Constructing a Risky Density Function

My colleague Haizhen Lin found a neat trick from someone in the math department. Suppose you have a density f(x) and you want to construct a pointwise less risky function, as in my paper cited below. You can use this: f(a, x) = (1/a) f( .5 - .5/a +x/a) If a=1, f(a,x) = f(x). If a is small, f(a,x) tends to get big because of the 1/a portion, and it gets very big for x=0, but for x far from 0, the f becomes small because the argument becomes very big, distant from 0. "When Does Extra Risk Strictly Increase the Value of Options?" The Review of Financial Studies, 20(5): 1647-1667 (September 2007). It is well known that risk increases the value of options. This paper makes that precise in a new way. The conventional theorem says that the value of an option does not fall if the underlying option becomes riskier in the conventional sense of the mean-preserving spread. This paper uses two new definitions of ``riskier'' to show that the value of an option strictly increases (a) if the underlying asset becomes ``pointwise riskier,'' and (b) only if the underlying asset becomes ``extremum riskier.'' Paper in tex or pdf ( http://www.rasmusen.org/published/Rasmusen07-RFS-options.pdf).