POWER UTILITY MAXIMIZATION IN CONSTRAINED EXPONENTIAL LÉVY MODELS

The paper studies connections between risk aversion and martingale measures in a discrete‐time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b= 1; that is, in the case of the log‐utility, the optimal dynamic portfolio defines the numéraire portfolio.

Suppose that discounted asset prices in a financial market are given by a P ‐semimartingale S . Among all probability measures Q that turn S into a local Q ‐martingale, the minimal entropy martingale measure is characterized by the property that it minimizes the relative entropy with respect to P . Via convex duality, it is intimately linked to the problem of maximizing expected exponential utility from terminal wealth. It also appears as a limit of p ‐optimal martingale measures as p decreases to 1. Like for most optimal martingale measures, finding its explicit form is easy if S is an exponential Lévy process and quite difficult otherwise.

In this article, we are interested in an investment problem with stochastic volatilities and Portfolio constraints on proportions. We model the risky assets by jump diffusion processes and we consider a power utility function. The objective is to maximize the expected utility from terminal wealth. The value function is known to be a viscosity solution of an integro-differential Hamilton-Jacobi-Bellman (HJB) equation. Thanks to an exponential transformation, we reduce the nonlinearity of the HJB equation to a semi-linear equation. We prove existence of a smooth solution to the latter equation and we state a verification theorem which relates this solution to the value function. We derive the backward stochastic differential equation (BSDE) associated with the semi-linear equation. We propose a numerical scheme for the resolution of the BSDE based on iterative regressions on function bases and Monte-Carlo method. We illustrate an example which shows the importance of this reduction for the optimal portfolio numerical study. Therefore, we compute the optimal investment strategy by solving the associated BSDE.

The paper studies connections between arbitrage and utility maximization in a discrete-time financial market. The market is incomplete. Thus one has several choices of equivalent martingale measures to price contingent claims. Davis determines a unique price for a contingent claim which is based on an optimal dynamic portfolio by use of a `marginal rate of substitution' argument. Here conditions will be given such that this price is determined by a martingale measure and thus by a consistent price system. The underlying utility function U is defined on the positive half-line. Then dynamic portfolios are admissible if the terminal wealth is positive. In case of the logarithmic utility function, the optimal dynamic portfolio is the numeraire portfolio.

We consider the problem of optimal investment in a market with random interest rate for borrowing. This represents an optimal stochastic control problem with a nonlinear system dynamics. We solve this problem under the assumption of coupled Hull-White type model for the interest rates. An explicit closed-form solution is derived for the power and logarithmic utility functions from terminal wealth.

We consider the problem of optimal investment in a market with borrowing and random coefficients. We assume that the bond interest rate, the borrowing interest rate, the appreciation rate and the volatility of stock, are random and possibly unbounded. Due to the possibility of borrowing, the formulated optimal investment problem is an optimal stochastic control problem with nonlinear system dynamics and possibly unbounded coefficients. Explicit closed-form solutions in terms of a linear backward stochastic differential equation are obtained for the power and logarithmic utility from terminal wealth. The optimal controls turn out to be of a linear state-feedback form.

We consider the problem of expected power utility maximization from terminal wealth in diffusion market models under partial information. After obtaining novel neat expressions for the value-process and for the optimal strategy, the issue of information sufficiency is addressed. In particular, necessary and sufficient conditions that guarantee that the partial information optimal strategy is still optimal when having access to all market information, are provided.

Mean–variance asset–liability management with asset correlation risk and insurance liabilities

An agent's optimization problem of the expected terminal wealth utility in a trinomial tree economy is solved. At each transaction date, the agent can trade in a riskless asset, a primitive asset subject to constant proportional transaction costs, and a contingent claim characterized by some parameter kappa whose bid and ask price is defined by allowing for different equivalent martingale measures. In addition to the classical portfolio choice problem, the characteristic of the contingent claim κ is determined endogenously in the optimization problem. Under suitable conditions, it is proved that the optimal demand of the agent in the primitive risky asset is zero independently of the choice of the terminal wealth utility function: the agent prefers not to trade in the asset subject to transaction costs, which prevents the market from being complete, rather than trading in both assets. Next, the optimal choice of the contingent claim is characterized and the results are applied to European call and put options with fixed maturity and varying exercise price κ.

Portfolio Optimization Under Convex Incentive Schemes

We consider the problem of maximizing expected utility from terminal wealth in models with stochastic factors. Using martingale methods and a conditioning argument, we determine the optimal strategy for power utility under the assumption that the increments of the asset price are independent conditionally on the factor process.

We consider a stochastic financial incomplete market where the price processes are described by a vector valued semimartingale that is possibly non locally bounded. We face the classical problem of the utility maximization from terminal wealth, with utility functions that are finite valued over $(a,\infty )$, $a\in \lbrack -\infty ,\infty )$, and satisfy weak regularity assumptions. We adopt a class of trading strategies that allows for non locally bounded stochastic integrals. The embedding of the utility maximization problem in Orlicz spaces permits to formulate the problem in a unified way for both the cases: $a\in \mathbb{R}$ or $a=-\infty $. By duality methods we prove the existence of the optimal solutions to the primal and dual problems and show that a singular component in the pricing functionals may occur also with utility functions finite on the entire real line.

We investigate portfolio optimization and pricing and hedging of the insurance payment process under the exponential utility function. First, we find the investment strategy under which the expected exponential utility of the insurer’s terminal wealth is maximized. We characterize the optimal value function of the optimization problem and the optimal investment strategy by a nonlinear BSDE. Next, we solve the exponential indifference pricing and hedging problem. We show that the indifference price and the indifference hedging strategy solve a nonlinear BSDE.

Utility Maximization, Risk Aversion, and Stochastic Dominance

In this paper, we solve an optimal reinsurance problem in the mathematical finance area. We assume that the surplus process of the insurance company follows a controlled diffusion process and the constant interest rate is involved in the financial model. During the whole optimization period, the company has a choice to buy reinsurance contract and decide the reinsurance retention level. Meanwhile, the bankruptcy at the terminal time is not allowed. The aim of the optimization problem is to minimize the distance between the terminal wealth and a given goal by controlling the reinsurance proportion. Using the stochastic control theory, we derive the Hamilton-Jacobi-Bellman equation for the optimization problem. Via adopting the technique of changing variable as well as the dual transformation, an explicit solution of the value function and the optimal policy are shown. Finally, several numerical examples are shown, from which we find several main factors that affect the optimal reinsurance policy.