Researchers propose a new way to use fractional Brownian motion for option pricing while avoiding a classic mathematical roadblock. Fractional Brownian motion (fBm) captures features seen in financial data, such as persistence and rough paths. But fBm on its own lacks a technical property called the semimartingale property, which is needed for the standard, no-arbitrage methods used to price options. The authors restore that property by evaluating fBm at a random "activity time" driven by a gamma process. Activity time represents accumulated trading activity rather than calendar time.

The time-changed process, written X = B_H(γ), keeps the key statistical features of fBm but becomes a semimartingale. The paper builds a fractional Variance Gamma (fVG) model by replacing the Brownian part of the classic Variance Gamma model with this time-changed fBm. The resulting model has five parameters that capture drift, volatility, skewness, kurtosis, and the dependence structure of returns. The authors show the process can be viewed as a Gaussian–Gamma mixture and prove the semimartingale result that makes standard arbitrage-free pricing techniques applicable.

Because the fVG process does not have independent increments and does not admit a simple closed-form density, the authors develop new tools for working with it. They construct a marked point process representation, derive the compensator (a technical object used to handle jumps and random times), and give closed-form expressions for raw and central moments. For practical use they propose a simulation approach that first simulates the gamma activity time on a primary grid and then evaluates the fractional Brownian motion on a finer grid at those activity times. For estimation they propose a feasible generalized method of moments (GMM) that uses return moments at multiple horizons.