On a Variance Gamma Model (VGM) in Option Pricing: A Difference of Two Gamma Processes

M.E. Adeosun, S.O. Edeki, O.O. Ugbebor

Abstract


The Variance-Gamma (VG) process is a three parameter stochastic process with respect to a Brownian motion. Here, we consider in our presentation, a detailed study of the VG process expressed as a difference of two gamma processes. As a result, we obtain the basic moments of the process using the characteristic function of the VG process with regard to the parameters of a differenced gamma processes. Also, the Levy-Khintchine formula for the process is derived via the Frullani’s integral. Finally, a modified European call option VG model incorporating a difference of two gamma processes is proposed.


Keywords


Option pricing; Variance gamma model; Levy processes; Levy-Khintchine formula

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References


S.O. Edeki, I. Adinya, O.O. Ugbebor, The Effect of Stochastic Capital Reserve on Actuarial Risk Analysis via an Integro-differential equation, IAENG International Journal of Applied Mathematics, 44 (2), (2014): 83-90.

P. Samuelson, Economics Theory and Mathematics-An appraisal, cowls foundation Paper 61, Reprinted, American Economic Review, 42 (1952).

L. Bachielier, Theorie de la speculative. Paris:Ganauthier-Villa Translated in cooler (1964).

F. Black and M. Scholes, The Pricing of options and cororporate liability, J. political economy, 81, (1973): 637-659.

M.E. Adeosun, S.O. Edeki, O.O. Ugbebor, Stochastic Analysis of Stock Market Price Models: A Case Study of the Nigerian Stock Exchange (NSE), WSEAS Transactions on Mathematics, 14, (2015): 353-363.

E.A. Owoloko and M.C. Okeke. Investigating the imperfection of the B-S Model; A case of Emerging Market, British Journal of Applied Science and Technology, 4(29), (2014): 4191-4200.

S. Trautmann and M. Beinert. Stock Price Jumps and their impact on Option Valuation, Johannes Gutenberg University, (1994), 59.

W. Schoutens. Levy Processes in Finance: Pricing Financial Derivatives, John Wiley and sons, Ltd,(2003) ISBN: 0-470-85156-2.

O.O. Ugbebor and S.O. Edeki, On Duality Principle in Exponentially Le ́vy Market, Journal of Applied Mathematics & Bioinformatics, 3 (2), (2013), 159-170.

D.B. Madan and E. Seneta, The variance Gamma (V.G) model for share market returns. The Journal of Business, 63 (4), (1990):511-524.

E. Eberlein, and U., Keller Hyperbolic distributions in Finance. Bernoulli, 1, (1995):281-299.

O.E. Barndorf-Nielson Normal Inverse Guassian Distributions and the Modeling of Stock Returns. Research Report no. 300, Department of Theoretical Statistics, Aarhus University, (1995).

P. Carr, H. Geman, D.H. Madan and M. Yor The Fine Structure of Asset Returns: An Empirical Investigation, Journal of Business, 75 (2002):305-332.

F. Milne and D. Madan. Option Pricing with V.G Martingale Components. Mathematical Finance, 1 (4) (1991):39-55.

D.B. Madan, P.P. Carr and E. C. Chang. The Variance Gamma Process and Option Pricing. European Finance Review 2 (1998):79-105.

G. Baski and Z. Chen. An Alternative Valuation Model for Contingent Claims, Journal of Financial Economics, 44 (1), (1997):123-165.

L. Scott. Pricing Stock Options in a Jump-Diffusion Model with Stochastic Volatility and Interest Rates: Application of Fourier Inversion Methods, Mathematical Finance, 7, (1997): 413-426.

D. Bates. Jumps and Stochastic Volatility: Exchange Rates Processes Implicit in Deutschemark Options, Review of Financial studies, 9, (1996): 69-108.

P. Carr and D.B Madan. Option Valuation using the Fast Fourier Transform. Journal of Computational Finance 2, (4), (1999): 61-73.

F. Fiorani, The Variance- Gamma for Option Pricing, Working paper, (2001).

A.E. Kyprianou. Introductory Lectures on Fluctuations of Levy Processes with Applications, Springer Berlin Heiderberg New York, (2006), ISBN-13 978-3-540-31342-7.

W. Schoutens and C. Jessica.Levy Processes in Credit Risk, John Wiley and sons, ltd., (2009), ISBN 978-0-470-74306-S.

B. De Finetti Sulle Funzioni and Increment Aleatorio. Read Acc. Naz. Lincei 10, 163-168.

K. Sato. Levy Processes and Infinitely Divisible distributions, Cambridge University Press; (1999), ISBN-10: 0521553024

A. C. Gourieroux, G. LeFol and B.Meyer, Analysis of Ordered Queues, working paper, CEST, Paris, Finance, (1996).

R. White Option Pricing with Fourier Methods; Open Gamma Quantitative Research, 7, (2012).

K.F. Riley, M.P. Hopson and S.J. Bance, Mathematical Methods for Physics and Engineering. 3rd Edition, ISBN: 9780521679718, (2006).

P. Carr, D.B.Madan, and R.H. Smith. Option valuation using the fast Fourier transform. Journal of Computational Finance, 2 (3), (1999): 61-73.




DOI: http://dx.doi.org/10.26713%2Fjims.v8i1.326

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