INTRODUCTION:
The Black-Scholes option pricing model, a Partial differential equation, is one of the most significant theories the area of financial engineering. They had published their paper in 1973 titled “The pricing of option and corporate Liabilities” in the Journal of Political Economy. The financial derivative tools are particularly developed for minimizing the impact of associated risks. Hence, Black-Scholes option pricing theory is used to price financial derivatives and develop hedging strategies to minimize the impact of risks written on European-style options.
Trading in derivative products is one of the most important principal opportunities of an effective securities market. The Black-Scholes option pricing model is a landmark in the history of Financial Derivative. This preferred model provides a closed analytical view for the valuation of European-style options. An option is a standardized financial contract, which gives the buyer the right, but not the obligation, to buy or sell specified quantity of the underlying assets, at a strike price on or before the expiration date. The underlying may be physical commodities like crude oil, wheat, rice, cotton etc. or financial instruments like equity stocks, equity index, bonds etc. There are two types of option- put and put option. The put option gives the buyer the right but not the obligation to buy whereas the put option gives the right but not the obligation to sell the underlying. Option allows people to bet on the future events and to reduce the associated financial risk. There are two kinds of options- American options and European options. The former may be exercised any time before its expiration date while the later can be exercised on its expiration date. The stock option contracts are priced by the Black-Scholes model which exhibits pricing errors on several occasions.
BLACK-SCHOLES MODEL:
Based on the above mentioned assumptions, Fisher Black and Myron Scholes have developed the following equation for the valuation option for no dividend paying stocks (Hull, 2007)-
P = X.e-rtN (-d2) - S.N (-d1)
Where,
In(S/X) + (r + 0.5б2) t
d1 =----------------------------
б√t
In(S/X) + (r - 0.5б2) t
d2 =----------------------------
б√t
The variables are-
P is the Put Price of stock option, S0 is the Current Stock price, X the is Exercise price, T the is Time remaining until expiration, expressed as a percent of a year, r the is Current Continuously compounded risk-free interest rate and б is the Standard deviation of stock’s return.
The unknown parameter of this model is б. The Black-Scholes Model says that the option price, no matter it is put or put, is a function of asset price, time to maturity, exercise price, compounded risk free interest rate and volatility of asset price. All those variables except for the volatility are easily obtainable from the market. б is the only unknown factor in the formula. б is assumed unchanged while calculating option prices. б is calculated through two approaches- historical volatility and implied volatility. The historical volatility is calculated by the annualized standard deviation of historical daily returns. The historical approach is much simpler than the other one. The implied volatility looks more on the future movements.
REVIEW OF LITERATURE:
A number of researches have been carried out by the researchers to judge the pricing accuracy of the Black-Scholes model. Blattberg and Gonedes (1974) studied the impact of volatility, as one of the basic parameters on the option price provided by the Black-Scholes model and concluded that volatility of the underlying stock is stochastic and random. Black (1975) had also identified that this model suffers from the pricing errors.
Some researchers have also examined the affect of option maturity. Time to maturity has also effect on the calculation of option prices under the Black-Scholes model. MacBerth and Merville (1979), using implied volatility instead of historical volatility, find that implied volatility is high for in-the-money options but the Black-Scholes model underprices these in-the-money options and secondly implied volatility is low for the out-of-the-money options but this model considerably overprices these out-of-the-money options.
Rubinstein (1985) has examined the implied volatility on the 30 options classes (Chicago Board of Option Exchange) for a period of two years (from August 1976 to August 1978) and found that the short maturity options had higher implied volatility than long maturity options.
However, some researches show that discrepancies between the market option prices and prices calculated under the Black-Scholes model are not large enough to be exploited. LauterBach and Schultz (2012) on pricing warrants, Jordan and Seale (1986) and Blomeyer and Boyd (1988) on futures options written on treasury bond have suggested that there is a very little difference between the market actual price and the Black-Scholes predicted price. Bailey (1987) studied on future option written on gold, Shastri and Tandon (1986) on Future (American options), and Jordan, Seale, McCabe and Kenyon (1987) on futures options written on soyabean have found discrepancies are not enough in the model predicted prices.
Varma (2003) has studied volatility, using data for a short period of time from June 2001 to February 2002, on Nifty Future and options prices under the Black-Scholes model. He suggests that the volatility is severely mispriced because of the imperfection of the Indian market and market is learning and the impact of learning effects can be seen over a long period of time.
Ramazan Gencay and Aslihan Salih (2003) compare the Black-Scholes model against the Feedforward Networks Model using S&P 500 option Index data from January1998 to December 1993. They suggest that the Black-Scholes Model exhibits pricing error at several occasions especially for the deeper out-of-the money options compare to the near out-of-the-money options and this pricing error worsens with increased in volatility. Hence, Feedforward networks provide less pricing error as compare to the Black-Scholes model for the deeper out-of-money options.
Rinalini Pathak Kakati (2006) studied the effectiveness of Black-Scholes option pricing model in the Indian context using 2342 put and 1280 put options written from July 2001 to March 2003. She found that the Black-Scholes model misprices options considerably on several occasions. Pricing errors are negative on an average and significantly different from zero. She further suggests that mispricing worsens with both increased in moneyness and increased in the volatility of the stocks. The Black-Scholes model, according to Rinalini Pathak Kakati (2006), overprices short-term options and underprices long-term options. Mckenzie, Gerace and Sbedar (2007) have studied the pricing errors produced by the Black-Scholes model using ASX 200 option Index and suggested that the use of a jump-diffusion approach and implied volatility instead of historical volatility increases the tail properties of the underlying lognormal distribution. Consequently, it increases the pricing accuracy of the Black-Scholes model1. Rubinstein (1994) studied extensively on implied volatility under the Black-Scholes model for S&P 500 option Index and states that the Black-Scholes model may under price options because the tail properties of underlying lognormal distribution are very small.
Emilia Vasile and Dan Armeanu (2009) have worked on the mispricing errors produced by the Black-Scholes Model for pricing options contracts. The operators take into consideration the moneyness of an option and the duration up to the due term thereof, when they calculate the volatility on account of which they evaluate the option. This is a direct consequence of the fact the Black-Scholes model cannot be applied in its original form: the prices of the financial assets do not follow log normal distribution.
Tripathi & Gupta (2010) examined the predictive accuracy of the Black-Scholes (BS) model in pricing the Nifty Index option contracts by examining whether the skewness and kurtosis adjusted BS model of Corrado and Su gives better results than the original BS model. They have also examined whether volatility smile in case of NSE Nifty options, if any, can be attributed to the non-normal skewness and kurtosis of stock returns. Based on data of S&P CNX NIFTY near-the-month put options for the period January 1, 2003 to December 24, 2008, their results show that BS model is misspecified as the implied volatility graph depicts the shape of a Smile‟ for the study period. There is significant under-pricing by the original BS model and that the mispricing increases as the moneyness increases. Even the modified BS model misprices options significantly. However, pricing errors are less in case of the modified BS model than in case of the original BS model. On the basis of Mean Absolute Error (MAE), they concluded that the modified BS model is performing better than the original BS model.
Subrata Kumar Mitra (2012) found that the total error in Black model was less than the Black-Scholes model. During his study, the mean error of the Black was 1.76 and the mean error of Black-Scholes was -10.49. The result was significant with the value of p = .000. He compared NIFTY Future Price with the theoretiputy calculated value that includes cost of carry and found that stock Index Futures sometime suffer from a negative cost of carry bias. He has not compared Future prices with the spot prices to see the deformities.
Sarbapriya Ray, (2012) carefully studies the Black-Scholes Model and makes a more detail analysis of the assumptions of the model. He states that the drawback to Black-Scholes is that it is based on simplistic assumptions such as constant volatility and a normal distribution function for the underlying asset return. This causes to substantial discrepancies between market prices and prices calculated under the Black-Scholes model. These discrepancies between market and prices calculated under the Black-Scholes model are most evident in the observation of different implied volatilities according to exercise prices and maturities. He further states that Inspite of several loopholes in Black Scholes option pricing model, there are several reasons for wide use of this model. The most important one is that the concepts behind the Black-Scholes analysis provide the framework for thinking about option pricing. All the research in option pricing since the Black-Scholes analysis has been done either to extend it or to generalize it. Another important reason for studying the Black-Scholes theory is that the financial world uses it as a standard. In fact, traders quote Black-Scholes volatility to each other, not the actual price of the options. However, Black-Scholes model gives very good approximations to the prices of options.
V. Panduranga (2013) using Banking sector stocks on testing the efficiency of the Black-Scholes Model found that the model is applicable for banking sector stocks in India. Paired sample T-test results indicate that this model can be applied for banking stock options. However, he has observed that in one out of four cases, there is a difference between expected price and market price of the option. Options may be under-priced or overpriced in the market.
Nilakantan and Jain (2014) found in their study in the context of Indian Stock market that the Black-Scholes model suffers from various deficiencies. They inferred that modified B-S Model is not able to produce efficient results for NIFTY index option in case of At-the-money, Out-of-the Money and Deep Out-of-the-Money options. Their paper aimed at applying the corrections suggested by Corrado-Su to the Black Scholes Model using Gram-Charlier (CG) expression for option pricing in Indian market. There is a definite improvement with the Corrado-Su modified formula for equity option pricing, as compared to the original B-S formula with implied volatility. The improvement in the case of NIFTY index option pricing is n not much and also not significant. The Corrado Su adjustments seem to work well with equity options but not with index options. Nagendran and Venketeswar (2014) have worked on 95,000 equity call options to test the validity of the Black-Scholes model in written on Indian National Stock Exchange. They found that there is an improvement in Black-Scholes model. Their results show the robustness of the Black-Scholes model in pricing stock options in India and that is further improved by incorporating implied volatility into the model.
OBJECTIVES OF THE STUDY:
The primary objectives of this research paper are:
a) To investigate the INDEX Nifty50 Put options pricing errors calculated under the Black-Scholes Model.
b) To investigate the Money and Maturity biasness of INDEX Nifty50 Put options calculated under the Black-Scholes Model.
PROPOSED HYPOTHESIS:
The study proposes to test the following hypothesis to meet the objectives of study:
H01 : The Black-Scholes Model do not produce pricing errors for the calculation of INDEX Nifty50 Put options, i.e., Mean pricing bias is zero.
H02: The Black-Scholes Model do not produce pricing bias for the Moneyness and Maturity, i.e., Mean pricing bias of Moneyness and Maturity are zeros.
RESEARCH ASSUMPTIONS:
(a) Research paper has tested the equity INDEX Nifty50 Put options quoted in the Indian National Stock Exchange (NSE) only.
(b) The historical Nifty50 Put option prices are considered as the closing prices.
(c) The Nifty50 Put option prices used are reliable and accurate.
DATA COMPILATION AND ANALYSIS:
This research paper examines 2826 put option contracts written on underlying INDEX Nifty50 during 2008 and 2010. For this purpose sample consists of closing prices of 2826 Put options (observations), written on underlying INDEX Nifty50 of Indian National Stock Exchange, have been collected from the website of exchange www.nseindia.com. To bring the uniformity among the data and to reduce the effect of the time gap in closing prices, this research paper considers only highly traded options and which are expected to trade until the last moment on the concern stock exchange.
Table 1- Year wise No. of observation
|
Observations/Year |
Jun.2008-May 09 |
Jun.2009-May 10 |
Jun.2010-May 11 |
Jun.2011-May 12 |
Total No. of Observations |
|
No. of Observations |
678 |
706 |
726 |
716 |
2826 |
(b) Types of Data: Secondary data of Put option have been collected and used for the purpose of the calculation of the theoretical predicted premium prices as well as for the standard deviation of the stock option. (c) Period: This research paper examines 2826 Put option prices written on underlying INDEX Nifty50 for a period of 4 years (from Jun. 2008 to May. 2012).
(d) Sources of data: The primary sources of data, time, contract month, option types, strike price and closing prices etc. are, for the purpose of this study paper have been taken from the Indian Stock Exchange website www.nseindia.com (assessed from Jun. 2008 to May. 2012). The risk free interest rate used in this analysis is the 90 days T. Bills of RBI (Reserve Bank of India) website www.rbi.org.in (assessed from Jun. 2008 to May. 2012).
(e) Processing of data: The collected data have been entered under the data base and processed under the Microsoft Excel for the final calculation of the standard deviation and theoretical prices of the stock option under the Black-Scholes model. The calculated prices of put options are also cross checked by the software FUTOP version 2.0.0 used for the calculation of option prices.
(f) Time to expiry: In the above mentioned formula term‘t’ is the time left for the option to expire. Here, calendar days has been used to calculate‘t’, irrespective of intervening holidays. Further,‘t’ is annualized by dividing ‘t’ by 365 days.
(g) Volatility: Volatility is a measure of the uncertainty on the return provided by the INDEX Nifty50. Here in the Black-Scholes model Volatility means standard deviation of the continuous compounded return. In this research researchers have used Hull’S suggested formula. Hull (2004, page no. 263) has suggested following formula to calculate annual volatility-
Volatility Per Annum = Volatility per trading day x √No. of trading days per annum.
(h) The pricing accuracy- The pricing accuracy of stock options provided by the Black-Scholes are compared by the Mean Error (ME), Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Thiel’s Inequality Coefficient:
(i) Mean Error (ME): ME is calculated by adding all error and dividing total error by the number of observations-
N
Mean Error = 1/N ∑ en
N=1
The result is acceptable when the all error data have the same sign (either all are positive or all are negative).
(ii) Mean Absolute Error (MAE)
The mean absolute error value is the average absolute error value. The closer this value is to zero, the better is the forecast. The neutralization of positive errors by negative errors can be avoided in this measure. MAE is computed using the formula-
n
MAE =1/N ∑ |en|
N=1
(iii) Mean Squared Error (MSE): Mean squared error is computed as the average of the squared error values. This is the commonly used error indicator in statistical fitting procedures.
As compare to the mean absolute error value, this measure is very sensitive to large outlier as it places more penalties on large errors than mean absolute error. MSE is computed using the following formula-
n
MSE = 1/N ∑ en2
N=1
(iv) Root Mean Squared Error (RMSE):
It is the square root of mean squared error and conceptually similar to the widely used statistic
called- Standard Deviation-
n
RMSE = √ 1/N ∑ en2
N=1
(v) Thiel’s U statistic (Inequality coefficient):
Henri Theil (1961) developed an inequality coefficient for measuring the degree to which one time series differs from another. Theil’s U statistic is computed as under-
Here, the two time series in question are (a) the actual value of options (yn) and (b) the value of option predicted by the models (fn). Thiel’s U will equal 1 if a forecasting method is found no better than using a naive forecast. If Thiel’s U is less than 1 it indicates that the method is superior to a naive forecast. A value close to zero indicates a good fit, whereas, value greater than 1 indicates that the technique is actually worse than using a naive forecast.