We consider a stochastic volatility stock price model in which the volatility is a non-centered continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen-Lo\`{e}ve expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the stock price density and implied volatility in these models in the limit of large or small strikes . Our main result provides explicit expressions for the first three terms in the expansion of the implied volatility , based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the L^{2} norm of the projection of the mean function on the top eigenspace. Strategies for using this expansion for calibration purposes are discussed .
We consider a stochastic volatility asset price model in which the volatility is the absolute value of a continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen-Lo\`{e}ve expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the asset price density for large or small values of the variable, and study the wing behavior of the implied volatility in these models . Our main result provides explicit expressions for the first five terms in the expansion of the implied volatility . The expressions for the leading three terms are simple, and based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the L^{2} norm of the projection of the mean function on the top eigenspace. The fourth term requires knowledge of all eigen-elements. We present detailed numerics based on realistic liquidity assumptions in which classical and long-memory volatility models are calibrated based on our expansion .
R
stock
asset
R
a non-centered
the absolute value of a
R
stock price density and
asset price density for large or small values of the variable, and study the wing behavior of the
D
in the limit of large or small strikes
R
three
five
R
,
. The expressions for the leading three terms are simple, and
R
Strategies for using this expansion for calibration purposes are discussed
The fourth term requires knowledge of all eigen-elements. We present detailed numerics based on realistic liquidity assumptions in which classical and long-memory volatility models are calibrated based on our expansion