For our numerical example we start with the same swing option setup that Meinshausen and Hambly [50] used for their illustrations. They investigate an in-the-money (itm) swing option with strike K = 0 and a single exercise right per hour. They defined a delivery period of t
= 0,..,1000 hours and allowed up to N = 100 swings. By setting the mean µ to zero they reduce their initial price process (see equation 2.7) to a discrete Ornstein-Uhlenbeck process with parameters
σ = 0.5, κ = 0.9, µ = 0, (2.54)
and x0= 1 (see Appendix C for sample paths). Meinshausen and Hambly rely on the Longstaff Schwartz algorithm and run a regular linear regression with the basis functions Ψ1,n(Xt) = 1
and Ψ2,n(Xt) = Xt
Yt+1(Xt, n) := α0,n,t+ α1,n,tXt. (2.55) In order to obtain the lower bound, they use 1000 price scenarios to pre-calculate the pa-rameters αr,n,t necessary to approximate the optimal exercise policy. Figure 2.3 illustrates the resulting exercise policy by plotting the indicator functions according to equation 2.16 for various marginal swing rights. The shape of the functions along the timeline resembles
0 100 200 300 400 500 600 700 800 900 1000
0 1 2 3 4 5 6
Hours
Price
Exercise Threshold for Various Swing Rights
Swing 1 Swing 5 Swing 10 Swing 50 Swing 100
Figure 2.3.: Indicator functions for different exercise rights
a concave curve and reflects the intuitive decision rule that a smaller exercise price will be accepted the closer one gets to the final expiration. At the last delivery hour any price above 0 will be accepted for the first exercise right. The same argument explains the nested struc-ture of the indicator functions which start at a lower price for every additional exercise right.
The more rights are still available the smaller the price threshold for the next exercise. The indicator functions touch the timeline at that time step where the remaining number of hours is identical to the remaining number of swings. This is a technical side-effect resulting from the simulation (formally the threshold functions are smooth curves). Any swing right larger than the remaining time period is automatically expired and therefore the threshold is 0. We can also observe that the concave curve becomes smoother with increasing number of swings.
High prices are rare and differ more in size than prices around the average (since x0 = 1 and µ = 0 the prices deviate around 1). Every swing right tries to capture the highest available price. Thus with every additional swing the remaining prices are more and more equal in size and therefore the jitter becomes weaker. Once the continuation values are approximated, the indicator functions and thus the set of stopping points πi0(N ) is fixed for one price scenario i. All swing options with intermediate upper swing number n < N will be based on a subset of π0i(N ). However, their stopping sequence in declining order according to the Meinshausen and Hambly definition (see section 2.3) varies with every n. In other words, the same exercise hour for the same price path can be the first, second or n-th stopping point for different n.
This is the very reason why we do not compute all N stopping points once for all n iterations of a single price path beforehand in our upper bound algorithm in section 2.3 and 2.4. Instead
Figure 2.4.: Sequence of stopping times for two and three swing rights
we have to re-compute the new sequence of stopping points πi0(n) with every intermediate maximum number of stopping rights n. We want to briefly illustrate the behavior with Figure 2.4. The left sub figure shows the exercise thresholds for two swing rights and the correspond-ing price path. The swcorrespond-ings will be exercised once the price exceeds one of the thresholds.
The marks on the trajectory indicate an exercise and the color corresponds to the threshold line that was the trigger. In the right sub figure we see the situation for three swing rights.
Now there is an extra trigger line for the third swing right located below the other two as explained in the previous paragraph. The new marks on the price trajectory demonstrate that the threshold for the third marginal swing right now triggers the second stopping time.
It also underlines that the initial two stopping times still belong to the optimal policy, only the sequence changes due to the additional swing right. In this case τ2(0, 2) = τ3(0, 4).
Let us now return to the lower bound calculation. After the approximation of the contin-uation values Meinshausen and Hambly generate another 1000 price paths and apply their exercise policy in a forward iteration to receive the actual lower bound C0(n) := C0(x0, n).
We follow exactly their approach. The second column of Table 2.3 shows our results for different numbers of initial swing rights n. These lower bound values differ from the results of Meinshausen and Hambly by less than ±0.5%. Next, we compute the corresponding upper
n C0(n) ∆D0(n) [C(99%)0 (n), C(99%)0 (n)]
1 4.777 0.004 [4.697 , 4.783]
2 9.029 0.022 [8.922 , 9.069]
3 13.051 0.028 [12.924 , 13.132]
4 16.842 0.024 [16.699 , 16.965]
5 20.463 0.033 [20.308 , 20.601]
10 37.346 0.023 [37.130 , 37.625]
15 52.668 0.028 [52.408 , 53.040]
20 66.981 0.022 [66.686 , 67.439]
30 93.670 0.020 [93.314 , 94.709]
40 118.452 0.015 [118.053 , 119.877]
50 141.799 0.039 [141.361 , 143.558]
60 164.044 0.012 [163.571 , 166.020]
70 185.414 0.021 [184.910 , 187.856]
80 205.983 0.036 [205.447 , 209.068]
90 225.876 0.021 [225.313 , 229.411]
100 245.154 0.032 [244.562 , 249.015]
Table 2.3.: Swing option with unit constraint
bounds for the swing option. Again we follow the setup by Meinshausen and Hambly and
use I = 20 price scenarios for the outer loop and J = 50 trajectories for the inner simula-tion. The third column presents the corresponding duality gaps of the marginal swing rights i.e. the difference of the upper and lower option value for the n-th marginal swing right
∆D0(n) = ∆C0(n) − ∆C0(n). Let us first look at the duality gap for the American option (n
= 1). With ∆D0(1) = 0.004 the gap is negligibly small which underlines the good approxi-mation of the exercise policy and thus the efficiency of the Longstaff-Schwartz approxiapproxi-mation procedure in case of an American option. Also note that the additional noise introduced by the nested loops does not have a significant numerical effect. However, the situation changes when we move to the swing option (n>1). Already in case of two swings the duality gap increases from 0.004 to 0.022. At least for any further number of swings this gap does not in-crease any more, but varies between 0.015 and 0.039 with an average around 0.031 throughout all swing rights. Meinshausen and Hambly present similar results. Their total duality gap for 100 swings for instance is 248.63 - 245.157 = 3.47 which breaks down to an average marginal duality gap of 0.0347 and lies within our computed ranges. There is no relation between gap size and number of swings. Our duality gap is rather stable around 0.025. We assume that the gap could be narrowed even further with a more refined set of basis functions Ψr,n,t for the Longstaff Schwartz regression. Recall that we currently only apply linear regression.
The last column in Table 2.3 shows the upper and lower bound of the swing option not on the marginal, but on the total level. Hence, for n = 30, the marginal gap of the 30th swing right is 0.020 and the option value for all 30 swing rights varies between 93.314 and 94.709. These intervals are based on the 1 − β = 99 % confidence level. For the purpose of direct comparison we follow Meinshausen and Hambly’s definition of the confidence interval [C(1−β)0 (n), C(1−β)0 (n)]. They apply the confidence level of a Normal distribution. The distri-bution of our value function is not necessarily Normal and the authors do not provide any other legitimation than the asymptotic convergence to the Normal distribution. They apply the two variances of the value functions for the lower and upper bound calculation separately
C(1−β)0 (n) := C0(x0, n) − βσ(n) defined in equation 2.34 and vi0(n) is the ACF of a single path (see equation 2.9). Our confi-dence intervals are slightly larger than the ranges that Meinshausen and Hambly presented.
We could only meet their figures if we increased the number of external scenarios from 20 to 200 which narrowed the volatility.
Finally we want to produce the same figures for the case of a swing option with volume
constraint. We return to our initial example of an off-peak swing option that allows to exercise the off-peak price twice on weekends since there are no peak hours. We introduce the number of maximum exercise rights Ut. Now we interpret each stage as days instead of hours and define t = 1 as Monday. Then, we set Ut = 2 for t = 6, 7, 13, 14, ..., 992, 993, 1000 and everywhere else Ut= 1. Again, we pre-calculate the relevant parameters αr,n,tto approximate the exercise policy by running our dynamic program in equation 2.41 on our first set of 1000 price scenarios. With our second set of 1000 trajectories we compute the lower bound value within a forward iteration (see second column of Table 2.4). By design, for a single right the values are identical to the unit constraint case. The more swing rights are available the more multiple exercises will occur and the higher the difference between unit and volume constraint swing options (compare second column of Tables 2.3 and 2.4). For two swings we start with a value difference of 9.15 - 9.029 = 0.121 ˆ= 1.3% and end with a spread of 258.706 - 245.154
= 13.55 ˆ= 5.53 % for 100 rights. However, this effect might be even stronger in reality where weekend off-peak prices are higher than weekday off-peak prices. Then multiple exercises on weekends would occur even more often than in our example where the average price does not vary throughout the entire delivery period. In order to compute the marginal duality gap in
n C0(n) ∆D0(n) [C(99%)0 (n), C(99%)0 (n)]
1 4.777 0.004 [4.697 , 4.783]
2 9.150 0.017 [9.028 , 9.288]
3 13.290 0.029 [13.144 , 13.414]
4 17.228 0.024 [17.057 , 17.372]
5 21.035 0.009 [20.844 , 21.242]
10 38.496 0.015 [38.239 , 39.141]
15 54.541 0.019 [54.227 , 55.184]
20 69.611 0.027 [69.251 , 71.245]
30 97.633 0.021 [97.202 , 98.882]
40 123.806 0.011 [123.316 , 125.965]
50 148.550 0.030 [148.009 , 151.008]
60 172.184 0.020 [171.602 , 174.815]
70 194.879 0.023 [194.258 , 197.948]
80 216.820 0.012 [216.161 , 220.088]
90 238.070 0.026 [237.376 , 241.198]
100 258.706 0.025 [257.981 , 262.018]
Table 2.4.: Swing option with volume constraint
the third column we apply the algorithm of the previous chapter. Like for the unit constraint swing option the duality gap does not significantly alter with more swing rights. If we compare the figures with the unit constraint case we can observe slightly smaller gaps. The average of the duality gap across all swing rights is still around 0.025, but the range is a bit smaller moving between 0.017 to 0.03. Consequently, the confidence intervals are smaller too. For 100 swing rights, for instance, the difference between the upper and lower bound is 262.218 257.981 = 4.237 which is slightly smaller compared to the unit constraint case of 249.015 -244.562 = 4.453. We can only explain this improvement with a better approximation of the continuation values as less different exercise points and thus less LS regressions are necessary to return the total option value. But the small difference could very well be only noise in the data. We therefore take another look at the variation of the marginal duality gaps. Figure 2.5 compares the 99 % confidence level of the marginal duality gap between the unit and volume constraint case. For consistency, we used again 200 instead of 20 external scenarios. In both
0 10 20 30 40 50 60 70 80 90 100 1.5