2.4.2. Calibrated Forward Curve

In this example, we use the trinomial-tree model described in Section 4 of Jaillet et al. (2004) from which price movements are simulated. This model is a 1-factor model with mean reversion that is seasonally adjusted and calibrated to the forward curve. The option we value is Example (a) of Section 4.2 in Jaillet et al. (2004). This option a two up right swing option with each right allowing the holder to take delivery of either 1 or 2 MMBTus of natural gas. We simplify to have four exercise opportunities and 4 months until expiry. Upon exercise the holder gets

$$\max\left(\mathcal{U}\_{l}\left(A\_{i}\mathcal{S}\_{l}-K\right),0\right) \tag{20}$$

where *Ui* is the volume chosen, *Ai* is the seasonality factor and *Si* is the deseasonalized spot price.

Figure 4 plots the option price estimates, including 95% confidence intervals, against branching factor. The number of repeated valuations used to generate prices with a branching factor of *b* is *R* = 160,000 *b* . We see that, with a branching factor of only 8, the confidence intervals for the highand low-biased estimators begin to overlap and quickly become almost indistinguishable for higher branching factors, numerical illustration of both estimators' consistency. Additionally, the high and low estimators converge to the true price computed using the trinomial model. We note that the serial computational times (for a single FOST valuation) for branching factors of 8 and 32 were approximately 4.5 and 110 s, respectively using a 2.1 Ghz Core 2 Duo processor. The pricing results shown in Figure 4 are consistent with the results in Jaillet et al. (2004) but we note that the valuation method in that publication breaks down in higher dimensions and in cases where the inclusion of more risk factors is desirable.

**Figure 4.** Option-value estimates versus (log) branching factor. Approximate pointwise 95% confidence intervals for each estimate are given by the vertical bars. The option and underlying model in this example is from Jaillet et al. (2004) with the Trinomial price given by their Forest of Trinomial Trees method.
