Detecting and measuring nonlinearity

This paper proposes an approach to measure the extent of nonlinearity of the exposure of a financial asset to a given risk factor. The proposed measure exploits the decomposition of a conditional expectation into its linear and nonlinear components. We illustrate the method with the measurement of the degree of nonlinearity of a European style option with respect to the underlying asset. Next, we use the method to identify the empirical patterns of the return-risk trade-off on the SP500. The results are strongly supportive of a nonlinear relationship between expected return and expected volatility. The data seem to be driven by two regimes: one regime with a positive return-risk trade-off and one with a negative trade-off.