Black-Scholes model under subordination

TL;DR

This paper extends the Black-Scholes model by incorporating a subordinated stochastic process with inverse -stable processes, introducing long-term memory effects into option pricing models.

Contribution

It introduces a novel mathematical framework for the Black-Scholes model using subordination with inverse -stable processes, capturing long-term memory effects.

Findings

The model incorporates long-term memory effects into option pricing.

It generalizes the classical Black-Scholes model with a subordinated process.

Potential implications for more accurate financial modeling.

Abstract

In this paper we consider a new mathematical extension of the Black-Scholes model in which the stochastic time and stock share price evolution is described by two independent random processes. The parent process is Brownian, and the directing process is inverse to the totally skewed, strictly \alpha-stable process. The subordinated process represents the Brownian motion indexed by an independent, continuous and increasing process. This allows us to introduce the long-term memory effects in the classical Black-Scholes model.