Jaime A. Londoño

A More General Valuation and Arbitrage Theory for Itô Processes

Abstract We specialize the results on characterization of arbitrage and valuation of European and American Contingent claims obtained in Londoño [7] for a model that is computational friendly, and include those whose price processes are of Itô type with coefficients that are only Hölder continuous. A unified framework for valuation of financial instruments of both European and American type is presented under some technical conditions. Also valuation of European and American contingent claims where the matrix of volatilities does not have maximal rank is provided, and valuation of contingent claims with random expiration date is developed.

State-Dependent Utilities and Incomplete Markets

The problem of optimal consumption and investment for an agent that does not influence the market is solved. The optimization criteria are based on a state-dependent utility functional as proposed in Londono (2009). The proposed solution is given in any market without state-tame arbitrage opportunities, includes several utilities structures, and includes incomplete markets where there are multiple state variables. The solutions obtained for optimal wealths consumptions, and portfolios are explicit and easily computable; the main condition for the result to hold is that the income process of each agent is hedgeable, requiring a natural condition on employer and employee to agree on a contract whose risk can be managed by both parties. In this paper we also developed a theory of markets when the processes are generalization of Brownian flows on manifolds, since this framework shows to be the natural one whenever the problem of intertemporal equilibrium is addressed.

An Approach of Randomness of a Sample Based on Its Weak Ergodic Limit

For a Polish Sample Space with a Borel -field with a surjective measurable transformation, we define an equivalence relation on sample points according to their ergodic limiting averages. We show that this equivalence relation partitions the subset of sample points on measurable invariant subsets, where each limiting distribution is the unique ergodic probability measure defined on each set. The results obtained suggest some natural objects for the model of a probabilistic time-invariant phenomenon are uniquely ergodic probability spaces. As a consequence of the results gained in this paper, we propose a notion of randomness that is weaker than recent approaches to Schnorr randomness.

A New Logistic-Type Model for Pricing European Options

We propose a family of models for the evolution of the price process

STATE-DEPENDENT UTILITY

S_t

Actuarial Sciences and Quantitative Finance

St of a financial market. We model share price and volatility using a two-dimensional system of stochastic differential equations (SDEs) driven by a single Wiener process. We prove that this family of models is well defined and that each model from this family is free of arbitrage opportunities, and it is (state) complete. We use option prices written over the S&P500 from December 2007 to December 2008 to calibrate a model of the proposed family and compare the calibration results with results of the Heston Model for the same data set. The empirical results achieved in both models show similarities for periods of low volatility, but the model studied shows a better performance during the period of higher volatility.

Actuarial Sciences and Quantitative Finance : ICASQF, Bogotá, Colombia, June 2014

We define an inter-temporal Duesemberry Equilibrium where agents are rational agents that optimize their consumption and investment decisions with respect to the relative incomes of their peers (relative income hypothesis). We characterize these markets, provide existence and uniqueness when a sufficient weak condition is met, and develop some simple examples. We propose and solve a maximization problem by every agent to choose their optimal consumption and portfolios. The solution achieved maximize the relative well-being with respect to other members of society and A posteriori the optimization problem maximize the satisfaction on the relative magnitude of consumption in society. The theoretical framework used is a generalization of markets when the processes are Brownian Flows on Manifolds.