Stochastic Models Based on the Time-Change Technique

In this paper, we study the time-change technique for L\'{e}vy jump processes and compare the effectiveness of different models based on these processes. The time-change technique is a promising idea for the theory of stochastic processes and its application to finance, which may allow modeling financial time series with real-world statistical features. More precisely, we consider two time-change approaches here: subordination and time-change via integrated CIR. In particular, we consider CGMY, Meixner and NIG processes as subordinated Brownian motion and analyze their performance on historical data. The main goals of this paper are to observe the ability of these models to behave like real-world financial time series. In modern literature, it is common to distinguish 2 approaches for model calibration: fitting it to historical data of a base asset and fitting ones to option prices in the so-called risk-neutral world. Since the second one is based on martingale theory which is beyond our research, we do not consider it in this paper.