This article introduces a class of distortion operators, g α (u) = Φ Φ [ () ] − + 1 u α , where Φ is the standard normal cumulative distribution. For any loss (or asset) variable X with a probability distribution S X (x) = 1-F X (x), g α [S X (x)] defines a distorted probability distribution whose mean value yields a risk-adjusted premium (or an asset price). The distortion operator g α can be applied to both assets and liabilities, with opposite signs in the parameter α. Based on CAPM, the author establishes that the parameter α should correspond to the systematic risk of X. For a normal (µ σ , 2) distribution, the distorted distribution is also normal with ′ = + ′ = µ µ ασ σ σ and. For a lognormal distribution, the distorted distribution is also lognormal. By applying the distortion operator to stock price distributions, the author recovers the risk-neutral valuation for options and in particular the Black-Scholes formula.

In this paper, we take an axiomatic approach to characterize insurance prices in a competitive market setting. We present four axioms to describe the behavior of market insurance prices. From these axioms it follows that the price of an insurance risk has a Choquet integral representation with respect to a distorted probability, (Yanri, 1987). We propose an additional axiom for reducing compound risks. This axiom determines that the distortion function is a power function.

This paper presents a universal framework for pricing financial and insurance risks. Examples are given for pricing contingent payoffs, where the underlying asset or liability can be either traded or not traded. The paper also outlines an application of the framework to prescribe capital allocations within insurance companies, and to determine fair value for insurance liabilities.2

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