A Malliavin-based Monte-Carlo Approach for Numerical Solution of Stochastic Control Problems: Experiences from Merton's Problem

The problem of choosing optimal investment and consumption strategies has been widely studied. In continuous time theory the pioneering work by Merton (1969) is a standard reference. In his work, Merton studied a continuous time economy with constant investment opportunities. Since then Merton's problem has been extended in many ways to capture empirically observed investment and consumption behavior. As more realism is incorporated into a model, the problem of optimal investment and consumption becomes harder to solve. Only rarely can analytical solutions be found, and only for problems possessing nice characteristics. To solve problems lacking analytical solutions we must apply numerical methods. Many realistic problems, however, are difficult to solve even numerically, due to their dimensionality. The purpose of this paper is to present a numerical procedure for solving high-dimensional stochastic control problems arising in the study of optimal portfolio choice. For expositional reasons we develop the algorithm in one dimension, but the mathematical results needed can be generalized to a multi-dimensional setting. The starting point of the algorithm is an initial guess about the agent's investment and consumption strategies at all times and wealth levels. Given this guess it is possible to simulate the wealth process until the investment horizon of the agent. We exploit the dynamic programming principle to break the problem into a series of smaller one-period problems, which can be solved recursively backwards. To be specific we determine first-order conditions relating the optimal controls to the value function in the next period. Starting from the final date we now numerically solve the first-order conditions for all simulated paths iteratively backwards. The investment and consumption strategies resulting from this procedure are used to update the simulated wealth paths, and the procedure can be repeated until it converges. The numerical properties of the algorithm are analyzed by testing it on Merton's optimal portfolio choice problem. The reason for this is that the solution to Merton's problem is explicitly known and can therefore serve as a benchmark for the algorithm. Our results indicate that it is possible to obtain some sort of convergence for both the initial controls and the distribution of their future values. Bearing in mind that we intend to apply the algorithm to a multi-dimensional setting, we also consider the possible complications that might arise. However, the state variables added will in most cases be exogenous non-controllable processes, which does not complicate the optimization routine in the proposed algorithm. Problems with computer storage could arise, but they should be solvable with clever computer programming

Utility-Based Hedging of Stochastic Income

In this dissertation, we study and examine utility-based hedging of the optimal portfolio choice problem in stochastic income. By assuming that the investor has a preference governed by negative exponential utility, we a derive a closed-form solution for the indifference price through the pricing methodology based on utility maximization criteria. We perform asymptotic analysis on this closed form solution to develop the analytic approximation for the indifference price and the optimal hedging strategy as a power series expansion involving the risk aversion and the correlation between the income and a traded asset. This gives a fast computation route to assess these quantities and perform our analysis. We implemented the model to perform simulations for the optimal hedging policy and produce the distributions of the hedging error at terminal time over many sample paths histories. In turn, we analyze the performance of the utility-based hedging strategy together with the strategy which arises from employing the traded asset as a substitute for the stochastic income

Investment, income, incompleteness

The utility-maximizing consumption and investment strategy of an individual investor receiving an unspanned labor income stream seems impossible to find in closed form and very dificult to find using numerical solution techniques. We suggest an easy procedure for finding a specific, simple, and admissible consumption and investment strategy, which is near-optimal in the sense that the wealthequivalent loss compared to the unknown optimal strategy is very small. We first explain and implement the strategy in a simple setting with constant interest rates, a single risky asset, and an exogenously given income stream, but we also show that the success of the strategy is robust to changes in parameter values, to the introduction of stochastic interest rates, and to endogenous labor supply decisions

Optimal funding and investment strategies in defined contribution pension plans under Epstein-Zin utility

A defined contribution pension plan allows consumption to be redistributed from the plan member’s working life to retirement in a manner that is consistent with the member’s personal preferences. The plan’s optimal funding and investment strategies therefore depend on the desired pattern of consumption over the lifetime of the member. We investigate these strategies under the assumption that the member has an Epstein-Zin utility function, which allows a separation between risk aversion and the elasticity of intertemporal substitution, and we also take into account the member’s human capital. We show that a stochastic lifestyling approach, with an initial high weight in equity-type investments and a gradual switch into bond-type investments as the retirement date approaches is an optimal investment strategy. In addition, the optimal contribution rate each year is not constant over the life of the plan but reflects trade-offs between the desire for current consumption, bequest and retirement savings motives at different stages in the life cycle, changes in human capital over the life cycle, and attitude to risk

Time is money: life cycle rational inertia and delegation of investment management : [Version November 2013]

We investigate the theoretical impact of including two empirically-grounded insights in a dynamic life cycle portfolio choice model. The first is to recognize that, when managing their own financial wealth, investors incur opportunity costs in terms of current and future human capital accumulation, particularly if human capital is acquired via learning by doing. The second is that we incorporate age-varying efficiency patterns in financial decisionmaking. Both enhancements produce inactivity in portfolio adjustment patterns consistent with empirical evidence. We also analyze individuals’ optimal choice between self-managing their wealth versus delegating the task to a financial advisor. Delegation proves most valuable to the young and the old. Our calibrated model quantifies welfare gains from including investment time and money costs, as well as delegation, in a life cycle setting

Modeling continuous-time financial markets with capital gains taxes

We formulate a model of continuous-time financial market consisting of a bank account with constant interest rate and one risky asset subject to capital gains taxes. We consider the problem of maximizing expected utility from future consumption in infinite horizon. This is the continuous-time version of the model introduced by Dammon, Spatt and Zhang [11]. The taxation rule is linear so that it allows for tax credits when capital gains losses are experienced. In this context, wash sales are optimal. Our main contribution is to derive lower and upper bounds on the value function in terms of the corresponding value in a tax-free and frictionless model. While the upper bound corresponds to the value function in a tax-free model, the lower bound is a consequence of wash sales. As an important implication of these bounds, we derive an explicit first order expansion of our value function for small interest rate and tax rate coefficients. In order to examine the accuracy of this approximation, we provide a characterization of the value function in terms of the associated dynamic programming equation, and we suggest a numerical approximation scheme based on finite differences and the Howard algorithm. The numerical results show that the first order Taylor expansion is reasonably accurate for reasonable market data