RELAXED UTILITY MAXIMIZATION IN COMPLETE MARKETS

For a relaxed investor—one whose relative risk aversion vanishes as wealth becomes large—the utility maximization problem may not have a solution in the classical sense of an optimal payoff represented by a random variable. This nonexistence puzzle was discovered by Kramkov and Schachermayer (1999) , who introduced the reasonable asymptotic elasticity condition to exclude such situations. Utility maximization becomes well posed again representing payoffs as measures on the sample space, including those allocations singular with respect to the physical probability. The expected utility of such allocations is understood as the maximal utility of its approximations with classical payoffs—the relaxed expected utility. This paper decomposes relaxed expected utility into its classical and singular parts, represents the singular part in integral form, and proves the existence of optimal solutions for the utility maximization problem, without conditions on the asymptotic elasticity. Key to this result is the Polish space structure assumed on the sample space.     

DUALITY IN OPTIMAL INVESTMENT AND CONSUMPTION PROBLEMS WITH MARKET FRICTIONS

In the style of Rogers (2001) , we give a unified method for finding the dual problem in a given model by stating the problem as an unconstrained Lagrangian problem. In a theoretical part we prove our main theorem, Theorem 3.1, which shows that under a number of conditions the value of the dual and primal problems is equal. The theoretical setting is sufficiently general to be applied to a large number of examples including models with transaction costs, such as Cvitanic and Karatzas (1996) (which could not be covered by the setting in Rogers [2001] ). To apply the general result one has to verify the assumptions of Theorem 3.1 for each concrete example. We show how the method applies for two examples, first Cuoco and Liu (1992) and second Cvitanic and Karatzas (1996) .     

PORTFOLIO OPTIMIZATION WITH JUMPS AND UNOBSERVABLE INTENSITY PROCESS

We consider a financial market with one bond and one stock. The dynamics of the stock price process allow jumps which occur according to a Markov-modulated Poisson process. We assume that there is an investor who is only able to observe the stock price process and not the driving Markov chain. The investor's aim is to maximize the expected utility of terminal wealth. Using a classical result from filter theory it is possible to reduce this problem with partial observation to one with complete observation. With the help of a generalized Hamilton–Jacobi–Bellman equation where we replace the derivative by Clarke's generalized gradient, we identify an optimal portfolio strategy. Finally, we discuss some special cases of this model and prove several properties of the optimal portfolio strategy. In particular, we derive bounds and discuss the influence of uncertainty on the optimal portfolio strategy.     

RISK-AVERSION BEHAVIOR IN CONSUMPTION/INVESTMENT PROBLEMS1

In this paper, we study the risk-aversion behavior of an agent in the dynamic framework of consumption/investment decision making that allows the possibility of bankruptcy. Agent's consumption utility is assumed to be represented by a strictly increasing, strictly concave, continuously differentiable function in the general case and by a HARA-type function in the special case treated in the paper. Coefficients of absolute and relative risk aversion are defined to be the well-known curvature measures associated with the derived utility of wealth obtained as the value function of the agent's optimization problem. Through an analysis of these coefficients, we show how the change in agent's risk aversion as his wealth changes depends on his consumption utility and the other problem parameters, including the payment at bankruptcy. Moreover, in the HARA case, we can conclude that the agent's relative risk aversion is nondecreasing with wealth, while his absolute risk aversion is decreasing with wealth only if he is sufficiently wealthy. At lower wealth levels, however, the agent's absolute risk aversion may increase with wealth in some cases.     

MEAN–VARIANCE HEDGING AND OPTIMAL INVESTMENT IN HESTON'S MODEL WITH CORRELATION

This paper solves the mean–variance hedging problem in Heston's model with a stochastic opportunity set moving systematically with the volatility of stock returns. We allow for correlation between stock returns and their volatility (so-called leverage effect). Our contribution is threefold: using a new concept of opportunity-neutral measure we present a simplified strategy for computing a candidate solution in the correlated case. We then go on to show that this candidate generates the true variance-optimal martingale measure; this step seems to be partially missing in the literature. Finally, we derive formulas for the hedging strategy and the hedging error.     

Optimal Investment and Consumption With Two Bonds and Transaction Costs1

An agent can invest in a high-yield bond and a low-yield bond, holding either long or short positions in either asset. Any movement of money between these two assets incurs a transaction cost proportional to the size of the transaction. the low-yield bond is liquid in the sense that wealth invested in this bond can be consumed directly without a transaction cost; wealth invested in the high-yield bond can be consumed only by first moving it into the low-yield bond. the problem of optimal consumption and investment on an infinite planning horizon is solved for a class of utility functions larger than the class of power functions.     

OPTIMAL LIQUIDATION OF DERIVATIVE PORTFOLIOS

We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.     

Bounds on Derivative Prices in an Intertemporal Setting with Proportional Transaction Costs and Multiple Securities

The observed discrepancies of derivative prices from their theoretical, arbitrage-free values are examined in the presence of transaction costs. Analytic upper and lower bounds on the reservation write and purchase prices, respectively, are obtained when an investor's preferences exhibit constant relative risk aversion between zero and one. The economy consists of multiple primary securities with stationary returns, a constant rate of interest, and any number of American or European derivatives with, possibly, path-dependent arbitrary payoffs.