Real option pricing with mean-reverting investment and project value

Abstract

In this work, we are concerned with valuing the option to invest in a project when the project value and the investment cost are both mean-reverting. Previous works on stochastic project and investment cost concentrate on geometric Brownian motions (GBMs) for driving the factors. However, when the project involved is linked to commodities, mean-reverting assumptions are more meaningful. Here, we introduce a model and prove that the optimal exercise strategy is not a function of the ratio of the project value to the investment V/I – contrary to the GBM case. We also demonstrate that the limiting trigger curve as maturity approaches traces out a nonlinear curve in (V, I) space and derive its explicit form. Finally, we numerically investigate the finite-horizon problem, using the Fourier space time-stepping algorithm of Jaimungal and Surkov [2009. Lev´y based cross-commodity models and derivative valuation. SIAM Journal of Financial Mathematics, to appear. http://www.ssrn.com/abstract=972837]. Numerically, the optimal exercise policies are found to be approximately linear in V/I; however, contrary to the GBM case they are not described by a curve of the form V*/I*=c(t). The option price behavior as well as the trigger curve behavior nicely generalize earlier one-factor model results.

Keywords:

• real options
• mean-reverting
• stochastic investment
• investment under uncertainty

JEL Classification::

• C6 (Mathematical Methods and Programming)
• C61
• C67

Acknowledgements

SJ was supported in part by NSERC of Canada. MOS was partially supported by CNPq and FAPERJ. JPZ was supported by CNPq grants 302161/2003-1 and 474085/2003-1. All authors acknowledge the IMPA-PETROBRAS cooperation agreement. The authors wish to thank Prof. Marco Antonio Dias (PUC-RJ, Brazil) for calling their attention to the issue of triggers in mean-reversion models.

Notes

The GMR process is also known as the Stochastic Logistic or the Stochastic Verhulst model (cf. Kloeden and Platen 1992).

Some early observations on homogeneity were made by Merton (1973) (see Theorem 9, p. 149) in the context of Warrants. Specifically, he noticed that the value of a warrant is homogeneous in the share price and strike price if the share price distribution is independent of the share's level. Here, however, we are dealing with two sources of uncertainty.

As usual we work on a complete filtered probability space where , is the natural filtration generated by the driving Brownian motions and is the statistical (real-world) probability measure.

Note that we have written the value in terms of the log-state variables and y=ln I−φ rather than V and I directly as the resulting PDE is simpler in these variables.

The Lambert-W function L(z) solves .