Option pricing under the KoBol model

Wenting Chen, Sha Lin

Abstract


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 L´evy 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.


doi:10.1017/S1446181118000196

Keywords


option pricing, fractional derivative, KoBol process, fractional partial differential equation.



DOI: http://dx.doi.org/10.21914/anziamj.v60i0.12154



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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.