We entertain the hypothesis that leverage considerations are relevant in describing the evolution of asset returns both statistically and risk neutrally. Adopting a constant elasticity of variance formulation in the context of a general Levy process as the driving uncertainty we show that the presence of leverage effects in this form has the implication that asset pricing satisfy a scaling hypothesis. Examples of continuous and pure jump Levy cases are constructed and explicit forms for the semigroups are obtained with empirical investigations.

In our study, we build in the leverage effect by introducing a time change dependent on the level of asset and hence affect the expected local volatility in an explicit manner. This is a fairly direct approach in the context of Levy processes. The continuous case in our study coincides with the development of the constant elasticity of variance models. We however, conduct our investigation in the continuous case through our incorporation of BESQ process as the semi-stable Markov process. In the pure jump case with underlying time changed Levy process being specified as CGMY process, we hope to engage the leverage effect as well as the ability of explaining long-tailedness and skewness as already being provided by using such pure jump Levy process with infinite activity. We show how to implement Generalized Method of Moments in this case to estimate parameters without the assumption of knowing the law of the process.

The development of forward Partial Integro-Differential Equations is under a general setup and shows great advantage over the backward ones. In both the continuous case and the pure jump case, we show how to calibrate our model parameters by solving such forward PIDEs and compare model prices to the market data. Although the numerical approach used in the pure jump case is discussed in the context of CGMY process, it is evident that the approach can be extended to a general frame work indifferent of the choice of Levy process and shall be similarly carried out where other Levy processes are specified in our model.