^{1}

^{2}

This study attempts to analyse one-day-ahead out-of-sample performance of the stochastic
volatility model of Heston (SVH) in the Indian context. Also, the study compares the ex-ante performance of the SVH with that of a Two-Scale-Realised-Volatility (TSRV)-based Black-Scholes model (BS) using the liquidity-weighted performance metrics. For the purpose, we utilise the tick-by-tick data of the CNX Nifty index and options thereon, the most liquid equity options in the world in terms of the number of contracts traded^{1}. Additionally, the study compares the two models across subgroups based on the moneyness, volatility of the underlying and time-to-expiration of the options. The results establish that the SVH model is better than the BS model in pricing equity index options. Further, the SVH model appears to be superior across all the subgroups, for both call options and put options.

The time-varying volatility is considered one of the most cited stylised facts revealed by the traded financial assets. The property is well documented in the financial economics literature. The practitioners and academia have contributed in development of a variety of econometric models that account for this characteristic. Options pricing is no exception. These models have made their way to the options pricing as well in order to capture market dynamics more closely. Option pricing has been studied and debated in the academic world. At the same time, practitioners have brought in their advancements and improvements to pricing models best suited to their needs. These factors have led to the development of numerous option pricing models. The seminal work amongst these is that of Black and Scholes [

As a result, new/modified models were proposed to overcome this assumption. A class of such models builds on the stochastic volatility framework. These models endeavour to relate volatility to a Cox-Ingersoll-Ross (CIR) process [

While these models have been studied in great detail in the developed markets, there are very few such studies in the developing markets. The number of similar studies in the Indian market is even less. Amongst such studies [

Based on the literature reviewed, it appears that only a few studies have been conducted in the Indian market using high-frequency data. In this regard, Singh and Vipul [

Given the literature gap mentioned above and in view of the fact that the CNX Nifty index options are the most liquid equity options in the world in terms of the number of contracts traded, it becomes imperative to conduct a comprehensive comparison of select option pricing models in this context. This study accomplishes it by performing an empirical analysis of the one-day-ahead out-of-sample performance of the SVH model. Further, we compare its performance to that of the TSRV-based BS model, using two years of tick data from the index options. Prices obtained by using these models are compared with the market prices of the options. Notably, the pricing errors are weighted with the respective liquidity of the options to compute the pricing errors for the entire index options market. The liquidity-weighted mispricing/pricing error is used to ascertain the best model for pricing the index options. The performance metrics are liquidity-weighted as options for some categories (e.g., in-the-money options) attract poor trading volumes. Using liquidity-weighted metrics would reduce any bias induced in the comparison due to liquidity issues. Additionally, to make the comparison robust, the performance of the models is evaluated across moneyness, maturity, and volatility of the underlying.

The remaining paper has been organised as follows: The details of the data and related issues are provided in Section 2. The formulation of the SVH model, the calibration approach, and the algorithms used are provided in Section 3. This section also includes the volatility calculation and the performance measures used for the comparison. Section 4 offers the results and discussions. The paper concludes with Section 5.

Index options were introduced in India in June 2001. This study uses data from the CNX Nifty index options to examine and compare the performances of the two option pricing models, viz., a TSRV-based BS model and the SVH model. Tick data of two years (486 trading days), from 03 January 2011 to 31 December 2012, is used for the purpose of this study. The F&O segment, as well as the equity segment in India, trade during 9:15 a.m. to 3:30 p.m. from Monday to Friday, other than the designated holidays. The data for both segments are sourced from the NSE.

Below we provide the details of how the various inputs to the models were obtained:

Time to Expiration: Only trading days (instead of the calendar days) have been counted towards measuring the time-to-expiration [

Risk-free rate: NSE provides the daily “zero-coupon yield curve” rates, based on the prices of the traded Treasury bills and bonds, on a weekly basis. In line with the approach followed by Vipul [

The BS model requires volatility of the underlying index to price options in addition to the other inputs. The details pertaining to the method and estimation of volatility are discussed in Section 3.

Data ScreeningThe following criteria are employed for data screening:

Options Data: Nifty index options with over 60 calendar days to maturity have poor liquidity (

Non-synchronous Trading: To reduce the effect of non-synchronous trading, timestamp of transactions in the equity segment and the derivatives segment are matched (up to the second, hh:mm:ss).

Moneyness: Literature proposes several measures to calculate moneyness. In this study, the measure proposed by Bakshi et al. [

Options which are far from the money lack sufficient liquidity, hence their price discovery may not be accurate. Such options are removed from the study and only options with moneyness range as 0.90 - 1.10 are considered [

Based on the above-mentioned screening criteria, the number of options analysed in the study are 2290714 for calls and 2201306 for puts.

Equity Data: In addition to the options data, certain filtering criteria are employed to screen the data for Nifty spot prices as well. Transactions during sessions which do not reflect the normal market behaviour, like the pre- open session (9:00 a.m.-9:15 a.m.) and the special extended trading session (beyond 3:30 p.m.), are removed. Furthermore, Nifty being an index, its value changes whenever there is a transaction for any of its constituent stock(s). Since the tick data is recorded up to second and not beyond it, there are numerous cases of more than one value for a single timestamp. To remove any selection bias, all the values for a given timestamp are averaged. This average is taken as the index value for that timestamp.

This section details the SVH model and the calibration methodology adopted. It also describes the procedure for TSRV calculation and the performance measures used for comparison of the pricing performance of the two models. The details of the BS model are not presented here.

Heston [

There are two Brownian motions at work in a stochastic volatility model, one for the drift of the underlying price process and the other for its variance, unlike the BS model, where only the former is present. The SVH model follows the basic assumption of the stochastic volatility models that the variance of the underlying is a random variable. Also, it accounts for the asymmetric contribution of new information (a stylised fact in the financial markets) by assuming that the two stochastic processes are correlated.

The bivariate Ito’s lemma is used to derive the fundamental partial differential equations. The derivation follows the no-arbitrage argument, similar to that of the BS model. However, as opposed to the BS model, two derivative assets are required here to make the resulting portfolio risk-free, as there are two sources of randomness. The two derivative assets are on the same underlying, but differ in strike price and maturity. Because of the second Brownian motion, it is not possible to find a closed-form solution for the European options. Here, the SVH model has an advantage over other models in its category that it has a semi-closed form (or quasi-closed- form) solution available for the plain vanilla European options. This, in turn, makes it feasible to calibrate the model to market prices.

The model builds on the following partial differential equations (the time subscript has been dropped from spot price and variance for better readability):

To take the leverage effect into account, the Wiener stochastic processes W_{1}, W_{2} should be correlated

The quasi-closed form of the Heston model for a European option on a non-dividend paying stock is as follows:

where,

For j = 1, 2 we have

where,

S―Spot price; K―Strike Price; v―Variance; W_{1,2}―Standard Brownian movements/Wiener processes; κ― Mean reversion rate; θ―Long run variance; σ―Volatility of variance; ρ―Correlation parameter; µ―Drift of the underlying; λ―Volatility risk; T―Maturity Date; τ-T-t (time remaining to maturity).

The model was implemented using the package “NMOF”, based on the work of Gilli et al. [

Due to the issues involved in the calculation of put option prices, studies of this nature generally concentrate on the call options for evaluating the performance of models, as the results of the put options are not a fair reflection on the performance of the model.

The SVH model does not have a closed form solution even for the call option. While some parameters of the model can be observed in the market (S, µ, K, T and τ), the other parameters need to be estimated. For this purpose, the model needs to be calibrated to the market price of the options. The calibration details, along with the algorithms considered for calibration, are provided in the next sub-section.

Given the available market price of a European call option, we try to find parameters of a model such that the price of the option obtained from the model is very close to the market price of the option. This process is called calibration. Different algorithms are available depending on the requirements of the problem at hand. What is common in all calibration algorithms is that all such algorithms search a region of parameter space in their specific method, by trying to minimise the error between the market price and model price. So, while the accuracy is important, we also have to consider the time taken by an algorithm to converge. For this reason, global optimizers are usually not recommended as they are very slow. Particularly, this disadvantage becomes more prominent in studies like the present work, where a large amount of data is involved. The general approach involves box constraints; that is, applying upper and lower limits to all the parameters need to be calibrated. It also requires a good initial value as the starting point for the parameter vector.

Five parameters of the SVH model need to be estimated (κ, θ, σ, ρ and v_{0}). The “inverse” method was adopted to estimate these parameters. It involves finding those parameters that produce the correct market prices of vanilla European options. To accomplish the same, an optimisation problem needs to be solved where the absolute differences between market prices and model prices of the vanilla European call options are minimised over the parameter space. We use the below equation for this minimisation procedure:

^{2} (12)

where ^{th} call option obtained using the parameters denoted by vector Ф, ^{th} call option, and N is the number of options used for calibration on any given day. K and τ are the strike price and time to maturity for the i^{th} option, and S is the spot price.

The choice of the optimisation algorithm to be used is discussed in the following subsection.

Ten optimisation algorithms were considered for this study. A ten day sample of the data was considered, and each algorithm was tested on the basis of accuracy and the execution time, to decide on the best algorithm. Due to time and resource constraints these algorithms could not be tested on the entire dataset. The following methods, used in this study, are implemented in R and have been modified as per the requirements:

(1) The two variants of the Differential Evolution (DE), a population based optimisation heuristic, proposed by Storn and Price [

Additionally, the two built-in R functions (namely, the “nlm” and the “nlminb”) were also considered. The “nlm”, proposed by Schnabel et al. [

Based on the analysis detailed above, “nlminb” proved to be the best method for accuracy. Even though it was not the quickest method to converge (3^{rd} out of the ten methods), it gave the best results in an acceptable timeframe. Hence, the subsequent analysis was conducted using this method.

The parameters of the SVH model have lower and upper bounds, beyond which the model is not defined. Despite the ability of “nlminb” to perform unconstrained optimisation, it is advisable to apply bounds to parameters, if available. This makes the convergence faster as the parameter space that an algorithm has to traverse, reduces. Hence, the lower and upper bounds were applied to the parameters.

The initial parameter vector (starting point) for the first day of the sample was obtained using a trial and error method. The model parameters were altered manually to minimise the “mean absolute error of the model price from the market prices”, and the vector thus obtained is used as the initial vector. For the subsequent days, the optimised parameter vector for day n is taken as the initial parameter vector for the day (n + 1). The assumption made here was the nature of the market would not change drastically over one day. Hence, the optimised parameters for the previous day can act as a good starting point for the current day.

The optimisation procedure results in parameter vectors Ф_{1}, Ф_{2}, ∙∙∙ Ф_{485}, which give the best results for the minimisation equation for days 1 - 485 of the sample. The parameter vector of day n is then used to price options for the day (n + 1). For instance, Ф_{1} is used to price options on day 2, Ф_{2} on day 3, and so on. The call option prices obtained are subsequently used to calculate the prices of the corresponding put options using put-call parity. Liquidity-weighted performance measures are then calculated to ascertain the efficiency and consistency of the SVH model. The same is further used for comparison across the different pricing models. The performance measures are detailed in Section 3.3. The next subsection provides details of TSRV calculation.

The issues related to the volatility calculated using closing prices are well documented. TSRV, proposed by Zhang et al. [

The variance at the low frequency

where

The TSRV estimator calculates the variance only for the trading hours, whereas most pricing models require the variance of the entire calendar day. Hence, the estimate obtained in the above equation is scaled up by the ratio of daily close-close to open-close historical variances. Below is the scaling factor that has been used in this study:

The same scaling factor has been used by Jacob and Vipul [

This study attempts to test both the models ex-ante. Therefore, in the case of BS model, the sum of the close-close TSRV estimates for the previous n-days has been used as a proxy for the actual variance to be experienced by the market in the next n-days. The variance estimate is annualised assuming 252 days in a year, as the BS formula requires annualised volatility. This approach makes the implementation easier as the market participants do not have the benefit of hindsight, and using this method does not require to forecast the volatility for the remaining time-to-expiration.

Regularising the SeriesIn any time series of tick data, there may be missing values for certain timestamps due to no trading at those instances of time. For TSRV calculation in this study, the series is regularised such that all timestamps have a corresponding index value. For ticks with no transaction, the last available price is considered as a proxy for the current tick price.

Liquidity-weighted performance measures are used to ascertain the one-day-ahead out-of-sample performance of the two option pricing models. It must be noted that the base is taken as the market price as it would ensure a fair comparison across different models by providing a common basis.

・ Liquidity-weighted Mean Percentage Error (MPE)

・ Liquidity-weighted Mean Absolute Percentage Error (MAPE)

where C_{i} and A_{i} are the calculated and actual prices of the i^{th} option, respectively. Q_{i} is the quantity traded for the i^{th} option and N is the total number of options analysed.

The results and analysis of this comparison are presented in the next section.

The one-day-ahead out-of-sample analysis of the two models shows that the SVH model performs better than the BS model, both in terms of the frequency as well as the magnitude of the pricing errors (

The results provided in

The overall results on the performance of both the models are provided in

The results provided in

Total No. of options | Cases of overpricing (market price > model price) | ||
---|---|---|---|

BS | Call | 2290714 | 2090177 (91.25%) |

Put | 2201306 | 2137066 (97.08%) | |

SVH | Call | 2290714 | 1468740 (64.12%) |

Put | 2201306 | 1908943 (86.72%) |

Note: This table contains the bias shown by each model in terms of frequency of overpricing. The value in parenthesis shows the % of overpriced options in relation to total options contracts studied.

Model | Mean absolute percentage error | Mean percentage error | Standard deviation | |||
---|---|---|---|---|---|---|

(MAPE) | (MPE) | (SD) | ||||

Call | Put | Call | Put | Call | Put | |

BS | 48.1510^{****} | 62.0758 | 47.8299 | 62.0758 | 30.9852 | 29.6548 |

SVH | 29.7929^{****} | 48.2165 | 18.6902 | 46.4818 | 37.3197 | 35.4226 |

Note: This table provides the overall performance measures for the BS Model and the SVH Model. All the metrics considered are liquidity-weighted and are based on trimmed data. The total no. of options analysed are 2290714 and 2201306 for the calls and puts, respectively. This table also reports the significance of the difference between the mispricing of call and put options, tested using the Wilcoxon Rank-Sum test. For both the models, the alternate hypothesis tested is that mispricing in call options is lower than that of the put options. “^{****}” denotes significance at α = 0.01%.

the lower (higher) SD in the case of the BS (SVH) model appears a result of the higher (lower) MPE and skewness of the pricing errors. It may be noted that the BS model produces one-sided bias (negative) for more than 90% of the cases; this leads to lower SD of the MPE.

Also, a statistical comparison of the mispricing magnitudes of both the models confirms that the SVH model outperforms the BS model comprehensively, for both option types. The results are provided in

Further, for a robust comparison, the mispricing has been examined across different subgroups formed by moneyness, volatility and time-to-expiration of an option. It would enable us to identify whether the BS model performs poorly only for certain cases of volatility, moneyness, and time-to-expiration; or, it is inferior to the SVH model across the subgroups.

The performance across various moneyness groups is provided in

The pattern across various subgroups is relevant once we establish that the mispricing in each subgroup is different from the remaining subgroups. To investigate this, the Kruskal-Wallis test (H-statistic) is used. This is a non-parametric test that compares the equality of medians for three or more subgroups. However, this test does not provide any information about the pair-wise differences. To overcome this, the Dunn’s test for the posthoc analysis was performed. The results validate that the subgroups indeed differ for both the option types, except for the ITM-deep ITM pair in the case of call options.

Model 1 | Model 2 | Test Stat. (p-value) |
---|---|---|

BS | SVH | 3.34817e+12 (1.0000) |

SVH | BS | 1.8992e+12^{****} (0.0000) |

Model 1 | Model 2 | Test Stat. (p-value) |
---|---|---|

BS | SVH | 2.8690e+12 (1.0000) |

SVH | BS | 1.9766e+12^{****} (0.0000) |

Notes for ^{****}” denotes significance at α = 0.01%. This is a one-sided test; therefore, it become necessary to conduct a two-way comparison.

Moneyness range | No. of records | Mean absolute percentage error (MAPE) | Mean percentage error (MPE) | Standard deviation (SD) | |||
---|---|---|---|---|---|---|---|

BS | SVH | BS | SVH | BS | SVH | ||

Deep OTM | 374923 | 74.2984 | 55.8299 | 74.1226 | 42.0319 | 27.8771 | 48.7923 |

OTM | 950352 | 61.0122 | 38.9083 | 60.7085 | 22.8575 | 29.1915 | 44.8022 |

ATM | 924630 | 27.2004 | 12.7289 | 26.7631 | 5.0707 | 17.5958 | 17.6362 |

ITM | 37192 | 7.0644 | 3.6435 | 6.9501 | 1.5580 | 4.7184 | 4.1708 |

Deep ITM | 3617 | 3.0501 | 1.8693 | 3.0501 | 1.5543 | 2.0107 | 1.7954 |

Note: This table provides the performance measures for all the models for various moneyness categories of NIFTY call options. All the metrics considered are liquidity-weighted metrics on the trimmed data. The moneyness range is taken from 0.90 - 1.10 with blocks of 0.04 points forming a group. For example, the “at-the-money” (ATM) range is 0.98 - 1.02.

Moneyness range | No. of records | Mean absolute percentage error (MAPE) | Mean percentage error (MPE) | Standard deviation (SD) | |||
---|---|---|---|---|---|---|---|

BS | SVH | BS | SVH | BS | SVH | ||

Deep OTM | 399575 | 94.8584 | 90.7328 | 94.8584 | 90.7328 | 7.5783 | 11.5977 |

OTM | 784955 | 79.9728 | 61.4171 | 79.9728 | 61.1088 | 17.2617 | 27.5145 |

ATM | 968587 | 35.1221 | 21.2730 | 35.1221 | 16.7311 | 17.9281 | 19.4275 |

ITM | 44341 | 8.8085 | 7.6556 | 8.7123 | 6.1742 | 6.3367 | 7.4749 |

Deep ITM | 3848 | 4.3303 | 4.5474 | 4.3258 | 4.1556 | 2.9167 | 3.4789 |

Note: This table provides the performance measures for all the models for various moneyness categories of NIFTY put options. All the metrics considered are liquidity-weighted metrics on the trimmed data. The meaning of moneyness for calls and puts is reversed. The moneyness range is taken from 0.90 - 1.10 with blocks of 0.04 points forming a group. For example, the “at-the-money” (ATM) range is 0.98 - 1.02.

al. [

For the deep OTM options, the performance of both the models appears to be poor and unreliable. In this regard, two possible reasons can be proposed. Firstly, such options are very cheap compared to other options with different moneyness. This leads to a large percentage error even for a small pricing error as the base is very small. Secondly, owing to the high demand for the OTM options, the option writers seem to charge a premium for providing such options. Furthermore, even the SD for all models is very high in the case of the OTM options. It indicates that the results in the OTM category may not be very reliable, as the SVH and the BS do not appear to be equipped with required inputs to deal with such options.

This section details the performance of the two models across four different subgroups based on the volatility of the underlying (TSRV-based estimates). The subgroups “0.05 - 0.10” and “0.20 - 0.25” refer to low and high volatility regimes, respectively. The remaining subgroups, viz., “0.10 - 0.15” and “0.15 - 0.20”, represent the normal market.

The Kruskal-Wallis test and the Dunn’s test for posthoc analysis were performed. The analysis shows that the subgroups formed based on the volatility subgroups are statistically different from each other, for both the models and the option types.

The analysis of the subgroups, presented in

Volatility range | No. of records | Mean absolute percentage error (MAPE) | Mean percentage error (MPE) | Standard deviation (SD) | |||
---|---|---|---|---|---|---|---|

BS | SVH | BS | SVH | BS | SVH | ||

0.05 - 0.10 | 370798 | 70.8821 | 63.1972 | 70.8821 | 44.2736 | 25.2402 | 74.4797 |

0.10 - 0.15 | 930104 | 54.1409 | 23.2749 | 54.1409 | 11.1616 | 28.5283 | 31.4509 |

0.15 - 0.20 | 704045 | 33.0342 | 22.2841 | 29.8520 | 8.1447 | 29.3576 | 30.5589 |

0.20 - 0.25 | 285767 | 37.4126 | 32.0323 | 34.3166 | 25.0277 | 29.9567 | 34.7098 |

Note: This table provides the performance measures for all the models for various volatility subgroups of NIFTY call options. The volatility presented in the table is the annualised sum of the daily realised volatility of the underlying, scaled to close-close timeframe, for the days remaining to expiration. Volatility presented is in ratio form and can be converted to percent terms. All the metrics considered are liquidity-weighted metrics on the trimmed data.

Volatility range | No. of records | Mean absolute percentage error (MAPE) | Mean percentage error (MPE) | Standard deviation (SD) | |||
---|---|---|---|---|---|---|---|

BS | SVH | BS | SVH | BS | SVH | ||

0.05 - 0.10 | 357794 | 74.3831 | 57.9888 | 74.3831 | 54.0769 | 27.4733 | 38.3063 |

0.10 - 0.15 | 946246 | 70.0622 | 49.6813 | 70.0622 | 48.2407 | 27.0095 | 36.6037 |

0.15 - 0.20 | 654398 | 50.0259 | 43.1687 | 50.0256 | 41.4129 | 28.3606 | 33.1657 |

0.20 - 0.25 | 242868 | 42.1069 | 41.1126 | 41.9557 | 38.3126 | 27.9002 | 33.1957 |

Note: This table provides the performance measures for all the models for various volatility subgroups of NIFTY put options. The volatility presented in the table is the annualised sum of the daily realised volatility of the underlying, scaled to close-close timeframe, for the days remaining to expiration. Volatility presented is in ratio form and can be converted to percent terms. All the metrics considered are liquidity-weighted metrics on the trimmed data.

However, the models perform poorly when the volatility is either very high or very low. The volatility- risk-premium, charged by the option writers for options in the high volatility regime, may explain this behaviour. This premium overstates the market price of the options. Given the fact that none of the models are designed to account for such a premium, we observe large deviations from the market prices when the volatility is high.

Both models follow similar patterns for the call options. Further, the put options have the same pattern; though, it is different from the one observed for the call options. The similarity in the performances of the two models is surprising as both models have different theoretical foundations and also have different estimation methods.

Lastly, this section examines the behaviour of the mispricing with respect to the time-to-expiration of the options. For the purpose, the subgroups for this analysis are based on the calendar days. Therefore, the maturity of the option contracts covered in this study would translate to 2 - 60 days calendar days as we have ignored option with maturity less than two days to contain expiration effect. The data is divided into eight subgroups based on time-to-expiration.

Similar to the preceding subgroups, the Kruskal-Wallis test and the Dunn’s test for posthoc analysis were performed. These tests confirm that, for both models and option types, all subgroups are significantly different from each other.

As can be observed from

While there are no clear patterns, except the first subgroup (≤5 days), the SVH model gives a consistently good performance. In fact, the MPE for time-to-expiration of 31 days and more is close to just 3%. Although the liquidity is not very high for these subgroups, it is still enough to suggest that the SVH model can be used as a viable method for pricing the call options with larger maturities. A vast majority of the trades in the Indian options market takes place in the category of 6 - 30 days to maturity. Notably, in this region, the SVH model gives a reasonably good performance. Therefore, the SVH model seems to capture the pricing dynamics in the Indian index options market fairly well.

In sum, the findings from the preceding subsections confirm that the SVH model is a far better choice for pricing Indian index options, as compared to the BS model; it comprehensively outperforms the BS model for all subgroups, across option types. It gives a reasonably good performance independently as well. In totality, the SVH proves to be a better model for pricing the index options in the Indian context.

It may be noted that all the results presented here are without considering the transaction costs. However, the same would not have any implications when mispricing is compared across different models.

Time-to-expiration range | No. of records | Mean absolute percentage error (MAPE) | Mean percentage error (MPE) | Standard deviation (SD) | |||
---|---|---|---|---|---|---|---|

BS | SVH | BS | SVH | BS | SVH | ||

≤5 | 353625 | 68.3102 | 44.4955 | 68.3102 | 28.6121 | 31.0013 | 49.1893 |

6 - 10 | 708027 | 55.8222 | 32.2794 | 55.8222 | 15.5565 | 30.6488 | 42.2656 |

11 - 15 | 349593 | 48.6490 | 26.3771 | 48.6490 | 16.9757 | 27.0676 | 34.0255 |

16 - 20 | 275752 | 39.0921 | 25.2858 | 37.0106 | 18.2660 | 28.8838 | 31.7347 |

21 - 25 | 293214 | 36.0255 | 24.4485 | 31.4689 | 17.7689 | 29.7643 | 31.6276 |

26 - 30 | 164185 | 31.9995 | 26.8742 | 27.6458 | 20.2429 | 29.1165 | 33.1840 |

31 - 40 | 96884 | 33.4891 | 22.0280 | 32.6871 | 3.3748 | 23.0099 | 31.0082 |

41 - 60 | 49434 | 32.1764 | 25.6592 | 26.7592 | 3.7892 | 27.4121 | 36.0873 |

Note: This table provides the performance measures for all the models for various subgroups of NIFTY call options, based on time-to-expiration. The time-to-expiration range is based on calendar days. All the metrics considered are liquidity-weighted metrics on the trimmed data.

Time-to-expiration range | No. of records | Mean absolute percentage error (MAPE) | Mean percentage error (MPE) | Standard deviation (SD) | |||
---|---|---|---|---|---|---|---|

BS | SVH | BS | SVH | BS | SVH | ||

≤5 | 359884 | 73.7870 | 55.9240 | 73.7870 | 48.9164 | 30.3493 | 43.9246 |

6 - 10 | 686077 | 65.0926 | 46.8797 | 65.0926 | 41.8778 | 31.1297 | 40.9364 |

11 - 15 | 323565 | 60.7841 | 43.5895 | 60.7841 | 43.0155 | 27.7266 | 33.4927 |

16 - 20 | 268104 | 57.0187 | 49.0960 | 57.0187 | 48.9592 | 28.0565 | 30.3546 |

21 - 25 | 261325 | 53.1039 | 48.5147 | 52.8436 | 48.4755 | 29.0431 | 29.8062 |

26 - 30 | 160541 | 55.0958 | 52.9967 | 55.0958 | 52.9967 | 26.1643 | 26.8838 |

31 - 40 | 94501 | 56.6399 | 44.3648 | 56.6399 | 44.3059 | 22.1895 | 26.9158 |

41 - 60 | 47309 | 55.2337 | 49.1082 | 55.2337 | 49.0415 | 23.0327 | 26.1230 |

Note: This table provides the performance measures for all the models for various subgroups of NIFTY put options, based on time-to-expiration. The time-to-expiration range is based on calendar days. All the metrics considered are liquidity-weighted metrics on the trimmed data.

The SVH model seems to be a popular choice amongst academicians and practitioners alike. Despite that, there has been no study in India which analyses the performance of the model using tick data. This study tries to fill the gap by conducting a one-day-ahead out-of-sample performance analysis of the SVH model, and compares its performance to that of a TSRV-based BS model, using liquidity-weighted performance metrics.

The findings of this paper establish that the SVH model comprehensively outperforms the BS model, for both option types. Notably, the superior performance of the SVH model gets corroborated across moneyness/volatility/ time-to-expiration subgroups. Also, the statistical comparison among the models shows that the SVH model is a far better model for pricing Indian index options. Even the bias exhibited by the SVH model is significantly lower than that of the BS model.

Remarkably, the pattern followed by both the models appears to be the same across subgroups based on moneyness and volatility, especially for the call options. However, it appears to be a bit counter-intuitive as both models build on completely different theoretical foundations, and even their estimation procedures are not the same. For the moneyness subgroups, both models show that the mispricing reduces as an option goes from deep OTM to deep ITM. In the case of volatility subgroups, both models perform the best in the normal volatility regime.

Prima-facie, the SVH model appears to perform poorly compared to the BS model with regard to the SD of MPE, especially for the low volatility and the deep OTM subgroups. However, it is important to note that the large (small) SD of MPE is attributable to the small (large) MPE. In the case of the BS, the MPE is highly skewed; this leads to the small SD of MPE compared to that in the case of the SVH model. In sum, it may be concluded that the SVH model outperforms the BS model comprehensively. In other words, it captures the pricing dynamics of the Indian index options market fairly well.

Regarding its contribution, the study offers a significant extension to the existing empirical option pricing literature in India. It bridges an important gap on option pricing as no study has been conducted in the Indian options market on the Heston model, using tick data. This would help the trades and other stakeholders in the options market in identifying a suitable model to price and hedge their positions with higher accuracy.

Shivam Singh,Alok Dixit, (2016) Performance of the Heston’s Stochastic Volatility Model: A Study in Indian Index Options Market. Theoretical Economics Letters,06,151-165. doi: 10.4236/tel.2016.62018