Optimization Problems in the Theory of Continuous Trading

Abstract

A unified approach, based on stochastic analysis, to the problems of option pricing, consumption/investment, and equilibrium in a financial market with asset prices modelled by continuous semi-martingales is presented. For the first of these problems, the valuation of both “European” and “American” contingent claims is discussed; the former can be exercised only at a specified time T (the maturity date), whereas the latter can be exercised at any time in $[0,T]$. Notions and results from the theory of optimal stopping are employed in the treatment of American options. A general consurption/ investment problem is also considered, for an agent whose actions cannot affect the market prices and whose intention is to maximize total expected utility of both consumption and terminal wealth. Under very general conditions on the utility functions of the agent, it is shown how to approach the above problem by considering separately the two, more elementary ones of maximizing utility from consumption only and of maximizing utility from terminal wealth only, and then appropriately composing them, The optimal consumption and wealth processes are obtained quite explicitly. In the case of a market model with constant coefficients, the optimal portfolio and consumption rules are derived very explicitly in feedback form (on the current level Qf wealth). The Hamilton–Jacobi–3ellman equation of dynamic programming associated with this problem is reduced to the study of two linear parabolic equations that are then solved in closed form. The results of this analysis lead to an explicit computation of the portfolio that maximizes capital growth rate from investment, and to a precise expression for the maximal growth rate. Finally, the results on the consumption/investment problem for a single agent are applied to study the question of equilibrium in an economy with several financial agents whose joint optimal actions determine the price of a traded commodity by “clearing” the markets. Some familiarity with stochastic analysis, including the fundamental martingale representation and Oirsanov theorems, is assumed. Previous exposure to financial economics and/or stochastic control theory is desirable, but not necessary.