Real Options Games in Complete and Incomplete Markets with Several Decision Makers

Abstract

Abstract. We consider optimal investment policies for irreversible capital investment projects under uncertainty in a monopoly situation and in a Stackelberg leader-follower game. We consider two types of payoffs: lump-sum and cash flows. The decisions are the times to enter into the market. The problems belong to the class of optimal stopping times, for which the right approach is that of variational inequalities (V.I.s). In the case of complete markets, payoffs are expected values with respect to the risk-neutral probability. In the case of incomplete markets, the risk-neutral probability is not defined. We consider an investor maximizing his/her utility function, and we consider the investment in the project as an additional decision, besides portfolio investment and consumption decisions. This decision remains a stopping time, conversely to the portfolio investment and consumption decisions (continuous controls). The game problem raises new difficulties. The leader's V.I. has a nondifferentiable obstacle. The weak formulation of the V.I. handles this difficulty. In some cases, the solution of the V.I. may be continuously differentiable although the obstacle is not. An additional difficulty occurs for lump-sum payoffs in the case of incomplete markets. We cannot compare gains and losses at different times. We propose an alternative approach, using equivalence (indifference) considerations. In the case of payoffs characterized by cash flows, this difficulty does not exist, but an intermediary problem arises which has a nice interpretation as a differential game. The solutions thus obtained for the Stackelberg game are not intuitive. Therefore, competition has important consequences on investment decisions.

Investment under uncertainty, competition and regulation

“…Finally, the introduction of risk-aversion should not be restrained to the coordination game. This issue is already tackled in Bensoussan et al [3], and Grasselli et al [7] for the incomplete market setting. But as a fundamentally different source of risk, the coordination game and the regulator's decision shall be evaluated with a different risk-aversion parameter, as we proposed in Section 5.…”

Game theoretic analysis of incomplete markets: emergence of probabilities, nonlinear and fractional Black–Scholes equations

“…Similar problems arise in the study of real options. We refer to Dixit and Pindyck [16] for a general background and to Bensoussan et al [7] for more recent mathematical results. A typical real option problem can be formulated as follows.…”

Real Options With Competition and Regime Switching

“…The properties of solutions are demonstrated in our numerical example. The form of two-interval solution has appeared in Dixit and Pindyck (1994); it was proved in Bensoussan et al (2010) for simpler models.…”

“…First, our work explicitly formulates the leader's payoff function in Stackelberg competition under regime-switching uncertainty. We face technical complexities not present in either Bensoussan et al (2012) due to the lack of competition, or in Bensoussan et al (2010) because of the lack of regime changes.…”

“…Valuation models developed along this line are often referred to as real options game models. These models have been explored at length by Smets (), Grenadier (), Weeds (), Lambrecht and Perraudin (), Paxson and Pinto (), Kong and Kwok (), Bensoussan, Diltz, and Hoe (), among others. Extensive reviews can be found in Smit and Trigeorgis (, ), Dias and Teixeira (), and Azevedo and Paxson ().…”