Luis H. R. Alvarez

Adoption of uncertain multi-stage technology projects: a real options approach

Abstract This study develops a real options approach in order to characterize the optimal timing of when to adopt an incumbent technology, incorporating as an embedded option a technologically uncertain prospect of opportunities for updating the technology to future superior versions. We develop a new mathematical approach based on the Green representation of Markovian functionals for finding the optimal exercise thresholds both of the ordinary real option associated with the updating decision and of the compound real option associated with the incumbent technology. We characterize how the real option values depend on the underlying market uncertainty and on the uncorrelated technological uncertainty regarding future new generations of the technology.

Reward functionals, salvage values, and optimal stopping

Abstract. We consider the optimal stopping of a linear diffusion in a problem subject to both a cumulative term measuring the expected cumulative present value of a continuous and potentially state-dependent profit flow and an instantaneous payoff measuring the salvage or terminal value received at the optimally chosen stopping date. We derive an explicit representation of the value function in terms of the minimal r-excessive mappings for the considered diffusion, and state a set of necessary conditions for optimal stopping by applying the classical theory of linear diffusions and ordinary non-linear programming techniques. We also state a set of conditions under which our necessary conditions are also sufficient and prove that the smooth pasting principle follows directly from our approach, while the contrary is not necessarily true.

Singular Stochastic Control, Linear Diffusions, and Optimal Stopping: A Class of Solvable Problems

We consider a class of singular stochastic control problems arising frequently in applications of stochastic control. We state a set of conditions under which the optimal policy and its value can be derived in terms of the minimal r-excessive functions of the controlled diffusion, and demonstrate that the optimal policy is of the standard local time type. We then state a set of weak smoothness conditions under which the value function is increasing and concave, and demonstrate that given these conditions increased stochastic fluctuations decrease the value and increase the optimal threshold, thus postponing the exercise of the irreversible policy. In line with previous studies of singular stochastic control, we also establish a connection between singular control and optimal stopping, and show that the marginal value of the singular control problem coincides with the value of the associated stopping problem whenever 0 is not a regular boundary for the controlled diffusion.

Optimal harvesting under stochastic fluctuations and critical depensation

We consider the derivation of the optimal harvesting strategy maximizing the expected cumulative yield from present up to extinction, under the assumption that the harvested population fluctuates stochastically and is subjected to an Allee-effect. By relying on both stochastic calculus and the classical theory of linear diffusions, we derive both the optimal harvesting thresholds at which harvesting should be initiated at full capacity and the value of the optimal strategy. In contrast to ordinary models which are absent of critical depensation, we show that the presence of an Allee-effect leads to the introduction of a lower harvesting threshold at which the population should be immediately depleted under the optimal policy. Moreover, we demonstrate that discounting increases the incentives to harvest and, therefore, increases the probability of a soon extinction of the harvested population.

The impact of delivery lags on irreversible investment under uncertainty

Abstract We consider the valuation and rational exercise of irreversible investment opportunities in the presence of revenue uncertainty and delivery lags. In order to capture supply side market imperfections in the markets for investment goods, we assume that the lag depends on the revenue process faced by the investor. We show that such imperfections have a pronounced decelerating impact on rational investment demand as they may increase the value of waiting in excess of the exercise payoff even for projects which otherwise would be perceived as highly remunerative. We also consider the comparative static properties of the optimal investment policy and its value, and demonstrate that typically increased uncertainty decreases the investment incentives by increasing the value of waiting.

A class of solvable singular stochastic control problems

We consider the singular stochastic control problem of a linear, time-homogeneous and regular diffusion process. By relying on a combination of stochastic calculus the classical theory of diffusions, and ordinary nonlinear programming techniques, we find the optimal policies and their value functions in the three most common cases appearing in the applications of singular controls. Especially we demonstrate that the smooth fit principle can be interpreted as an ordinary first order necessary condition for optimality

Tax policy uncertainty and corporate investment: A theory of tax-induced investment spurts

Abstract The anticipatory effects of a corporate tax reform of the tax-cut cum base-broadening variety are analyzed in a dynamic stochastic adjustment model of firm behavior, focusing on the case where the firm is uncertain both about the timing and the contents of the expected reform. The value of the firm is solved prior to and after the reform. The existence of investment spurts prior to the implementation of the tax reform is established. Rigorous results are derived under constant returns and the effects of diminishing returns are explained. The expectation of a future tax cut causes the firm to accelerate optimal investment, while the expectation of a reduction in the tax base (the rate of fiscal depreciation) has the opposite effect. For a firm which updates information, timing uncertainty interacts with the expectation effect; moreover, increased timing uncertainty may accelerate or decelerate investment as an optimal response to an expected tax cut. Futhermore, for reasonable assumptions, it is shown that a rate-cut cum base-broadening tax reform of the type implemented in several OECD-countries in the 1980s and 1990s cannot be revenue neutral.

Singular stochastic control in the presence of a state-dependent yield structure

We consider the determination of the optimal singular stochastic control for maximizing the expected cumulative revenue flows in the presence of a state-dependent marginal yield measuring the instantaneous returns accrued from irreversibly exerting the singular policy. As in standard models of singular stochastic control, the underlying stochastic process is assumed to evolve according to a regular linear diffusion. We derive the value of the optimal strategy by relying on a combination of stochastic calculus, the classical theory of diffusions, and non-linear programming. We state a set of usually satisfied conditions under which the optimal policy is to reflect the controlled process downwards at an optimal threshold satisfying an ordinary first-order necessary condition for an optimum. We also consider the comparative static properties of the value and state a set of sufficient conditions under which it is concave. As a consequence, we are able to state a set of sufficient conditions under which the sign of the relationship between the volatility of the process and the value is negative.

Optimal exit and valuation under demand uncertainty: A real options approach

This paper considers both the optimal exit strategy and the valuation of stochastic cash flows of a firm facing demand uncertainty and potential excess supply. By relying on the standard theory of linear diffusions and ordinary nonlinear programming, we derive the value of the rationally managed firm, and state the necessary condition for optimal exit. In contrast to the standard approaches in the real options literature, our analysis is completely independent of both dynamic programming and the smooth-fit principle. I demonstrate that irreversible exit is optimal only when the value of the future productive opportunities becomes smaller than the value of irreversibly exercising the option to exit and in this way avoid further cumulative losses. I also present the comparative static properties of the optimal exit threshold and demonstrate that increased uncertainty may increase or decrease the optimal exit threshold depending on the sign of the net convenience yield.

Stochastic Forest Stand Value and\linebreak Optimal Timber Harvesting

We consider a Faustmann timber harvesting problem arising in the literature on rational forest management by modeling the value of the harvested resource as a time homogeneous, regular, and linear diffusion. We state a set of easily verifiable general conditions under which the existence and uniqueness of an optimal cutting value and, consequently, an optimal impulse control, are guaranteed. We also present a set of conditions under which increased volatility increases both the value and the optimal harvesting threshold at which the irreversible harvesting strategy is exercised.