A class of distortion operators for pricing financial and insurance risks

This article introduces a class of distortion operators, ga(t) = D[44-(u) + a], where D is the standard normal cumulative distribution. For any loss (or asset) variable X with a probability distribution Sx(x) = 1- Fx(x), ga [Sx(x)] defines a distorted probability distribution whose mean value yields a risk-adjusted premium (or an asset price). The distortion op- erator ga can be applied to both assets and liabilities, with opposite signs in the parameter a. Based on CAPM, the author establishes that the pa- rameter ca should correspond to the systematic risk of X. For a normal (L,aU2) distribution, the distorted distribution is also normal with '= u + aa and a5' = a. For a lognormal distribution, the distorted dis- tribution is also lognormal. By applying the distortion operator to stock price distributions, the author recovers the risk-neutral valuation for op- tions and in particular the Black-Scholes formula.