Conjoint Analysis of Option and Volatility Models

Abstract

This study endeavours to find an impeccable option-pricing model to meet the requirements of ‘Options’ practitioners during a tumultuous period. It encompasses ‘smile’ and ‘skew’ characters exhibiting price bias across moneyness and maturity. For the same, we compared and contrasted the classical Black–Scholes model with deterministic and stochastic volatility models. In order to make applicability of models more prominent, the hypothetical model has been put into practical implication of Nifty index options of India. Also, to ensure the model’s all-round applicability, they all have been passed through the most dramatic phase of the Indian financial economy spanning 2006–11, an ideal time to examine the sustainability of such models. Accuracy of model prices has been testified relative to the market, using the well-known error metrics. This research suggests that the deterministic volatility function (DVF) is the most suitable framework to price the Nifty index options. It not only out passes the benchmark Black–Scholes model but also dominates its stochastic counterpart the stochastic alpha, beta and rho (SABR) model.

JEL Classification: C01, C13, C52, C53, G17

Keywords

Black–Scholes call options deterministic volatility function implied volatility Nifty index options SABR stochastic

Introduction

The history of the approach to the model smile/smirk/parabolic shape of Black–Scholes-implied volatility goes back to the application of scientific approaches since 1973 (Backus et al. 1997). Researchers have been trying to incorporate dynamic characteristics of underlying financial assets into option pricing models. The works of Amin and Victor (1993), Bates (1995), Cox and Ross (1976), Cox et al. (1979), Derman and Kani (1994), Dumas et al. (1998), Dupire (1994), Heston (1993), Heston and Nandi (2000), Hull and White (1987), Johnson and Shannon (1987), Scott (1987), Stein and Stein (1991) and Wiggins (1987) have made a significant contribution to literature. They characterised option pricing models as deterministic and stochastic. In deterministic models, volatility can be determined from the historical data of related underlying assets, while in stochastic models, it cannot be; thus it is considered to be more complicated than the former (Brockman and Chowdhury 1997). As seen earlier, there are many models to model the uncertainty of financial assets, but only few are successful in actual implementation.

To contrast the forecasting competitiveness of deterministic and stochastic models, various empirical studies have been done, but none points to a single winner. Bakshi et al. (1997) also analysed and compared the pricing performance of the models incorporating stochastic volatility, stochastic interest rate and jumps with the benchmark Black–Scholes model and found that pricing bias is not significant compared to the classical Black–Scholes model. Thus, for the purpose of this research, we have ignored the set of complex stochastic models.

By virtue, if the volatility—the only unknown parameter of the Black–Scholes model—could replicate the implied skew pattern of option smile, then undoubtedly, the model (Black–Scholes) would be the most appropriate model to price the index and stock options (Glosten et al. 1993). Figure 1 depicts the variation of volatility across maturity and strike price, commonly known as the volatility smile. This smile pattern has become the genesis of options pricing research. It proves that the most basic constant volatility assumption of the Black–Scholes model is untenable. This violation led to the development of more complex models and laid down the foundation of financial engineering. The stochastic alpha (a), beta (b) and rho (t) (SABR) model is one such stochastic volatility model, which attempts to capture the volatility smile.

Figure 1.

Source: Author’s calculations.

Thus, instead of focusing on the development of more complex framework of stochastic models to price plain vanilla options, we have focused on the selection of volatility models of both types, deterministic and stochastic. Further, to provide a level playing field, the models have been selected in a fashion which ensures that other than pronouncing the volatility smile of the Black–Scholes model they should also map the observed financial characteristics of the underlying (Nifty here). Going forward, we centred on only two volatility functions, one deterministic (DVF’s) and the other stochastic in nature (SABR). Eventually, to substantiate our research, we have used both the models as an input into the classical Black–Scholes model and computed the theoretical option prices, which would then be compared and contrasted with the actual market price of the Nifty index options.

Since inception, whether or not the formula provides a good approximation of the market prices of options underlying indices or stocks, it has been the focal point of debate, but then onwards, without any hitch, unanimously the formula is used inversely to determine the value its own unknown: volatility. Out of several implied versions, at-the-money implied volatility is the most popular (Derman and Kani 1994; Mayhew 1995). Feinstein (1989) asserted that as the Black–Scholes formula is nearly linear in sigma for at-the-money options, it is therefore only virtually unbiased estimate of volatility for pricing of options across moneyness–maturity. Thereafter, research work of Canina and Figlewski (1993), Christensen and Prabhala (1998), Ederington and Guan (2002), Edey and Elliot (1992) and Poteshman (2000) reaffirmed the work of Feinstein (1989). All of them concluded that the implied volatilities, extracted from market option prices, constitutes a forward-looking estimate of the real volatility of the underlying asset and thus reflects the future expectations of the market participants. Motivated by the same, we have also decided to utilise at-the-money implied volatility to figure out the prices of Nifty index options via the Black–Scholes model. Similar to the previous works, this would act as a benchmark to testify the competitiveness of the hypothetical models, discussed later.

Whereas, most of the researchers were focusing on development of complex models, Dumas et al. (1998) developed the simplest version to model the implied volatility. They constituted a simple framework which circumvented Black–Scholes’ constant volatility by non-constant volatility through a set of liner quadratic equations encompassing moneyness and maturity. Through this combination they succeeded in modelling the parabolic smile and smirk shape of Black–Scholes to a great extent. They named it the deterministic volatility function (DVF). The DVF of Dumas et al. proved to be a milestone as, other than handling the smile bias, it also improved the pricing bias of Black–Scholes significantly, as an input. Most importantly, calibrations of parameters of the DVF model were not only easy to compute, but also self-consistent and analytically tractable. There is an extensive list of empirical research papers proving the superiority of DVF models, compared to other species of option pricing models. However, based on the utilisation of sample period and estimation methods, their result tends to differ.

Following the deterministic framework of Dumas et al. (1998) to model the options underlying forward/futures, Hagan et al. (2002) developed a similar stochastic framework. To obtain the fair prices of European call options they simply relaxed the constant volatility assumption of the Black (1976), but contrary to DVF, planted volatility as a stochastic process. They named it SABR. In addition to smile, the framework of SABR also incorporated two main characteristics of financial assets return; leverage effect and non-lognormal distribution (see Figures 2 and 3).

Figure 2.

Source: National Stock Exchange of India and author’s calculations.

To testify the applicability of SABR and DVF in the Indian context, the models have been put into practical implication of Nifty index options, the most heavily traded instrument of the electronic bourse of the National Stock Exchange (NSE) of India. The upgraded and advanced formulations of the financial and economic factors change continuously, thus it becomes important to testify the forecasting capability of DVF and SABR in the same scenario. Therefore, to serve the best to the options market participants in such highly unpredictable conditions, this article endeavours to investigate the out-of-sample moneyness–maturity forecasting performance of the above mentioned option pricing models during the recent waves of economic imbalance of Indian capital market. The period of the last six years is the most ideal time to examine and observe the capacity of a model. This particular phase is not only an extreme period of phenomenal unpredictability but also ranges the high and low tides of financial flux and thus provides the most apt situation for testifying the comparative competitiveness of the models in question. As the models deal with the extreme range of wide highs and lows of index movements, the applicability/usability of stochastic and deterministic SABR models becomes more dominant. Figure 2 clearly depicts that the span of the last six years is the most dynamic phase of Indian capital market and it gave rise to a host of financial experts to estimate the efficiency of various financial models. For this reason, we also banked on this specific period to test the usability of models in question.

Figure 3.

Source: Author’s calculations.

Since we are dealing with Nifty index options, this empirical work also emphasises on incorporating the time series properties of the underlying Nifty in it. However, the core objective of this study is to find an alternative of Black–Scholes for pricing Nifty index options. Ironically, despite knowing the well-known deficiencies of Black–Scholes, NSE and all other exchanges of the world still use it for fixing the base prices of options contracts underlying indices, stocks and futures, etc., owing to its computational simplicity and analytical tractability. Therefore, this article attempts to search out an apt model which can provide better forecast estimates of the Nifty index options and thus enhance the returns of the options traders.

The rest of the article is organised as follows. The second section briefly describes the competing models. The third section details the Nifty index options data and briefly explains the requirements of filter statistics and various filters techniques being applied. The fourth section provides parameter-estimation procedures. The fifth section analyses the structure of pricing errors and also discusses the pricing performance of the Black–Scholes, DVF and SABR models. The sixth section finally concludes the research work.

Option Pricing Methods

Classical Black–Scholes Option Pricing Model

In 1973, utilising the mathematical frameworks of risk neutrality, geometric Brownian motion and the Wiener process, Black and Scholes had developed a formula for pricing European call options, published in the Journal of Political Economy, May–June edition of 1973. Coincidently, in the same year, in the month of April, the Chicago Board of Trade also established the Chicago Board Options Exchange (CBOE), the first exchange, that started the trading of standardised, exchange-traded stock options on the 125th birthday of the Chicago Board of Trade. The formula of Black–Scholes was like an elixir for option traders, struggling hard with the options pricing. The formula not only revolutionised the trading of options but had also strengthened the foundation of development of financial engineering, laid down by Bachelier in 1900. The researchers, practitioners and traders admired the formula of Black–Scholes for its closed-form solution, which enhances its applicability and makes it more tractable. The formula for pricing European stock call option paying non-dividend is

C_{B S} = S N (d_{1}) - K e^{- r t} N (d_{2})

d₁ and d₂ is defined as

d_{1} = \frac{\ln [S / K] + [r + 0.5 σ^{2}] t}{σ \sqrt{t}}

d_{2} = \frac{\ln [S / K] + [r - 0.5 σ^{2}] t}{σ \sqrt{t}} = d_{1} - σ \sqrt{t}

where C denotes the price of a given European call option with exercise price K and time to maturity t, S is the spot price of the underlying asset (Nifty here), r the riskless asset return, $σ^{2}$ is the variance of Nifty returns and N(d) is the standard normal distribution function.

Deterministic DVF Volatility Model

Motivated by the parabolic shape of implied volatility (manifested by at-the-money options of the Black–Scholes model (Figure 1), Dumas et al. (1999) modelled a set of quadratic functions with varying combinations of moneyness and maturity, only. The set of equations were utilised further to price options underlying indices and stocks. In contrast to Black–Scholes, they found that their model led to lower valuation errors across moneyness and maturity. A few years later, Christoffersen and Jacobs (2004) did some improvement and renamed the same as the Practitioner Black–Scholes (PBS) model. The framework of Dumas et al. was so simple that it got the widespread attention of researchers and practitioners. Since it is practically not possible to explore each of the combinations of DVF, we restricted ourselves to the following five unique specifications of DVF’s (0–4)

\begin{array}{l} \begin{matrix} D V F 0 σ_{I V} = a_{0} \\ D V F 1 σ_{I V} = a_{0} + a_{1} M^{2} \\ D V F 2 σ_{I V} = a_{0} + a_{1} T^{2} \end{matrix} \\ \begin{matrix} D V F 3 σ_{I V} = a_{0} + a_{1} M T \\ D V F 4 σ_{I V} = a_{0} + a_{1} M^{2} + a_{2} T^{2} \end{matrix} \end{array}

where σ_iv is the implied volatility, M is the call moneyness ratio ((S/K) − 1), T is the time to maturity, a₀, a₁ and a₂are model parameters.

In case if a₀, a₁ and a₂ all become zero then DVF would converge to Black–Scholes, that is, to market-linked implied volatility, which is possible only on the last trading day of the contract and that too when it expires. This calls for stopping trading on the exchange, as after expiry, the question of pricing terminates automatically. In no condition, we would have zero values for a₁ as the parameter M (moneyness) can never be zero. The other parameter T (time to maturity) can have a zero value, but that is also possible only on the last day of the expiry month after the expiration of the month contract. For such contracts trading would stop automatically. Whereas, the at-the-money implied volatility method of pricing options via Black–Scholes only incorporates information content of at-the-money option for forecasting of option prices, the DVF method is found to incorporate the information content of rest others as well.

Stochastic SABR Volatility Model

Hull and White (1987, 1988) saw the first success in ensuring a stochastic approach to value options, modelled movement of stock price and its volatility as conspicuous stochastic process and laid down the foundation of second-generation stochastic models. Following their footwork Hagan et al. (2002) extended the literature and developed a stochastic volatility model, named SABR. The SABR model is a two-factor stochastic process model defined as:

d F = α F^{β} d z

d α = ξ α d w

where F is the forward/future price, b is a constant deciding the distributional characteristics of the asset price, a is the volatility of forward price, p is the volatility of the volatility, dz and dw are two correlated Wiener processes.

In order to obtain an analytical solution, they employed singular perturbation technique and defined volatility as

σ_{​} = \frac{α}{{(F X)}^{\frac{(1 - β)}{2}} (1 + \frac{{(1 - β)}^{2}}{24} \ln {(\frac{F}{X})}^{2} + \frac{{(1 - β)}^{4}}{1920} \ln {(\frac{F}{X})}^{4})} \frac{z}{χ (z)}

\times [1 + (\frac{{(1 - β)}^{2}}{24} \frac{α^{2}}{{(F X)}^{1 - β}} + \frac{1}{4} \frac{ρ β ξ α}{{(F X)}^{(1 - β) / 2}} + \frac{2 - 3 ρ^{2}}{24} ξ^{2})] T + ........

where

z = \frac{ξ}{α} {(F X)}^{(1 - β) / 2} \ln (F / X)

χ (z) = \ln (\frac{\sqrt{1 - 2 ρ z + z^{2}} + z - ρ}{1 - ρ})

The volatility obtained by Hagan et al. is a function of the current forward/future price, and thus seems to be equivalent to implied volatility of the Black–Scholes model. To price the option contracts underlying futures and forwards, Hagan et al. directly substituted this analytical volatility as an input into the Black (1976) model.

Data Description

Figure 2 clearly exhibits that the span of year 2006–2011 is the most dynamic movement on the canvas of the Indian capital market. The period challenges the applicability of all financial models depicting capital market and related instruments. Therefore, this period provides the best laboratory conditions to test the pricing efficiency of models in question. Since Nifty index options are the heaviest-traded instrument in the category of derivative options contracts available for trade on the bourse of NSE (Figure 4), we utilised the same for this research work.

Figure 4 depicts the trading popularity of index options. During the period of the study, the volume has gone up exponentially and roughly accounts for three-fourths (75 per cent) of the total turnover of the NSE derivative segment now. The data of Nifty index options has been collected from the official data sources of NSE. The interest rate data required for this study has been collected from the Reserve Bank of India (RBI). The data set, namely risk-free interest rate, equal to yield of 91-day T-bill securities, strike prices, index price and time to maturity is collected and collated for the sample period spanning 1 January 2006 to 31 December 2011, that is, 1,487 trading days.

Figure 4.

Source: Author’s calculations.

Data Screening Procedure

Realising the linear relationship of put–call parity of the Black–Scholes model, and the fact that the put option replicates inversely the theoretical characteristics of the call options, we have only focused on the data set of call options for the purpose of this research. On an average, around 200 options contracts are available for trade on the bourse of NSE, but not all of them are of interest to traders and practitioners. Only high liquid sets of data are of their interest. Globally, it has been observed that out of three moneyness categories, only two, namely, out-of-the-money and at-the-money contracts remain highly liquid; the others are not. The same concept also applies to pricing of options contracts. Therefore, it becomes highly essential to filter the data to get the best raw material. To testify the effectiveness of the models in question, we transformed the raw data through the following four exclusionary filtering channels: (a) the call options contracts which are either not traded at all or traded with less than 50 numbers of contracts for a day; (b) the options with strike price less than 5 days and more than 90 days of remaining maturity; (c) options with moneyness ratio higher than 15 per cent and lower than 15 per cent; and (d) the options data not satisfying the arbitrage opportunity cum lower boundary condition

S_{t}^{}_{}^{​} - K e^{- r (T - t)} \leq C (S_{t}, t) \leq S (t) .

Option data not satisfying the above four filters has been discarded from the data set. The objective behind applying such rigorous filtering technique is to ensure that only relevant data (free of price and volatility biases) should be used for parity analysis and for finding the cross-competence of the models. Ironically, the above four discretionary filters have jointly ruled out 490,926 (93.6 per cent) data of the original sample, leaving only a marginal 33,576 (6.4 per cent) call options for the empirical research. Table 1 provides the complete descriptive statistics of the data of the Nifty index call option.

Table 1.

Filter Statistics

Total Call Contracts	Year						Sub Total
	2006	2007	2008	2009	2010	2011	2006–11
	44,555	38,890	72,494	93,860	95,008	17,9695	524,502
Criteria	Data Rejected
No. Trading Volume/Open Interest	35,304	29,052	56,835	75,785	76,514	157,009	430,499
No. of Traded Contracts #50	5,182	4,163	7,216	9,137	6,481	7,394	39,573
Moneyness > +15%	170	366	211	1,459	1,055	1,132	4,393
Moneyness < −15%	84	9	1,866	738	263	2,673	5,633
Maturity > 90 Days	0	2	543	390	1,533	2,029	4,497
Maturity < 3 Days	385	490	551	486	502	520	2,934
No Arbitrage Relationship	584	573	123	196	932	989	3,397
Rejected Data	41,709	34,655	67,345	88,191	87,280	171,746	490,926
Rejected Data (%)	93.61	89.11	92.90	93.96	91.87	95.58	93.60
Remaining Data	2,846	4,235	5,149	5,669	7,728	7,949	33,576
Remaining Data (%)	6.39	10.89	7.10	6.04	8.13	4.42	6.40

Source: National Stock Exchange of India, author’s calculations.

Option Categories

The final set of filtered data was then placed categorically to testify the cross-comparative competitiveness of the models in question. In order to find the cross competence of models, we categorically divided the data into 15 categories of moneyness and maturity, defined mathematically as

\begin{matrix} S h o r t M a t u r i t y \\ M e d i u m M a t u r i t y \\ L o n g M a t u r i t y \end{matrix}} i f T i m e t o M a t u r i t y (D a y s) \in {\begin{matrix} (5, 30] \\ (30, 60] \\ (60, 90] \end{matrix}

\begin{array}{l} D e e p - o u t - o f - t h e - m o n e y \\ O u t - o f - t h e - m o n e y \\ A t - t h e - m o n e y \\ I n - t h e - m o n e y \\ D e e p - i n - t h e - m o n e y \end{array}} \begin{matrix} i f C a l l \\ M o n e y n e s s (M) \end{matrix} \in {\begin{matrix} [- 15 %, - 10 %) \\ [- 10 %, - 5 %) \\ [- 5 %, + 5 %] \\ (+ 5 %, + 10 %] \\ (+ 10 %, + 15 %] \end{matrix}

Table 2.

Nifty Index Call Option Statistics for the Years 2006–11 (Post-Filtration)

		Call Moneyness ((S/K)–1)					Total/Sub Total
		DOTM	OTM	ATM	ITM	DITM	Total/Sub Total
Maturity	Short	2511	3287	8169	1498	607	16072
		7.48%	9.79%	24.33%	4.46%	1.81%	47.87%
	Medium	1737	2706	6049	1011	259	11762
		5.17%	8.06%	18.02%	3.01%	0.77%	35.03%
	Long	869	1681	2939	210	43	5742
		2.59%	5.01%	8.75%	0.63%	0.13%	17.10%
Total/Sub Total		5117	7674	17157	2719	909	33576
		15.24%	22.86%	51.10%	8.10%	2.71%	100.00%

Source: National Stock Exchange of India, author’s calculations.

Table 2 provides the categorical descriptive statistics of the filtered data. Table 2 clearly exhibits that OTM¹ and ATM options would remain the focal point of this study, as, together they constitute 74 per cent of the filtered data.

Methodology

In order to test the applicability of SABR model in Indian context, we need to have the historical data of options contracts underlying forward or future. But, so far in India, trading of such contracts has been not started. Thus, to testify the applicability of SABR model we need to do some technical adjustments. For this, we replaced the options contracts underlying futures with contracts underlying Nifty. Also, we replaced F (future) by S (spot), and instead of using Black-76 we have used the original version of the Black–Scholes (version 73).

After evaluating the conceptual framework of models and carrying forward the same, we arrived at the conclusion that for evaluating the performance of different models, the same loss functions should be used to provide a level playing field to models.

Thus, to judge the empirical performance of the models we have set two yardsticks. First, for estimating models parameters we employed technique of optimisation and utilised linear and non-linear least squares loss functions. Second, to find the inter competence and statistical parities of the models, instead of comparing the prices of DVF and SABR models directly to the prices of benchmark Black–Scholes we had compared it with the corresponding prices of the market values (price quotes of Nifty index options). For the same we applied the two basic techniques of error metrics, defined as

Mean Percentage Error (MPE)

M P E = \frac{1}{n} \sum_{i = 1}^{n} (C_{i}^{}_{}^{M o d e l} - C_{i}^{}_{}^{M a r k e t}) / C_{i}^{M a r k e t}

Mean Absolute Percentage Error (MAPE)

M A P E = \frac{1}{n} \sum_{i = 1}^{n} [A B S ((C_{i}^{M o d e l} - C_{i}^{M a r k e t}) / C_{i}^{M a r k e t})]

where $C_{i}^{M a r k e t}$ is the actual/market price and $C_{i}^{M o d e l}$ is the theoretical/expected/model price of the option for observation i, n is the number of observations.

Negative/positive relative PME (expressed in percentage) means that the model under-prices/over-prices the specific option, while large/small MAPE provides deviation between model and market price in the absolute sense. A small relative error of MAPE means the model provides a good approximation of the market, while a large relative error means the model is providing a poor approximation of the market.

Calibration of Model Parameters

To infer the options related to structural parameters, governing the underlying asset return distribution, we trusted the most dominant method of parameter estimation: the technique of optimisation. When statistical significance of the estimation is not a major concern, this technique is undoubtedly the most effective for finding the parameters of models. The technique minimises the price bias of model and market and reveals the parameters which remain statically stable over the time horizon.

Technically, comparative to DVF, the calibration of the SABR model is not an easy task as the latter requires concurrent estimation of a set of four non-liner parameters, which is not easy to calibrate, while DVF only requires calibration of a set of linear combinations of parameters. Thus, to provide a level playing field to both the models, the unknown parameters of the models are obtained by applying a common objective loss function f(Ω), defined as

f (Ω) = \min_{Ω} \sum_{i = 1}^{n} {(C_{i}^{M o d e l} (Ω) - C_{i}^{M a r k e t})}^{2}

where Ω is a set of parameters, which will determine implicitly here from options data set of Nifty index options on a daily basis. The set of optimal parameters, obtained for day (i) is then utilised to compute the model price of options of day (i + 1).

The same process is repeated for the entire sample, from the beginning to the end. The advantage of this process is that it ensures the transformation of volatility from one period to another, that is, from the past to the future. The at-the-money implied volatility would be calibrated using numerical analysis techniques. Newton Raphson and bi-section are the two most popular techniques one can use for the same. The process is simple in its approach. For the set of strike price equal to exercise price (ATM option—to the closet proximity), we iterate volatility till the time difference of option prices of market and model reduces to zero.

Pricing Performance

To analyse the parity analysis of the SABR, DVF and Black–Scholes models, the article explores the multilayered inter-relations of options prices, implied volatility and price errors given explicitly in Tables 3 to 10. The emphasis is on finding the performance of the models relative to market. Instead of comparing the prices of DVF and SABR with the classical Black–Scholes, we have compared them with the market, which seems more practical. Practically, every options trader wants to see the performance of the model against the market in which they are trading, not against the Black–Scholes which is only used once at the beginning of day for fixing the base prices of the options.

Table 3.

Price Statistics of Nifty Index Call Options (Years 2006–11)

		Moneyness Statistics
		Models
		Market	BS (ATM)	DVF 0	DVF 1	DVF 2	DVF 3	DVF 4	SABR
DOTM	Average	15.16	34.58	15.63	13.47	15.71	15.50	13.89	15.24
	Std.Dev.	21.92	47.39	25.13	22.40	24.44	24.38	22.27	20.26
OTM	Average	38.37	69.21	39.59	37.90	39.80	39.83	38.30	36.32
	Std.Dev.	37.52	67.00	40.33	38.94	39.22	39.73	38.48	31.38
ATM	Average	145.49	176.85	144.02	144.72	144.82	145.20	145.26	146.86
	Std.Dev.	88.51	102.87	88.47	89.59	87.81	88.75	89.07	86.92
ITM	Average	377.23	385.99	372.87	376.41	373.79	374.56	376.87	352.71
	Std.Dev.	94.85	106.08	93.79	95.15	93.57	93.81	94.87	88.20
DITM	Average	547.62	539.10	539.50	543.47	540.04	540.88	543.73	533.21
	Std.Dev.	124.88	132.50	124.25	123.45	123.97	123.81	123.20	119.67
Sub Total	Average	130.80	157.31	129.82	129.86	130.38	130.63	130.34	133.25
	Std.Dev.	138.82	143.55	137.39	139.05	137.24	137.81	138.83	148.24

Source: National Stock Exchange of India, author’s calculations.

Tables 3 and 4 provide the moneyness–maturity statistics of the Nifty index call options for the sample period spanning 1 January 2006 to 31 December 2011. Prices displayed in Tables 3 and 4 validate the theoretical framework of options pricing that price follows a definite pattern and moves in ascending order understood by maturity and moneyness series, short-term < medium-term < long-term and DOTM <OTM<ATM<ITM<DITM.

Table 4.

Price Statistics of Nifty Index Call Options (Years 2006–11)

			Maturity Statistics
			Models
			Market	BS (ATM)	DVF 0	DVF 1	DVF 2	DVF 3	DVF 4	SABR
Short Term	DOTM	Average	6.2	14.0	5.1	4.3	5.8	5.8	5.0	5.4
		Std.Dev.	10.1	26.7	11.2	10.1	12.2	12.1	10.8	11.7
	OTM	Average	17.4	34.9	16.1	15.3	17.6	17.9	16.7	16.3
		Std.Dev.	22.1	53.0	22.6	21.7	24.0	24.1	22.9	22.8
	ATM	Average	106.5	127.5	103.4	104.0	106.0	106.4	106.1	104.3
		Std.Dev.	76.8	87.9	75.0	75.9	75.8	76.1	76.4	74.6
	ITM	Average	352.2	350.6	346.3	348.8	347.9	348.6	349.9	346.8
		Std.Dev.	89.7	98.9	87.5	88.3	87.6	87.9	88.3	89.3
	DITM	Average	539.2	523.3	530.6	532.9	531.5	532.2	533.6	534.2
		Std.Dev.	121.4	127.1	119.8	118.8	119.3	119.1	118.4	116.6
	Sub Total	Average	111.9	126.6	109.0	109.3	110.9	111.3	110.9	112.5
		Std.Dev.	142.4	143.0	140.2	141.3	140.5	140.8	141.4	140.0
Medium Term	DOTM	Average	21.9	48.7	23.2	20.1	23.4	23.2	20.6	21.3
		Std.Dev.	26.6	56.0	30.6	27.0	29.7	30.3	26.8	27.9
	OTM	Average	47.7	85.1	50.1	47.9	50.4	50.5	48.2	50.3
		Std.Dev.	39.0	66.4	41.4	40.1	40.6	42.5	39.7	41.7
	ATM	Average	171.1	208.2	170.1	170.9	170.3	170.5	170.8	169.6
		Std.Dev.	83.7	94.7	83.1	84.6	82.8	84.1	84.2	83.6
	ITM	Average	396.5	416.1	393.9	398.4	394.4	395.2	398.3	398.7
		Std.Dev.	86.5	92.2	85.3	86.3	85.7	85.9	86.7	86.9
	DITM	Average	549.2	554.6	543.0	549.8	542.7	544.1	549.1	542.8
		Std.Dev.	121.5	129.1	122.7	121.3	123.0	122.9	121.4	120.3
	Sub Total	Average	148.4	181.8	148.3	148.2	148.5	148.7	148.3	148.4
		Std.Dev.	137.2	141.7	135.8	138.0	135.6	136.4	137.7	136.6
Long Term	DOTM	Average	27.7	65.7	31.0	26.8	29.0	28.2	26.1	29.5
		Std.Dev.	25.3	48.1	28.5	26.4	27.4	26.3	25.9	27.8
	OTM	Average	64.3	110.7	68.5	65.8	66.1	65.5	64.5	66.1
		Std.Dev.	36.8	59.0	39.7	38.6	37.7	37.1	38.1	38.0
	ATM	Average	201.1	249.6	203.1	203.8	200.2	201.0	201.5	202.8
		Std.Dev.	79.2	87.9	79.9	81.4	79.8	81.9	82.0	78.4
	ITM	Average	462.6	493.8	461.3	467.9	459.6	460.0	466.2	460.3
		Std.Dev.	98.2	100.9	96.5	98.4	95.6	96.2	97.1	96.0
	DITM	Average	656.6	668.8	644.5	654.7	644.5	644.5	655.0	647.0
		Std.Dev.	144.1	147.3	147.4	145.1	146.7	147.2	144.4	145.1
	Sub Total	Average	147.8	193.2	150.4	149.7	147.8	148.0	147.9	149.7
		Std.Dev.	124.3	130.1	123.5	126.1	123.2	124.3	125.9	122.3

Source: National Stock Exchange of India, author’s calculations.

Tables 5 and 6 provide the descriptive statistics of the most crucial parameter of the study: implied volatility. Whether or not the formula provides a good estimate of the market will continue to remain a matter of empirical debate. But it can be inversely used to estimate the value of volatility, termed as implied volatility, which is a good forecast of future volatility (Canina and Figlewski 1993; Christensen and Prabhala 1998; Day and Craig 1992; Ederington and Guan 2002; Edey and Elliot 1992). As explained earlier, against the basic intuition of Black–Scholes that the volatility remains constant throughout expiry, Tables 5 and 6 clearly demonstrate that the implied volatilities of Black–Scholes (ATM), DVF 1–4 and SABR varies systematically from DOTM to DITM options and makes a systematic upward trend towards both the ends, that is, from ATM to DITM and to DOTM call options. Surprisingly, contrary to the theoretical assumption that the volatility bears a positive correlation with time to maturity, Table 6 reveals that volatility of short maturity Nifty index options is higher than the medium and long maturity options. Technically, this can be rooted to the statistical fact that long maturity options are least traded/popular followed by medium-term options on the bourse of NSE. Table 2 statistically justifies this. It reveals that the long-term contracts constitutes only 17 per cent of the total liquid trades while medium and short-term contracts accounts 35 per cent and 48 per cent, respectively. Hence, the pricing performance of short- and medium-term contracts will remain the focal point of this section.

Table 5.

Implied Volatility Statistics of Nifty Index Call Options (Years 2006–11)

		Moneyness Statistics
		Models
		BS (ATM)	DVF 0	DVF 1	DVF 2	DVF 3	DVF 4	SABR
DOTM	Average	0.28	0.21	0.20	0.22	0.22	0.21	0.23
	Std.Dev.	0.13	0.09	0.09	0.10	0.12	0.09	0.11
OTM	Average	0.26	0.20	0.20	0.20	0.20	0.20	0.21
	Std.Dev.	0.12	0.09	0.08	0.09	0.10	0.09	0.10
ATM	Average	0.25	0.19	0.19	0.19	0.19	0.19	0.20
	Std.Dev.	0.10	0.08	0.08	0.08	0.09	0.08	0.08
ITM	Average	0.27	0.21	0.23	0.22	0.22	0.23	0.22
	Std.Dev.	0.11	0.08	0.09	0.09	0.10	0.09	0.11
DITM	Average	0.31	0.25	0.27	0.26	0.26	0.27	0.27
	Std.Dev.	0.12	0.09	0.10	0.10	0.13	0.11	0.11
Sub Total	Average	0.26	0.20	0.20	0.20	0.20	0.20	0.21
	Std.Dev.	0.11	0.08	0.08	0.09	0.10	0.09	0.11

Source: National Stock Exchange of India, author’s calculations.

Tables 7 and 8 exhibit the mean percentage error (MPE) of the Nifty index call options, and provides the detail of model undervaluing and overvaluing. Tables 9 and 10 provide the moneyness–maturity statistics of mean absolute percentage price error (MAPE) of the same. Whereas, former two tables provides the relative pricing statistics of the models and reveals the degree of overvaluing and undervaluing of the models, the latter two provides the statistics of deviation of models and market in absolute sense.

Table 6.

Implied Volatility Statistics of Nifty Index Call Options (Years 2006–11)

			Maturity Statistics
			Models
			BS (ATM)	DVF 0	DVF 1	DVF 2	DVF 3	DVF 4	SABR
Short Term	DOTM	Average	0.29	0.23	0.22	0.23	0.24	0.23	0.25
		Std.Dev.	0.13	0.10	0.09	0.11	0.11	0.10	0.11
	OTM	Average	0.28	0.22	0.21	0.23	0.23	0.22	0.22
		Std.Dev.	0.12	0.09	0.09	0.10	0.10	0.10	0.10
	ATM	Average	0.26	0.20	0.20	0.21	0.21	0.21	0.20
		Std.Dev.	0.11	0.08	0.08	0.09	0.09	0.09	0.09
	ITM	Average	0.29	0.22	0.24	0.23	0.24	0.24	0.22
		Std.Dev.	0.11	0.08	0.09	0.09	0.09	0.09	0.08
	DITM	Average	0.32	0.25	0.27	0.26	0.26	0.28	0.26
		Std.Dev.	0.13	0.10	0.11	0.10	0.11	0.11	0.11
	Sub Total	Average	0.27	0.21	0.21	0.22	0.22	0.22	0.22
		Std.Dev.	0.12	0.09	0.09	0.10	0.10	0.09	0.09
Medium Term	DOTM	Average	0.28	0.22	0.21	0.22	0.21	0.21	0.22
		Std.Dev.	0.13	0.10	0.09	0.09	0.14	0.09	0.10
	OTM	Average	0.26	0.20	0.20	0.20	0.20	0.20	0.20
		Std.Dev.	0.12	0.09	0.08	0.08	0.11	0.08	0.09
	ATM	Average	0.24	0.19	0.19	0.19	0.19	0.19	0.18
		Std.Dev.	0.10	0.08	0.08	0.08	0.10	0.08	0.08
	ITM	Average	0.26	0.21	0.22	0.21	0.21	0.22	0.21
		Std.Dev.	0.11	0.08	0.09	0.08	0.12	0.09	0.09
	DITM	Average	0.32	0.26	0.28	0.26	0.24	0.28	0.26
		Std.Dev.	0.12	0.09	0.10	0.09	0.18	0.09	0.11
	Sub Total	Average	0.25	0.20	0.20	0.20	0.20	0.20	0.22
		Std.Dev.	0.11	0.08	0.08	0.08	0.11	0.08	0.09
Long Term	DOTM	Average	0.23	0.17	0.16	0.17	0.17	0.16	0.16
		Std.Dev.	0.08	0.06	0.05	0.06	0.05	0.05	0.05
	OTM	Average	0.21	0.16	0.16	0.16	0.16	0.16	0.16
		Std.Dev.	0.07	0.06	0.05	0.05	0.06	0.05	0.05
	ATM	Average	0.21	0.16	0.16	0.16	0.16	0.16	0.16
		Std.Dev.	0.07	0.05	0.05	0.06	0.06	0.06	0.06
	ITM	Average	0.22	0.17	0.18	0.17	0.17	0.18	0.17
		Std.Dev.	0.08	0.06	0.06	0.06	0.06	0.07	0.06
	DITM	Average	0.24	0.20	0.22	0.20	0.20	0.22	0.21
		Std.Dev.	0.07	0.06	0.07	0.07	0.06	0.07	0.07
	Sub Total	Average	0.21	0.16	0.16	0.16	0.16	0.16	0.16
		Std.Dev.	0.07	0.06	0.06	0.06	0.06	0.06	0.05

Source: National Stock Exchange of India, author’s calculations.

Table 7.

Mean Percentage Price Error Statistics of Nifty Index Call Options (Years 2006–11)

		Moneyness Statistics
		Models							No. of Observations
		BS (ATM)	DVF 0	DVF 1	DVF 2	DVF 3	DVF 4	SABR	No. of Observations
DOTM	Average	1.40	−0.28	−0.40	−0.23	−0.23	−0.35	−0.26	5117
	Std.Dev.	4.73	0.61	0.69	0.61	0.63	0.61	0.62
OTM	Average	1.06	−0.07	−0.12	−0.03	−0.03	−0.09	−0.05	7674
	Std.Dev.	2.66	0.45	0.42	0.45	0.46	0.41	0.43
ATM	Average	0.31	−0.01	−0.01	0.00	0.00	0.00	0.03	17157
	Std.Dev.	0.53	0.17	0.16	0.17	0.20	0.16	0.16
ITM	Average	0.03	−0.01	0.00	−0.01	−0.01	0.00	−0.02	2719
	Std.Dev.	0.13	0.04	0.04	0.04	0.04	0.04	0.03
DITM	Average	−0.01	−0.02	−0.01	−0.01	−0.01	−0.01	−0.03	909
	Std.Dev.	0.12	0.02	0.02	0.02	0.02	0.02	0.03
Sub Total	Average	0.62	−0.06	−0.09	−0.04	−0.04	−0.07	−0.05	33576
		Std.Dev.	2.32	0.35	0.38	0.35	0.37	0.35	0.33

Source: National Stock Exchange of India, author’s calculations.

Table 8.

Mean Percentage Price Error Statistics of Nifty Index Call Options (Years 2006–11)

			Maturity Statistics
			Models							No. of Observations
			BS (ATM)	DVF 0	DVF 1	DVF 2	DVF 3	DVF 4	SABR	No. of Observations
Short Term	DOTM	Average	1.07	−0.56	−0.64	−0.48	−0.47	−0.57	−0.50	2511
		Std.Dev.	6.50	0.61	0.83	0.63	0.63	0.71	0.65
	OTM	Average	1.18	−0.24	−0.30	−0.17	−0.14	−0.24	−0.21	3287
		Std.Dev.	3.96	0.57	0.54	0.58	0.59	0.53	0.59
	ATM	Average	0.36	−0.03	−0.02	0.00	0.01	0.00	0.04	8169
		Std.Dev.	0.71	0.22	0.21	0.21	0.22	0.20	0.23
	ITM	Average	0.00	−0.02	−0.01	−0.01	−0.01	−0.01	−0.03	1498
		Std.Dev.	0.15	0.03	0.03	0.03	0.03	0.03	0.04
	DITM	Average	−0.02	−0.02	−0.01	−0.01	−0.01	−0.01	−0.05	607
		Std.Dev.	0.13	0.02	0.02	0.02	0.02	0.02	0.02
	Sub Total	Average	0.59	−0.15	−0.18	−0.11	−0.10	−0.14	−0.14	16072
		Std.Dev.	3.20	0.43	0.49	0.43	0.43	0.45	0.45
Medium Term	DOTM	Average	1.69	−0.06	−0.24	−0.01	−0.02	−0.17	−0.04	1737
		Std.Dev.	1.85	0.46	0.34	0.47	0.59	0.33	0.45
	OTM	Average	1.04	0.06	0.00	0.08	0.08	0.01	0.06	2706
		Std.Dev.	0.91	0.29	0.21	0.29	0.34	0.21	0.28
	ATM	Average	0.28	0.00	0.00	0.00	0.00	0.00	0.02	6049
		Std.Dev.	0.31	0.09	0.09	0.09	0.17	0.09	0.10
	ITM	Average	0.05	−0.01	0.01	0.00	0.00	0.01	0.04	1011
		Std.Dev.	0.11	0.04	0.04	0.04	0.04	0.04	0.04
	DITM	Average	0.01	−0.01	0.00	−0.01	−0.01	0.00	0.05	259
		Std.Dev.	0.10	0.03	0.02	0.03	0.02	0.02	0.02
	Sub Total	Average	0.63	0.00	−0.03	0.02	0.02	−0.02	−0.03	11762
		Std.Dev.	1.02	0.24	0.20	0.24	0.30	0.19	0.23
Long Term	DOTM	Average	1.76	0.12	−0.04	0.06	0.02	−0.04	0.08	869
		Std.Dev.	1.41	0.42	0.44	0.48	0.45	0.49	0.43
	OTM	Average	0.85	0.08	0.04	0.05	0.04	0.02	0.05	1681
		Std.Dev.	0.61	0.23	0.21	0.25	0.24	0.23	0.22
	ATM	Average	0.28	0.01	0.01	0.00	0.00	0.00	0.03	2939
		Std.Dev.	0.23	0.09	0.08	0.14	0.20	0.16	0.13
	ITM	Average	0.07	0.00	0.01	0.00	0.00	0.01	−0.03	210
		Std.Dev.	0.07	0.04	0.04	0.05	0.04	0.04	0.04
	DITM	Average	0.02	−0.02	0.00	−0.02	−0.02	0.00	−0.06	43
		Std.Dev.	0.04	0.04	0.03	0.03	0.04	0.03	0.03
	Sub Total	Average	0.66	0.05	0.01	0.02	0.02	0.00	0.02	5742
		Std.Dev.	0.85	0.22	0.22	0.25	0.26	0.26	0.23

Source: National Stock Exchange of India, author’s calculations.

Table 9.

Mean Absolute Percentage Price Error Statistics of Nifty Index Call Options (Years 2006–11)

		Moneyness Statistics
		Models							No. of Observations
		BS (ATM)	DVF 0	DVF 1	DVF 2	DVF 3	DVF 4	SABR	No. of Observations
DOTM	Average	1.74	0.52	0.51	0.50	0.50	0.47	0.51	5117
	Std.Dev.	4.61	0.42	0.61	0.41	0.46	0.52	0.43
OTM	Average	1.17	0.32	0.28	0.31	0.31	0.26	0.30	7674
	Std.Dev.	2.61	0.33	0.34	0.32	0.34	0.32	0.34
ATM	Average	0.35	0.10	0.09	0.09	0.09	0.09	0.10	17157
	Std.Dev.	0.51	0.14	0.13	0.14	0.18	0.14	0.14
ITM	Average	0.07	0.03	0.03	0.03	0.03	0.02	0.04	2719
	Std.Dev.	0.11	0.03	0.03	0.03	0.03	0.03	0.03
DITM	Average	0.05	0.02	0.02	0.02	0.02	0.02	0.03	909
	Std.Dev.	0.11	0.02	0.02	0.02	0.02	0.02	0.02
Sub Total	Average	0.72	0.20	0.19	0.20	0.20	0.18	0.19	33576
		Std.Dev.	2.29	0.30	0.34	0.29	0.32	0.31	0.30

Source: National Stock Exchange of India, author’s calculations.

Table 10.

Mean Absolute Percentage Price Error Statistics of Nifty Index Call Options (Years 2006–11)

			Maturity Statistics
			Models							No. of Observations
			BS (ATM)	DVF 0	DVF 1	DVF 2	DVF 3	DVF 4	SABR	No. of Observations
Short Term	DOTM	Average	1.73	0.70	0.74	0.68	0.68	0.69	0.69	2511
		Std.Dev.	6.36	0.45	0.75	0.39	0.41	0.60	0.42
	OTM	Average	1.43	0.48	0.46	0.46	0.47	0.43	0.46	3287
		Std.Dev.	3.87	0.39	0.41	0.38	0.39	0.39	0.39
	ATM	Average	0.41	0.13	0.13	0.12	0.13	0.12	0.14	8169
		Std.Dev.	0.68	0.18	0.17	0.17	0.18	0.16	0.18
	ITM	Average	0.07	0.03	0.02	0.02	0.02	0.02	0.03	1498
		Std.Dev.	0.13	0.03	0.03	0.02	0.02	0.02	0.03
	DITM	Average	0.05	0.02	0.02	0.02	0.02	0.02	0.02	607
		Std.Dev.	0.12	0.02	0.02	0.02	0.02	0.02	0.02
	Sub Total	Average	0.78	0.28	0.28	0.27	0.27	0.26	0.27	16072
		Std.Dev.	3.16	0.37	0.44	0.35	0.36	0.39	0.38
Medium Term	DOTM	Average	1.74	0.36	0.33	0.36	0.36	0.29	0.35	1737
		Std.Dev.	1.80	0.29	0.25	0.31	0.46	0.24	0.30
	OTM	Average	1.06	0.21	0.15	0.22	0.22	0.15	0.20	2706
		Std.Dev.	0.88	0.20	0.14	0.21	0.26	0.14	0.18
	ATM	Average	0.30	0.06	0.06	0.06	0.07	0.06	0.07	6049
		Std.Dev.	0.29	0.07	0.06	0.07	0.15	0.07	0.07
	ITM	Average	0.08	0.03	0.03	0.03	0.03	0.02	0.04	1011
		Std.Dev.	0.09	0.03	0.03	0.03	0.03	0.03	0.03
	DITM	Average	0.05	0.02	0.02	0.02	0.02	0.02	0.02	259
		Std.Dev.	0.09	0.02	0.02	0.02	0.02	0.02	0.02
	Sub Total	Average	0.66	0.14	0.12	0.14	0.14	0.11	0.14	11762
		Std.Dev.	1.01	0.19	0.16	0.20	0.27	0.15	0.17
Long Term	DOTM	Average	1.77	0.29	0.21	0.27	0.25	0.22	0.28	869
		Std.Dev.	1.40	0.32	0.39	0.40	0.38	0.44	0.39
	OTM	Average	0.86	0.16	0.11	0.15	0.15	0.11	0.15	1681
		Std.Dev.	0.60	0.18	0.18	0.20	0.20	0.21	0.21
	ATM	Average	0.29	0.06	0.06	0.06	0.06	0.06	0.06	2939
		Std.Dev.	0.22	0.06	0.06	0.12	0.19	0.15	0.12
	ITM	Average	0.08	0.03	0.03	0.03	0.03	0.03	0.03	210
		Std.Dev.	0.07	0.03	0.03	0.04	0.03	0.03	0.03
	DITM	Average	0.04	0.03	0.02	0.03	0.03	0.02	0.03	43
		Std.Dev.	0.03	0.03	0.02	0.03	0.03	0.02	0.03
	Sub Total	Average	0.67	0.12	0.09	0.12	0.11	0.10	0.12	5742
		Std.Dev.	0.84	0.19	0.19	0.22	0.24	0.24	0.21

Source: National Stock Exchange of India, author’s calculations.

Prima facie, the analyses of Tables 3 and 4 signal the poor pricing performance of the Black–Scholes at-the-money implied volatility. The price statistics of the Black–Scholes seems as an outlier across moneyness and maturity with prices being almost double as compared to its counterparts. Descriptive statistics of Tables 5 and 6 are also in line with the results of Tables 3 and 4. Therefore, the rest of the analysis focuses on finding the quantitative hypothesis: whether or not the DVF and SABR models provide materialistic improvement over the two classical versions of Black–Scholes (ATM) and DVF 0 (market-implied constant volatility).

Contrary to the empirical evidence that the Black–Scholes (BS) model underprices DOTM and OTM options, Tables 7 and 8 reveal that the BS (ATM) severely overprices DOTM, OTM and ATM options across moneyness and maturity, and pattern as DOTM>OTM>ATM> ITM>DITM. The degree of overpricing also follows the maturity pattern depicted as long-term > medium-term > short-term. This implies that the predicting capacity of the ATM volatility is limited to the short period, and hence it is not a reliable tool for pricing options of higher periods. The degree of overpricing ranges from 0 per cent to +172 per cent, across moneyness–maturity.

On the other hand, DVF 0–4 and SABR models support the empirical evidence and underprices short-term DOTM (−56 per cent), OTM (−24 per cent), ATM (−3 per cent), ITM (−2 per cent) and DITM (−2 per cent) call options and overprices long-term DOTM (12 per cent) and OTM (8 per cent) call options. Whereas, the DVF 0 model overprices long-term DOTM and OTM call options, the DVF models exhibits a mix of trends. DVF 2 and 3 perform in-line with DVF 0 and overpriced DOTM (6 per cent) and OTM (2 per cent), while DVF 1 and DVF 4 underprices DOTM (−4 per cent) and OTM (−4 per cent). This analysis also supports the empirical evidence that not all versions of DVF models provide material improvement over the traditional BS (ATM) model for pricing options (Christoffersen and Jacobs 2004). Out of the five versions discussed here, only three, DVF 2, DVF 3 and DVF 4, have been found to significantly improve the price bias of the Black–Scholes model. Also, their average percentage pricing error is lower in 12 out of 15 moneyness–maturity groups compared to DVF 1. They are also found to outperform deterministic Black–Scholes and stochastic SABR in 11 (out of 15) categories of moneyness–maturities. This further implies that only quadratic versions of DVF models portray the skew of option smile and dynamics of financial index, well.

On an average, the out-of-sample forecast ability of the SABR model is superior to the BS (ATM) model in 13 moneyness–maturity groups while out-of-sample forecast ability of the DVF (DVF 2, DVF 3 and DVF 4) model is superior to the SABR model in 10 out of 15. Although, there is very close competition between the two, yet the DVF model prevails in terms of ease of implementation, analytical tractability and is closed from solution. Compared to DVF 0–4, cross-sectional performance of BS (ATM) is highly dissatisfactory; however, it can be utilised for measuring short-, medium- and long-term ITM and DITM options, as BS (ATM) measures them with price error +7 per cent.

Tables 9 and 10 depict that the mean absolute percentage price error (MAPE) of the SABR and DVF model is lower than the BS (ATM) model in 13 and 12 moneyness–maturity groups respectively. Whereas, compared to SABR, the MAPE of the DVF model is superior in only 6 out of 15 moneyness–maturity groups (Table 10). For DVF models, MAPE bias is high for short-term call options and low for long-term call options, thereby implying that the pricing of short-term call options is quite tough compared to medium and long-term options.

The price error sequence based on the behaviour pattern of Tables 9 and 10 put the option pricing model in a sequence jumbled as DVF 3 > DVF 4 > SABR > BS. Results of Tables 9 and 10 provide additional support to the claim that the efficiency of the DVF models is superior to SABR and BS across moneyness and maturity. Similar to MPE, MAPE also shows systemic reduction in the price errors of the models, from long maturity to short maturity and from DOTM to DITM call options.

So far the quintessence of this empirical research is concerned we have deduced that the DVF model, especially DVF 2 and 3 outperforms all others in most categories of maturity and moneyness. However, at the same time, after critically examining the three models (DVF, SABR and BS) we have safely deduced that there is no single model that dominates the others in all categories of moneyness–maturity. Thus, we ended up with the remark that no single model supports the multi-farious moneyness–maturity dimension that comprises of or best caters to all the requirements of traders and practitioners. Several reasons can be stated for same but the two important ones are that either parameter values were incorrectly estimated or models are not fully capable of capturing the dynamics of the market.

Conclusion

The most astonishing feature of the empirical finding of this research work is that compared to the classical BS (ATM-IV), DVF and SABR, it improves the pricing bias significantly across moneyness and maturity. During the impression of being virtually ineffective, DVF is the most suitable model to price the Nifty index options. It not only out passes the classical Black–Scholes model, but also outperforms its stochastic counterpart. The simplicity of its usability makes its most dominant model for various applications and dependability in matter of reliability.² Though, the outcome of this rigorous empirical research reveals that the DVF and SABR perform better than the BS (ATM) model across all the phases of recent financial upheavals, yet the models need to be juxtaposed with more calculative and dynamic approach of other sophisticated models to substantiate the all round applicability (pricing and hedging effectiveness) of DVF models. In the realm of option pricing, success of BS model owes to its ease of use and closed form analytical solution. But its limitations to price complex options, not having closed form solution have restricted its uses all across the board of option pricing. With a highly challenging objective of safeguarding the investor’s interest, the idea was to figure out a volatility framework which could compete on the canvass of such oft-repeated financial hardships and to keep the investments protected.

Footnotes

Notes

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