Abstract: Stochastic time change is a well-used tool for construction of stochastic models which are able to represent the so-called stylized features of stock prices. From mathematical point of view, the main idea is to change the deterministic time t of a stochastic process X(t) (usually - of a Levy process) by another increasing process T(s). As a result, one obtains a process Y (s) = X(T(s)), which is referred to as a time-changed process. The economical interpretation of this model is based on the idea that the "business" time T(s) may run faster than the physical time in some periods, for instance, when the amount of transactions is high. Due to this interpretation, Ys represents the log-returns of a stock price, and a natural candidate for T(s) is a cumulative number of trades till time s. The most popular choice of a process X is a Brownian motion with or without drift. This choice is mainly based on the Monroe theorem, which says that the class of time-changed Brownian motions in fact coincides with the class of all semimartingales. In this research, we consider another case, when the class of stable processes is used for X: Empirically it turns out that the considered model is more appropriate than the subordinated Brownian motion for describing the stock returns. This can be explained by the observation that in our model rapid changes in log-returns are made not only due to jumps in number of trades (as in time-changed Brownian motion), but also due to stochastic factors, which are incorporated in X: I will also present a a multivariate time-changed process such that each component is a subordinated stable process and the dependence between subordinators is described via some Levy copula. For the considered class, I will show a simulation method based on the series representation. Moreover, I will describe a method of semiparametric estimation of the parameters of copula and related distributions, and show some properties of the considered estimates. The performance of the proposed method will be illustrated by the examples related to the description of dependence between stock prices.
Link to abstract (in PDF)
Video
Slides