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Stochastic Volatility and Multifractional Brownian Motion

2011; Springer Nature; Linguagem: Inglês

10.1007/978-3-642-22368-6_6

2190-5622

Antoine Ayache, Qidi Peng,

Complex Systems and Time Series Analysis

In order to make stochastic volatility models more realistic some authors (see for example 3.Ayache Technique et science informatiques 20–29:1133–1152, 2001; Comte and Renault J. Econom. 73:101–150, 1996; Comte and Renault Math. Financ. 8:291–323, 1998; 21. Gloter, A., Hoffmann, M.: Stochastic volatility and fractional Brownian motion. Prépublication du laboratoire de Probabilités & Modèles Aléatoires des Universités Paris 6 & Paris 7, 746 2002; Rosenbaum Stoch. Proc. Appl. 118:1434–1462 2008) have proposed to replace the Brownian motion governing the volatility by a more flexible stochastic process. This is why, we introduce multifractional stochastic volatility models; their main advantage is that they allow to account variations with respect to time of volatility local roughness. Our definition of multifractional stochastic volatility models is inspired by that of the fractional stochastic volatility models previously introduced by Gloter and Hoffmann (Ayache Technique et science informatiques 20–29:1133–1152, 2001; Gloter, A., Hoffmann, M.: Stochastic volatility and fractional Brownian motion. Prépublication du laboratoire de Probabilités & Modèles Aléatoires des Universités Paris 6 & Paris 7, 746 2002). The main goal of our article is to extend to these new models some theoretical results concerning statistical inference which were obtained in (Ayache Technique et science informatiques 20–29:1133–1152, 2001; Gloter, A., Hoffmann, M.: Stochastic volatility and fractional Brownian motion. Prépublication du laboratoire de Probabilités & Modèles Aléatoires des Universités Paris 6 & Paris 7, 746 2002). More precisely, assuming that the functional parameter H( ⋅) of multifractional Brownian motion is known, we construct, in a general framework, an estimator of integrated functional of the volatility, and we derive from it, in the linear case, an estimator of a parameter θ related to the volatility.