I survey and assess the development of continuous-time methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuous-time models. Capital market frictions and bargaining issues are being increasingly incorporated in continuous-time theory. THE ROOTS OF MODERN CONTINUOUS-TIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuous-time modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.

Job Market Paper This paper analyzes the optimal investment decisions of heterogeneous Þrms in a competitive, uncertain environment. We characterize Þrms ’ optimal investment strategy explicitly, and derive a closed form solution for Þrm value. We show that in the strategic equilibrium real option premia are signiÞcant. As a result Þrms delay investment, choosing optimally not to undertake some positive NPV projects. The model predicts that Þrm returns vary over the business cycle, with returns negatively skewed during expansions but positively skewed in recessions.

This paper examines the investment behaviour of an incumbent and a potential entrant that are competing for a patent with a stochastic payogg. We incorporate asymmetric information into the model by assuming that the challenger has complete information about the incumbent whereas the latter does not know the precise value of its opponent's investment cost. We find that even a small probability of being preempted gives the informationally-disadvantaged firm an incentive to invest at the breakeven point where it is indifferent between investing and being preempted. By investing ine ciently early to protect its market share, the incumbent gives up not only its option to delay the investment, but also reduces the value of the firm by an amount that increases with the investment cost incurred and the potential loss of market share. The investment behaviour of the challenger is the same as under complete information, namely the challenger `epsilon preempts' the incumbent, if optimal to do so.

This article provides a uni…ed equilibrium approach to valuing a wide variety of commercial real estate lease contracts. Using a game-theoretic variant of real options analysis, the underlying real estate asset market is modeled as a continuous-time Nash equilibrium in which developers make construction decisions under demand uncertainty. Then, using the economic notion that leasing simply represents the purchase of the use of the asset over a speci…ed time frame, I use a contingent-claims approach to value many of the most common real estate leasing arrangements. In particular, the model provides closed-form solutions for the equilibrium valuation of leases with options to purchase, pre-leasing, gross and net leases, leases with cancellation options, ground leases, escalation clauses, lease concessions and sale-leasebacks.

Abstract. In this article we develop a model to analyze patent-protected R&D investment projects when there is (imperfect) competition in the development and marketing of the resulting product. The competitive interactions that occur substantially complicate the solution of the problem since the decision maker has to take into account not only the factors that affect her/his own decisions, but also the factors that affect the decisions of the other investors. The real options framework utilized to deal with investments under uncertainty is extended to incorporate the game theoretic concepts required to deal with these interactions. Implementation of the model shows that competition in R&D not only increases production and reduces prices, but also shortens the time of developing the product and increases the probability of a successful development. These benefits to society are countered by increased total investment costs in R&D and lower aggregate value of the R&D investment projects. 1.

This article considers the value of information and optimal timing to acquire it in a model of irreversible investment. There are two types of uncertainties: first, the value of the investment project depends on an observable stochastic process, and second, it depends on an ex-ante uncertain parameter, whose true value may be learnt at a cost. The former type of uncertainty implies that the opportunity to acquire information has an option-like character: besides owning a real option on the actual project, the firm also owns an option to learn. We derive and illustrate the value of such learning options and the optimal timing to learn. Choosing the timing involves a trade-off: by postponing it the firm benefits from the delayed cost, but on the other hand suffers from increased likelihood that the revealed information would have been more beneficial earlier.

Preemption games are widely used to model patent races, innovation adoption and market entry problems. A previously neglected feature of these problems is that the agents ’ states (e.g. R&D …rms ’ technological improvements) are kept secret and stochastically change over time. We fully characterize equilibrium in preemption games where private information evolves according to Poisson processes, and provide a strategic rationale for the common wisdom that ‘big things happen fast. ’ In the context of patent races we surprisingly …nd that strengthening patent rights need not increase innovation disclosure. Furthermore, we clarify a basic welfare tradeo ¤ between duplication costs and preemption: the former likely take place in early stages of the race, and preemption in later stages.

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In this paper we study, for a duopoly market, the combined effect of uncertainty, competition and “technological complementarity ” on firms ’ investment behaviour for a game-choice setting where it is assumed that there is a first-mover advantage. Firms do often use inputs whose qualities are complements such as computer and modem, equipment and structure, train and track, and transmitter and receiver and, therefore, on such cases, investment decisions on upgrades or replacements must consider the degree of complementarity between investments. We derive analytical or quasi-analytical solutions for the leader and the follower value functions and their respective investment thresholds. At the beginning of the investment game firms have two technologies available, whose functions are complement, and the option to adopt both technologies at the same time or at different times, in a context where the evolution of the gains that can be made through the adoption of the technology(ies) and the price of the technologies are uncertain. Our results contradict the conventional wisdom which says that “when a production process requires two extremely complementary inputs, a firm should upgrade (or replace) them simultaneously”. We found that when uncertainty about revenues and the price of the two technologies is considered it might be optimal for the leader and the follower to adopt the two technologies asynchronously, first, the technology whose price is decreasing at a lower rate and then the technology whose price is decreasing more rapidly. Some of the illustrated results show nonlinear and complex investment criteria and sensitivities to the expected rate of change in the price of the technologies and the degree of complementarity between the two technologies. 2

This paper investigates an interaction between the managerial flexibility and the competition in a dynamic situation. The value of the flexibility can be valued as a real option while the competition can be analyzed with the game theory. We consider a multi-stage game with two firms under demand uncertainty. In the model, One firm called firm L firstly makes an investment decision, and the other firm called firm F decides secondly after observing firm L’s decision at each stage. The model developed here is an extension of the two-stage investment game of Imai and Watanabe(2005), which fully characterize the equilibrium strategies for the two competitive firms by their investment costs. We show that the project values for both firms can be considered as a special example of switching options, and hence these values can be evaluated by the extended switching option model. We apply a binomial or a trinomial lattice to the underlying demand process. Although the lattice model is discrete it is a well-known fact that the trinomial process can converge efficiently to the continuous-time process if parameter values of the lattice model are carefully chosen and the number of trading periods tends to infinity. Hence our model can be also considered as an approximation of the continuous-time model. This paper analyzes equilibrium strategies and project values of the two competitive firms quantitatively under more realistic situations. In addition, by comparing our model with the two-stage game we can investigate the effects of multiple decision opportunities under a more realistic demand process.