9.1 Monitoring of the Volatility
We have seen that volatility of the market or of the asset prices plays a crucial role in many aspects. One of the underlying assumptions in the standard Black and Scholes market of Chapter 6 is that volatility is supposed to be constant. We have seen many examples (see, e.g. Section 6.6) about the fact that this assumption is simply unrealistic when we go to analyze real financial data. We have also seen deviations from the standard geometric Brownian motion model which allow for nonconstant volatility. Change point analysis was initially introduced in the framework of independent and identically distributed data by these authors: Hinkley (1971), Csörg and Horváth (1997), Inclan and Tiao (1994), Bai (1994, 1997) and quickly applied to the analysis of time series: Kim et al. (2000), Lee et al. (2000), Chen et al. (2005). For continuous time diffusion models Kutoyants (1994, 2004) and Lee et al. (2006) studied change point in the drift term from continuous time observations. Due to the fact that volatility can be estimated without error in continuous time, the change point analysis in this setup is not very interesting. Recently, De Gregorio and Iacus (2008) considered least squares estimation for the volatility of a one-dimensional stochastic differential equation and later Iacus and Yoshida (2009) consider the problem under ...