Abstract
In the classical approach the price of an asset is described by the celebrated Black-Scholes model. In this paper we consider a generalization of this model, which captures the subdiffusive characteristics of financial markets. We introduce a subdiffusive geometric Brownian motion as a model of asset prices exhibiting subdiffusive dynamics. We find the corresponding fractional Fokker-Planck equation governing the dynamics of the probability density function of the introduced process. We prove that the considered model is arbitrage-free and incomplete. We find the corresponding subdiffusive Black-Scholes formula for the fair prices of European options and show how these prices can be evaluated using Monte-Carlo methods. We compare the obtained results with the classical ones.
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Magdziarz, M. Black-Scholes Formula in Subdiffusive Regime. J Stat Phys 136, 553–564 (2009). https://doi.org/10.1007/s10955-009-9791-4
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DOI: https://doi.org/10.1007/s10955-009-9791-4