Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
the calculations illustrate that by converting speed volatility to delay-equivalents, we can
consistently measure the cost of unreliability and the benefits of eliminating unreliability.
When one wishes to value unreliability over an arbitrarily-long evaluation horizon, the fixed-life
feature of the European put option is inappropriate. First, there is no assumed fixed life, so a
perpetual option formulation is required. In addition, the American put option, allowing exercise of
the option any time during the (perpetual) life of the option, is the only exercise feature that makes
sense in the context of a perpetual valuation horizon.
The value of an American put option with perpetual life can be calculated from Equation 2, which
has been adapted from McDonald (2002).The option in Equation 2 can be valued in a case where
the log-normal speed variability can be used to parameterize the option. The valuation would yield
the certainty-equivalent value of various speed guarantees, I, associated with various average
speed measures, V.
Equation 1 - Valuing a Perpetual American Put Option
m
I ⎛ m −1 V ⎞
P( I,∞)=
⎜ ⎟
1− m ⎝ m I ⎠
where
m = 1 2 − r
σ − r
2 σ − 1 2
⎛ ⎞
⎜ ⎟ + 2r
⎝
2 2⎠
σ 2
and where
P I,∞
V = the (unknown) speed experienced traversing the link, in mph
I = the guaranteed speed, in mph
r = the annualized, risk - free continuously - compounded interest rate
σ = variability of V; the square root of the log - value variation process of V
( ) = the value of the perpetual American put option in mph, as a function of the speed guarantee
The perpetual American put option is useful in valuing a policy intended to control speed
variability of the morning commuting period over a long period of time. The parameters of the lognormal
speed distribution must then be estimated in a manner that is consistent with this long-lived
`option (i.e., using long histories of the morning commuting speeds). Because the reliability
measure applies to a long time interval, the certainty-equivalent value of unreliability is higher
than in the finite, European put option.
Parameterizing the Options Model
The examples provided above illustrate how options theory can be used in a setting in which the
unreliability problem is caused by recurring events and the speed performance metric is distributed
log-normally. The same method can be applied to circumstances other than the volatility of speed
measured in five-minute intervals.
The parameters of the options formulation should be measured so that they are consistent with the
network performance along the facility under study. For example, performance measures can be
derived that are specific to a longer period, such as an entire weekday, the a.m. peak hour,
weekend travel, etc. In all cases, the log-mean and log-standard deviations need to be estimated
DETERMINING THE ECONOMIC BENEFITS OF IMPROVING TRAVEL-TIME RELIABILITY Page B-15