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Keywords:

  • diffusion;
  • non-Markov processes;
  • option pricing theory

Abstract

Diffusion processes play an important role in physics as well as in financial sciences. The usual description by stochstic differential equations considers the Wiener process as external stochastic term, while the deterministic drift term is local in time. However, the complexity of the interaction between the observed diffusive degree of freedom and the hidden dynamics of the whole system, e.g., the coupling between a certain particle and the surrounding liquid or the relations between the price of a financial asset and the global market, requires the consideration of possible memory effects.

In the present paper we analyze the effects of a non-Markovian asset price model on the corresponding European option prices. This model considers drift terms which are non-local in time, so that memory effects appear. As the main result we present a generalized Black-Scholes equation considering the whole history of the asset price evolution.