Utility functions and the St. Petersburg Paradox

Abstract

The objective of this undergraduate thesis is to understand and review the fundamental aspects of the standard Expected Utility Theory. The expected utility theory is a model of the behaviour of the economic agent when choosing among uncertain or risky decisions. The roots of this theory can be found in Daniel Bernoulli’s famous paper Exposition of a New Theory on the measurement of Risk. Inspired by The St. Petersburg Paradox, Bernoulli switched from the belief in an objective value of the money to the more subjective utility, which allowed to account personal differences in tastes, wealth and risk aversion in Economics, Finance and Actuarial Sciences. The theory is built over a set of axioms that define what is a preference relation in a set. In the first chapter, we expose those axioms and discuss under which circumstances there exists a numerical representation of the preference relation. In the second part, we define a special kind of numerical representations that are better suited to work with, specially when adopting the monetary point of view. That is, when we restrict the set of choices to lotteries with a monetary outcome. Those representations are called von Neumann-Morgenstern representations and require further axioms to guarantee its existence. We end this part studying the continuous case and its relationship with the weak topology. The third part of the work defines the key concept of risk aversion and studies its relationship with concave functions. Also, we present the Arrow-Pratt Coeficient of Absolute Risk Aversion and use it to rank lotteries and obtain widely used utility functions. The fourth part is devoted to see how the expected utility theory modifies the portfolio optimization problem. We construct the martingale and the dynamic programming methods and use them to compute the optimal terminal wealth of binomial markets. In particular, we use the binomial approximation to the Black-Scholes model to obtain the Merton’s solution to the problem of maximizing the terminal utility of a portfolio. Then we devote a whole part to study modern application of the expected utility theory. In particular, • We analyse the mean-variance analysis under the prism of the expected utility theory. The main result of the section is an implicit definition of certainty equivalent level curves that modify the Feasible Area and the Optimal Frontier. • We introduce the Indifference Price Method of valuing derivatives and see that it is an extension of the risk neutral pricing in the sense that coincide with it in complete markets and allows us to set a range of buyer-seller prices in the incomplete ones. • We end the part showing that, under the expected utility theory, the path dependent derivatives are suboptimal to risk averse agent with a fixed investment horizon. The sixth part studies the main drawbacks of the expected utility theory and, when possible, tries to solve them refining the model. The guide of the exposition are the experimental Allais and Ellberg paradoxes, as well as the Markowitz hypothesis, and the proposed solutions are the Savage’s Theory of Subjective Probabilites and the Machina’s theorems on Fr ́chet differentiable enumerical representations. The last part of the text recovers the historical motivation of the expected utility theory, The St. Petersburg Paradox, and applies the expected utility theory to solve it, as Daniel Bernoulli did. Also, we show the insufficiency of the historical solution, via the construction of a Menger’s Super-Petersburg Paradox, when not using bounded utility functions. We end discussing the implications of the boundedness hypothesis and how we obtain new paradoxes. In particular, we study signs of boundedness, without an explicit determination of the utility function, like Rabin’s Calibration Theorem. Methodologically, the sources of this work are both primary and secondary. The secondary sources are the main contributors to the exposition of the fundamentals of the theory while the primary are intensively used in the application and drawbacks of the model. Most of the results came directly from the bibliographic sources and were adapted to a common notation to keep the inner coherence of the text. The original results are limited to small propositions, expansions of known proofs, footnotes and the most part of the remarks. As a final note on the methodology, it is worth to note that the results of this work use the techniques of a rich variety of mathematical fields like Topology, Real Analysis and Measure Theory, Set Theory, Functional Analysis, Financial Engineering and Probability Theory. Hence, the understanding of the whole thesis requires some mathematical baggage.