Distribution-Based Pricing Formulas Are Not Arbitrage-Free [Discussion]

David Ruhm is entirely correct that risk load formulas based on transforming probability distributions of contract outcomes cannot guarantee arbitrage-free prices. This is what he illustrates by a clever and entertaining example. But the title of the paper seems to assert that no method of transforming distributions is arbitrage-free. This is not the case, as transforms of the probabilities of the underlying events that generate the outcomes are well known to produce arbitrage-free prices. In fact, Ruhm illustrates this by showing that the Black-Scholes formula arises from such a transform. He also shows that this formula builds in risk-adjustments to prices, thus addressing the misapprehension that since the options prices come from a risk-neutral valuation they do not incorporate risk adjustments. To illustrate the application of probability transforms to fundamental events in insurance, this discussion provides an example of using an alternative transform of underlying frequency and severity distributions to price loss layers.