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## 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).

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