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Testing pricing efficiency of index options using Black-Scholes model: evidence from Indian index options market.

Introduction

Since the inception of exchange traded derivatives in June 2000 in India, the Indian derivatives market has witnessed a tremendous augmentation in the volume of trade with total number of derivative contracts traded jumped from 90,580 in June 2000 to 657,390,497 in 2008-09. This escalation in the volume of trade has been attributed to index options market, with more than 63 percent of the total derivative trading recorded under index options in 2010-11. Therefore, it is necessary that options market should perform its functions in the finest possible way because well functioning of financial markets is critical to the orderly development of any economy. Financial markets facilitate price discovery, risk hedging and allocation of capital to its most productive uses (Ackert and Tian, 2000). However, for well functioning of options market, which is a significant segment of the market, it is essential that the financial market should be efficient. Market efficiency implies that economic profits from trading are zero, where economic profits are risk-adjusted returns net of all costs (Jensen, 1978). This indicates the inability of any trader to consistently generate an above-normal average rate of return. Ackert and Tian (2000) defined efficiency of an options market, i.e., the correctness of option prices denotes that it is working well at its well-identified functions.

One of the strategies to test the options market efficiency is to evaluate market prices of options with the theoretical prices implied by Black and Scholes (1972) option pricing model. If any mispricing is observed, a riskless profit can be generated by dynamic hedging approach that consists of creating risk-neutral hedge positions i.e. buy (sell) one option and sell (buy) a fraction of the underlying stock. But, a major disadvantage of this strategy is, it engages a joint test of two hypotheses, (i) the Black and Scholes pricing model is valid and, (ii) the options market is efficient. Therefore, to test these two hypotheses in a distinguished way; we need to assume any one out of these two hypotheses i.e. model validity or market efficiency holding true.

However, it has been observed from previous researches that Black and Scholes model is reasonably efficient in explaining the option prices in several countries markets (Latane and Rendleman 1976; Mac B eth and Merville 1979; Manaster and Rendleman, 1982). Hence, we make an assumption that Black and Scholes (1972) option pricing model is valid and therefore, in this study the pricing efficiency of index options market in India is examined by using Black and Scholes model for valuing the index options. National Stock Exchange (NSE) of India also recommends Black and Scholes option pricing model for determining the theoretical price of options. The other alternative methods that test the efficiency of the options market are pure arbitrage test' (Jensen 1978), which includes test of lower boundary condition and put-call parity conditions.

Most of the studies on market efficiency are based on developed markets like, US options and futures market and European options markets, which are efficiently traded (Merton 1973; Black and Scholes 1972; Klemkosky and Resnick 1980; Evnine and Rudd 1985; Kamara and Miller 1995; Ackert and Tian 2000, 2001; Puttonen 1993; Chesney et al, 1995, Berg et al, 1996; Cavallo and Mammola 2000, and Brunetti and Torricelli 2003; Mittnik and Rieken, 2000a,b; Capelle-Blancard and Chaudhury 2001). In contrast, the Indian derivative markets are still at the developing stage in terms of volume traded. However, since 2008, the volume of trade on derivatives has jumped significantly and making Indian derivative market as a new emerging market. Although Indian derivative markets cannot be compared with US derivative markets in terms of large volume of trade, which gives enough arbitrage risk-free opportunities, the question of market efficiency in developing markets is equally important as developed markets. In fact, the new emerging markets are more susceptible to market inefficiency (Capelle-Blancard and Chaudhury, 2001). Thus, it is essential to examine the market efficiency of the Indian options market regularly.

The pricing efficiency of options market can be stated to be prevailing, when there is no significant deviation exists between the theoretical option prices obtained from the Black and Scholes model and the observed market prices of the options. However, one important feature about the Black and Scholes option pricing model is that all its key factors are observable except volatility of the underlying asset. The volatility of the underlying asset thus needs to be forecasted to determine the theoretical price of an option. The volatility can be estimated by two approaches, either it can be computed by taking the standard deviation of the historical returns (HSD) over the recent past (Taleb, 1997) or by calculating implied volatility (IV) (Latane and Rendleman, 1976) from the observed spot option prices by solving Black and Scholes pricing model. However, the precision with which these volatility estimation methods (HSD and IV) forecast theoretical option price is always a matter of controversy in practice as well as in academics. Hence, the present study assesses both the volatility methods (HSD and IV) to compute the theoretical price of an option using Black and Scholes model.

The research paper is organized as follows. In the next section, a review of literature on the market efficiency by using Black and Scholes (1 972) option pricing model has been discussed, followed by the objectives of the study, the description of the data and the procedure to test the pricing efficiency using Black and Scholes option pricing model, the data analysis and empirical evidences and lastly, the concluding observations with the implications have been presented.

Review of Literature

Black and Scholes (1972, 1973) tested the market efficiency of stock options for US market and concluded that options prices which are close to the theoretical value of options derived from option valuation formula will not provide a definite profit by dynamic hedging approach. They later empirically found that the options valuation formula is a precise estimator for predicting the variance of the returns of the underlying asset till the maturity of the contract. After the proposition of Black and Scholes options valuation model, Dan Galai (1977) conducted one of the first tests of market efficiency by identifying mispriced options using Black-Scholes model on Chicago Board of Options Exchange (CBOE). To exploit the misprice in options a hedge portfolio was constructed of undervalued or overvalued options and underlying stock. The ex-post and ex-ante hedging test generated average returns that were statistically significant from zero. Hence, Galai (1977) suggested that CBOE was not efficient during the period of his study and abnormal returns did exist. However, these average returns were generated by ignoring the transaction cost. Later on Phillips and Smith (1980) in their study showed that these abnormal average returns were completely removed when transaction cost was considered.

Blomeyer and Klemkosky (1983) further tested the efficiency of the CBOE using a hedging strategy. They used the Chiras and Manaster (1978) weighted implied standard deviation method to forecast the expected volatility of underlying asset to calculate the theoretical price of an option. The study showed that all the abnormal mean returns produced by ex-post and ex-ante hedging test turned negative when transaction cost was considered. Thus, Blomeyer and Klemkosky concluded that the option market appears to be efficient using the Black-Scholes hedging strategy. Other similar studies using the Black-Scholes hedging strategy were done by Krausz (1985) on CBOE, Castagna and Matolcsy (1982) on Australian traded options market, Mittnik and Rieken (2000) on German DAX options market, Cavallo and Mammola (2000) on Italian index option market. All these investigations suggested that the options market appears to be efficient under the consideration of transaction cost. However, Van der Hilst (1980) studied the efficiency of the European Options Exchange (EOE) using the Black-Scholes hedging strategy and found that EOE was not completely efficient under the period of study.

There were only few studies of market efficiency done on Indian options market (Varma 2002; Misra and Sangeeta 2005; Sehgal and Vijaykumar 2009; Dixit et al, 2009, 2011). Varma (2002) showed severe under-pricing of volatility and overpricing of deep in the money call options. Misra and Sangeeta (2005) examined the violations on the basis of put-call parity relationship for S&P CNX Nifty index options traded on NSE. Sehgal and Vijaykumar (2009) found that both call and put options are fairly priced in Indian options market using historical volatility measures subject to trading asymmetry condition and weighted implied volatility underestimates the option prices. Thus, showing that option pricing in India is conditionally efficient and historically volatility is a good predictor of true volatility of the underlying asset. Dixit et al (2009) found frequent violations of the lower boundary condition of option prices using spot prices of Nifty index. These violations were more prominent in case of call options than that of put options. Thus, they indicated the existence of arbitrage opportunities on account of violations of lower boundary conditions. However, these results were interpreted in the absence transaction cost as well as under the restriction of short-selling. Hence, Dixit et al (2011) made another attempt by incorporating transaction cost and using future prices. The future prices was taken in a view that futures market will help in doing away with the short-selling constraint as short position can easily be taken in futures. They concluded that majority of violations in call as well as put options cannot be exploited on account of existing market-microstructure in India.

Another important issue while testing the market efficiency of options market using Black-Scholes model is of accurate estimation of volatility of the underlying asset till the maturity of the options contract. The reason is all the key factors of Black-Scholes model are observable except volatility of the underlying asset and therefore, it needs to be forecasted to determine the theoretical price of an option. One of the commonly used methods to estimate volatility is the basic historical standard volatility estimator (HSD). However, historical standard deviation would be a good estimator of actual standard deviation provided that the actual standard deviation of the underlying asset should be constant over a period till maturity, which is one of the assumptions of Black-Scholes model. Kemna (1987) studied the practicality of this assumption on European Options Exchange (EOE) by subdividing the sample of twenty weeks into two sub-samples and found that there was a significant difference between the average implied standard deviations of the two sub-samples. This concluded that the implied standard deviation is not constant over a period of time till maturity. Thus, unfortunately the given assumption made in Black-Scholes model is not a very good approximation of actuality.

Latane and Rendleman (1976) and Chiras and Manaster (1978) showed in their studies that weighted implied volatility of the return of the stocks is a better estimator of future volatility than historical standard deviation. Chiras and Manaster regressed the actual standard deviations on the historical standard deviations (HSD) and the weighted implied standard deviations (WISD) independently, and found that R2 (Coefficient of Determination) in WISD was 0.63 as compared to 0.31 in HSD. Thus, they concluded that weighted implied standard deviation (WISD) is considerably better estimator of future volatility than historical standard deviation (HSD). Latnane and Rendleman in their study found a much stronger correlation of 0.82 between the actual standard deviation and the weighted implied standard deviation (WISD) as compared to correlation of 0.55 between the actual standard deviation and historical standard deviation (HSD). Hence, they also concluded that WISD is better predictor of future volatility. Further, Latane and Rendleman observed that certain options are more sensitive to changes in the standard deviations and therefore, these options should be given more weight than low sensitive options with respect to changes in standard deviations. Thus, a simple averaging of implied volatilities would not be an appropriate method to measure the expected volatility of the underlying asset. Latane and Rendleman, therefore proposed a weighing technique by using the partial derivative of option prices with respect to its implied volatility. Chiras and Manaster also used weighted implied standard deviation by assigning the weights to each of the implied volatilities on the basis of price elasticity of each option with respect to its implied volatility.

From the above review of literature, it seems that the abnormal profit opportunities did exist in these options market. However, some studies indicate that these profits were completely removed when transaction cost was considered. Hence, the hypothesis that markets were efficient cannot be rejected. But this conclusion of past literature does not rule out the need for further research particularly in developing markets. Keane (1983) emphasised the importance of regular investigation of financial markets in terms of efficiency. According to Keane regular scrutiny serves two purposes, firstly, it provides a continuous attestation of efficiency of financial market and secondly, it keeps a watch on the process so that any violations could be quickly identified and eliminated. The review of literature also shows that till date there is no definite prescribed method for accurate estimation of volatility for pricing an option and it remains pretty much an art rather than a science (Figlewski, 2004).

Objectives of the Study

The present paper attempts to test the pricing efficiency of S&PCNX Nifty index options traded on National Stock Exchange (NSE) India, by using Black and Scholes (1 972) model for valuing the index options.

The main objective of the study is, to examine the pricing efficiency by empirically testing the theoretical option price of S&P CNX Nifty index options obtained from Black and Scholes (1 972) model with the observed market price of S&P CNX Nifty index options.

Thus, in the light of the above objective, examine the following research questions:

* Is there any significant deviation between the theoretical option price of S&P CNX Nifty index options obtained from Black and Scholes (1972) model and the market price of S&P CNX Nifty index options

* Which one out of these three volatility measures: (i) Historical Standard Deviation (HSD) (ii) Weighted Implied Standard Deviation (^VISD), and (iii) Average Implied Standard Deviation (AISD), is the best predictor of true volatility of the underlying asset.

Data and Methodology

The data collected in this study consists of daily closing prices of S&P CNX Nifty index options contracts from April 01, 2008 to March 31, 2012. The daily closing prices of S&P CNX Nifty index spot values has also been considered for this period. As a substitute for risk-free interest rate, yield on 91-day Treasury bills has been taken for the same period to test the lower boundary condition. The data related to S&P CNX Nifty index options contracts and S&P CNX Nifty index spot values have been taken from the website of National Stock Exchange (NSE). The yield on 91-day Treasury bills has been taken for the same period to test the lower boundary condition. The data related to S&P CNX Nifty index options contracts and S&P CNX Nifty index spot values have been taken from the website of National Stock Exchange (NSE). The yield on 91-day Treasury bills has been taken from the website of Reserve Bank of India (RBI). The yields on 91-days Treasury bills are then converted into continuously compounded annual rate of return.

The main objective of the study is to examine the pricing efficiency of S&P CNX Nifty index options traded on National Stock Exchange (NSE) India by using Black and Scholes (1972) model for valuing the index options. The pricing efficiency of options market can be stated to be prevailing, when there is no significant deviation exists between the theoretical option price obtained from Black and Scholes model and the observed market price of the options. If any mispricing is observed, a riskless profit can be generated by dynamic hedging approach consists of creating risk-neutral hedge positions i.e. buy (sell) one option and sell (buy) a fraction of the underlying stock (Black and Scholes 1972; Galai 1977; MacBeth and Merville 1979). T hus, the first step towards the objective is to compute the theoretical price of the index options using the Black and Scholes model.

The Black and Scholes model is used to calculate the theoretical value of an option by utilizing five factors that affect the price of a stock option. The formula of Black and Scholes model is as follows;

European Call Option:

c = [S.sub.0]N([d.sub.1])-K [e.sup.-rT]N([d.sub.2])

European Put Option.

p = K [e.sup.-rT] N(-[d.sub.2]) - [S.sub.0] N(-[d.sub.1])

Where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The notations are. [S.sub.o] = stock price today, K = strike price, T = time remaining until expiration, expressed in terms of year, r = current continuously compounded risk-free interest rate (91 day T--bill yield rate), s--annual volatility of stock price (the standard deviation of the short-term returns over one year). The statistical terms are: ln--natural logarithm, N(x)--standard normal cumulative distribution function, e--the exponential function.

One important feature about the Black and Scholes option pricing model is that all its key factors are observable except volatility of the underlying asset. The volatility of underlying asset thus needs to be forecasted to determine the theoretical price of an option. However, as the volatility changes frequently over time, precise forecasting of volatility is a complex activity. Hull (2006) defines volatility as a measure of the uncertainty of the return realised on an asset. Statistically volatility is referred to as the standard deviation of the returns of the underlying asset from spot date till expiration date of the options contract. Volatility can be estimated by two approaches, either it can be computed by taking the standard deviation of the historical returns (HSD) over the recent past (Taleb 1997) or by calculating implied volatility (IV) (Latane and Rendleman 1976) from the observed spot option prices by solving Black and Scholes pricing model. The idea behind implied volatility is, as the current option price incorporates all the information available in the market therefore, volatility implied by current observed option price is an accurate measure of expected volatility of an underlying asset (Fontanills 2003).

However, the precision with which these volatility estimation methods (HSD and IV) forecast theoretical option price is always a matter of controversy in practice as well as in academics. The studies that support implied volatility as a true measure of future volatility are. Latane and Rendleman 1976, Chiras and Manaster 1978; Germill 1986; Shastri and Tandon 1986, Scott and Tucker 1989. According to these studies, standard deviation of historical returns is not an appropriate method as historical returns do not contain the prevailing current information about the market. Whilst implied volatility is more accurate method of estimating true volatility as it is derived from the current option value which incorporates all the market information (Figlewski 2004). However, studies like, Canina and Figlewski 1991, Day and Lewis 1992, Lamoureux and Lastrapes 1993; X u and Taylor 1995 rejects the null hypothesis that standard deviation of historical returns does not contain additional information that is already been there in implied volatility.

Thus, from the results of the previous studies, it is observed that till date there is no definite prescribed method for accurate estimation of volatility for pricing an option and it remains pretty much an art rather than a science.

Hence, the present study assesses both the volatility methods (HSD and IV) to compute the theoretical price of an option using Black and Scholes model. Further, the predictive ability of these volatility estimators will be statistically tested by Mann-Whitney-U test.

Historical Standard Deviation (HSD) Volatility Measure

One of the commonly used methods to estimate volatility is the basic historical standard volatility estimator (HSD). HSD is measured by taking the standard deviation of the moving averages of daily logarithmic returns of index values. Sehgal and Vijaykumar (2009) showed in their study that moving average of 21 day window is a better volatility estimation parameter than a window of 7 day. Thus, a moving average of 21 day period has been taken for forecasting the volatility. The HSD is defined using the following formula (Hull, 2003)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The annualized historical standard deviation = [sigma] x [square root of (252)]

Where, n = number of observations under defined moving average

[[mu].sub.i] = Daily logarithmic returns of index values

[S.sub.i] = Index Value at the end of ith interval for i = 1,2, ..., n

Implied Volatility

In practice, traders price an option by using a volatility measure known as Implied Volatility. Implied volatility is a volatility measure which is implied by the current option price observed in the market i.e. implied volatility when employed in Black-Scholes formula, will equate the market price of the option with the computed price of the Black-Scholes formula. The major difficulty in calculating implied volatility is that the Black-Sc holes pricing formula cannot be reversed, so that IV (s) is represented as a function of observed spot option price (c or p), S0, K, r and T. However, this problem can be resolved by using Newton-Raphson method with the help of Visual Basic Application (VBA) software. Here, the option pricing model is solved backwards with the help of VBA software, i.e. volatility is fed using Newton-Raphson method so that it must equate the pricing equation to observed market price of the option.

According to Black-Scholes model, the implied volatility of the option class must remain same at any given time. However, this is not the case in practice, the implied volatility tends to vary across different strike prices and different expiration dates on the same underlying asset. This brings another crucial problem for the estimation of implied volatility. This is known as implied volatility smile. In implied volatility smile a sneer appears because implied volatility decreases monotonically as the strike prices and the maturity rises (Dumas et al, 1998). However, this issue can be resolved by applying the principle of averaging on implied volatilities of different strike prices and different expiration dates. The two averaging methods that are commonly used for estimation of implied volatility are. (i) Average Implied Standard Deviation (AISD), and (ii) Weighted Implied Standard Deviation (WISD).

Average Implied Standard Deviation (AISD) Volatility Measure

Kerruish (1984) and Chiras and Manaster (1978) proposed a simple way to resolve the problem of volatility smile by summarizing the implied volatilities of all the option class using simple arithmetic mean on a particular given date.

AISD = [N.summation over (j=1)] [ISD.sub.j]/N

Where, N = the number of options recorded under an option class on a given date, [ISD.sub.j] = implied standard deviation of option j.

Weighted Implied Standard Deviations (WISD) Volatility Measures

One major limitation of AISD method is equal weights have been assigned to each of the implied volatilities under the option class on a given date. However, in practice, some options with particular strike price and time to maturity are traded frequently in the market and the re by these options are expected to incorporate more valid information than the less traded options. Further, Latane and Rendleman (1976) observed that certain options are more sensitive to changes in the standard deviation and therefore, these options should be given more weight than low sensitive options with respect to changes in standard deviations. Thus, a simple averaging of implied volatilities would not be an appropriate method to measure the expected volatility of the underlying asset. Therefore, another method used to compute expected volatility is by weighted average of implied volatilities proposed by Chiras and Manaster (1978). Chiras and Manaster (1978) assigned the weights to each of the implied volatilities on the basis of price elasticity of each option with respect to its implied volatility. According to them the price elasticity of options is defined as percentage change in option price with respect to percentage change in implied volatility, this is also known as VEGA. VEGA is one of the important Greek letters, which is widely used for hedging purpose in derivatives. The equation for WISD proposed by Chiras and Manaster (1978) is as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where, N = the number of options recorded under an option class on a given date, Wj = the current option price of the index, ISDj = implied standard deviation of option j, (dWj x ?j / d?j x Wj) = price elasticity of option j with respect to its implied standard deviation (?j).

Data Analysis and Empirical Evidences

Now the authors are in a position to calculate the theoretical price of an options using Black-Sc holes model by considering all the above volatility estimates for a given day. However, as the volatility is computed from observed spot option prices, therefore, cannot employ these current volatility estimates directly to compute the theoretical price of an option using Black-Scholes model on the same given date. The reason is these volatilities are not available before the transaction occurs. Instead authors use these volatilities as a forecast of an expected volatility for calculating next day's theoretical price of an option using Black-Scholes model i.e. authors use previous day s volatility as an expected volatility for the current date. Thus, after obtaining the ex-ante measure of volatility of the underlying asset the theoretical option price of an option has been computed using Black-Scholes model.

The objective of the present study is to find the significant deviation between the theoretical price of an option and the market price of an option and also to observe that which one out of the three volatility measures HSD, WISD, and AISD is the best predictor of true volatility of the underlying asset. Therefore, after calculating the theoretical prices of options, authors are now interested in comparing the theoretical prices of options obtained from three different volatility estimates with the market price of options by applying Student's t-test. (Hereafter, call theoretical price of an option obtained from Black-Scholes model as Model Price). But Student's t-Test test has an underlying assumption that the samples are drawn from normally distributed population. Now to test this assumption Kolmogorov-Smirnov test (K-S test) has been used. From the p-value of K-S test shown in Table-I, reject the null hypothesis (i.e. H0: sample population is normally distributed) at 5 percent level of significance. The deviation from normality of the sample data is further supported by the skewness and kurtosis values, which are not close to the value zero as shown in Ta ble-1 . Thus, the sample data has not been drawn from normally distributed population and therefore Student's t'test test cannot be applied. So, a non parametric test Mann-Whitney-U will be applied.

After running the Mann-Whitney-U test on the model prices and the market prices of the options, obtained the results which are depicted in Table-II. It can be observed from the results of the significance test that there is no significant difference between the mean ranks of model prices of options and market prices of options at 5 percent level of significance when call options are considered. But, when put options are considered the mean ranks of model prices (HSD) and model prices (AISD) show a significant difference from the mean rank of the market prices of options at 5 percent level of significance. However, there is no significant difference between the mean rank of model prices (WISD) and the mean rank of market prices of options at 5 percent level of significance. Thus, from the above findings it can be suggested that Black-Scholes model is good predictor of market price when call options are considered. Whereas, the Black-Scholes model overall suffers in closely predicting the market price when put options are considered, except model prices (WISD), which shows insignificant difference when compared with the market prices of put options. However, this does not mean that Black-Scholes model lacks its predictive ability when put options are considered or put market is inefficient. The significant deviation may be because of trading cost and other trading limitations like, short selling restrictions. Although, Securities Lending and Borrowing (SLB)" mechanism is allowed by Securities and Exchange Board of Indian (SEBI) but, it is not very popular among the market players. Further short selling is restricted for foreign institutional investors. This creates trading asymmetry in the market. Moreover, the accuracy of estimated volatility can be a matter of question, as there is no definitive way of measuring volatility exists so far in the literature. Therefore some deviation between model price and market price should not be a matter of surprise.

As our second objective is to identify the best predictor of true volatility of the underlying asset out of the three volatility measures HSD, WISD, and AISD, we compare the absolute difference of mean rank of each model price of options with the market price of options. The results are shown in Table-III. However, the absolute differences of mean rank of options are inconclusive in determining the best volatility predictor because model prices (AISD) performs best by showing minimum absolute mean rank difference in call options whereas model prices (WISD) shows minimum absolute mean rank difference in put options. But one important thing to notice from the results is model prices (HSD) shows maximum absolute mean rank difference in both the call and put options. Thus, to conclude we finally take the arithmetic mean of absolute difference of mean ranks by clubbing the call and put options. From the results shown in Table-III, it can be observed that there is not much of difference in the arithmetic mean of absolute mean rank difference, based on WISD and AISD. However, among the three volatility estimators, the arithmetic mean is minimum in case of WISD and therefore, we can suggest that during the period of the study WISD volatility measure is the best predictor of true volatility of the underlying asset applied in Black-Scholes model.

Conclusions

The present paper attempts to test the pricing efficiency of S&PCNX Nifty index options traded on National Stock Exchange (NSE) India , by using Black and Scholes (1972) model for valuing the index options. The data collected in this study consist of daily closing prices of S&PCNX Nifty index options contracts from April 01,2008 to March 31, 2012.

From the findings of the study, it can be suggested that the Black-Scholes model is a good predictor of market prices of call options by utilizing the following volatility estimators: (i) Historical Standard Deviation (HSD) (ii) Weighted Implied Standard Deviation (WISD), and (iii) Average Implied Standard Deviation (AISD). Whereas, the Black-Scholes model overall suffers in closely predicting the market prices of put options, except when utilizing WISD volatility estimator. However, this does not mean that Black-Scholes model lacks its predictive ability when put options are considered or put market is inefficient. The significant deviation may be because of trading cost and other trading limitations like, short selling restrictions that create trading asymmetry in the market.

Further, the mean rank analysis is done to identify the best predictor of true volatility of the underlying asset out of the three volatility measures; HSD, WISD, and AISD. The results show that the absolute differences of mean rank of options are inconclusive in determining the best volatility predictor. Thus, to conclude, the arithmetic mean of absolute difference of mean ranks has been taken by clubbing the call and put option prices. The results suggest that during the period of the study WISD volatility measure is the best predictor of true volatility of the underlying asset applied in Black-Scholes model.

Thus, the study indicates that despite the trading asymmetry Black-Scholes model is a good predictor of option prices and WISD volatility measure is the best estimator of volatility of the underlying asset in Indian options market. Therefore, it can be suggested that options traded in Indian options market from the period April 01, 2008 to March 31, 2012 were efficiently priced. The present study is very important for National Stock Exchange (NSE), Securities and Exchange Board of India (SEBI), brokerage houses and institutional investors, as the study suggest that options are efficiently priced in Indian options market. Further, as the pricing in the options market is aligning with the theoretical principles, it facilitates the mechanism of price discovery in the underlying market. Lastly, the study attempts to contribute to the literature on market efficiency of index options particularly in the case of Indian index options.

There are some issues that can be considered as a further scope of research on this topic. First, as the study focuses on regular investigation (Kean, 1983) by testing the pricing efficiency of Indian index options market during its most active period in terms of volume of trade from 2008, therefore, the test of pricing efficiency of Indian index options market is restricted to limited time frame. However, a more comprehensive test of pricing efficiency is made possible by considering an entire business cycle from April, 2008, which can be considered as an extended future scope of the present research. Second, issue is to circumvent the trading asymmetry problem. This can be made possible by using future prices instead of spot prices in Black-Sc holes option pricing model. Futures will help in overcoming the problem of short-selling as short position can easily be taken in futures market.

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P. K. Priyan

Professor

G.H Patel Post Graduate Institute of Business Management, Vallabh Vidyanagar, Sardar Patel University.

Debaditya Mohanti

Assistant Professor

S.K Patel Institute of Management and Computer Studies, Gandhinagar.
Table I

Summary of Kolmogorov-Smirnov Statistics

                       MOD        MOD        MOD      MOD
                      (WISD)     (AISD)     (HSD)    (WISD)
                       Call       Call      Call      Put

Mean                  240.16     240.65    262.53    343.26
Median                101.74     103.84    114.66    183.79
Std. Deviation        362.79     361.06    383.56    442.38
Skewness               2.76      2.7 7      2.60      2.10
Kurtosis               9.37       9.46      8.21      4.90
Kolmogorov-Smirnov     0.25       0.25      0.25      0.22
Sig.                   0.00       0.00      0.00      0.00

                       MOD        MOD     MKT Call   MKT Put
                      (AISD)     (HSD)
                       Put        Put

Mean                  340.96    324.12     239.00    340.24
Median                179.80    148.65     95.88     172.00
Std. Deviation        442.45    445.22     363.81    443.98
Skewness               2.11      2.13       2.75      2.16
Kurtosis               4.93      5.00       9.27      5.15
Kolmogorov-Smirnov     0.22      0.23       0.26      0.22
Sig.                   0.00      0.00       0.00      0.00

Note: 1. MOD indicates "Model Price" of Black-Scholes model
using a particular volatility estimates shown in the parenthesis
2. MKT indicates the observed "Market Price" of options.

Table II

Summary of Mann-Whitney U Statistics

                   Number of     Mean Rank   Mann-Whitney U
                  Observations

MOD (WISD) Call      10044        9987.5        49870000
MKT Call             10044        10101.5

MOD (AISD) Call      10043       10022.41       50220000
MKT Call             10043       10064.59

MOD (HSD) Call       10043       10107.35       49790000
MKT Call             10043        9979.65

MOD (WISD) Put       10017       995 2.38       49520000
MKT Put              10017       10082.62

MOD (AISD) Put        9999        9898.29       48980000
MKT Put               9999       10100.71

MOD (HSD) Put        10065        9727.57       47250000
MKT Put              10065       10403.43

                  Wilcoxon W   Asymp. Sig.
                               (2-tailed)

MOD (WISD) Call   100300000       0.16
MKT Call

MOD (AISD) Call   100700000       0.61
MKT Call

MOD (HSD) Call    100200000       0.12
MKT Call

MOD (WISD) Put     99690000       0.11
MKT Put

MOD (AISD) Put     98970000       0.01
MKT Put

MOD (HSD) Put      97910000       0.00
MKT Put

Note: 1. MOD indicates "Model Price" of Black-Scholes model
using a particular volatility estimates shown in the parenthesis
2. MKT indicates the observed "Market Price" of options.

Table III

Summary of Mean Rank Analysis of Call and Put Options

                  Mean Rank    Absolute Mean    Arithmetic Mean
                              Rank Difference   of Absolute Mean
                                                Rank Difference
                                                of Call and Put
                                                    Options

MOD (WISD) Call    9987.5           114              122.12
MKT Call           10101.5

MOD (WISD) Put     9952.38        130.24
MKT Put           10082.62

MOD (AISD) Call   10022.41         42.18             122.3
MKT Call          10064.59

MOD (AISD) Put     9898.29        202.42
MKT Put           10100.71

MOD (HSD) Call    10107.35         127.7             401.78
MKT Call           9979.65

MOD (HSD) Put      9727.57        675.86
MKT Put           10403.43

Note: 1. MOD indicates "Model Price" of Black-Scholes model
using a particular volatility estimates shown in the parenthesis
2. MKT indicates the observed "Market Price" of options.
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Author:Priyan, P.K.; Mohanti, Debaditya
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