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

David Ruhm's paper is a welcome addition to the actuarial literature. It illustrates some difficult concepts in a refreshing way. As actuaries are increasingly faced with the need to price non-traditional risks, it is important that they understand how to do so. One of the paper's main points is to emphasize the important finding from financial economics that the probability distribution of risk outcomes does not always contain enough information to produce arbitrage-free prices for that risk. However, the probability distribution of outcomes can, and indeed must, be used to determine the expected cost of that risk. This discussion uses Ruhm's examples to underscore the distinction between price and cost, and the potential implications for the seller of a derivative. Ruhm's paper also seeks to generalize about the arbitrage-free prices of calls and puts compared to their expected value payoff. Ruhm concludes that calls are priced at a discount and puts at a premium, at least when the underlying security has an expected return E that is greater than the risk-free rate r. He then seeks to explain why investors would buy puts, given that htey are priced at a premium to expected value. He concludes that some risks have a qualitative nature as either insurance or investment. This pattern of discounted calls and surcharged puts it true ONLY if E > r. Under the condition E r, which Ruhm did not discuss, calls are surcharged and puts are discounted. As a result investor behavior can be accounted for in a simpler way than by appealing to investor risk aversion or the "qualitative nature of a risk."