The theory of real options determines the optimal time to invest in a project of given size. As a main result, it is found that in a more uncertain environment, it is optimal for a firm to delay its investment. In other words, uncertainty generates a "value of waiting." Recently, contributions appeared that in addition determine the optimal size of the investment. This paper surveys this literature. As a general result, it is obtained that more uncertainty results in larger investments taking place at a later point in time. So, where from the traditional real options literature one can conclude that uncertainty is bad for growth, this is not so clear anymore when also the size of the investment needs to be determined. The survey consists of two parts. First, we present single firm models, and second, we give an overview of the oligopoly models that have appeared up until now.
We present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black–Merton–Scholes model.
This paper considers investment problems in real options with non-homogeneous two-factor uncertainty. We derive some analytical properties of the resulting optimal stopping problem and present a finite difference algorithm to approximate the firm’s value function and optimal exercise boundary. An important message in our paper is that the frequently applied quasi-analytical approach underestimates the impact of uncertainty. This is caused by the fact that the quasi-analytical solution does not satisfy the partial differential equation that governs the value function. As a result, the quasi-analytical approach may wrongly advise to invest in a substantial part of the state space.
This article studies strategic investment behavior of firms facing an uncertain demand in a duopoly setting. Firms choose both investment timing and the capacity level while facing additional uncertainty about market participants, which is introduced via the concept of hidden competition.We focus on the analysis of possible strategies of the market leader in terms of its capacity choice and on the influence of hidden competition on these strategies.We show that due to hidden competition the follower is more eager to invest. As a result the entry deterrence strategy of the leader becomes more costly and it can only be implemented for smaller market size, leaving additional room for entry accommodation. The leader has incentives to prevent entry of the hidden competitor stimulating simultaneous investment if the hidden firm has a large capacity, and has more incentives to apply entry deterrence in the complementary case of a small capacity of the hidden player. In the first case overinvestment aimed to deter the follower's entry does not occur for a wide range of parameters values.
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