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Option pricing in subdiffusive Bachelier model. (English) Zbl 1269.82053
Summary: 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 \(\alpha \)-stable and tempered \(\alpha \)-stable distributions of waiting times.

MSC:
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
91B24 Microeconomic theory (price theory and economic markets)
60J65 Brownian motion
82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
60J60 Diffusion processes
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