Initial Gas Price: £0.667 Intrinsic Value Full Value Extrinsic Value %4 Extrinsic Value
MRVG 10.983 11.2105 0.2275
-MRVG-2x 10.983 11.9642 0.9812 431.3%
MRVG-3x 10.983 15.1477 4.1647 1830.64%
All values are expressed in pence/therm Table 3.4.1: Two Factor Valuation Results
The valuation results for both two factor models along with the results of the single factor MRVG model.
Table 3.4.1 presents the results for the two multifactor models along with the results for the single factor MRVG model. The valuations for the MRVG-2x and MRVG-3x models were carried out using the FFT-m approach outlined in Chapter 2, with grid sizes of 512 × 512 and 256 × 2048 respectively. For these valuations, the factor domains (in log-space) were approximated as [−3.4539, 3.4471]. The MRVG-2x value is 11.9642 pence/therm with 0.9812 pence/therm extrinsic value, this represents a 431% increase in extrinsic value over the single factor MRVG valuation. The MRVG-3x model value is much higher at 15.1477 pence/therm with 4.1677 pence/therm extrinsic value, a 1831% increase over the MRVG model.
This increase highlights the sensitivity of the storage asset to the model implied seasonal decorrelation, which is highest in the case of the MRVG-3x model.
Figure 3.4.2 displays the initial value for the MRVG-2x model for different levels of the individual factors.
These graphs are helpful in understanding the differences between the models and identifying the drivers of the extrinsic value in each case. We can see that for the MRVG-2x model, the value reaches a maximum when both factors are at their positive extremes, which is also where the initial gas price is at a maximum for this sample of the value grid. Figure 3.4.3 displays the implied forward curve given by the factor values at these extreme points. At y1, y2 = [0.4, 0.4] the whole curve and therefore the initial intrinsic value would increase due to the proportional shifts to the forward curve caused by the second factor and to a lesser extent the first factor. At y1, y2 = [−0.4, 0.4], the curve is anchored at the short end by the decrease in the first factor which causes an increase in the summer/winter spread and therefore an increase in intrinsic value.
Figure 3.4.2: MRVG-2x Initial Value Grid
The two dimensional storage value grid for the MRVG-2x model. The value is expressed in pence/therm and displayed as function of the underlying factor levels.
Figure 3.4.3: MRVG-2x Implied Forward Curves
The factor implied forward curves for the MRVG-2x model at each of the extreme points of the storage value grid presented in Figure 3.4.2.
For the MRGV-3x model both factors contribute significantly to the value of the storage asset through their effect on the forward curve as the individual factor levels change. The value is maximized in Figure
3.4.4 when the first factor is at its maximum and the second factor is at its minimum at y1, y2 = [0.4, −0.4].
The model implied forward curve at this level is displayed in Figure 3.4.5. We see that the changes in both factors cancel each other out initially, so that the initial gas price remains unchanged. The positive change in the first factor would persist along the forward curve whereas the negative change in the second factor would decay quite quickly given their respective mean reversion rates. The net effect is a large increase in the forward curve relative to the prompt gas price and therefore a large change in the intrinsic value.
A similar effect is seen when both factors are at their minimum values at y1, y2 = [−0.4, −0.4]. Here the large mean reversion rate of the second factor leads to a much larger decrease in the forward curve at the front end, leading to more profitable injection opportunities.
Figure 3.4.4: MRVG-3x Initial Value Grid
The two dimensional storage value grid for the MRVG-3x model. The value is expressed in pence/therm and displayed as function of the underlying factor levels.
Figure 3.4.5: MRVG-3x Implied Forward Curves
The factor implied forward curves for the MRVG-3x model at each of the extreme points of the storage value grid presented in Figure 3.4.4.
It is clear from the above results that incorporating an accurate representation of the forward curve dynamics in conjunction with a market based calibration of the general level of curve variability leads to substantially greater storage values than market calibration alone. This is of course due entirely to market incompleteness with respect to time-spread optionality. As such, a model which is capable of representing the dynamics observable from historical returns gives traders at least some comfort in the level of extrinsic being bid/offered. Not only does the MRVG-3x model meet this requirement, the par-simony of the model allows one to adjust ones price levels significantly by adjusting a single parameter, ε , without significantly impacting the models ability to calibrate to market. Given the growing liquidity observable in the over-the-counter storage and 100% take-or-pay markets however, this requirement of utilizing historical information may not be necessary for much longer. In such a situation, a model which can easily and quickly be calibrated to market prices for storage and take-or-pay contracts would allow a trader to infer the price of time-spread optionality directly from these products. Again, the MRVG-3x, where the entire curve correlation structure is controlled by two parameters, would be considerably more suited to this purpose than a market model of the forward curve with a full correlation matrix.
In conclusion, in this chapter we aimed to present an unique class of multifactor forward curve models that were reflective of the statistical dynamics of the natural gas forward curve and allowed for accurate calibration to the options market. To that end, we first conducted a Principal Component Analysis of the historical relative maturity NBP gas forward curve returns in order to inform our model specification.
We presented a three factor model with a volatility specification given by the shape of the eigenvector values for the first two principal components as a function of relative maturity. Given concerns over the
potential computational burden of using three stochastic factors, we went on to introduce a range of two factor models which approximated the volatility specification of the three factor model. The conditional characteristic functions for this family of models were derived in Sections B.2 and B.3 of the Appendix, along with a range of utility methods which allowed us to further analyze the dynamics of each model.
We introduced an implied moment based calibration technique in order to calibrate these models to the options market and went on to present results of the joint calibration and statistical estimation of the model parameters. In order to value storage contracts we extended the valuation algorithm presented in Chapter 2 to a multidimensional setting and finished with the presentation of valuation results and anal-ysis using two of the proposed multifactor models. In summary, the main contributions to the literature have been
• The extension of the MRVG model to a multifactor setting under the Cheyette model framework.
• The derivation of the conditional characteristic functions for a number of multifactor model speci-fications.
• The presentation of a detailed derivation of the implied spot price drift for each model specification and the conditional characteristic function of the resulting log spot price process.
• The presentation of an innovative implied moments calibration technique applicable to any finite dimensional instantaneous forward curve model. This was an extension of the work of Guillaume and Schoutens (2013) specifically tailored to the calibration of energy forward curve models but equally applicable to interest rate models.
• The derivation of a number of utility methods for each model specification including the forward curve covariance function and the moments of each process, which we used to analyze the model implied factor dynamics.
• The derivation of the spot factor implied forward curve for each model specification and using these results to derive a general Fourier based swaption pricing algorithm.
• The extension of the Fourier based valuation algorithm for processes with state dependent incre-ments to an arbitrary number of dimensions.
Given the innovative model development work carried out in Chapters 2 and 3, we feel that a detailed analysis surrounding the model risk inherent in our proposed modelling framework is important at this juncture, particularly for the models to take traction in both industry and academia. We address this in the next chapter, where a selection of the models set out thus far are analyzed under an innovative calibration and parameter estimation risk framework, with a particular focus on the impact of model risk on the storage valuation results presented in Chapters 2 and 3.