Methodology for stochastic volatility process calibration application to the CAC 40 index

Abstract

In considering volatility as a stochastic, the aim of this paper is to estimate the four parameters related to a particular stochastic process named P1 and based on a Wiener–Levy process. We present the methodology to estimate its four parameters. We calibrate this theoretical model P1 to the CAC 40 index real data. In the same time, we test the normality of the random variables related to the two Wiener–Levy processes. The calibration is performed using the implemented aforesaid algorithm. We compare the stochastic process P1 with another process named P2 and to the Heston [Closed form solution for options with stochastic volatility with application to bonds and currency options, Rev. Financ. Stud. 6(2) (1993), pp. 327–343] process named H0 and to two other improved Heston processes named H1 and H2. For the empirical study, the same algorithm is used to calibrate the five processes. The calibration is based on a database including the CAC 40 index daily ‘closing fixing’ values for the time period from 3rd January 2005 to 22nd January 2007. The data are divided into 18 classes relative to 18 different contracts of European calls on the CAC 40 index. As a result, we find that, the normality test of the CAC 40 index is rejected which is in accordance with the previous original works dealing with this problem. For the five volatility processes, the normality test is verified almost for the same contracts. We also find that according to the used data, the process P1 and its equivalent H1 are the best for calibration.

Keywords:

• stochastic volatility
• Heston process
• implicit volatility
• calibration
• adjustment test
• CAC 40 index option