Optimal Dynamic Basis Trading

We study the problem of dynamically trading a futures contract and its underlying asset under a stochastic basis model. The basis evolution is modeled by a stopped scaled Brownian bridge to account for non-convergence of the basis at maturity. The optimal trading strategies are determined from a utility maximization problem under hyperbolic absolute risk aversion (HARA) risk preferences. By analyzing the associated Hamilton-Jacobi-Bellman equation, we derive the exact conditions under which the equation admits a solution and solve the utility maximization explicitly. A series of numerical examples are provided to illustrate the optimal strategies and examine the effects of model parameters.Comment: 27 pages, 10 figure

Structure of Defective Crystals at Finite Temperatures: A Quasi-Harmonic Lattice Dynamics Approach

In this paper we extend the classical method of lattice dynamics to defective crystals with partial symmetries. We start by a nominal defect configuration and first relax it statically. Having the static equilibrium configuration, we use a quasiharmonic lattice dynamics approach to approximate the free energy. Finally, the defect structure at a finite temperature is obtained by minimizing the approximate Helmholtz free energy. For higher temperatures we take the relaxed configuration at a lower temperature as the reference configuration. This method can be used to semi-analytically study the structure of defects at low but non-zero temperatures, where molecular dynamics cannot be used. As an example, we obtain the finite temperature structure of two 180^o domain walls in a 2-D lattice of interacting dipoles. We dynamically relax both the position and polarization vectors. In particular, we show that increasing temperature the domain wall thicknesses increase

Minimizing the Expected Lifetime Spent in Drawdown under Proportional Consumption

We determine the optimal amount to invest in a Black-Scholes financial market for an individual who consumes at a rate equal to a constant proportion of her wealth and who wishes to minimize the expected time that her wealth spends in drawdown during her lifetime. Drawdown occurs when wealth is less than some fixed proportion of maximum wealth. We compare the optimal investment strategy with those for three related goal-seeking problems and learn that the individual is myopic in her investing behavior, as expected from other goal-seeking research.Comment: This paper is to appear in Finance Research Letter

Optimal Investment to Minimize the Probability of Drawdown

We determine the optimal investment strategy in a Black-Scholes financial market to minimize the so-called {\it probability of drawdown}, namely, the probability that the value of an investment portfolio reaches some fixed proportion of its maximum value to date. We assume that the portfolio is subject to a payout that is a deterministic function of its value, as might be the case for an endowment fund paying at a specified rate, for example, at a constant rate or at a rate that is proportional to the fund's value.Comment: To appear in Stochastics. Keywords: Optimal investment, stochastic optimal control, probability of drawdow

Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates

We consider the optimal dividend problem under a habit formation constraint that prevents the dividend rate to fall below a certain proportion of its historical maximum, the so-called drawdown constraint. This is an extension of the optimal Duesenberry's ratcheting consumption problem, studied by Dybvig (1995) [Review of Economic Studies 62(2), 287-313], in which consumption is assumed to be nondecreasing. Our problem differs from Dybvig's also in that the time of ruin could be finite in our setting, whereas ruin was impossible in Dybvig's work. We formulate our problem as a stochastic control problem with the objective of maximizing the expected discounted utility of the dividend stream until bankruptcy, in which risk preferences are embodied by power utility. We semi-explicitly solve the corresponding Hamilton-Jacobi-Bellman variational inequality, which is a nonlinear free-boundary problem. The optimal (excess) dividend rate $c^*_t$ - as a function of the company's current surplus $X_t$ and its historical running maximum of the (excess) dividend rate $z_t$ - is as follows: There are constants $0 < w_{\alpha} < w_0 < w^*$ such that (1) for $0 < X_t \le w_{\alpha} z_t$, it is optimal to pay dividends at the lowest rate $\alpha z_t$, (2) for $w_{\alpha} z_t < X_t < w_0 z_t$, it is optimal to distribute dividends at an intermediate rate $c^*_t \in (\alpha z_t, z_t)$, (3) for $w_0 z_t < X_t < w^* z_t$, it is optimal to distribute dividends at the historical peak rate $z_t$, (4) for $X_t > w^* z_t$, it is optimal to increase the dividend rate above $z_t$, and (5) it is optimal to increase $z_t$ via singular control as needed to keep $X_t \le w^* z_t$. Because, the maximum (excess) dividend rate will eventually be proportional to the running maximum of the surplus, "mountains will have to move" before we increase the dividend rate beyond its historical maximum.Comment: To appear in SIAM J. Financial Mathematics, 34 pages, 11 figure

On the existence of chaotic circumferential waves in spinning disks

We use a third-order perturbation theory and Melnikov's method to prove the existence of chaos in spinning circular disks subject to a lateral point load. We show that the emergence of transverse homoclinic and heteroclinic points respectively lead to a random reversal in the traveling direction of circumferential waves and a random phase shift of magnitude $\pi$ for both forward and backward wave components. These long-term phenomena occur in imperfect low-speed disks sufficiently far from fundamental resonances.Comment: 8 pages, 5 figures, to appear in CHAOS (Volume 17, Issue 2, June 2007