Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
Comment #2:
I have read and pondered your email of yesterday and the portion of the options theoretic that you
included. The question is whether the Black/Scholes option pricing equation can be used to value a
decrease in the variability of travel times over a specified road segment. As I read your report I had
two questions in mind: First, is the Black/Scholes model applicable? Second, have you applied it
correctly? My answers are yes and yes.
Response #2:
We agree with the reviewer that the Black-Scholes model is applicable to the problem of travel time
variability and we believe that this model has been appropriately applied.
Comment #3:
How to justify applying any interest rate (growth rate) to travel speed (or time) in the way used in the
model and how to select the value of this interest rate for travel speed (or time)?
Response #3:
A finite and fixed option life (insurance contract life) is necessary to derive a value of the insurance
premium/option value. The assumption to apply an interest rate (growth rate) is arbitrary (although
less so than assumptions of other approaches that assume that all that matters in measurement of
reliability is the probability of tail events.)
Comment #4:
It appears that the value of r is arbitrarily set and there does not seem to be any justification for it.
The example sets the value of r to be 5% (or 3%). A 3-5% interest rate is commonly used when
considering money because these are close to the historical interest rates people could get by putting
their money in something like a bank account with guaranteed interest rate. However, it is not clear
how to justify the assumption that the travel speed will grow at any rate with certainty over T-t time
along the target road segment. If we can properly justify applying a guaranteed interest/growth rate r
to travel speed, then the value of r will need to be determined carefully because the result of the
model depends heavily on the value of r.
Response #4:
The value of the interest rate used in the formulation should be the real, annual riskless rate of
return. This rate varies somewhat with macroeconomic conditions, but should reflect the real
discount rate the market is applying to value funds received in the future versus today. This is also
called the "time value of money" in finance parlance. The reviewer is correct that the interest rate in
the implementation of the Black-Scholes formula should be a low, single-digit annual rate in the vast
majority of macroeconomic settings.
DETERMINING THE ECONOMIC BENEFITS OF IMPROVING TRAVEL-TIME RELIABILITY Page B-34