2.4.1. Single Dimension

Beginning with the one dimensional case, we have based our simulations on an underlying asset with a risk neutralized price process that satisfies the following stochastic differential equation,

$$dS\_i = S\_i \left[ \left( r - \delta \right) dt + \sigma dZ\_i \right]. \tag{17}$$

In this equation, *r* is the riskless interest rate, *Zi* is a standard Brownian motion process, *σ* is a constant volatility parameter and the underlying asset itself pays a continuous dividend yield *δ*. The parameter values for the underlying asset are specified as *r* = 0.05, *δ* = 0.1, and *σ* = 0.2.

The swing options considered have both up and down swing rights and the payoff upon exercise is

$$\max\left[\max\left(S - K\_{\nu}, 0\right), \max\left(K\_{d} - S, 0\right), 0\right],\tag{18}$$

where *u* is the volume exercised, *S* is the price of the underlying asset at the exercise time, and *Ku* and *Kd* are the up and down swing strike prices, respectively. For the examples considered here, we set *Ku* = *Kd* = *K* which simplifies the payoff function to

$$
\mu \times \max \left( S - K, K - S \right). \tag{19}
$$

For all examples in Section 2.4.1 the option expiry is 3.0 years and the options have both up and down swing rights with strike prices *Ku* = *Kd* = 40.0, respectively. In examples where the holder controls the amount exercised, a list of volume choices is given. The volume choices are consecutive integer multiples of a base amount and the up swing and down swing volumes have the same

magnitude. For comparison purposes the results in this subsection include a binomial value which is calculated using the Forest of Trees Lari et al. (2001).

All simulations in this subsection were completed on the SHARCNet cluster Whale. Whale is located at the University of Waterloo and consists of Opteron 2.2 GHz processors (four per node) with a Gigabit Ethernet interconnect. Timing results listed below are given in total cpu time accumulated which is approximately equal to (program runtime) × (number of processors used).

**Example 1.** *(Illustration of Bias and Convergence) The swing option in this example has one up and one down swing right, three exercise opportunities, exercise volume of 60 units of the underlying and there is no penalty. The initial price is USD 40. Here we illustrate the effect of the branching factor b on the value estimates. Specifically, we perform R repeated valuations of the FOST with a branching factor of b and hold the total sample size fixed using the relation R* = 32,000 10*b . Figure 2 plots the FOST estimates versus branching factor (the estimates are the averages of the R repeated valuations). Taking the Binomial estimate as the true option value, it is clear that the high estimator overestimates the true price while the low estimator underestimates the true price. Furthermore, as the branching factor increases, the high estimator decreases and the low estimator increases to the true option price, clearly illustrating estimator convergence. Estimator standard errors are approximately 0.07% of estimator value.*

**Figure 2.** Option value estimates (USD) vs. log branching factor (*b*) with a single underlying asset. The option has one up and one down swing right, 3 exercise opportunities, exercise volume of 60 units of the underlying and there is no usage penalty. The initial price is USD 40. The number of repeated valuations *R* = 32000 10*b* results in standard errors ≈ 0.07% of option value.

**Example 2.** *(Effect of Usage Penalty and Initial Asset Price) In this example, the option has two up and two down swing rights and, upon exercise there are three volume choices—20, 40 and 60 units of the asset. Should the final net volume exercised exceed 90 units or be below* −*90 units a penalty is imposed. The penalty is calculated by multiplying the terminal asset price by ten times the excess usage above 90 units or below* −*90 units. To see the effect of the penalty on option value, we also turn the penalty term off and value the corresponding option with no penalty. The initial asset value ranges from USD 20 to USD 60 in steps of USD 10. There are m* = 5 *exercise opportunities and we use a branching factor of b* = 20 *with R* = 4000 *repeated valuations.*

*The pricing results are presented in Table 2. In each row of the table, we see that the high and low estimators bound the true price. Unsurprisingly, imposing a penalty on the cumulative volume reduces the option value. Furthermore, as the initial stock price increases, the increase in an up swing right's value is more than the*

*decrease in a down swing right's value. The opposite is true as the initial stock price decreases. The end result is that as the initial stock price moves away from being at-the-money, the option value increases.*

*The average computing time per row (not including the binomial forest valuation) was 5.6 h for the cases with usage penalty and was 1.1 h without usage penalty. The reduction in runtime for the case with no penalty can be described as follows. If there are no constraints (e.g., penalty, storage) on the option holder then upon exercise it is always optimal to choose the maximum amount. Therefore with no penalty this option is equivalent to that of an otherwise identical swing option with no volume choices and an exercise volume of 60 units. The latter has fewer trees in its forest and is therefore quicker to evaluate. In Table 2 we have chosen to exploit this as a convenient way to save computational time. For the binomial method run times were on the order of a few seconds.*

**Table 2.** Swing option values as a function of initial asset price and usage penalty with a single underlying asset. Parameter values used are N*u* = N*d* = 2, U*i* = {20, 40, <sup>60</sup>}, *b* = 20, *R* = 4000, *m* = 5, *Umin* = −90, and *Umax* = 90.


**Example 3.** *(Effect of Number of Exercise Rights) In this example we illustrate that the option value increases with the number of exercise rights and compare the swing option value with that of a corresponding basket of American options. The option has m* = 5 *exercise opportunities, an exercise volume of 60 units, and there is no usage penalty. Additionally, we set the initial price to S*0 = 40 *and use a branching factor of b* = 20 *with R* = 4000 *repeated valuations. We consider options having an equal number of up and down swing rights. Table 3 gives the option price estimates for* N*u* = N*d* = 1, 3, 5 *along with prices computed using the Forest of Trees. First notice that the high and low estimates bound the true option from the binomial model. Next note that with* N*u* = N*d* = 5 *exercise rights, the high and low estimates are exactly the same. In this case the number of exercise opportunities is equal to the numbers of up and down swing rights and since the up and down swing strikes are equal, exactly one of these rights will be exercised at each opportunity (see Equation (19)). This makes both the exercise payoff and the continuation value estimates exactly the same at all times and along all branches for both the low and high estimators, yielding identical prices.*

*Second, the option value increases with the number of swing rights. However, the price increases by a factor that is less than the increase in the number of swing rights. For example, when the number rights increases by a factor of 3 (e.g., going from* N*u* = N*d* = 1 *to* N*u* = N*d* = 3*) the option value increases by a factor of 2.5 and when the number of rights increases by a factor of* 53 *(e.g., going from* N*u* = N*d* = 3 *to* N*u* = N*d* = 5*), the option value increases by a factor of 1.2. This result matches with the intuition that a swing option with a given number of rights is less valuable than a basket of American put and call options with otherwise identical parameters and Kd* ≤ *Ku.*

**Table 3.** Swing option values as a function of the number of exercise rights with a single underlying asset. Parameter values used are exercise volume of 60 units, *S*0 = 40, *b* = 20, *R* = 4000, *m* = 5, and no usage penalty.


*For the case of one up and one down swing right the basket of American options would contain a single call corresponding to the up swing right and a single put corresponding to the down swing right, with equal strike prices for the call and put. Changing the number of up and down swing rights invokes a change in the corresponding number of call and put options in the basket. Figure 3 shows a comparison between the values of a basket of American options and a swing option with a comparable number of exercise rights. The value of the basket is linear in the number of exercise rights and the swing option value is below the value of the corresponding basket when the number of rights is greater than one. This follows from the restriction that only one swing right may be exercised at any exercise opportunity whereas all American options of a particular type could be exercised at a given time. In the case of one up and one down right the two are equal since it would never be optimal to exercise both the put and call style rights at the same time. As the number of rights increases, the difference in values increases due to the swing option restriction allowing only a single right to be exercises at each opportunity. The low-biased estimator is used for both the basket and swing option values in Figure 3.*

**Figure 3.** Basket of American calls and puts and swing option values versus the number of exercise rights using a single underlying asset. Parameter values used are exercise volume of 60 units, *S*0 = 40, *b* = 20, *R* = 4000, *m* = 5, and no usage penalty.
