day and its price in the time of pricing can be described:

while the price of European style call option is described by the discounted value of the option price on the maturity day:

where ^{Q}

In practice, the pricing is done through Monte Carlo simulation. In the first stage, we estimate the parameters of the model (14), (15) and (16), and then on the basis of (17), (18), (19), and (20), we simulate N realization of basis instrument price. Call and put option prices are then calculated in the following way:

We use GARCH-M(1,1) model in this study.

In the results section we will refer to this model as to GARCH(1,1).

Many research papers show that this order of the model defines the dynamics of stock index returns in the most adequate way (Hansen and Lunde 2004 or Zivot 2008). Similarly, like in the case of the BSM model, we use index returns. We estimate the parameters of the equations (14), (15) and (16) on the basis of data from 1/1/2007 until the moment of option pricing. As a result, the size of sample used to estimate the GARCH parameters varies from one year (for pricing done on 2nd January, 2008) to 1.5 year (for pricing done on 30th June, 2008). In order to eliminate problems with instability of GARCH model parameters, we have decided to delete overnight returns from our data sample.The number of replications in Monte Carlo simulations is another important choice to be made. Finance literature suggests strongly that N = 10,000 gives an adequate precision of estimates. However, due to the very large number of pricing (5-minute data) we need, we have to limit the number of replication to N = 1000. In order to minimize possible negative effects of that choice, we use two popular variance reduction techniques: antithetic variables sampling and control variates. The data we use in this study are described in detail in section 3.

Log returns volatility in stochastic volatility models is represented by a given stochastic volatility process with dynamics set a priori. Hull and White (1987) are among the pioneers of applying stochastic volatility for option pricing. They assume that variance dynamics can be described with the following differential equation:

Under additional assumption – that volatility is not correlated with the basis instruments – Hull and White present the analytical formula for European style call option. One of the main conclusions of their research is that the BSM model systematically underestimates the prices of ITM and OTM options, and overestimates the prices of ATM options.

The constant volatility assumption is responsible for this drawback of the BSM model.

The Heston model we used in our research is an extension of Stein and Stein (1991). Their option pricing formula assumes that volatility is described by the Ornstein-Uhlenbeck process and is not correlated with the basis instrument. On the other hand, Heston (1993) presents the call option pricing formula with no assumption on correlation of volatility with the basis instrument. His model assumes that the dynamics of underlying asset price _{t}_{t}

where {_{t}_{t}_{≥0} and {_{t}_{t}_{≥0} indicate the price and the variance of the basis instrument, and _{t}_{t}_{≥0} is a mean reverting process, with long memory expected value θ and mean reverting coefficient

One of the main reasons, why the Heston (1993) model has become so popular is the fact that it is possible to obtain its closed-form solution for the European style call option pricing for an asset not paying dividend, which is given by:

where

for

Formula 28 is not difficult to implement in practice. The only problem is to calculate the limit of the integral therein. This limit is often approximated by an adequate quadrature (Gauss-Legendre or Gauss-Lobatto), what can be done in many statistical software packages.

Practical implementation of the Heston model is done in two stages. First, we have to calibrate the model in order to find its parameters from equation (25)(26) and (27) Calibration can be done on the basis of call transactional prices observed in every one-hour interval. We choose parameter values in such a way as to minimize the difference between market and theoretical prices. Next, we use formulas (28) and (29) to calculate theoretical prices.

The calibration of the Heston model can be conducted in two ways – via global or local optimization. Global optimization guarantees that we find the true global minimum of our target function. The disadvantage of this method is that it is time-consuming and the parameters obtained here tend to be very unstable. On the other hand, local optimization gives only local minima but it is very fast and the parameters derived in this way are stable.

In our study, the global optimization is used for the first period and its results are the starting point for the local optimization in the second period. Then, the further iterations of the local optimization are being performed, for which the starting point is set to the local minimum from the previous stage.

In the second stage, the parameters found previously are used to calculate the theoretical prices in the next hourly interval. The prices of call options are calculated according to the formulas (28) and (29), while put option prices are found on the basis of call-put parity:

where _{t}_{t}_{t}_{f}

The calibration of the Heston model in our study for the Japanese market has been done on the basis of an hourly interval. It means that in the time of calibration, we use transactional prices from the previous hourly interval, and then we use those results to price options for the current interval. The calibration of the Heston model was based on all the available transactional prices in one-hour interval.

Measuring option pricing error

To assess the accuracy option pricing models, we compare the option transactional prices with the theoretical prices obtained from each model. To measure the average pricing error, we use the median absolute percentage error (MdAPE). Since the distribution of errors is relatively strongly positively skewed, we claim that it is better to use the median rather than the mean value to express the average pricing error. The MdAPE statistic is defined as:

where

We also calculate the percentage of overprediction (OP) in order to see whether a given model on average over- or under predicts the transactional price of an option:

where _{i}_{i}_{i}_{i}_{i}

Both statistics were calculated for all the models, for different TTM and MR classes, and for both call and put options.

Thus, we study the properties of the following models:

BHV – the Black model with historical volatility (

BRV – the Black model with realized volatility (realized volatility as an estimate of

BIV – the Black model with implied volatility (implied volatility as an estimate of

Heston – the Heston option pricing model

GARCH – GARCH (1,1) option pricing model based on the Duan’s methodology.

Initially, we calculate the BRV models with four different Δ values: 5, 10, 15 and 30 minutes. Then, we check the properties of averaged RVs with different values of parameter n in pricing models. We find, like Kokoszczyński

Our choice is confirmed by results in Sakowski (2011) and Kokoszczyński

We use transactional data

Some papers that test alternative option pricing models and include the Black-Scholes model among models tested therein use instead of transactional data bid-ask quotes (midquotes), as they allow to avoid microstructural noise effects (Dennis and Mayhew 2009). Ait-Sahalia and Mykland (2009) state explicitly that quotes ‘contain substantially more information regarding the strategic behaviour of market makers’ and they ‘should be probably used at least for comparison purposes whenever possible’ (p. 592). On the other hand, Beygelman (2005) and Fung and Mok (2001) argue that midquote is not always a good proxy for the true value of an option.

for Nikkei 225 index options, Nikkei 225 index and Nikkei 225 index futures, which have been provided by the Reuters companyThanks to the financial support of the government, we were able to buy all the necessary data (5 minutes intervals) from Reuters Datascope company.

The market for Nikkei 225 index option started in this period at 1.00 CET and ended at 7.00 CET

In practice, the market session lasted from 1.00 CET to 3.00 CET, then there was a pause, and later session lasted from 4.30 CET to 7.00 CET. Therefore, we get 56 5-minutes intraday returns.

. For that reason, we have 6745 observations (122 session days with 56 5-minutes intervals eachSome days, close to the most important national holidays, the market session finished before 7.00 CET.

As a result, our data set for Nikkei 225 index options comprises transactional prices for 160 call options and 160 put options maturing in January, February, March, April, May, June and July 2008.

Maturity days of these options and their symbols for each call and put series are as follows: 11.01.2008 (

The results of our analysis will be presented with respect to 2 types of options, 5 classes of MR and 5 classes of TTM:

2 types of options (call and put)

5 classes of moneyness ratio, for call options: deep OTM (0–0.85), OTM (0.85–0.95), ATM (0.95–1.05), ITM (1.05–1.15) and deep ITM (1.15+), and for put options in the opposite order

Moneyness ratio is usually calculated according to the following formula:

where _{f}

5 classes for time to maturity: [0–15 days], [16–30 days], [31–60 days], [61–90 days], [91+ days).

This categorization allows us to compare the different pricing models along several dimensions.

The number of transactional prices, theoretical prices and pricing errors are presented in Tab. 2 and Fig. 5.

The descriptive statistics for Nikkei 225 index returns for samples with and without opening jump effects

sample with opening jump effects | sample without opening and mid-session jump effects | ||||
---|---|---|---|---|---|

N | 6745 | 6504 | |||

Mean | -0,000025394 | -0,000014111 | |||

Median | 0,000032644 | 0,000036116 | |||

Standard Deviation | 0,0030907 | 0,0028429 | |||

Minimum | -0,0319108 | -0,0319108 | |||

Maximum | 0,0216127 | 0,0216127 | |||

Kurtosis | 10,4364219 | 12,7560437 | |||

Skewness | -0,6227228 | 0,7206586 | |||

Kolmogorov-Smirnov | Statistic | 0,093349 | 0,086497 | ||

Jarque-Berra | Statistic | 30995,9195 | 44584,7971 |

Number of theoretical premiums for different classes of MR and TTM for BRV model*

option | moneyness | 0–15 days | 16–30 days | 31–60 days | 61–90 days | 91+ days | Total |
---|---|---|---|---|---|---|---|

deep OTM | 372 | 4327 | 27089 | 23799 | 10494 | 66081 | |

OTM | 6501 | 11635 | 22572 | 19567 | 8959 | 69234 | |

ATM | 8199 | 9681 | 17385 | 12141 | 5368 | 52774 | |

ITM | 3880 | 4510 | 5373 | 1484 | 761 | 16008 | |

deep ITM | 1205 | 1935 | 3032 | 1044 | 1335 | 8551 | |

total CALL | 20157 | 32088 | 75451 | 58035 | 26917 | 212648 | |

deep OTM | 6964 | 20580 | 44831 | 31225 | 7768 | 111368 | |

OTM | 6109 | 8142 | 15466 | 12674 | 5631 | 48022 | |

ATM | 8028 | 9669 | 17014 | 12001 | 6413 | 53125 | |

ITM | 4278 | 4826 | 7427 | 1790 | 1096 | 19417 | |

deep ITM | 2411 | 3002 | 3098 | 1161 | 1962 | 11634 | |

total PUT | 27790 | 46219 | 87836 | 58851 | 22870 | 243566 | |

47947 | 78307 | 163287 | 116886 | 49787 | 456214 |

*456 thousand for BIV, Heston and GARCH(1,1) model and 445 thousand for BHV

We begin our study with the basic analysis of the time series of returns of the basis instrument. Tab. 1 presents the descriptive statistics for 5-minute interval data. They are calculated for two samples: with and without opening jump effects.

By

Both samples have high kurtosis and are asymmetric. The distribution for the full sample has negative skewness, while removing jump effects makes the distribution right skewed. Overall, both Jarque-Bera and Kolmogorov-Smirnov statistics indicate that returns in both samples are far from normal. Nevertheless, we observe interesting feature that – contrary to the data from the Polish market (Kokoszczyński

Fig. 1 and Fig. 2 additionally confirm this observation showing high negative and positive returns in both time series with and without jump effects. Formally, the lack of normality of the basis instrument means that the standard BSM model should not be applied for option pricing with these data. Accordingly, we transform this model varying its assumption about the nature of the volatility process. Moreover, we also

apply the Heston and GARCH option pricing models to the same data.

We consider three different volatility measures: historical, realized and implied volatility for the Black option pricing model and – in addition to that – stochastic volatility and GARCH model. Obviously, the volatility process assumed in pricing is one of important reasons for differences among theoretical option prices we compare.

In the case of the historical volatility estimator N _{Δ} = 1 for every ri,t (daily log returns) and Ci,t in formulas (5), (6) and (7). Moreover, we use the constant value of parameter n being equal to 63, because we want to reflect historical volatility from the last three trading months.

On the basis of similar studies for the Polish market (Kokoszczyński

On the other hand, the implied volatility time series exhibits substantially different trajectories than RV or HV time series. Fig. 4 presents how IV estimates (for ATM call options) evolve in time.

IV estimates for ATM put options show similar pattern.

Similar to KokoszczyńskiLiquidity constraints are typical features of an emerging derivatives market and they put severe limits for conducting such a study as we have done for the Polish market. To make our comparisons of both markets (WIG20 and Nikkei 225) as comprehensive as possible, we have also decided to present a detailed discussion of developed market liquidity on the example of the Nikkei 225 index option market with respect to (1) the number of transactional prices available in the sample, (2) the volume and (3) the turnover of option transactions.

The number of transactional prices is shown in Tab. 2 and Fig. 5. Their distribution suggests that the activity of market participants (measured by the number of single trades and not by their volume) concentrates on call ATM, OTM and deep OTM and put deep OTM option with the TTM between 16 and 90 days.

On the other hand, observing the distribution of volume for call options, presented in Fig. 6 (left panel), we notice that the highest volume is observed for the MR

equal to ATM and OTM for the TTM up to 60 days. The lowest volume we see for the low TTM and the MR equal to ITM, deep ITM, and deep OTM. This suggests that investors rarely trade highly valued options (deep ITM and ITM) or options with the long TTM.

The distribution of volume for put options (Fig. 6, right panel) is very similar. The only difference is that the volume is also high for deep OTM options with the TTM less than 60 days. However, this is mostly due to the fact that put options are used as an insurance against sharp downward movement of the basis instrument.

One buys the right to sell the basis instrument in the case of an extreme financial catastrophe, e.g., financial crash, for a relatively low cost (put option premium).

Generally, we could say that the volume distribution for call and put options is very similar and that investors focus their trades on low-valued options with the short TTM.Fig. 7 addresses the liquidity issue from another perspective by focusing on the turnover volume that increases the importance of traded options’ value. We observe significant shift from deep OTM to ATM and then to OTM options. It obviously means that most investors involved in the option trades concentrate in the ATM-OTM range. The same results are observed for call and put options with only slightly higher turnover volume for put options. However, this latter feature can be tied to the behaviour of the basis instrument in the period we study.

We observe sharp downward movement of Nikkei 225 index in the time of research.

The most important outcome from the liquidity analysis is that we can indicate where the volume of options concentrates. We notice that after dividing the set of options into different MR and TTM classes, we can distinguish options with the low TTM, which are ATM, OTM or ITM that cumulate more than 90% of the total volume, both in case of call and put options. Similar situation has been observed in the case of the Polish market. Finally, it is worth noticing that the accumulation of volume in the given class of the MR and the TTM is partly conditional on the availability of options with the specified MR or TTM ratio.

We divide this section into two subsections containing results presented separately for call and put options (section 5.1), and the comparison of joint results with respect to different dimension (section 5.2). This enables us to present a multidimensional comparative analysis of option pricing models.

For all the available transactional and theoretical prices, we calculate two pricing error measures (MdAPE and OP), separately for six different pricing models (1. Heston, 2. GARCH(1,1), 3. BRV5m, 4. BRV5m_63, 5. BHV, and 6. BIV). The discussion of our results is based on two-dimensional charts containing five panels (Fig. 8 to 11). Each panel presents results for the separate MR class. Values of statistics have been joined with dashed or solid lines for a given TTM class.

Fig. 8 presents MdAPE statistics for call options. We can observe that the Black model with the implied volatility estimator (BIV) has the smallest average pricing errors for the majority of option classes. Slightly higher errors we got for the Heston model, but on the other hand, it is the best model for ATM options with the TTM less than 60 days. In the next place, depending on the given option class, we can rank BRV5m_63, BRV5m and BHV model (despite relatively large errors for deep OTM for the latter one). The worst results are observed for the GARCH(1,1) model with very large pricing errors for OTM options with time to maturity below 15 days.

Moreover, for all the models, we can observe distinctive relationship between average pricing errors and MR and TTM classes. Values of MdAPE statistics decline when we go from deep OTM through deep ITM options. The influence of TTM classes looks a little bit different. For deep OTM and OTM options, pricing errors are higher for short times to maturity, whereas for ITM and deep ITM options, we observe higher errors for longer times to maturity. Smallest differences between all the models, and simultaneously, most precise theoretical premiums we obtain for deep ITM options with time to maturity below 15 days.

Fig. 9 – with OP values for call options – indicates that the BIV model (number 6) is the best one. It is characterised by almost the same level of over- and underprediction (the value of OP is approximately equal to 0.5). Results for other models differ. The second best models according to this metric are the Heston model (number 1) and the BHV model. On average, GARCH, BRV5m and BRV5m_63 models underpredict the market values of options, especially for ATM, ITM and deep ITM options. Nevertheless, we can see that results – with the exception of the BIV model – vary strongly with changes in the TTM.

MdAPE statistics for put options are shown in Fig. 10. The results are similar to those for call options. Once again we see that the best model is BIV. The Heston model performs only slightly worse. On the next places, we can rank BHV, BRV5m_63 and BRV5m. The highest average pricing errors are produced by the GARCH model.

Moreover, we can observe the same relationship between average pricing errors and MR and TTM classes as previously described for call options. Errors are significantly smaller for the higher-valued options (ITM and deep ITM), while the effect of TTM classes depends on the MR class. For deep OTM and OTM options, we get smaller pricing errors for longer times to maturity, whereas for ITM and deep ITM options, the errors are smaller for shorter times to maturity.

Fig. 11 with OP statistics for put options confirms the ranking of models derived from results for call options. The BIV model is the best one, then the Heston model is the second one and as the third one, we have the BHV model. The results for models other than the BIV depend strongly on TTM classes. We also observe strong underestimation of market prices for all the models with the only exception of the BIV model.

Finally, it has to be emphasized that our results are based on all the available theoretical prices of the analysed options. We did not remove any observations from the original sample. In order to omit the problem of possible outliers among the pricing errors, we used median (absolute percentage) error, which is a robust measure to the possible large deviations of errors from their average value. We argue that this approach, as opposed to removing from the sample problematic observations (low-valued options or options with few days to maturity), is a better solution of the problem of outliers, since it allows analysing models’ properties in all the option classes.

In this closing subsection, we present our conclusions in a more formal way. Fig. 12 presents the frequency of best pricing for all the tested models in 5 diagrams for each moneyness ratio for call and put options together. Our initial conclusions from section 5.1 are confirmed here by this aggregated approach. BIV is clearly the best model, the Heston model is the next one, and the third one is BHV. Additionally, we see that the Heston model seems to behave much better for OTM, and especially for ATM options, while the BIV model is the worst model for ATM and then for ITM options. Finally, we noticed that the BRV and the GARCH models are the worst models for every MR.

Fig. 13 shows next the frequency of best pricing for all the tested models, but for each TTM class for call and put together. The BIV model is – as expected – the best one, and the Heston model is ranked as the second one, the BHV model follows. Additionally, we see that the BIV model gains on efficiency, while the Heston model worsens its performance when we go from the lowest TTM to the highest one. Other models do not change their performance with respect to the TTM class.

The final Fig. 14 that presents the frequency of best pricing for all the tested models with respect to the type of options obviously does not change the model ranking. However, we see a very interesting pattern concerning the two best models. The BIV model performs much better for put options, while the Heston model is better for call options. Here again we do not observe any significant differences for other models.

In this study, we present a detailed analysis of option pricing models’ performance using 5-minutes transactional data for the Japanese Nikkei 225 index options. We compare 6 different types of option pricing models: the Black model with different assumptions about the volatility process (BRV – two cases, BHV, BIV), the Heston model and the GARCH model. Then, we present detailed error statistics describing how efficient in option pricing are the models we test. Furthermore, we focus on the analysis of liquidity for option market in order to better understand different behaviours of options within various classes of the TTM and the MR. Here, we try to summarize our conclusions from this study and we formulate some thoughts concerning further research.

First of all, when we consider the performance of models we have tested, the model ranking, from the most efficient to the least efficient one, is as follows: BIV, Heston, BHV, BRV5m_63, BRV5, and GARCH(1,1). The BIV model comes out as the best in majority of option classes. The model of Heston occurred to be only slightly worse. Next places, with similar results, belong to the Black model with historical volatility and the Black model with realized volatility averaged across last 63 trading days. Average pricing errors for the Black model with realized volatility (not averaged) were higher, due to the more volatile estimates of RV compared to HV estimates. We obtained the worst results for the GARCH(1,1) model. Generally, these results confirm the previous findings for the Polish and Brazilian emerging option markets (Kokoszczyński 2010a, Sakowski 2011). The only exception is the Heston model, which performed significantly worse for the less developed markets. The probable reason is that calibration of the Heston model is strongly dependent on the number of options with different maturities. Nevertheless, to some extent, we can claim that this model ranking is not only a feature of an individual market, but can also be regarded as robust to the level of development, liquidity or various other market characteristics.

Secondly, for both call and put options, we observe the clear relation between average pricing errors and option moneyness: high error values for deep OTM options and the best fit for deep ITM options. We can explain this pattern by noting that highly valued options (ITM or deep ITM) are relatively better priced because of the more active participation of market makers and institutional investors in this market segment, where we do not observe strong under- or overreaction to new information as it happens with individual investors. The concentration of liquidity for low-valued options with short maturities may mean high error for options that are traded more frequently. Such error distribution can explain higher interest of speculative investors for deep OTM and OTM options, where information noise, responsible for larger departure of transactional prices from the theoretical ones, is of greater importance. All these outcomes confirm our previous results for the Polish WIG20 index option market (Kokoszczyński 2010a).

Thirdly, focusing on parameter n (RV averaging parameter) for BRV models, we observe that much lower error values are obtained for n = 63 than in the case of the non-averaged RV, what confirms our initial hypothesis that the non-averaged RV estimator (Fig. 3) is rather a poor choice considering the efficiency of option pricing model. This is the confirmation of results presented in the literature on the efficiency and accuracy of various volatility estimators (Ślepaczuk and Zakrzewski 2009).

Fourthly, we would like to focus on two models with the most time-consuming estimation process (the Heston model and GARCH models). Results we have presented earlier make us doubt whether there is any gain from using them, especially in the case of the GARCH model, which comes out as the worst one, when better models are formally much less complicated, and additionally, less time-consuming in the process of estimation.

Analysing the liquidity issues, we observe several interesting features of the Japanese index option market data. First of all, the volume of calls and puts concentrates in ATM, OTM and deep OTM options, with hardly any volume noticed for deep ITM and ITM options. Secondly, the turnover volume peaks around ATM and ITM options, indicating that the highest (in terms of transaction value) liquidity is observed for ATM options, and then for ITM options. Thirdly, the liquidity – however measured – is significantly higher for put options. Nevertheless, we are aware of the fact that the latter conclusion could result from the sharp downward movement of the market in the time we study and the high demand for put options for hedging purposes.

This final observation shows clearly how important are the liquidity issues for patterns we get while comparing performance of various option pricing models. They should be certainly the subject of further studies. Our intention is thus to conduct a similar study for other markets.

There are suggestions in the literature that notwithstanding unrealistic assumptions of the BSM or the Black model, they can produce results of the same quality than much more sophisticated models do. Our paper constitutes an argument supporting this opinion, because superiority of this model is shown for majority of option classes.

#### Number of theoretical premiums for different classes of MR and TTM for BRV model*

option | moneyness | 0–15 days | 16–30 days | 31–60 days | 61–90 days | 91+ days | Total |
---|---|---|---|---|---|---|---|

deep OTM | 372 | 4327 | 27089 | 23799 | 10494 | 66081 | |

OTM | 6501 | 11635 | 22572 | 19567 | 8959 | 69234 | |

ATM | 8199 | 9681 | 17385 | 12141 | 5368 | 52774 | |

ITM | 3880 | 4510 | 5373 | 1484 | 761 | 16008 | |

deep ITM | 1205 | 1935 | 3032 | 1044 | 1335 | 8551 | |

total CALL | 20157 | 32088 | 75451 | 58035 | 26917 | 212648 | |

deep OTM | 6964 | 20580 | 44831 | 31225 | 7768 | 111368 | |

OTM | 6109 | 8142 | 15466 | 12674 | 5631 | 48022 | |

ATM | 8028 | 9669 | 17014 | 12001 | 6413 | 53125 | |

ITM | 4278 | 4826 | 7427 | 1790 | 1096 | 19417 | |

deep ITM | 2411 | 3002 | 3098 | 1161 | 1962 | 11634 | |

total PUT | 27790 | 46219 | 87836 | 58851 | 22870 | 243566 | |

47947 | 78307 | 163287 | 116886 | 49787 | 456214 |

#### The descriptive statistics for Nikkei 225 index returns for samples with and without opening jump effects

sample with opening jump effects | sample without opening and mid-session jump effects | ||||
---|---|---|---|---|---|

N | 6745 | 6504 | |||

Mean | -0,000025394 | -0,000014111 | |||

Median | 0,000032644 | 0,000036116 | |||

Standard Deviation | 0,0030907 | 0,0028429 | |||

Minimum | -0,0319108 | -0,0319108 | |||

Maximum | 0,0216127 | 0,0216127 | |||

Kurtosis | 10,4364219 | 12,7560437 | |||

Skewness | -0,6227228 | 0,7206586 | |||

Kolmogorov-Smirnov | Statistic | 0,093349 | 0,086497 | ||

Jarque-Berra | Statistic | 30995,9195 | 44584,7971 |

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