Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T21:28:08.795Z Has data issue: false hasContentIssue false
  • English
  • Français

OPTION PRICING UNDER THE KOBOL MODEL

Part of: Parabolic equations and systems Applications

Published online by Cambridge University Press:  12 September 2018

WENTING CHEN  and
SHA LIN Open the ORCID record for SHA LIN [Opens in a new window]
WENTING CHEN
Affiliation:
School of Business, Jiangnan University, Wuxi, Jiangsu Province, China email cwtwx@163.com
SHA LIN*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email sl945@uowmail.edu.au
*
✉sl945@uowmail.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the pricing of European options under a modified Black–Scholes equation having fractional derivatives in the “spatial” (price) variable. To be specific, the underlying price is assumed to follow a geometric Koponen–Boyarchenko–Levendorski process. This pure jump Lévy process could better capture the real behaviour of market data. Despite many difficulties caused by the “globalness” of the fractional derivatives, we derive an explicit closed-form analytical solution by solving the fractional partial differential equation analytically, using the Fourier transform technique. Based on the newly derived formula, we also examine, in theory, many basic properties of the option price under the current model. On the other hand, for practical purposes, we impose a reliable implementation method for the current formula so that it can be easily used in the trading market. With the numerical results, the impact of different parameters on the option price are also investigated.

Keywords

option pricingfractional derivativeKoBol processfractional partial differential equation

MSC classification

Primary: 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 35K25: Higher-order parabolic equations
Type
Research Article
Information
The ANZIAM Journal , Volume 60 , Issue 2 , October 2018 , pp. 175 - 190
Copyright
© 2018 Australian Mathematical Society 

References

Bates, D. S., “Maximum likelihood estimation of latent affine processes”, Rev. Financial Studies 19 (2006) 909–965; doi:10.1093/rfs/hhj022.Google Scholar
Bates, D. S., “Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options”, Rev. Financial Studies 9 (1996) 69–107; doi:10.1093/rfs/9.1.69.Google Scholar
Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Political Economy 81 (1973) 637–654; doi:10.1086/260062.Google Scholar
Boyarchenko, S. and Levendorski, S., Non-Gaussian Merton ‘Black–Scholes’ theory (World Scientific, Singapore, 2002).Google Scholar
Bracewell, R. N. and Bracewell, R. N., The Fourier transform and its applications (McGraw-Hill, New York, 1986).Google Scholar
Butzer, P. L., Kilbas, A. A. and Trujillo, J. J., “Mellin transform analysis and integration by parts for Hadamard-type fractional integrals”, J. Math. Anal. Appl. 270 (2002) 1–15;doi:10.1016/S0022-247X(02)00066-5.Google Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M., “The fine structure of asset returns: an empirical investigation”, J. Business 75 (2002) 305–333; doi:10.1086/338705.Google Scholar
Carr, P. and Wu, L., “The finite moment log stable process and option pricing”, J. Finance 58 (2003) 753–778; doi:10.1111/1540-6261.00544.Google Scholar
Cartea, Á. and del-Castillo-Negrete, D., “Fractional diffusion models of option prices in markets with jumps”, Phys. A: Statistical Mechanics Applications 374 (2007) 749–763;doi:10.1016/j.physa.2006.08.071.Google Scholar
Chen, W.-T., Du, M.-Y. and Xu, X., “An explicit closed-form analytical solution for European options under the CGMY model”, Commun. Nonlinear Sci. Numer. Simul. 42 (2017) 285–297; doi:10.1016/j.cnsns.2016.05.026.Google Scholar
Chen, W.-T., Xu, X. and Zhu, S.-P., “Numerically pricing American options under the modified Black–Scholes equation with a spatial-fractional derivative”, Appl. Numer. Math. 97 (2015) 15–29; doi:10.1016/j.apnum.2015.06.004.Google Scholar
Chen, W.-T., Xu, X. and Zhu, S.-P., “Analytically pricing European-style options under the modified Black–Scholes equation with a spatial-fractional derivative”, Quarter. Appl. Math. 72 (2014) 597–611; doi:10.1090/S0033-569X-2014-01373-2.Google Scholar
Heston, S. L., “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Financial Studies 6 (1993) 327–343; doi:10.1093/rfs/6.2.327.Google Scholar
Hildebrand, F. B., Introduction to numerical analysis (McGraw-Hill, New York–Toronto–London, 1956).Google Scholar
Hull, J. and White, A., “The pricing of options on assets with stochastic volatilities”, J. Finance 42 (1987) 281–300; doi:10.1111/j.1540-6261.1987.tb02568.x.Google Scholar
Koponen, I., “Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process”, Phys. Rev. E 52 (1995) 1197; doi:10.1103/PhysRevE.52.1197.Google Scholar
Madan, D. B., Carr, P. and Chang, E. C., “The variance Gamma process and option pricing”, Eur. Finance Rev. 2 (1998) 79–105; doi:10.1023/A:1009703431535.Google Scholar
Mantegna, R. N. and Stanley, H. E., “Scaling behavior in the dynamics of an economic index”, Nature 376 (1995) 46–49; doi:10.1038/376046a0.Google Scholar
Merton, R. C., “Option pricing when underlying stock returns are discontinuous”, J. Financial Economics 3 (1976) 125–144; doi:10.1016/0304-405X(76)90022-2.Google Scholar
Metzler, R. and Klafter, J., “The random walk’s guide to anomalous diffusion: a fractional dynamics approach”, Phys. Rep. 339 (2000) 1–77; doi:10.1016/S0370-1573(00)00070-3.Google Scholar
Peiro, A., “Skewness in financial returns”, J. Banking Finance 23 (1999) 847–862;doi:10.1016/S0378-4266(98)00119-8.Google Scholar
Rachev, S. T., Menn, C. and Fabozzi, F. J., Fat-tailed and skewed asset return distributions: implications for risk management, portfolio selection, and option pricing (John Wiley & Sons, Hoboken, NJ, 2005).Google Scholar
Tankov, T., Financial modelling with jump processes (CRC Press, Boca Raton, FL, 2007).Google Scholar