Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution

Abstract

We determine the optimal portfolio management rules for a portfolio selection problem with consumption 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 the driving noise instead of the more frequently used Brownian motion. The optimization problem is a state constrained singular stochastic control problem and the associated Hamilton-Jacobi-Bellman equation is a non linear 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. Also for the classical Merton problem, which is a special case of our optimization problem, we provide explicit policies. Instead of relying on a classical verification theorem, we verify our results within a viscosity solution framework. This framework is an adaption of the one used in our companion paper [4], which is devoted to a characterization of the value function as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation in the case of general Utilities and pure-jump Lévy processes.

Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance

“…Lemma 6.1 (Sign of RHS of Equation (6.8)) If A n+1 (Q k )V k is given by equation (4.8), with the control parameter determined by 12) then every element of the right hand side of equation (6.8) is nonnegative, that is,…”

“…Examples where such nonlinear PDEs arise include transaction cost/uncertain volatility models [28,4,38], passport options [3,43], unequal borrowing/lending costs [13], large investor effects [2], risk control in reinsurance [32], pricing options and insurance in incomplete markets using an instantaneous Sharpe ratio [51,31,11], and optimal consumption [12,15]. A recent survey article on the theoretical aspects of this topic is given in [35].…”

Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

“…The difficulty of dealing with a boundary where no data are assigned has been overcome in [6] by investigating constrained solutions. With respect to equations of the type (1.1) the use of constrained solutions is not necessary.…”

“…Empirical work shows that the normal distribution poorly fits the logreturn data for, e.g., stock prices. Among other things the data show heavier tails than predicted by the normal distribution, and it has in recent years been suggested to model logreturns by generalized hyperbolic distributions (see the references in [6,7,8,10,26,9] for relevant works).…”

“…Various existence and comparison/uniqueness results for viscosity solutions of integropartial differential equations of first order (i.e., no local second order term) can be found in [30,31,27,28,6]. When the Lévy measure is bounded, general existence and comparison/uniqueness results for semicontinuous unbounded viscosity solutions of second order degenerate parabolic integro-partial differential equations are given in [1,2,3].…”