Option Value of the MLs Using the Black-Scholes Method

This research estimated the option value of the MLs in 2012 using the Black- Scholes method. A total of $1,117,755 and $221,518 were estimated as the option value of the MLs in 2012 in Cases 1 and 2, respectively (see Table 31). Recall, for Case 1, 214,859 travelers who used the MLs at least once in 2012 were selected and their

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11,721,575 GPL trips were used to estimate the option value of MLs in 2012. For Case 2, 1,561,879 option users (any traveler with a transponder who used Katy Freeway in 2012) were selected and their 31,234,266 GPL trips were used to estimate the option value of the MLs in 2012. The total option value estimate in Case 2 was about one fifth of the total option value estimate in Case 1 even though the number of GPL trips made by option users in Case 2 was approximately three times the number of GPL trips in Case 1. This was because the larger value of travel time estimated from Case 1 option users served to estimate the option value of the MLs (i.e., $14/hour in Case 1 and $5.97/hour in Case 2). The average option value per person-trip was estimated as $0.095 and $0.007 in Cases 1 and 2, respectively. The average option value for each option user for the year was estimated as $5.20 and $0.14 in Cases 1 and 2, respectively (see Table 31).

Table 31 Option Value of the MLs using the Black-Scholes Method

Statistical Measure Case 1 Case 2

Total Number of GPL Trips 11,721,575 31,234,266

Total Number of Option Users 214,859 1,561,879

Total Option Value $1,117,755 $221,518

Average Option Value Per Person-Trip $0.095 $0.007

Standard Deviation of Option Value Per Person-Trip $0.337 $0.049

Average Option Value Per Person $5.202 $0.142

Standard Deviation of Option Value Per Person $18.205 $0.875

About 99 percent of the option values per person-trip were less than $1.7 in both cases (see Table 32). However, a few large option value estimates were observed in Case

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1, and the maximum option value per person-trip in Case 1 was $15.02 (see Table 32). Thus, this research needed to identify why the large option values were estimated from the Black-Scholes method. The option value of the MLs partially depends on the standard deviation of the generalized trip costs on the GPLs. For the trip that had an option value of $15.02, the standard deviation of the generalized trip costs on the GPLs in 2012 was $2.74. This cost is equivalent to 11.7 minutes of variability (standard deviation) in travel times on GPLs using the value of travel time of $14/hour in Case 1. Whereas, for the trips that had almost no option value per person-trip, the standard deviation of the generalized trip costs on the GPLs in 2012 was very small (e.g., $0.02, which is equivalent to 0.1 minutes of the variability in travel times). These results coincide with the finding in the sample estimations using the Black-Scholes method in Section 5.6: the higher the standard deviation of the generalized trip costs on the GPLs in 2012, the higher the option value of MLs. Specifically, from the option value

estimates using the Black-Scholes method, the correlation coefficient between the option value per person-trip and the standard deviation of the generalized trip costs on the GPLs in 2012 was estimated as 0.58. Thus, those unusually large option value estimates could be caused by very high variability in the generalized trip cost (travel time) on the GPLs because the high variability increases the chances that the MLs may be needed (valued) by GPL travelers.

About 99% of the option users valued the option of having the MLs less than $72 for the year in both cases. The unusually large option value per person (the maximum option value per person in Case 1 in Table 32) is due to the large number of GPL trips in

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2012, i.e., 434 trips in 2012. In addition, the traveler who had the maximum option value per person in Case 1 frequently traveled on the GPL section where the generalized trip cost is unreliable. Specifically, s/he traveled 86 times (out of 434 times) on the section where the standard deviation of the generalized trip costs on the GPLs was greater than $2 (which is equivalent to 8.6 minutes of the variability in travel times).

Table 32 Quantiles of the Option Value of the MLs Using the Black-Scholes Method

Quantiles Case 1 Case 2

Per Person-Trip Per Person Per Person-Trip Per Person

100% Max $15.019 $1165.799 $3.514 $204.477 99% $1.645 $71.810 $0.203 $2.178 95% $0.536 $20.300 $0.016 $0.622 90% $0.231 $10.715 $0.002 $0.292 75% Q3 $0.025 $3.795 $0.000 $0.038 50% Median $0.000 $1.100 $0.000 $0.001 25% Q1 $0.000 $0.182 $0.000 $0.000 10% $0.000 $0.000 $0.000 $0.000 5% $0.000 $0.000 $0.000 $0.000 1% $0.000 $0.000 $0.000 $0.000 0% Min $0.000 $0.000 $0.000 $0.000

Figure 17 provides the distributions of the option value estimates using the Black-Scholes method. As in the previous section, this research randomly selected five percent of the option value estimates to draw the distributions. As shown in Figure 17, most option value estimates for each trip were less than $0.9 in Case 1 and most option value estimates for each trip were less than $0.2 in Case 2. Most option value estimates for each option user were less than $35 per year in Case 1 and most option value

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estimates for each option user were less than $2 per year in Case 2. The next section compares the option values of the MLs in 2012 using both methods.

<Distribution of Option Value per Person-Trip>

Case 1 Case 2

(n=586,079, 5% of sample) (n=1,561,713, 5% of sample) <Distribution of Option Value per Person>*

Case 1 Case 2

(n=10,743, 5% of sample) (n=78,094, 5% of sample)

*: The option value estimate samples that were greater than $35 per year were excluded in the distributions. Those were 2.4% and 0.01% of the samples in Cases 1 and 2, respectively. Figure 17 Distribution of the Option Value Using the Black-Scholes Method

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