# The fractional and mixed-fractional CEV model

## TL;DR

This paper develops fractional and mixed-fractional extensions of the CEV model to better capture long-range dependence and leverage effects in financial markets, providing analytical formulas for option pricing and Greeks.

## Contribution

It introduces novel fractional and mixed-fractional CEV models, deriving transition densities and valuation formulas using fractional calculus and non-stationary Feller processes.

## Key findings

- Derived analytical transition probability density functions.
- Provided explicit European Call option valuation formulas.
- Compared Greeks with standard models, showing differences.

## Abstract

The continuous observation of the financial markets has identified some stylized facts which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss-Wiener process (Brownian motion), characterized by stationary independent increments, by a fractional version. On the other hand, the CEV model addresses the Leverage effect and smile-skew phenomena, efficiently. In this paper, these two insights are merging and both the fractional and mixed-fractional extensions for the CEV model, are developed. Using the fractional versions of both the Ito's calculus and the Fokker-Planck equation, the transition probability density function of the asset price is obtained as the solution of a non-stationary Feller process with time-varying coefficients, getting an analytical valuation formula for a European Call option. Besides, the Greeks are computed and compared with the standard case.