Volume 6, Issue 5, October 2017, Page: 215-221
Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model
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
Received: Jun. 9, 2017;       Accepted: Jun. 26, 2017;       Published: Sep. 7, 2017
DOI: 10.11648/j.acm.20170605.11      View  419      Downloads  62
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.
Keywords
Geometric Brownian Motion (GBM), Alternating Direction Implicit (ADI) Scheme Crank Nicolson Scheme, Black-Scholes Model, European Put Option
To cite this article
Kolawole Imole Oluwakemi, Mataramvura Sure, Ogunlade Temitope Olu, Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model, Applied and Computational Mathematics. Vol. 6, No. 5, 2017, pp. 215-221. doi: 10.11648/j.acm.20170605.11
Copyright
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[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.
Browse journals by subject