A test for a parametric form of the volatility in second-order diffusion models

Second-order diffusion models have been found to be promising in analyzing financial market data. Based on nonparametric fitting, Nicolau (Stat Probabil Lett 78(16):2700-2704, 2008) suggested that the quadratic function may be an appropriate specification of the volatility when a second-order diffusion model is used to analyze some European and American financial market data sets, which motivates us to develop a formal statistical test for this finding. To achieve the task, a generalized likelihood ratio test is proposed in this paper and a residual-based bootstrap is suggested to compute the p value of the test. The analysis of many real-world financial market data sets demonstrates that the quadratic specification of the volatility function is in general reasonable.