Debt Management and the Developing Nationts Economy: A Stochastic Optimal Control Analysis

This paper explores the usefulness of a stochastic optimal control analysis to discourage the less developed nations from borrowing funds from the more developed ones to service their investments (or worst still to service their yearly budget). The lenders of these funds are only interested in evaluating whether a borrower is likely to default.  So they make policies to regulate and monitor the risk of an excessive debt that significantly increases the probability of default. The borrowing nation invests part of the loan into a cash account and a stock account in order to make more money. The net effect of the loan on the economy of the nation given that it must be repaid at a nominal interest rate compounded over a period of time is determined herein. Amazingly, this study discovered that as the debt is serviced by the borrower, the principal amount borrowed decreases as time increase but the interest rate (though fixed) increases as time increases, thereby sending the net worth of the borrowers'economy (income) towards a big crash. &nbsp

A Literature Review: Modelling Dynamic Portfolio Strategy under Defaultable Assets with Stochastic Rate of Return, Rate of Inflation and Credit Spread Rate

This research aims to find an optimal solution for dynamic portfolio in finite-time horizon under defaultable assets, which means that the assets has a chance to be liquidated in a finite time horizon, e.g corporate bond. Besides investing on those assets, investors will also have benefit in the form of consumption. As a reference in making investment decisions the concept of utility functions and volatility will play a role. Optimal portfolio composition will be obtained by maximizing the total expected discounted utility of consumption in the time span during the investment is executed and also to minimize the risk, the volatility of the investment. Further the reduced form model is applied since the assets prices can be linked with the market risk and the credit risk. The interest rate and the rate of inflation will be allowed as a representation of market risk, while the credit spread will be used as a representation of credit risk. The dynamic of asset prices can be derived analytically by using Ito Calculus in the form of the movement of the three risk factors above. Furthermore, this problem will be solved using the stochastic dynamic programming method by assuming that market is incomplete. Depending on the chosen utility function, the optimal solution of the portfolio composition and the consumption can be found explicitly in the form of feedback control. This is possible since the dynamic of the wealth process of the control variable is linear. To apply dynamic programming as well as to find solutions we use Backward Stochastic Differential Equation (BSDE) where the solution can be solved explicitly, especially where the terminal value of the investment target is chosen random. Further, it will be modeled with Monte Carlo simulation and, calibrated using Indonesia data of stock and corporate bond

Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem

In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical H\"older smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method

Continuous-Time Markowitz's Model with Transaction Costs

A continuous-time Markowitz's mean-variance portfolio selection problem is studied in a market with one stock, one bond, and proportional transaction costs. This is a singular stochastic control problem,inherently in a finite time horizon. With a series of transformations, the problem is turned into a so-called double obstacle problem, a well studied problem in physics and partial differential equation literature, featuring two time-varying free boundaries. The two boundaries, which define the buy, sell, and no-trade regions, are proved to be smooth in time. This in turn characterizes the optimal strategy, via a Skorokhod problem, as one that tries to keep a certain adjusted bond-stock position within the no-trade region. Several features of the optimal strategy are revealed that are remarkably different from its no-transaction-cost counterpart. It is shown that there exists a critical length in time, which is dependent on the stock excess return as well as the transaction fees but independent of the investment target and the stock volatility, so that an expected terminal return may not be achievable if the planning horizon is shorter than that critical length (while in the absence of transaction costs any expected return can be reached in an arbitrary period of time). It is further demonstrated that anyone following the optimal strategy should not buy the stock beyond the point when the time to maturity is shorter than the aforementioned critical length. Moreover, the investor would be less likely to buy the stock and more likely to sell the stock when the maturity date is getting closer. These features, while consistent with the widely accepted investment wisdom, suggest that the planning horizon is an integral part of the investment opportunities.Comment: 30 pages, 1 figur

Optimal Investment with Transaction Costs and Stochastic Volatility

Two major financial market complexities are transaction costs and uncertain volatility, and we analyze their joint impact on the problem of portfolio optimization. When volatility is constant, the transaction costs optimal investment problem has a long history, especially in the use of asymptotic approximations when the cost is small. Under stochastic volatility, but with no transaction costs, the Merton problem under general utility functions can also be analyzed with asymptotic methods. Here, we look at the long-run growth rate problem when both complexities are present, using separation of time scales approximations. This leads to perturbation analysis of an eigenvalue problem. We find the first term in the asymptotic expansion in the time scale parameter, of the optimal long-term growth rate, and of the optimal strategy, for fixed small transaction costs.Comment: 27 pages, 4 figure

Dynamic optimal reinsurance and dividend-payout in finite time horizon

This paper studies a dynamic optimal reinsurance and dividend-payout problem for an insurer in a finite time horizon. The goal of the insurer is to maximize its expected cumulative discounted dividend payouts until bankruptcy or maturity which comes earlier. The insurer is allowed to dynamically choose reinsurance contracts over the whole time horizon. This is a mixed singular-classical control problem and the corresponding Hamilton-Jacobi-Bellman equation is a variational inequality with fully nonlinear operator and with gradient constraint. The $C^{2,1}$ smoothness of the value function and a comparison principle for its gradient function are established by penalty approximation method. We find that the surplus-time space can be divided into three non-overlapping regions by a risk-magnitude-and-time-dependent reinsurance barrier and a time-dependent dividend-payout barrier. The insurer should be exposed to higher risk as surplus increases; exposed to all the risks once surplus upward crosses the reinsurance barrier; and pay out all reserves in excess of the dividend-payout barrier. The localities of these regions are explicitly estimated.Comment: 7 figure

Finite horizon optimal investment and consumption with transaction costs

10.1137/070703685SIAM Journal on Control and Optimization4821134-1154SJCO

FINITE HORIZON OPTIMAL INVESTMENT AND CONSUMPTION WITH TRANSACTION COSTS

Bachelor'sBACHELOR OF SCIENCE (HONOURS