2.4.3. Five Dimensions

Due to the computationally intensive nature of this method it only becomes truly useful in cases where PDE or tree based methods fail. In this subsection we present high-dimensional versions of the examples presented in Section 2.4.1. The asset prices are assumed to follow a risk neutralized correlated geometric Brownian motion described by the stochastic differential equations,

$$dS\_i^k = S\_i^k \left[ \left( r - \delta^k \right) dt + \sigma^k dZ\_i^k \right], \quad k = 1, \ldots, d,\tag{21}$$

where *Z<sup>k</sup> i* is a standard Brownian motion process and the instantaneous correlation between *Z<sup>k</sup>* and *Zs* is *<sup>ρ</sup>ks*. Here the parameter values are specified as *r* = 0.05, *δk* = *δ* = 0.1, *σk* = *σ* = 0.2 for all *k* and that *ρks* = 0 for all *k* = *s*. In addition all assets have the same initial value, *S*0, and we take the number of assets to be *d* = 5.

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

$$\mu \times \max \left[ \left( \max\_{k=1,\ldots,d} S^k - K\_{\nu}, K\_d - \max\_{k=1,\ldots,d} S^k \right), 0 \right],\tag{22}$$

where *Sk* is the price of the *k*th underlying asset at the exercise time. This payoff is an extension of the example given in Broadie and Glasserman (1997) and Broadie and Glasserman (2004) for single-exercise American-style options. For the examples considered here, we set *Ku* = *Kd* = *K* which simplifies the payoff function to

$$\forall u \times \max\left(\max\_{k=1,\ldots,d} S^k - K, K - \max\_{k=1,\ldots,d} S^k\right). \tag{23}$$

As 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. Note that we present results from our FOST methodology without comparisons to other methods as there is no generally accepted benchmark for the examples considered here.

**Example 4.** *(Illustration of Bias and Convergence) This 5-dimensional example corresponds with Example 1. The swing option 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. 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* = 32000 10*b . This results in standard errors* ≈ *0.09% of option value. Figure 5 plots the FOST estimates versus branching, showing that the high estimator overestimates the option price while the low estimator underestimates the price. Furthermore, as the branching factor increases, the high estimator decreases and the low estimator increases and they appear to be converging to the same value, illustrating estimator convergence. These findings are consistent with those in Example 1.*

**Figure 5.** Option value estimates (USD) vs. log branching factor (*b*) with a five-dimensional underlying. The option has one up and one down swing right, three exercise opportunities, exercise volume of 60 units and there is no usage penalty. The number of repeated valuations *R* = 32000 10*b* results in standard errors ≈ 0.09% of option value.

**Example 5.** *(Effect of Usage Penalty and Initial Asset Price) This 5-dimensional example corresponds with Example 2, with the option specifications the same as presented there, modulo the adjustment to the payoff function to five dimensions. The pricing results are in Table 4. The effects of a usage penalty and initial stock*

*price are qualitatively the same compared with the 1-dimensional results. We note that the option value estimates in this example are higher than those in Example 2 due to the payoffs depending on the maximum of the five asset prices. The computing times for the 5-dimensional case are similar to those for the 1-dimensional asset.*

**Example 6.** *(Effect of Number of Exercise Rights) This 5-dimensional example corresponds with Example 3, with the option specifications the same as presented there, modulo the adjustment to the payoff function to five dimensions. The results are given in Table 5 and Figure 6. The results, intuition, and interpretation are qualitatively the same as the 1-dimensional results.*

**Table 4.** Swing option values as a function of moneyness and penalties with a five-dimensional 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.


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


Option Estimate vs. Exercise rights - Five Assets - Low Estimator

**Figure 6.** Basket of American calls and puts and swing option values versus the number of exercise rights using a 5-dimensional underlying asset. Parameter values used are exercise volume of 60 units, *S*0 = 40, *b* = 20, *R* = 4000, *m* = 5, and no usage penalty.

### *2.5. Algorithmic Enhancement via Parallel Processing*

One method for enhancing the computational efficiency of this algorithm is by taking advantage of multi-processor computing techniques. The simplest and most obvious implementation would be to parallelize across repeated valuations of the forest resulting in serial farming of the repeated valuations. Since each repeated valuation results in an iid random value for the option estimate, the generation of all the results may be completed independently of one another, removing the need for communication between processors. This method is simple and effective. However we state here without numerical evidence that it results in a near perfect speed up without the need for expensive interconnections. With this method the minimum run time that can be produced is determined by the number of processors available, the number of repeated valuations necessary for the desired accuracy and the run time of a single forest.

A variation on the aforementioned parallel implementation is to parallelize the FOST computations internally within the forest. In the results shown in Figure 7 the FOST algorithm has been modified so that the computation of the individual trees within the forest is done using multiple processors. Here we have begun the parallelization after the first time step by dividing up the computation of the remaining subtrees across different processors. Upon completion, the results are gathered and the option value at the initial time step is determined. In Figure 7 we see that this method results in a near perfect speed up due to the small ratio of communication time versus computational time. This implementation may be combined with serial farming resulting in further computational time efficiency. This is discussed more fully in Marshall et al. (2011).

In Figure 7 the swing option is identical one in Example 4 and pricing is done with a branching factor *b* = 160. The computational times were generated using the SHARCNET cluster Hound which comprises 2.2 GHz Opteron processors with 4 GB per core and Infini-Band interconnections. Run times are normalized to the run time of a single processor.

**Figure 7.** Normalized runtime using MPI versus number of CPUs (*np*). The option is identical to the no volume choice swing option with a five-dimensional underlying considered in Section 2.4.3. The branching factor used is *b* = 160.
