Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model

Kolawole Imole Oluwakemi; Mataramvura Sure; Ogunlade Temitope Olu

doi:doi:10.11648/j.acm.20170605.11

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Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model

Kolawole Imole Oluwakemi, Mataramvura Sure, Ogunlade Temitope Olu

Published in Applied and Computational Mathematics (Volume 6, Issue 5)

Received: 9 June 2017 Accepted: 26 June 2017 Published: 7 September 2017

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Abstract

Stochastic volatility models were introduced because option prices have been mis-priced using Black-Scholes model. In this work, focus is made on pricing European put option in a Geometric Brownian Motion (GBM) stochastic volatility model with uncorrelated stock and volatility. The option is priced using two numerical methods (Crank-Nicolson and Alternating Direction Implicit (ADI) finite difference). Numerical schemes were considered because the closed form solution to the model could not be obtained. The change in option value due to changes in volatility, maturity time and market price of volatility risk are considered and comparison between the efficiency of the numerical methods by computing the CPU time was made.

Published in	Applied and Computational Mathematics (Volume 6, Issue 5)
DOI	10.11648/j.acm.20170605.11
Page(s)	215-221
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Geometric Brownian Motion (GBM), Alternating Direction Implicit (ADI) Scheme Crank Nicolson Scheme, Black-Scholes Model, European Put Option

References

[1]	K. Ajayi and T. O Ogunlade. Global Journal of Computer Science and Technology 11(10) pp 5-10, 2011.
[2]	M. Bellalah. Derivatives, Risk Management & Value. World Scientific Publishing Co. Pte. Ltd., 2010.
[3]	D. J. Duffy. Finite Difference Methods in Financial Engineering A Partial Differential Equation Approach. John Wiley & Sons, Ltd, 2006.
[4]	K. I. Houtand S. Foulon. ADI Finite Difference Schemes for Option Pricing in the Heston Model With Correlation. International Journal of Numerical Analysis and Modeling, 7:2303-320, 2010.
[5]	J. Hull and A. White. Pricing of Options on Assets with Stochastic Volatilities. The Journal of Finance, 42:281-300, 1987.
[6]	S. Ikonen and J. Toivanen. Operator Splitting Methods for American Options with Stochastic Volatility. European Congress on Computational Methods in Applied Sciences and Engineering, 2004.
[7]	M. Musiela and M. Rutkowski. Martingale Methods in Financial Modelling. Springer, 2004.
[8]	C. R. Nwozo and S. E. Fadugba. On Stochastic Volatility in the Valuation of European Options. BritishJournal of Mathematics & Computer Science, 5:104-127, 2014.
[9]	F. D. Rouah. The Heston Model and Its Extensions in Matlab and C#. John Wiley & Sons, Inc., 2013.
[10]	S. E. Shreve. Stochastic Calculus for Finance II: Continuous-time models, volume 11. Springer Science& Business Media, 2004.
[11]	U. F. Wiersema. Brownian Motion Calculus. John Wiley & Sons Ltd, 2008.
[12]	S.-P. Zhu and W.-T. Chen. A New Analytical Approximation for European Puts with Stochastic Volatility. Applied Mathematics Letters., page 23 (2010) 687-692, 2010.

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APA Style

Kolawole Imole Oluwakemi, Mataramvura Sure, Ogunlade Temitope Olu. (2017). Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model. Applied and Computational Mathematics, 6(5), 215-221. https://doi.org/10.11648/j.acm.20170605.11

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ACS Style

Kolawole Imole Oluwakemi; Mataramvura Sure; Ogunlade Temitope Olu. Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model. Appl. Comput. Math. 2017, 6(5), 215-221. doi: 10.11648/j.acm.20170605.11

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AMA Style

Kolawole Imole Oluwakemi, Mataramvura Sure, Ogunlade Temitope Olu. Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model. Appl Comput Math. 2017;6(5):215-221. doi: 10.11648/j.acm.20170605.11

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@article{10.11648/j.acm.20170605.11,
  author = {Kolawole Imole Oluwakemi and Mataramvura Sure and Ogunlade Temitope Olu},
  title = {Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {5},
  pages = {215-221},
  doi = {10.11648/j.acm.20170605.11},
  url = {https://doi.org/10.11648/j.acm.20170605.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170605.11},
  abstract = {Stochastic volatility models were introduced because option prices have been mis-priced using Black-Scholes model. In this work, focus is made on pricing European put option in a Geometric Brownian Motion (GBM) stochastic volatility model with uncorrelated stock and volatility. The option is priced using two numerical methods (Crank-Nicolson and Alternating Direction Implicit (ADI) finite difference). Numerical schemes were considered because the closed form solution to the model could not be obtained. The change in option value due to changes in volatility, maturity time and market price of volatility risk are considered and comparison between the efficiency of the numerical methods by computing the CPU time was made.},
 year = {2017}
}

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TY  - JOUR
T1  - Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model
AU  - Kolawole Imole Oluwakemi
AU  - Mataramvura Sure
AU  - Ogunlade Temitope Olu
Y1  - 2017/09/07
PY  - 2017
N1  - https://doi.org/10.11648/j.acm.20170605.11
DO  - 10.11648/j.acm.20170605.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
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EP  - 221
PB  - Science Publishing Group
SN  - 2328-5613
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AB  - Stochastic volatility models were introduced because option prices have been mis-priced using Black-Scholes model. In this work, focus is made on pricing European put option in a Geometric Brownian Motion (GBM) stochastic volatility model with uncorrelated stock and volatility. The option is priced using two numerical methods (Crank-Nicolson and Alternating Direction Implicit (ADI) finite difference). Numerical schemes were considered because the closed form solution to the model could not be obtained. The change in option value due to changes in volatility, maturity time and market price of volatility risk are considered and comparison between the efficiency of the numerical methods by computing the CPU time was made.
VL  - 6
IS  - 5
ER  -

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Author Information

Kolawole Imole Oluwakemi

School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, Scotland
Mataramvura Sure

Department of Mathematical Sciences, University of Cape Town, Cape Town, South Africa
Ogunlade Temitope Olu

Department of Mathematics, Ekiti State University, Ado-Ekiti, Nigeria

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APA Style

Kolawole Imole Oluwakemi, Mataramvura Sure, Ogunlade Temitope Olu. (2017). Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model. Applied and Computational Mathematics, 6(5), 215-221. https://doi.org/10.11648/j.acm.20170605.11

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ACS Style

Kolawole Imole Oluwakemi; Mataramvura Sure; Ogunlade Temitope Olu. Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model. Appl. Comput. Math. 2017, 6(5), 215-221. doi: 10.11648/j.acm.20170605.11

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AMA Style

Kolawole Imole Oluwakemi, Mataramvura Sure, Ogunlade Temitope Olu. Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model. Appl Comput Math. 2017;6(5):215-221. doi: 10.11648/j.acm.20170605.11

Copy | Download

@article{10.11648/j.acm.20170605.11,
  author = {Kolawole Imole Oluwakemi and Mataramvura Sure and Ogunlade Temitope Olu},
  title = {Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {5},
  pages = {215-221},
  doi = {10.11648/j.acm.20170605.11},
  url = {https://doi.org/10.11648/j.acm.20170605.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170605.11},
  abstract = {Stochastic volatility models were introduced because option prices have been mis-priced using Black-Scholes model. In this work, focus is made on pricing European put option in a Geometric Brownian Motion (GBM) stochastic volatility model with uncorrelated stock and volatility. The option is priced using two numerical methods (Crank-Nicolson and Alternating Direction Implicit (ADI) finite difference). Numerical schemes were considered because the closed form solution to the model could not be obtained. The change in option value due to changes in volatility, maturity time and market price of volatility risk are considered and comparison between the efficiency of the numerical methods by computing the CPU time was made.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model
AU  - Kolawole Imole Oluwakemi
AU  - Mataramvura Sure
AU  - Ogunlade Temitope Olu
Y1  - 2017/09/07
PY  - 2017
N1  - https://doi.org/10.11648/j.acm.20170605.11
DO  - 10.11648/j.acm.20170605.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 215
EP  - 221
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20170605.11
AB  - Stochastic volatility models were introduced because option prices have been mis-priced using Black-Scholes model. In this work, focus is made on pricing European put option in a Geometric Brownian Motion (GBM) stochastic volatility model with uncorrelated stock and volatility. The option is priced using two numerical methods (Crank-Nicolson and Alternating Direction Implicit (ADI) finite difference). Numerical schemes were considered because the closed form solution to the model could not be obtained. The change in option value due to changes in volatility, maturity time and market price of volatility risk are considered and comparison between the efficiency of the numerical methods by computing the CPU time was made.
VL  - 6
IS  - 5
ER  -

Copy | Download