An Empirical Comparison of Alternative Stochastic Volatility
Option Pricing Models in an Emerging Market: Evidence from
Indian Nifty 50 Index Option Market
Adam Wilson
201707012
University of Strathclyde: Department of Accounting and Finance
Honours Dissertation
Submitted in partial fulfilment of the requirements for the degree of
BSc (Hons) Mathematics, Statistics & Finance
,ABSTRACT
This paper investigates the improvement in the pricing of the Indian emerging market NIFTY 50 index
call options when stochastic volatility is taken into account. The empirical performances of the Dumas
et al. (1998) Practitioners Black Scholes model, the Heston (1993) Stochastic Volatility model, the
Heston and Nandi (2000) GARCH model and the Bates (1996) Stochastic Volatility Jump Diffusion
model are compared in an emerging market from three different angles: (1) model specification, (2) in-
sample pricing performance, (3) out-of-sample pricing performance. First, it is found that even after the
inclusion of stochastic volatility to a model and the inclusion of stochastic volatility coupled with jump-
diffusion to a model, all the models are still significantly misspecified, as none of the models can
mitigate the “volatility smiles”. However, there is a noticeable improvement in the specification of the
models that include stochastic volatility compared to the Dumas et al. (1998) Practitioners Black
Scholes Model. Second, it is concluded that Bates (1996) Stochastic Volatility Jump Diffusion model
outperforms the other models for both in-sample and out-of-sample pricing. Third, it is found that in an
emerging market, the inclusion of stochastic volatility offers a significant improvement over the Dumas
et al. (1998) Practitioners Black Scholes model, and the inclusion of jump-diffusion to a stochastic
volatility model provides further improvement in pricing performance.
ii
,TABLE OF CONTENTS
1.0 INTRODUCTION 1
1.1 History of Options......................................................................................................................... 1
1.2 History of the Black-Scholes Model ............................................................................................. 1
1.3 Motivation & Contribution ........................................................................................................... 2
1.4 Structure of Dissertation ............................................................................................................... 3
2.0 LITERATURE REVIEW 4
2.1 Geometric Brownian Motion & the Black-Scholes Model ........................................................... 4
2.2 Alternative Models........................................................................................................................ 6
2.3 Jump-Diffusion Process ................................................................................................................ 7
2.4 Stochastic Volatility ...................................................................................................................... 8
2.5 GARCH Process ......................................................................................................................... 11
3.0 THE MODELS 14
3.1 Practitioners Black Scholes Model ............................................................................................. 14
3.2 Heston Stochastic Volatility Model ............................................................................................ 15
3.3 Heston and Nandi GARCH Model ............................................................................................. 17
3.4 Bates Stochastic Volatility Jump Diffusion Model ..................................................................... 20
4.0 RESEARCH METHODOLOGY 22
4.1 Calibration of the Models ........................................................................................................... 22
4.1.1 Loss Function ...................................................................................................................... 22
4.1.2 Minimisation Algorithm ...................................................................................................... 23
4.2 Implied Volatilities ..................................................................................................................... 24
4.3 Pricing Performance & Error Structure....................................................................................... 25
4.4 Summary of Methodology .......................................................................................................... 27
4.5 Research Limitations .................................................................................................................. 27
5.0 DATA 28
5.1 Options Contract Structure.......................................................................................................... 28
5.2 Data Sources ............................................................................................................................... 29
5.3 Filtered Data & the Categories.................................................................................................... 30
5.4 Discussion of Data ...................................................................................................................... 31
6.0 EMPIRICAL RESULTS 32
6.1 Implied Volatility Smile of the BS Model .................................................................................. 32
6.2 Parameter Estimation .................................................................................................................. 33
6.2.1 PBS Model ........................................................................................................................... 33
6.2.2 Heston SV Model & Bates SVJD model .............................................................................. 33
6.2.3 Heston and Nandi GARCH Model ...................................................................................... 34
6.3 Model Implied Volatilities .......................................................................................................... 35
iii
, 6.4 In-Sample Pricing Performance .................................................................................................. 37
6.5 Out-of-Sample Pricing Performance ........................................................................................... 41
7.0 CONCLUSION 45
Appendix 1 – Tables & Figures 48
Appendix 2 – MATLAB Code 71
Model Functions ............................................................................................................................... 71
PBS Model .................................................................................................................................... 71
Heston SV Model .......................................................................................................................... 71
Heston and Nandi GARCH Model ............................................................................................... 72
Bates SVJD Model ........................................................................................................................ 74
Calibration of the Models ................................................................................................................. 75
PBS Model .................................................................................................................................... 75
Heston SV Model .......................................................................................................................... 76
Heston and Nandi GARCH Model ............................................................................................... 78
Bates SVJD Model ........................................................................................................................ 80
Implied Volatilities Functions ........................................................................................................... 82
PBS Model .................................................................................................................................... 82
Heston SV Model & Bates SVJD Model ....................................................................................... 82
Heston and Nandi GARCH Model ............................................................................................... 82
References 84
ACKNOWLEDGEMENTS 91
DECLARATION 92
iv
Option Pricing Models in an Emerging Market: Evidence from
Indian Nifty 50 Index Option Market
Adam Wilson
201707012
University of Strathclyde: Department of Accounting and Finance
Honours Dissertation
Submitted in partial fulfilment of the requirements for the degree of
BSc (Hons) Mathematics, Statistics & Finance
,ABSTRACT
This paper investigates the improvement in the pricing of the Indian emerging market NIFTY 50 index
call options when stochastic volatility is taken into account. The empirical performances of the Dumas
et al. (1998) Practitioners Black Scholes model, the Heston (1993) Stochastic Volatility model, the
Heston and Nandi (2000) GARCH model and the Bates (1996) Stochastic Volatility Jump Diffusion
model are compared in an emerging market from three different angles: (1) model specification, (2) in-
sample pricing performance, (3) out-of-sample pricing performance. First, it is found that even after the
inclusion of stochastic volatility to a model and the inclusion of stochastic volatility coupled with jump-
diffusion to a model, all the models are still significantly misspecified, as none of the models can
mitigate the “volatility smiles”. However, there is a noticeable improvement in the specification of the
models that include stochastic volatility compared to the Dumas et al. (1998) Practitioners Black
Scholes Model. Second, it is concluded that Bates (1996) Stochastic Volatility Jump Diffusion model
outperforms the other models for both in-sample and out-of-sample pricing. Third, it is found that in an
emerging market, the inclusion of stochastic volatility offers a significant improvement over the Dumas
et al. (1998) Practitioners Black Scholes model, and the inclusion of jump-diffusion to a stochastic
volatility model provides further improvement in pricing performance.
ii
,TABLE OF CONTENTS
1.0 INTRODUCTION 1
1.1 History of Options......................................................................................................................... 1
1.2 History of the Black-Scholes Model ............................................................................................. 1
1.3 Motivation & Contribution ........................................................................................................... 2
1.4 Structure of Dissertation ............................................................................................................... 3
2.0 LITERATURE REVIEW 4
2.1 Geometric Brownian Motion & the Black-Scholes Model ........................................................... 4
2.2 Alternative Models........................................................................................................................ 6
2.3 Jump-Diffusion Process ................................................................................................................ 7
2.4 Stochastic Volatility ...................................................................................................................... 8
2.5 GARCH Process ......................................................................................................................... 11
3.0 THE MODELS 14
3.1 Practitioners Black Scholes Model ............................................................................................. 14
3.2 Heston Stochastic Volatility Model ............................................................................................ 15
3.3 Heston and Nandi GARCH Model ............................................................................................. 17
3.4 Bates Stochastic Volatility Jump Diffusion Model ..................................................................... 20
4.0 RESEARCH METHODOLOGY 22
4.1 Calibration of the Models ........................................................................................................... 22
4.1.1 Loss Function ...................................................................................................................... 22
4.1.2 Minimisation Algorithm ...................................................................................................... 23
4.2 Implied Volatilities ..................................................................................................................... 24
4.3 Pricing Performance & Error Structure....................................................................................... 25
4.4 Summary of Methodology .......................................................................................................... 27
4.5 Research Limitations .................................................................................................................. 27
5.0 DATA 28
5.1 Options Contract Structure.......................................................................................................... 28
5.2 Data Sources ............................................................................................................................... 29
5.3 Filtered Data & the Categories.................................................................................................... 30
5.4 Discussion of Data ...................................................................................................................... 31
6.0 EMPIRICAL RESULTS 32
6.1 Implied Volatility Smile of the BS Model .................................................................................. 32
6.2 Parameter Estimation .................................................................................................................. 33
6.2.1 PBS Model ........................................................................................................................... 33
6.2.2 Heston SV Model & Bates SVJD model .............................................................................. 33
6.2.3 Heston and Nandi GARCH Model ...................................................................................... 34
6.3 Model Implied Volatilities .......................................................................................................... 35
iii
, 6.4 In-Sample Pricing Performance .................................................................................................. 37
6.5 Out-of-Sample Pricing Performance ........................................................................................... 41
7.0 CONCLUSION 45
Appendix 1 – Tables & Figures 48
Appendix 2 – MATLAB Code 71
Model Functions ............................................................................................................................... 71
PBS Model .................................................................................................................................... 71
Heston SV Model .......................................................................................................................... 71
Heston and Nandi GARCH Model ............................................................................................... 72
Bates SVJD Model ........................................................................................................................ 74
Calibration of the Models ................................................................................................................. 75
PBS Model .................................................................................................................................... 75
Heston SV Model .......................................................................................................................... 76
Heston and Nandi GARCH Model ............................................................................................... 78
Bates SVJD Model ........................................................................................................................ 80
Implied Volatilities Functions ........................................................................................................... 82
PBS Model .................................................................................................................................... 82
Heston SV Model & Bates SVJD Model ....................................................................................... 82
Heston and Nandi GARCH Model ............................................................................................... 82
References 84
ACKNOWLEDGEMENTS 91
DECLARATION 92
iv
