Interest rate paradox

Maximization of result from operations with securities is not always the ultimate goal of participants. For example, result can be exchanged into different currencies. There can be different utility functions that transform result into some asset. Different risk-neutral probability densities could be derived from one set of option prices by participants using different utility functions. Integral of derived density function must be equal to one. There have to be no such utility function for which this condition is not met. Otherwise, derived function is not a probability density. This allows using of risk-free profitable arbitrage strategies. However it was shown that such utility function almost always exist. It is hard to use on nowadays markets. By this reason such opportunity was called “weak arbitrage”