Arbitrage in Fractal Modulated Markets When the Volatility is Stochastic

TL;DR

This paper develops an arbitrage strategy for a modified Black-Scholes model driven by fractional Brownian motion with stochastic volatility, extending previous models to account for heavy tails and long-range dependence in stock returns.

Contribution

It introduces a novel arbitrage approach for stochastic volatility models driven by fractional Brownian motion or its time change, addressing the non-semimartingale nature of fractional Brownian motion.

Findings

Arbitrage strategy applicable to fractional Brownian motion driven models.

Shows that integrals with zero quadratic variation can serve as integrators.

Discusses the suitability of fractional Brownian motion for modeling stock returns.

Abstract

In this paper an arbitrage strategy is constructed for the modified Black-Scholes model driven by fractional Brownian motion or by a time changed fractional Brownian motion, when the volatility is stochastic. This latter property allows the heavy tailedness of the log returns of the stock prices to be also accounted for in addition to the long range dependence introduced by the fractional Brownian motion. Work has been done previously on this problem for the case with constant `volatility' and without a time change; here these results are extended to the case of stochastic volatility models when the modulator is fractional Brownian motion or a time change of it. (Volatility in fractional Black-Scholes models does not carry the same meaning as in the classic Black-Scholes framework, which is made clear in the text.) Since fractional Brownian motion is not a semi-martingale, the Black-Scholes differential equation is not well-defined sense for arbitrary predictable volatility processes. However, it is shown here that any almost surely continuous and adapted process having zero quadratic variation can act as an integrator over functions of the integrator and over the family of continuous adapted semi-martingales. Moreover it is shown that the integral also has zero quadratic variation, and therefore that the integral itself can be an integrator. This property of the integral is crucial in developing the arbitrage strategy. Since fractional Brownian motion and a time change of fractional Brownian motion have zero quadratic variation, these results are applicable to these cases in particular. The appropriateness of fractional Brownian motion as a means of modeling stock price returns is discussed as well.