Real Option Pricing with Mean-Reverting Investment and Project Value

Here, the expected value is taken under an appropriate risk-adjusted measure. Furthermore, VT and IT represent the project’s value and amount to be invested, respectively, at time T . If the project can be started at anytime, then (1.1) is modified to its American counterpart. In this case, the maturity date T is replaced by a stopping time τ (0 ≤ τ ≤ T ) and the investor chooses the stopping time to maximize the option’s value. As such, the problem becomes a free boundary problem in which the optimal strategy is computed simultaneously with the option’s value. Traditionally, the project value is assumed to be a geometric Brownian motion (GBM) and the investment amount is constant or deterministic, as in the pioneering work of Tourinho (1979). Stochastic investment amounts have also been investigated: the case of an investment that is driven by a GBM, in the special situation that the opportunity to invest does not expire in time i.e. a perpetual option, is treated in McDonald and Siegel (1986) See also Berk, Green, and Naik (1999). More recently, Elliott, Miao, and Yu (2007) have investigated the case of regime switching investment costs. It should be also pointed out that similar problems arise in swap options, as in Margrabe (1978), and in uncertain payoffs, as in Fischer (1978). (MAX REFERENCE). However, much of this work—e.g. McDonald and Siegel (1986) and Blenman and Clark (2005)—also assumes that the amount to be invested is also a GBM. A GBM may be a good model for the project value in certain circumstances, since in many cases it represents a net present value. On the other hand, as already noticed in McDonald and Siegel (1986), the investment costs are typically prices of commodities, and thus ∗Department of Statistics, University of Toronto,100 St. George Street, Toronto,Ontario, Canada M5S 3G3. †Departamento de Matemática Aplicada, UFF, R. Mario Santos Braga, s/n, Niteroi, RJ 22240-120, Brazil. ‡IMPA, Est. D. Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil