Option Pricing in Subdiffusive Bachelier Model |
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Authors: | Marcin Magdziarz Sebastian Orzeł Aleksander Weron |
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Institution: | 1.Hugo Steinhaus Center, Institute of Mathematics and Computer Science,Wroc?aw University of Technology,Wroc?aw,Poland |
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Abstract: | The earliest model of stock prices based on Brownian diffusion is the Bachelier model. In this paper we propose an extension
of the Bachelier model, which reflects the subdiffusive nature of the underlying asset dynamics. The subdiffusive property
is manifested by the random (infinitely divisible) periods of time, during which the asset price does not change. We introduce
a subdiffusive arithmetic Brownian motion as a model of stock prices with such characteristics. The structure of this process
agrees with two-stage scenario underlying the anomalous diffusion mechanism, in which trapping random events are superimposed
on the Langevin dynamics. We find the corresponding fractional Fokker-Planck equation governing the probability density function
of the introduced process. We construct the corresponding martingale measure and show that the model is incomplete. We derive
the formulas for European put and call option prices. We describe explicit algorithms and present some Monte-Carlo simulations
for the particular cases of α-stable and tempered α-stable distributions of waiting times. |
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