A Class of Distortion Operators For Pricing Financial and Insurance Risks

This paper introduces a class of distortion operators, ga(u) = F[F-1(u)+a], where F 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 operator ga can be applied to both assets and liabilities, with opposite signs in the parameter a. Based on CAPM, we establish that the parameter a should correspond to the systematic risk of X. For a normal (m,s2) distribution, the distorted distribution is also normal with m¢ = m+as and s¢ = s. For a lognormal distribution, the distorted distribution is also lognormal. By applying the distortion operator to stock price distributions, we recover the risk-neutral valuation for options and in particular the Black-Scholes formula.