Proofs for the New Definitions in Financial Markets

Constructing theorems can help to determine the shape of certain utility curves that make up the new definitions in financial markets. The aim of this study was to present proofs for these theorems. Basic thoughts of new alternative definitions emerge from the decision-making under uncertainty in economics and finance. Shape of the certain utility curve is central to standard definitions in determining risk attitudes of investors. Shape alone determines risk behavior of investors in standard theory. Although the terms “risk-averse,” “risk-loving,” and “risk-neutral” are equivalent to “strict concavity,” “strict convexity,” and “linearity,” respectively, in standard theory, strict concavity or strict convexity, or linearity are valid for certain new definitions, not being the same as standard theory. Hence, it can be stated that new alternative definitions are broader than standard definitions from the viewpoint of shape. For instance, the certain utility curve of a risk-averse investor can be strict concave or strict convex, or linear in alternative definitions.