Lower and upper pricing of financial assets

Modeling of uncertainty by probability errs by ignoring the uncertainty in probability. When financial valuation recognizes the uncertainty of probability, the best the market may offer is a two price framework of a lower and upper valuation. The martingale theory of asset prices is then replaced by the theory of nonlinear martingales. When dealing with pure jump compensators describing probability, the uncertainty in probability is captured by introducing parametric measure distortions. The two price framework then alters asset pricing theory by requiring two required return equations, one each for the lower upper valuation. Proxying lower and upper valuations by daily lows and highs, the paper delivers the first empirical study of nonlinear martingales via the modeling and simultaneous estimation of the two required return equations.