INTRODUCTION

The research paper entitled ‘Options and Corporate Liabilities’ by Black and Scholes (1973) was a trend-setting and revolutionary publication, which reoriented the trading of options and the price mechanism. The work was an extension of the earlier research work by Bachelier (1900), Samuelson (1965 and 1952) and Merton (1973). Previous research explicitly displays that the constant volatility-based Black–Scholes model for option pricing is very simple and elegant; however, post the 1987 crash, practitioners find that it exhibits some systematic pricing biases with respect to strike price and time to maturity. Thereafter, two important assumptions of the Black–Scholes model, namely, constant volatility and log normal distribution (Fama, 1965), were empirically challenged by a host of researchers. Enforced by wrong distributional assumptions and its implications, researchers focused on determining the alternative model defining the volatility smile aligned with non-lognormal distributional assumptions of Black–Scholes (Amin and Victor, 1993a, 1993b; Derman and Kani, 1994; Duan, 1996; Backus et al, 1997; Heston and Nandi, 2000). When implied volatility (volatility obtained by inverting the Black–Scholes model is known as implied volatility) is plotted against the time to maturity and moneyness (ratio of stock price and strike price), it manifests systematic pricing bias and formed a ‘smile’ or ‘skew’ pattern (theoretically it should remain neutral for all maturity-moneyness on the same underlying assets). Academicians and researchers forced upon the concept of the variants of Black–Scholes model to capture the smile smirk phenomena. The parabolic shape of the volatility smile, and its dependence on moneyness and maturity, has motivated Dumas et al (1998) to model implied volatility as a quadratic function of moneyness and maturity: they called it ad hoc Black–Scholes model. The same was also analyzed and improved upon by Christoffersen and Jacobs (2004) and they named it Practitioner Black–Scholes model. The Deterministic Volatility Function (DVF) approach has been extensively studied in Monte Carlo settings by Dumas et al (1998) and Pena et al (1999), Heston and Nandi (2000) and Christoffersen and Jacobs (2004). In recent times, the model has been evaluated by many. Ahoniemi and Lanne (2009) tested joint modeling of call and put implied volatility. Andreou et al (2010) generalized the parameter functions of DVFs. Berkowitz (2010) provided a detailed analysis on justifications for the uses of ad hoc Black–Scholes method in option pricing. Since its inception, the DVF approach of option pricing has been empirically tested to prove its effectiveness (Brandt and Wu, 2002; Becker and Clements, 2008; Constantinides et al, 2009). They all conceptually agreed upon that the DVF approach outperforms complex and theoretically sound approaches of the rest.

The research as projected by Christoffersen and Jacobs explains that the performance of DVF is higher when it has quadratic combinations of time to maturity and strike price. However, with the addition of the higher-order terms (cubic and quadruple), performance of the same deteriorates with the order. Considering this specific nature of DVFs, we have not included the higher-order variants of it for modeling the volatility smile. In order to explain the parabolic shape of volatility smile/smirk pattern, we have identified a set of DVFs of Dumas et al; besides that, we have done its restructuring and examined its performance empirically along with frequent updating of DVF parameters.

In order to substantiate this research work, we compared and contrasted the log normal distribution of return of Nifty 50 with its global counterparts such as ASX 200, HANGSENG, FTSE 100, Nikkei 225 and S&P 500. Figure 1 clearly displays that the return distribution of Nifty is most unique: it exhibits highest deviation on either side of mean. Thus, it provides a concrete platform for analyzing the pricing bias of Black–Scholes amid distributional characteristics of Nifty 50 causing smile effect. This article further investigates out-of-sample forecasting performance of the classical Black–Scholes model along with DVF adjustments. The core and foremost objective of this research is to accurately model the volatility smile. The article is also an attempt to empirically examine the price performance of the benchmark Black–Scholes model with DVF (implied volatility) as an input parameter. Directly or indirectly this research work also gives an informative view on whether the discrepancies in option prices occur as a result of the market mispricing or because of incorrect theoretical assumptions of the Black–Scholes model.

Figure 1
figure 1

Log non-normal distribution of return of global indices.

Therefore, the objective of this article is to provide further empirical evidence on the performance of the Black–Scholes and its DVF variants, in Indian context, utilizing Nifty index options of India. This article also includes an evaluation of the Black–Scholes model with DVF variants for predicting the price of Nifty index options traded heavily on the bourse of National Stock Exchange (NSE), one of the prominent stock exchanges of Asia. In 2012, NSE was ranked as the world’s third-largest derivative exchange by contract volume, putting it below Eurex and above NYSE EuronextFootnote 1. The volume of index option has grown exponentially over the years, and now it accounts three-fourth (75 per cent) of the total turnover of the F&O segment traded on NSE. Figure 2 depicts the same.

Figure 2
figure 2

F&O turnover statistics of NSE.

In order to find out the best researched model, we used the technique of Error Metrics with the help of which we managed to give a systematic comparison and contrast of the Black–Scholes model (with constant volatility assumption, that is, DVF 0) and its DVF variants, relative to market. The hypothecated DVF model, which remains in the question of this study, has been put into practical implication to testify the most turbulent financial vicissitudes. This particular phase rows as an extreme of phenomenal unpredictability ranging the high and low tides of financial flux. The model becomes more dominant because it deals with the extreme range of wide highs and lows of Nifty index movements (Figure 3 displays the same).

Figure 3
figure 3

Nifty movement.

The remainder of the article is organized as follows: the next section explains the volatility smile of Nifty index options. Then the data and source are described. The section after that demonstrates the research methodology. The subsequent section discusses the Black–Scholes and DVFs. After that, the structural parameter estimation procedure used to compute the implied volatility of Black–Scholes and DVF is described. The penultimate section provides the empirical testing results, and the last section finally concludes.

VOLATILITY SMILE

The Figures 4 and 5 depict a correlation of implied volatility with strike price and time to maturity. It is important to know that all the figures clearly indicate the existence of volatility smile pattern, not captured by Black–Scholes. Figures 4 and 5 depict typical volatility smile skewed pattern; implied volatility skewed pattern is initially very steep, but it flattens very rapidly with time. Figures 4 and 5 are obtained from calculating implied volatilities across a range of strike prices and time to maturities for Nifty index option series. This research work attempts to model volatility smile pattern of Nifty index options through a set of quadratic DVFs.

Figure 4
figure 4

Nifty implied volatility smile/smirk. (Contract date 21 January 2008; Expiry date 31 January 2008).

Figure 5
figure 5

3-D smile/smirk surface of Nifty-implied volatility. (Contract date 21 January 2008; Parameters: S=5208.8, K=4700: 5700, r=7.1 per cent, q=0, T=10: 95 days).

DATA AND SOURCE

Whatever methodology we adopt, in the absence of suitable data, no research is possible or believable in terms of conceptual approach. For this research, we banked on official websites of NSE (www.nseindia.com) and Reserve Bank of India (www.rbi.org.in) to collect and collate the historical data of Nifty index options, Nifty index (underlying asset) and risk-free interest rate. As we conducted a survey, the period between 1 January 2007 and 31 December 2009 was full of various vicissitudes, tumultuous and full of unpredicted ups and downs affecting globally, comprising financial services, products and strategy. This particular period rows an extreme of phenomenal unpredictability ranging the high and low tides of Nifty and thus provides the most apt situation for the modeling of volatility smile of the model in question. During this period, Nifty marked its zenith, high of 6357.10 on 8 January 2008, but soon thereafter it fell to the dirt of crises in the same year; on 24 October 2008, it fell to its multi-year low 2525.05. But thereafter, it rebounded and went up to the level of 5000 by the end of 2009. Although Nifty managed to recover from its bear mode very swiftly, in 2009 other global indices were struggling hard to recover.

Therefore, we collected data of this specific duration because this phase could provide us the most tested upon time for our research. The following data have been collected for the option contracts: Option type (only European call options), index price, strike price, time to maturity and risk-free rate of interest (equal to 91 days Treasury bill rate issued by Government of IndiaFootnote 2). In order to examine the various interplays of these components and their dynamics on the pattern of cause and effect, we have collected data on a daily basis. For the purpose of this research, we have analyzed the data of call options only. To reduce the computational stress, and as the put options only replicates the price characteristics of call options, we have not included them in the study.

Data screening procedure

Before putting the data into the practical use, the same have been filtered through a set of filtering conditions. This will help in extracting the data that are either highly illiquid or highly sensitive to the pricing of options, thus not of our use. Filtration of such data is essential, if not done carefully they could distort the study. The specific identified methods that have been rigorously involved into the filtration process are the call option prices taken from the market to be checked for lower boundary condition

where S is the current underlying asset price, K is the strike price, r is the risk-free interest rate, C(S, t) is the market call price at time t and t is the time to maturity in year. Call prices not satisfying the lower boundary condition is considered as an invalid observation, and hence they are discarded. As very deep out-of-the-money (DOTM) and very deep in-the-money (DITM) options are less actively traded on NSE, their price quotes may not reflect the true option value. Therefore, data of call moneyness, ((S/K)−1) greater/less than +15 per cent/−15 per cent, are excluded. Option strikes with less than 3 days of maturity are very sensitive to the underlying asset price and its volatility, thus ruled out amid price bias. After applying the rigorous filtering process, the data figure in 15 053 call options. Tables 1 and 2 exhibit the filter statistics of Nifty index option.

Table 1 Filter statistics of Nifty index call options
Table 2 Nifty index call option statistics for the year 2007–09 (after filtration)

Option categories

After having been through the process of skimming of filtered out data, we categorized them in 15 different sections based on time to expiration and ratio of asset price to strike price. Then, we further investigated and examined the same by applying them in the three different frameworks to confirm the concept in question.

Considering the fact that option prices are very sensitive to their exercise prices and time to maturity, for the purpose of this research work, we divide the option data into 15 categories according to moneyness (five) and time to maturity (three). Defined moneyness categories are as follows:

The extent to which a call option is in-the-money is measured as: M i =(S i K i )/K i where M i is the moneyness, S and K are the index and strike price of ith observation, respectively. While three ranges of time to expiration are categorized as:

METHODOLOGY

This research attempts to find out the parameters of the DVFs on analytical basis, and compare their forecasting performance for pricing day-ahead option prices, specifying corresponding market values. Concentrating to provide methodical fundamental to substantiate the concept and to establish apt techniques and tools, this article adopts a common method for evaluating the performance of option pricing models. The method involves statistical calculation of the Error Metrics measuring its deviation, oscillating between market and model. To determine how well Black–Scholes and its DVF version performs, we correlated it with the relative error generated by them. The Black–Scholes and its DVF versions are computed using implied volatility as an input (obtained analytically by optimization techniques). The article then focuses on the effectiveness of the classical Black–Scholes and deterministic DVF models.

Measuring model performances

In order to establish an effective outcome and maintain the simplicity of its usability, we adopted a common method that encompasses calculation of Error Metrics. To make it more definite and to justify fundamentally the concept that how well DVFs perform, we analyzed their relative errors. To find out the relative performance of DVFs we banked upon the following two error metrics:

Mean Percentage Error (MPE)

Absolute Mean Percentage Error (AMPE)

where C i Model is the predicted price of the option, and C i Market is the actual price for observation i. Whereas positive/negative relative MPE means that the model overprices/underprices the specific option, small (large) AMPE provides deviation between model and market price in absolute sense. Small (large) relative deviation/error means that the model provides a good (poor) approximation to the market.

OPTION PRICING MODELS

The classic Black–Scholes option pricing model

Chronologically, the scientific approaches to value financial instruments are being attempted since 1900. The model introduced by Black and Scholes (1973) is just an extension of the previous works of Bachelier (1900), Wiener (1938), Levy (1948), Itˆo (1951) and Samuelson (1965). In the series, the French mathematician Bachelier (1900) was the first who succeeded in value options based on the concept of random movement of stock price (further conceptualized as Brownian motion by Elbert Einstein (1905)). Despite the fact that the assumptions made in the benchmark Black–Scholes formula are unrealistic in nature, the model is still used by all the stock exchanges of the world to fix the base price of the European style call and put options. Its popularity is tremendous and unchallengeable so far. The model of Black–Scholes is famous because of its closed-form solution, analytical tactility and simplicity. The Black–Scholes formula for pricing European call option on an index/stock paying no dividends is

where

C denotes the price of a call option, S denotes the underlying Index price, K denotes the option exercise price, t is the time to expiry in years, r is the risk-free rate of return, N(d 1 ) and N(d 2 ) are the standard normal distribution functions, and σ2 is the variance of returns on the Index. However, the Black–Scholes model is designed to price European call options only; put options can also be analyzed utilizing famous put-call parity equation

where C and P represent prices of European call and put options, respectively, S is the price of the underlying asset (index or stock) and PV(K) is the present value of the strike price K.

Deterministic volatility functions (DVFs)

Modeling of dependence of implied volatility on strike price and time to maturity is useful because it provides estimates of volatility for a combination of moneyness and maturity, which is not available in observed option prices (Christoffersen et al, 2009). However, if a set of observed option prices contained continuum information of strike prices and time to maturity, there would be no need for the DVF functions. To explain the parabolic smile/smirk pattern of Black–Scholes implied volatility, Dumas et al (1998) modeled implied volatility as DVFs with strike price and maturity adjustments. The model was also analyzed and improved upon by Christoffersen and Jacobs (2004). Dumas, Fleming and Whaley furnished modeling of implied volatility, through various DVFs. For comparing and contrasting variants of DVFs, we will simply extract implied volatility from a cross-section of option prices with varying strike price and time to maturity. For modeling the volatility smile, we have diagnosed the following specifications of DVF displaying quadratic combinations of strike price and time to maturity, with and without multiples:

Where σ IV = Black–Scholes implied volatility, K=strike price, T=time to maturity, and a0, a1,… … …, a5 are model parameters.

PARAMETER ESTIMATION

The process of the estimation of parameters of DVF models is very exhaustive. The method of ordinary least square is considered the most appropriate for the formulations involving quadratic combinations of parameters, one such as DVF. The process needs to iterate for the entire sample size on a daily basis. Therefore, it has always been an exhaustive challenge to decide and determine the quality of parameters of DVF to be taken into account to estimate the study in question. We did come across the same kind of challenge. As defined earlier, the model of Black–Scholes is itself the most right tool to determine the implied volatility, knowing the market value of the options. However, unlike the simpler algebraic counterparts, the formula of Black–Scholes cannot invert directly/algebraically. This needs to be done numerically. To determine the at-the-money implied volatility of Black–Scholes and implied volatility of variants of DVFs, we have used a square loss objective function f, which will minimize the difference of the model and market for a given set of parameters, for both. The function is

In normal course, the implied volatility is a value of volatility (σ=σ IV ), which produces a zero difference between the observed price and the Black–Scholes price f(σ)=0. To determine the Black–Scholes standard deviation, we have applied minimization algorithm such as Newton–Raphson method. Squaring the difference of model and market ensured that minimization algorithm does not produce negative values for f(σ).

Whereas the determination of at-the-money implied volatility of Black–Scholes requires a single set of parameters, implied volatilities of DVFs utilizes the full set of strike price available for the trade on a day (remained after filtration). The objective function f(σ), for DVFs is

The step mentioned above has been a process to identify and obtain the empirically valued parameters of DVF. The estimated parameters of the DVFs are then implanted back into them to identify σ IV , which will be used further in the Black–Scholes model to make a forecast, determining the expected option pricing for the next day, based on the logical sequencing and interpreting the data collected out of previous day pricing.

EMPIRICAL ANALYSIS AND RESULT

In order to have a consolidated study on the various patterns describing the specific, definite and comparative behavior of various DVF models, this section examines the cross-sectional, comparative and analytical study of the given tables.

Table 3 exhibits the moneyness statistics of Nifty index call options for the period 1 January 2007 to 31 December 2009. Prices displayed in Table 3 supports the fundamentals of call options pricing. The call option prices move in ascending order as we advance from DOTM and till we finally reach DITM, which can easily be understood by the series DOTM<OTM<ATM<ITM<DITM. Tables 4, 5 and 6 also validate the fundamental principal of call option pricing, as an effect of which long maturity options have higher time value when compared with short maturity options.

Table 3 Moneyness statistics of S&P CNX Nifty 50 index call options (1 January 2007 –31 December 2009)
Table 4 Moneyness-maturity statistics of S&P CNX Nifty 50 index call options (1 January 2007 – 31 December 2009)
Table 5 Moneyness-maturity statistics of S&P CNX Nifty 50 index call options (1 January 2007 – 31 December 2009)
Table 6 Moneyness-maturity statistics of S&P CNX Nifty 50 index call options (1 January 2007 – 31 December 2009)

In Tables 7, 8, 9 and 10, we have enlisted the dependence of implied volatility on maturity and strike price, which have been extracted through the collected sample data. Tables 7, 8, 9 and 10 further exemplify that DVF-implied volatilities vary systematically with respect to strike prices and time to maturities. Implied volatility tends to vary from DOTM to DITM options. It is also to be noticed simultaneously that implied volatility makes a systematic movements toward upward trend when it moves either from ATM to DITM or toward DOTM call options. In Tables 7, 8, 9 and 10 the figure highlighted in bold relates the DVF models having lowest implied volatility moneyness and moneyness-maturity wise, whereas the figures highlighted in italics underline display the DVF models having the highest volatility moneyness and moneyness-maturity wise.

Table 7 Implied volatility moneyness bias
Table 8 Implied volatility moneyness-maturity bias
Table 9 Implied volatility moneyness-maturity bias
Table 10 Implied volatility moneyness-maturity bias

Table 11 exhibits the MPE statistics of Nifty index call options moneyness wise, whereas Tables 12, 13 and 14 demonstrate the MPE statistics of Nifty index call options moneyness-maturity wise. In Tables 11, 12, 13 and 14 data highlighted in bold, underline, and italics (with asterisk) represents the three distinguished categories of Deterministic Volatility Functions. Bold indicates the DVF models having lowest pricing error connoting to correct pricing of Nifty call options (to a great extent) while the data highlighted in underline, and italics (with asterisk) indicates the DVF models having highest positive and negative pricing error connoting to over pricing and under pricing of Nifty index call options.

Table 11 Mean Percentage Error (MPE) moneyness bias
Table 12 Mean Percentage Error (MPE) Moneyness-maturity bias
Table 13 Mean Percentage Error (MPE) moneyness-maturity bias
Table 14 Mean Percentage Error (MPE) moneyness-maturity bias

Table 15 explains the Absolute Mean Percentage Error (AMPE) statistics of Nifty index call options moneyness wise, whereas Tables 16, 17 and 18 demonstrate the AMPE statistics of Nifty index call options moneyness-maturity wise. In Tables 15, 16, 17 and 18, figures highlighted in bold and underlines represents the DVF models having low and high price ends connoting to models pricing Nifty call options correctly. The lowest AMPE seems to correctly define the market price, whereas its high end goes to the extent of mispricing Nifty call options.

Table 15 Absolute Mean Percentage Error (AMPE) moneyness bias
Table 16 Absolute Mean Percentage Error (AMPE) moneyness-maturity bias
Table 17 Absolute Mean Percentage Error (AMPE) moneyness-maturity bias
Table 18 Absolute Mean Percentage Error (AMPE) moneyness-maturity bias

Hence, having viewed the data in Table 11, it can be confidently said that the DVF 0 overprices DOTM, OTM and ATM Nifty index call options. We noticed a systematic reduction in price error going from DOTM to ATM options. The sequence based on the price behavior pattern of DVF 0 is DOTM (23 per cent)<OTM (19 per cent)<ATM (5 per cent). We also deduced that of all available DVF models, when put together and examined, DVF 1 model is the only model that underprices DOTM call options across moneyness. Of all the DVF models, DVF 8 can be identified as the one that prices ITM call options correctly with pricing error 0 per cent. OTM call options are correctly priced by DVF 6, whereas very close marginal case is repeated by DVF 7, 9 and 11 models with negligible pricing error. Table 15 reveals that on the one hand the pricing error of DVF 7 is lowest, whereas the pricing error of DVF 0 is the highest, estimating DOTM, OTM and ATM category options.

The models DVF 0 and DVF 4 overprice the short-term DOTM call options, whereas rest other DVF models tend to underprice it. DVF 13 proves strong enough to price short-term OTM and DITM options and the rest other three categories (DOTM, ATM and ITM) of short-term call options are strongly supported by DVF 3, 9 and 11 models in pricing. DVF 1 underprices short-term DOTM and OTM options with pricing error –47 per cent and –17 per cent, whereas DVF 2 underprices short-term ITM and DITM options with pricing error 2 per cent, respectively.

As the pattern resulted in short-term maturity options, we had the same model (DVF 0) that was causing overpricing of short-term DOTM, OTM and ATM call options. The same sequence and the effect seem to be repeated in medium-term DOTM, OTM, ATM and ITM options adopting the pattern like: DOTM (40 per cent)<OTM (25 per cent)<ATM (6 per cent)<ITM (2 per cent)<DITM (1 per cent). The pattern of sequential effect in the representation of short- and medium-term moneyness is because of a definite and specific configuration of DVF 0.

Table 19 empirically acknowledges the fact that the incorporation of DVF into the option pricing formula reduces pricing bias and gives values closer to market prices compared with the classical Black–Scholes, that is, DVF 0. Figures 6 and 7 graphically depict the same. It also shows that, of all, the price performance of DVF 0 (Black–Scholes) is the worst and as expected the price bias percentage follows the definite order patterned as DOTM>OTM>ATM>ITM>DITM. Juxtaposition of Figures 6 and 7 and Figure 8 reveals that the DVF models reduce the pricing bias of DVF 0 (BS) as DVF 0 and all other DVF family models overprice Nifty index call options compared with market, while at the same time DVF models underprice them when compared with BS (DVF 0).

Table 19 Price statistics of DVFa and S&P CNX Nifty index options
Figure 6
figure 6

Price bias of DVF and Black–Scholes models with respect to market.

Figure 7
figure 7

Price bias of DVF models with respect to Black–Scholes.

Figure 8
figure 8

Implied volatility pattern of mode.

On the basis of our result, and considering the ease of implementation of DVF models, we suggest that DVF model is a better substitute of the traditional Black–Scholes model. However, the limitation of DVF can better be explained in terms of its various versions, which again remains a difficulty to filter the best version to be applied for pricing options. In addition, the challenge lies in finding one such specific version that caters to and collaborates with option maturities and moneyness.

Of all cross-calculations and cross-empirical analyses, it can definitely be stated that we do not have any single outstanding DVF model performing or emerging as a positive tool to be applied among the set of all DVF models. This specific reason beats the idea of using it widely and universally concerned to the specific problem of option pricing.

Evidentially, we also found that DVF superiorly explains/captures the volatility smile/smirk phenomenon manifest in implied volatility of Black–Scholes and provides better forecast about future movement of asset volatility, consequently prices Nifty index call options better than the traditional Black–Scholes model (Table 20 and Figure 8 disclose the same). Tables 19 and 20 and Figures 6, 7 and 8 computed with the help of the following parameters: index/underlying price 5200.70, risk-free rate of interest 7.30 per cent, volatility of index return 30 per cent, time to maturity 31 days and range of strike price 4600–6100 (in multiples of 100).

Table 20 Implied volatility statistics of DVF and Black–Scholes a

The above results clearly reveal that the incorporation of volatility smile features into the classic Black–Scholes formula significantly improves its pricing performance and reduces the pricing bias between model and market. In general, pricing performance is improved by the modeling of DVF, but because of a certain reason it remains ineffective in eliminating the pricing bias completely. On the basis of the results of our findings, we found that the DVF not only captures the volatility smile phenomena but also improves the pricing performance of the most trusted and elegant Black–Scholes model. On the basis of its simplicity, analytical tractability and robustness, DVFs have become a benchmark for evaluating and strengthening the forecasting accuracy of the Black–Scholes option pricing model.

CONCLUSION

The result procured through the empirical research toward the analytical treatment of the topic clearly mentions that the DVF approach performs significantly far better than the traditional Black–Scholes model in most of the moneyness-maturity groups. As the DVF models landmark better than the classic Black–Scholes model during the most turbulent phase of Indian financial economy, we can deduce that as the DVF model ran successfully in the period of high turbulent phase, which generally is an idealist time to examine and observe the effectiveness of any mathematical/financial model, it can perform far smoothly and effectively during the normal course of time as well. The prominent finding is that the performance of DVFs is comparatively a better proposition than the benchmark Black–Scholes, but DVFs have not been juxtaposed with other sophisticated models (as with stochastic volatility models), as such cross-examining can raise some other far-fetched complexities that may result into dilution of the unified direction of this study. It can be represented that DVFs outperform the Black–Scholes model on theoretical explanations; however, although undiscovering the in-depth reasons of the nature of the DVF model behind its behavior is yet to be found out, that too based on analytical and empirical basis, it still requires a separate research with more calculative and dynamic approach.