Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
in this interval—not at the end of the interval. This makes the use of the European option
inappropriate, since it assumes a finite horizon and exercise at the end of that interval. In such
cases, it is more logical to use an American option with a perpetual life.
Coupling a perpetual life option with EV distributional assumptions complicates the arithmetic of
the unreliability-valuation process considerably. There are only a handful of published papers that
address this concept. Work by Koh and Paxson (2005) has been adapted to value decisions to
invest in rare-event ventures. This work provides some guidance regarding placing a value on
unreliability associated with rare events that may play out over a long period of time. However, the
adaptation likely can be better refined to traffic-related concerns with further research.
The approach taken by Koh and Paxson assumes that one embarks on a program to produce
benefits or reduce costs recognizing that potential benefits of the strategy are highly uncertain.
Only few events occur even after having had the strategy (option) in place for a long time. The
Koh and Paxson example is similar in spirit to highway situations in which there is insufficient
information to parameterize the network performance (e.g., speed) distribution directly, but one
knows that rare events affect network performance.
Sufficient data must be available to parameterize the EV distribution, but only for the rare-event
process. The connection between the rare event process and the economic consequences of the
event occurring is determined by characterizing the value of the event process separately. Koh and
Paxson do this by postulating a Wiener process, with a log-mean and log-standard deviation. The
Koh and Paxson method also allows the cost, K, of facilitating the mitigating strategy to be
incorporated, so that the option is a project value (net benefit) concept. Incorporating project
valuation directly into the options-theoretic framework with the Generalized EV is a potentially
valuable way to derive certainty-equivalent economic values of highway management strategies.
There is limited literature on the use of perpetual American options, so the Koh and Paxson paper
is particularly interesting. The mathematics become doubly complex due to both rare-event and
perpetual-option considerations. The Koh and Paxson option is a perpetual American call option.
However, put-call parity allows us to use this formulation by restating the problem slightly.
Equation 4 presents the Koh and Paxson method for the Gumbel EV distribution.
VALUATION OF TRAVEL-TIME RELIABILITY FOR RARE EVENTS Page C-8