Ioannis Karatzas

Optimal portfolio and consumption decisions for a “small investor” on a finite horizon

A general consumption/investment problem is considered for an agent whose actions cannot affect the market prices, and who strives to maximize total expected discounted utility of both consumption ...

Martingale and duality methods for utility maximization in a incomplete market

The problem of maximizing the expected utility from terminal wealth is well understood in the context of a complete financial market. This paper studies the same problem in an incomplete market containing a bond and a finite number of stocks whose prices are driven by a multidimensional Brownian motion process W. The coefficients of the bond and stock processes are adapted to the filtration (history) of W, and incompleteness arises when the number of stocks is strictly smaller than the dimension of W. It is shown that there is a way to complete the market by introducing additional “fictitious” stocks so that the optimal portfolio for the thus completed market coincides with the optimal portfolio for the original incomplete market. The notion of a “least favorable” completion is introduced and is shown to be closely related to the existence question for an optimal portfolio in the incomplete market. This notion is expounded upon using martingale techniques; several equivalent characterizations are provided...

On the pricing of American options

The problem of valuation for contingent claims that can be exercised at any time before or at maturity, such as American options, is discussed in the manner of Bensoussan [1]. We offer an approach which both simplifies and extends the results of existing theory on this topic.

Explicit Solution of a General Consumption/Investment Problem

This paper solves a general consumption and investment decision problem in closed form. An investor seeks to maximize total expected discounted utility of consumption. There are N distinct risky investments, modelled by dependent geometric Brownian motion processes, and one riskless deterministic investment. The analysis allows for a general utility function and general rates of return. The model and analysis take into consideration the inherent nonnegativity of consumption and consider bankruptcy, so this paper generalizes many of the results of Lehoczky, Sethi, and Shreve Lehoczky, J., S. Sethi, S. Shreva. 1983. Optimal consumption and investment policies allowing consumption constraints and bankruptcy. Math. Oper. Res.8 613--636.. The value function is determined explicitly, as are the optimal consumption and investment policies. The analysis is extended to consider more general risky investments. Under certain conditions, the value functions derived for geometric Brownian motion are shown to provide upper and lower bounds on the value functions in the more general context.

Optimization problems in the theory of continuous trading

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

A generalized clark representation formula, with application to optimal portfolios

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The numéraire portfolio in semimartingale financial models

. 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 maxim...

Connections between Optimal Stopping and Singular Stochastic Control I. Monotone Follower Problems

A modification of J. M. C. Clarks formula is established for the stochastic integral representation of Wiener functionals under an equivalent (Girsanov) change of probability measure. It is shown how this modified Clark formula leads to the representation of optimal portfolios fora variety of situations in the modern theory of financial economics.

Irreversible investment and industry equilibrium

Abstract We study the existence of the numéraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numéraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numéraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a no-free-lunch-type notion. In particular, the full strength of the “No Free Lunch with Vanishing Risk” (NFLVR) condition is not needed, only the weaker “No Unbounded Profit with Bounded Risk” (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks.

Hedging American contingent claims with constrained portfolios

The stochastic control problem of tracking a Brownian motion by a nondecreasing process (Monotone Follower) is related to a question of Optimal Stopping. Direct probabilistic arguments are employed to show that the two problems are equivalent, and that both admit optimal solutions.