Small-Time Asymptotics for Gaussian Self-Similar Stochastic Volatility Models

We consider the class of Gaussian self-similar stochastic volatility models, and characterize the small-time (near-maturity) asymptotic behavior of the corresponding asset price density, the call and put pricing functions, and the implied volatility. Away from the money, we express the asymptotics explicitly using the volatility process’ self-similarity parameter H, and its Karhunen–Loève characteristics. Several model-free estimators for H result. At the money, a separate study is required: the asymptotics for small time depend instead on the integrated variance’s moments of orders 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}$$\end{document} and 32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{3}{2}$$\end{document}, and the estimator for H sees an affine adjustment, while remaining model-free.