Fear of the Market or Fear of the Competitor? Ambiguity in a Real Options Game

In this paper we study an investment game between two firms with a first--mover advantage, where payoffs are driven by a geometric Brownian motion. At least one of the firms is assumed to be ambiguous over the drift, with maxmin preferences over a strongly rectangular set of priors. We develop a strategy and equilibrium concept allowing for ambiguity and show that equilibria can be preemptive (a firm invests at a point where investment is Pareto dominated by waiting) or sequential (one firm invests as if it were the exogenously appointed leader). Following the standard literature, the worst--case prior for an ambiguous firm in the follower role is obtained by setting the lowest possible trend in the set of priors. However, if an ambiguous firm is the first mover, then the worst--case drift can fluctuate between the lowest and the highest trends. This novel result shows that ``worst--case prior'' in a setting with drift ambiguity does not always equate to ``lowest trend''. As a consequence, preemptive pressure reduces. We show that this results in the possibility of firm value being increasing in the level of ambiguity. If only one firm is ambiguous, then the value of the non--ambiguous firm can be increasing in the level of ambiguity of the ambiguous firm

The effectiveness of carbon pricing: The role of diversification in a firm's investment decision?

It is often argued that compared to a carbon tax, a volatile carbon price under an emissions trading system poses a problem in the transition towards a low carbon economy. However, this paper shows that, when sufficiently positively correlated with the electricity price, carbon price uncertainty diminishes overall volatility because of a diversification effect. To get this result, we develop a dynamic real options model to analyze the impact of positively correlated price uncertainty on the timing of an investment decision. In contrast to static models, we show that even when the carbon price is initially the same under both policy instruments, the timing of the investment decision will typically be different. More importantly, we find that multiple correlated price uncertainties under an emissions trading system encourages investment more than less uncertainty under a carbon tax. Hence, to stimulate a low carbon (or discourage a carbon intensive) investment, an emissions trading system (carbon tax) is preferred. The policy reverts for higher levels of uncertainty and low correlations

Optimal double stopping problems for maxima and minima of geometric Brownian motions

We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima of a geometric Brownian motion. It is shown that the optimal stopping times are the first times at which the underlying process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original double optimal stopping problems to sequences of single optimal stopping problems for the resulting three-dimensional continuous Markov process. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state space. We show that the optimal stopping boundaries are determined as the extremal solutions of the associated first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual real double lookback options with floating sunk costs in the Black-Merton-Scholes model

Investment decisions with two-factor uncertainty

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