Wang transform

The Wang transform distorts a risk's survival function before taking its expectation. Shaun Wang published it in "A class of distortion operators for pricing financial and insurance risks," Journal of Risk and Insurance, vol. 67, no. 1, 2000, pp. 15-36. For a loss X with survival function S(x) = P(X > x), the distorted probability is S*(x) = Φ(Φ⁻¹(S(x)) + λ), where Φ is the standard normal distribution, and the premium is the expectation under S*, that is the integral of S* over [0, +∞) for a positive loss. The parameter λ is the price of risk. At λ = 0 one recovers the pure premium; the higher λ goes, the more the tail swells. It belongs to the family of distortion measures, where Wang had already proposed, in the mid-1990s, the proportional hazard transform S(x)^(1/ρ). Two properties got it adopted. On a normal distribution with standard deviation σ, it returns a normal distribution with its mean shifted by λσ, which is the standard deviation principle. On the lognormal distribution of a stock price, it gives back Black-Scholes, which brings reinsurance pricing close to option pricing; Wang ties λ to systematic risk through the CAPM. For a reinsurance layer or a cat bond, the consequence shows in the multiple, the ratio of spread to expected loss: the same λ multiplies the expected loss of a low layer a little and that of a high layer a lot. Since the distortion function is concave, the Wang premium is also a spectral measure, and so coherent. Antoon Pelsser (2008) showed, however, that it does not, in general, give time-consistent, arbitrage-free prices for a stochastic process, geometric Brownian motion aside.