Estimating dynamic geometric fractional Brownian motion and its application to long-memory option pricing (Q2869760)

Geometric fractional Brownian motion (GFBM) is a geometric version of the fractional Brownian motion (FBM), which was first studied by Kolmogorov [\textit{A. N. Kolmogoroff}, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 26, 115--118 (1940; Zbl 0022.36001; JFM 66.0552.03)]. Application of FBM and GFBM to dynamic financial option pricing is a natural way of extending the Black-Scholes option theory based on Brownian motion and geometric Brownian motion.NEWLINENEWLINEThe authors of the paper under review follow an approach reported in [\textit{Y. Mishura}, Stochastic calculus for fractional Brownian motion and related processes. Berlin: Springer (2008; Zbl 1138.60006)], where a formula for European option under dynamic fractional Black-Scholes market is given. They address the problem of estimating the unknown drift, volatility, and index of self-similarity parameters, involved in the GFBM, based on discrete-time observations. The proposed complete maximum likelihood estimation (CMLE) approach allows deriving simultaneously estimators of all these parameters for risky assets in the dynamic fractional Black-Scholes market governed by GFBM.NEWLINENEWLINEThe presented simulation study results indicate that the proposed CMLE approach in estimating GFBM is performing better than the previous methods introduced in the literature. Finally, some empirical work on pricing European call options is also reported.