When Does Extra Risk Strictly Increase an Option's Value?

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 asset 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