Karatzas(1989)

SIAM J. CONTROL
AND
OPTIMIZATION
1989 Society

for Industrial and Applied
Vol. 27, No. 6, pp. 12211259, November 1989
Mathematics 001
Invited Expository Article
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 semimartingales 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 consumption/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 of wealth). The Hamilton-Jacobi-Bellman 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 Girsanov theorems, is assumed. Previous exposure to financial economics and/or stochastic control theory is desirable, but not necessary.
OPTIMIZATION PROBLEMS IN THE THEORY OF CONTINUOUS TRADING*
IOANNIS
KARATZAS"
This paper is dedicated to Dr,. Vdclav E. Beneg on the occasion of his 60th birthday.
* Received by the editors July 12, 1988; accepted for publication (in revised form) January 22, 1989. This research was supported in part by National Science Foundation grant DMS-87-23078. t Department of Statistics, Columbia University, New York, New York 10027.
1. Introduction and summary. Our aim in this article is to report, hopefully to a wider audience than the already well-informed, on certain recent advances in the theory
Key words, option pricing, consumption/investment optimization, equilibrium, stochastic analysis and
control
AMS(MOS) subject classifications, primary 93E20, 90A09; secondary 60G44, 90A16, 49B60, 60G40, 90A14
1221
1222
IOANNIS KARATZAS
of continuous trading which have been made possible thanks to the methodologies of stochastic analysis. All the questions treated here are formulated in the context of a financial market which includes a risk-free asset called the bond, and several risky assets called stocks; the prices of these latter are driven by an equal number of independent Brownian motions, which model the exogenous forces of uncertainty that influence the market. The interest rate of the bond, the appreciation rates of the stocks as well as their volatilities, constitute the coefficients of the market model; we allow them to be arbitrary bounded measurable processes, adapted to the Brownian filtration, but require that a certain nondegeneracy (or "completeness") condition (2.3) be satisfied. The questions that we address include the following: (i) A general treatment of the pricing of contingent claims such as options, both European (to be exercised only at maturity) and American (which can be exercised any time before or at maturity); (ii) The resolution of consumption problems for a "small investor" (i.e., an economic agent whose actions cannot influence the market prices) with quite general utility functions; and (iii) The associated study of equilibrium models. These are formulated in the context of an economy with several small investors and one commodity, whose price is determined by the joint optimal actions of all these agents in a way that "clears" the markets (i.e., equates supply and demand for the commodity at all times). Instrumental in the approach that we adopt are two fundamental results of stochastic analysis: the Girsanov change of probability measure and the representation of Brownian martingales as stochastic integrals. The former constructs processes that are independent Brownian motions under a new, equivalent probability measure which, roughly speaking, "equates the appreciation rates of all the stocks to the interest rate of the bond." The latter of these results provides the "right portfolios" (investment strategies) for the investors in the above-mentioned problems. We assume that the reader is familiar with both these results; they are discussed in several monographs and texts dealing with stochastic analysis, such as Ikeda and Watanabe (1981) and Karatzas and Shreve (1987). Here is an outline of the paper. Sections 2 and 3 set up the model for the financial market and for the small investor, respectively; the latter has at his disposal the choice of a portfolio (investment strategy) and a consumption strategy, which determine the evolution of his wealth. The notion of admissible portfolio/consumption strategies, which avoid negative terminal wealth with probability one, is introduced and expounded on in 4, which can be regarded as the cornerstone of the paper. Based on the results of 4, we treat the pricing of European contingent claims in 5; we provide the fair price and the subsequent values for such instruments, and derive the celebrated Black and Scholes (1973) formula for European call options as a special case of these results. The analogous problems for American contingent claims are taken up in 6; predictably, their treatment requires notions and results from the theory of optimal stopping. Sections 7-11 are concerned with optimization problems for a small investor. We introduce the concept of utility function in 7, and treat first a problem in which utility is derived only from consumption ( 8); based on the methodology of 4, we provide quite explicit expressions for the optimal consumption and wealth processes, as well as for the associated value Vl(x) of this problem, as a function of the initial wealth x > 0. The "dual" situation, with utility derived only from terminal wealth, is discussed in 9; again, explicit expressions are obtained for the above-mentioned quantities,