On Statistical Aspects in Calibrating a Geometric Skewed Stable Asset Price Model

Estimation of an asset price process under the physical measure can be regarded as the first step of the calibration problem, hence is of practical importance. In this article, supposing that a log-price process is expressed by a possibly skewed stable driven model and that a high-frequency dataset over a fixed period is available, we provide practical procedures of estimating the dominating parameters. Especially, the scale parameter may be time-varying and possibly random as long as it is independent of the driving skewed stable Lévy process. By means of the scaling property and realized bipower variations, it is possible to estimate the index and positivity (skewness) parameters without specific information of the scale process. When the target scale parameter is constant, our estimators are asymptotically normally distributed, the rate of convergence being $\sqrt{n}$. When the scale is actually time-varying, we focus on estimation of the integrated scale, which is an analogue to the integrated volatility in the Brownian-semimartingale framework. In this case we show that estimation of the integrated scale exhibits a kind of asymptotic singularity with respect to the unknown index parameter, with the rate of convergence being the slower $\sqrt{n}/{\rm log}\, n$.