2015/02/18 by Archil Gulisashvili, Gulisashvili, Archil, Frédéri Viens +3
Economics, Econometrics and Finance · #40E05 #60G15 #91G20 #Complex Systems and Time Series Analysis #FOS: Economics and business #Financial Risk and Volatility Modeling #Mathematical Finance (q-fin.MF) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1502.05442
openalex publication_date 2015/02/18 · openalex created_date 2021/02/01 · openalex updated_date 2026/07/28
We consider a stochastic volatility asset price model in which the volatility\nis the absolute value of a continuous Gaussian process with arbitrary\nprescribed mean and covariance. By exhibiting a Karhunen-Lo `eve expansion\nfor the integrated variance, and using sharp estimates of the density of a\ngeneral second-chaos variable, we derive asymptotics for the asset price\ndensity for large or small values of the variable, and study the wing behavior\nof the implied volatility in these models. Our main result provides explicit\nexpressions for the first five terms in the expansion of the implied\nvolatility. The expressions for the leading three terms are simple, and based\non three basic spectral-type statistics of the Gaussian process: the top\neigenvalue of its covariance operator, the multiplicity of this eigenvalue, and\nthe L2 norm of the projection of the mean function on the top eigenspace.\nThe fourth term requires knowledge of all eigen-elements. We present detailed\nnumerics based on realistic liquidity assumptions in which classical and\nlong-memory volatility models are calibrated based on our expansion.\n