On a Constrained Fractional Stochastic Volatility Model

TL;DR

This paper extends the Black-Scholes model by incorporating a fractional Brownian motion-driven volatility model, ensuring positivity and market completeness, and deriving a pricing formula.

Contribution

It introduces a new fractional stochastic volatility model with viability conditions, ensuring no arbitrage and market completeness, extending classical models.

Findings

Market is arbitrage-free and complete under the model.

A closed-form pricing formula is derived.

Volatility remains positive due to viability conditions.

Abstract

This paper deals with an extension of the so-called Black-Scholes model in which the volatility is modeled by a linear combination of the components of the solution of a differential equation driven by a fractional Brownian motion of Hurst parameter greater than $1/4$. In order to ensure the positiveness of the volatility, the coefficients of that equation satisfy a viability condition. The absence of arbitrages, the completeness of the market and a pricing formula are established.