Kristin Reikvam

Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach

Abstract. We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump Lévy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear first order integro-differential equation subject to gradient and state constraints. We characterize the value function of the singular stochastic control problem as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. This characterization is obtained in two main steps. First, we prove that the value function is a constrained viscosity solution of an integro-differential variational inequality. Second, to ensure that the characterization of the value function is unique, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-differential variational inequalities. In the case of HARA utility, it is possible to determine an explicit solution of our portfolio-consumption problem when the Lévy process posseses only negative jumps. This is, however, the topic of a companion paper [7].

OPTIMAL PORTFOLIO MANAGEMENT RULES IN A NON-GAUSSIAN MARKET WITH DURABILITY AND INTERTEMPORAL SUBSTITUTION

Abstract. We consider an optimal portfolio-consumption problem which incorporates the notions of durability and intertemporal substitution. The logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve Lévy processes as driving noise instead of the more frequently used Brownian motion. The optimization problem is a singular stochastic control problem and the associated Hamilton-Jacobi-Bellman equation is a nonlinear second order degenerate elliptic integro-differential equation subject to gradient and state constraints. For utility functions of HARA type, we calculate the optimal investment and consumption policies together with an explicit expression for the value function when the Lévy process has only negative jumps. For the classical Merton problem, which is a special case of our optimization problem, we provide explicit policies for general Lévy processes having both positive and negative jumps. Instead of following the classical approach of using a verification theorem, we validate our solution candidates within a viscosity solution framework. To this end, the value function of our singular control problem is characterized as the unique constrained viscosity solution of the Hamilton-Jacobi-Bellman equation in the case of general utilities and general Lévy processes.

Merton's portfolio optimization problem in a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein-Uhlenbeck type

We study Mertons classical portfolio optimization problem for an investor who can trade in a risk‐free bond and a stock. The goal of the investor is to allocate money so that her expected utility from terminal wealth is maximized. The special feature of the problem studied in this paper is the inclusion of stochastic volatility in the dynamics of the risky asset. The model we use is driven by a superposition of non‐Gaussian Ornstein‐Uhlenbeck processes and it was recently proposed and intensively investigated for real market data by Barndorff‐Nielsen and Shephard (2001). Using the dynamic programming method, explicit trading strategies and expressions for the value function via Feynman‐Kac formulas are derived and verified for power utilities. Some numerical examples are also presented.

Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs

We investigate an infinite horizon investment-consumption model in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks) whose prices are governed by Lévy (jump-diffusion) processes. We suppose that transactions between the assets incur a transaction cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption under Hindy-Huang-Kreps intertemporal preferences. This portfolio optimisation problem is formulated as a singular stochastic control problem and is solved using dynamic programming and the theory of viscosity solutions. The associated dynamic programming equation is a second order degenerate elliptic integro-differential variational inequality subject to a state constraint boundary condition. The main result is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation. Emphasis is put on providing a framework that allows for a general class of Lévy processes. Owing to the complexity of our investment-consumption model, it is not possible to derive closed form solutions for the value function. Hence, the optimal policies cannot be obtained in closed form from the first order conditions for the dynamic programming equation. Therefore, we have to resort to numerical methods for computing the value function as well as the associated optimal policies. In view of the viscosity solution theory, the analysis found in this paper will ensure the convergence of a large class of numerical methods for the investment-consumption model in question.

A semilinear Black and Scholes partial differential equation for valuing American options

Abstract. Using the dynamic programming principle in optimal stopping theory, we derive a semilinear Black and Scholes type partial differential equation set in a fixed domain for the value of an American (call/put) option. The nonlinearity in the semilinear Black and Scholes equation depends discontinuously on the American option value, so that standard theory for partial differential equation does not apply. In fact, it is not clear what one should mean by a solution to the semilinear Black and Scholes equation. Guided by the dynamic programming principle, we suggest an appropriate definition of a viscosity solution. Our main results imply that there exists exactly one such viscosity solution of the semilinear Black and Scholes equation, namely the American option value. In other words, we provide herein a new formulation of the American option valuation problem. Our formulation constitutes a starting point for designing and analyzing “easy to implement” numerical algorithms for computing the value of an American option. The numerical aspects of the semilinear Black and Scholes equation are addressed in [7].

A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence

In [7], we proved that the American (call/put) option valuation problem can be stated in terms of one single semilinear Black and Scholes partial differential equation set in a fixed domain. The semilinear Black and Scholes equation constitutes a starting point for designing and analyzing a variety of “easy to implement” numerical schemes for computing the value of an American option. To demonstrate this feature, we propose and analyze an upwind finite difference scheme of “predictor‐corrector type” for the semilinear Black and Scholes equation. We prove that the approximate solutions generated by the predictor‐corrector scheme respect the early exercise constraint and that they converge uniformly to the American option value. A numerical example is also presented. Besides the predictor‐corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed.

A NOTE ON PORTFOLIO MANAGEMENT UNDER NON-GAUSSIAN LOGRETURNS

We calculate numerically the optimal allocation and consumption strategies for Mertons optimal portfolio management problem when the risky asset is modelled by a geometric normal inverse Gaussian Levy process. We compare the computed strategies to the ones given by the standard asset model of geometric Brownian motion. To have realistic parameters in our studies, we choose Norsk Hydro quoted on the New York Stock Exchange as the risky asset. We find that an investor believing in the normal inverse Gaussian model puts a greater fraction of wealth into the risky asset. We also investigate the limiting investment rate when the volatility increases. We observe different behaviour in the two models depending on which parameters we vary in the normal inverse Gaussian distribution.

On the existence of optimal controls for a singular stochastic control problem in finance

We prove existence of optimal investment-consumption strategies for an infinite horizon portfolio optimization problem in a Levy market with intertemporal substitution and transaction costs. This paper complements our previous work [4], which established that the value function can be uniquely characterized as a constrained viscosity solution of the associated HamiltonJacobi-Bellman equation (but [4] left open the question of existence of optimal strategies). In this paper, we also give an alternative proof of the viscosity solution property of the value function. This proof exploits the existence of optimal strategies and is consequently simpler than the one proposed in [4].