The economics of time as it is embedded in the prices of options§

Abstract

Risk neutral variance term structures are characterized by their time elasticities. They are synthesized by space scaling and time changing self-decomposable laws at unit time. Monotone concave or convex time elasticities are modeled using exponential functions while gamma functions permit changes in curvature. Results for both cases as time changes are followed by those with simultaneous space-scaling and time-changing. Space scaling contributes towards front end options while time changing works on the back end. Splitting the space scaling and time changing for the positive and negative moves delivers models with rising absolute skewness and kurtosis. The space scaled and time changed densities are those of additive processes. The space scaled process scales a solution to a time varying OU equation driven by a time changed Lévy process taken at log time. The mean reversion rates for the OU process are the variance time elasticities. The two processes are termed the space scaled and time change components and their relative contributions, space to time are determined to be twice the ratio of their variance elasticities. In particular, the space scaling elasticity synthesizes the effects of perpetual motion as captured by mean reversion in the underlying OU equation.