Research On S&P500 Index Analysis And Its Option Pricing

With the improvement and development of the financial market,the existing research has also produced a lot of results about option pricing.Based on previous studies,we select the S&P500 index and its options,which can significantly represent the overall market dynamics.Through the analysis of the basic statistical properties,the central recovery,asymmetry,sharp peak and thick tail distributions of the S&P500 index are obtained.Based on the above characteristics of the data,after detrending the logarithmic processing,we choose two typical affine-jump models-OU and CIR models to grasp the data.On this basis,we introduce the stochastic volatility or mean process,and three typical tempered steady processes—the VG,CGMY and the CTS processes into the models.After modeling,we apply affine and martingale properties to establish the Riccati equation.The explicit expression of the characteristic functions are expressed,which lays a foundation for deriving the option pricing formula.In the option pricing section,the pricing expressions of European call options for S&P500 stock indexes are successfully derived through transformations of two martingale measure and characteristic functions.Finally,the option pricing formula is empirically fitted with actual data.Parameters are estimated and results are compared.Then the final conclusion is drawn: both stochastic volatility and tempered steady processes can improve the model’s fitting effect.At the same time,the pricing effect of the CIR environment is significantly better than that of the OU environment,and subsequent research can be further explored on the basis of CIR.Compared with existing research,the research in this paper makes up for the problem of insufficient fitting of financial data and provides a certain reference for the study of options market.