[Paper Review] Arbitrage in Fractal Modulated Markets When the Volatility is Stochastic

Erhan Bayraktar, H. Vincent Poor|ArXiv.org|Jan 22, 2005

Stochastic processes and financial applications27 references3 citations

TL;DR

This paper constructs an arbitrage strategy in fractal-modulated Black-Scholes models with stochastic volatility, where the underlying price process is driven by fractional Brownian motion (fBm) or a time-changed fBm. By leveraging the zero quadratic variation property of such processes, the authors establish a stochastic integral framework that supports continuous, adapted integrators, enabling explicit arbitrage without requiring estimation of the Hurst parameter H.

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.

Motivation & Objective

• To extend existing arbitrage results in fractional Brownian motion models to cases with stochastic volatility.
• To develop a stochastic integration framework compatible with fBm and time-changed fBm, which are not semi-martingales.
• To demonstrate that continuous, adapted processes with zero quadratic variation can serve as integrators for both functions of the integrator and continuous semi-martingales.
• To show that such integrals also have zero quadratic variation, enabling recursive use as integrators in arbitrage construction.
• To provide an economically interpretable arbitrage strategy that does not require estimating the Hurst parameter H.

Proposed method

• Defining stochastic integrals via convergence in probability of Stieltjes sums over shrinking partitions, rather than using Wick-type integrals.
• Proving that any almost surely continuous, adapted process with zero quadratic variation can act as an integrator for continuous adapted semi-martingales and for functions of the integrator.
• Establishing that the resulting integral also has zero quadratic variation, allowing recursive use in constructing trading strategies.
• Applying this framework to fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1], and to time-changed fBm.
• Constructing an explicit arbitrage strategy based on the dynamics of the stock price under stochastic volatility and fBm-driven diffusion.
• Demonstrating that the strategy is independent of the Hurst parameter H, as it enters only through the price process, not the trading rule.

Experimental results

Research questions

• RQ1 Can arbitrage be constructed in a modified Black-Scholes model driven by fractional Brownian motion with stochastic volatility?
• RQ2 Does the zero quadratic variation property of fBm and time-changed fBm enable a consistent stochastic integration framework for non-semimartingale processes?
• RQ3 Can such integrators be used recursively to build a self-financing trading strategy with guaranteed profit?
• RQ4 Is the resulting arbitrage strategy robust to the value of the Hurst parameter H, or does it require estimation of H?
• RQ5 How does the proposed Stieltjes-type integration framework compare to Wick-type integrals in terms of economic interpretability and arbitrage generation?

Key findings

• An explicit arbitrage strategy exists in the modified Black-Scholes model driven by fractional Brownian motion or its time-changed version when volatility is stochastic.
• The stochastic integral defined via convergence of Stieltjes sums is well-defined for any almost surely continuous, adapted process with zero quadratic variation.
• Such integrals preserve the zero quadratic variation property, enabling recursive use as integrators in the construction of trading strategies.
• The arbitrage strategy does not require estimation of the Hurst parameter H, as it depends on the stock price process but not on H directly.
• The framework supports continuous trading strategies that generate riskless profit, demonstrating the presence of arbitrage in fractal-modulated markets with stochastic volatility.
• The results show that fBm-based models, despite their non-semimartingale nature, can support economically interpretable arbitrage strategies under the chosen integration theory.