Storage Model Risk
4.2 Storage Parameter Risk: Numerical Examples
N g θhN − g (θh) ∼ N (0, Σ)
Applying the delta method we then have
√
For a storage payoff given by X , applying the delta method again gives the sampling error induced storage value density
This last result gives the storage value distribution induced by the uncertainty, represented by Σ, of the forward curve covariance matrix . The relationship between the two can be understood as first, weighting the matrix Σ by the sensitivity of the model parameters to the forward curve covariance matrix represented by ∇g−1 and then secondly, weighting the result by the sensitivity of the storage value to the model parameters, represented by ∇Eθh. The steps required to derive the parameter risk variance are summarized below, further details can be found in Section C.1 of the Appendix
1. Derive the sample covariance matrix, C, of the relative maturity forward curve returns.
2. Using the sample Fischer Information Matrix associated with C, derive the sample parameter co-variance matrix, Σ.
3. Derive the gradient of the model parameter estimation function, ∇g−1. The estimation function takes the matrix C as input and returns a vector of model parameters.
4. Derive the sample model parameter covariance matrix given by, ∇g−1|
Σ∇g−1.
5. Finally, using the storage value parameter sensitivities ∇Eθh, derive the parameter risk variance as per Equation 4.1.5.
4.2 Storage Parameter Risk: Numerical Examples
We will now present numerical results utilizing three of the forward curve models detailed in Chapters 2 and 3. These provide canonical examples of the different approaches to calibration and parameter estima-tion risk measurement outlined above. We will begin with a non-technical review of the pricing models to be tested and relate them to the information we wish to incorporate into a storage pricing model, namely
the volatility surface and the scaled covariance matrix of forward curve returns 6. As discussed previ-ously, the choice of models at our disposal is limited due to the fact the there is very little in the literature which approaches the storage valuation problem with a view to maintaining consistency with the volatil-ity smile. Thus although price models developed for storage valuation have the abilvolatil-ity to capture the forward curve time-spread dynamics well, for example Parsons (2013) and Boogert and De Jong (2011), they are generally driven by diffusion processes and thus fail to replicate the volatility smile observable in gas option markets. Conversely, traditional price models which are capable of replicating the smile, Heston7or SABR8 for example, will yield perfectly correlated forward curve returns and therefore little or no extrinsic value to the storage.
We will proceed by first constructing push forward densities for the storage values associated with each model specification and show how a trading desk can use this information to construct calibration and parameter estimation risk adjusted bid-offer levels capable of providing a buffer against p&l risk owing to model calibration/estimation. Finally, we will discuss how the AVaR model uncertainty measure can be used by a risk or model validation team to rank the individual model specifications by focusing on a practical example of a proposed model being tested against an incumbent.
The first of our test models is the Mean Reverting Variance Gamma (MRVG) model detailed in Section 2.1. Recall, the log-spot price model is specified by the following dynamics.
dx(t) =
∂ f (0, t)
∂ t − κj(exp (−αt)) + α f (0,t) − α ˆ t
0
κj(exp (−α(t − s))) ds − αx(t)
dt+ dX (t) (4.2.1) where f (0,t) is the initial log forward price for time t and dX (t) is a driftless Variance-Gamma pro-cess with κjits cumulant.
As demonstrated in Chapter 2, this model can be calibrated directly to market option prices and thus the measure distribution R can be obtained using the approach of Bannör and Scherer (2013b). The model is specified such that the instantaneous variance of different maturities along the forward curve shows exponential decay as time to maturity increases. This property, known commonly as the Samuelson Ef-fect Serletis (1992), is a stylized feature of many commodity markets and particularly natural gas. In the case of constant σ , the exponential decay rate, α, fully controls the structure of the covariance amongst different points on the curve. This results in non-parallel shifts in the forward curve due to changes in the underlying stochastic driver. It is this last element which is crucial for storage valuation as it results in the operator exercising their optionality to switch planned injection and withdrawals thus creating extrinsic
6The scaled covariance matrix is defined as the covariance of returns matrix relative to the volatility of the shortest maturing contract.
7Heston (1993)
8Hagan et al. (2002)
value. In the spot price representation of the model, this parameter is more commonly referred to as the mean reversion rate as it controls the speed at which the spot price reverts to the expected forward level.
The ν parameter controls the implied volatility smile attenuation and ensures that the model is consistent with the initial volatility surface
The second model is the MRJD detailed in Section 2.1. The model, which was first specified by Deng (2000) and later used in swing contract valuations by Kjaer (2008), is equivalent to Equation 4.2.1. The only difference is in the choice of stochastic driver, dX (t), which in this case is a jump-diffusion pro-cess with Compound Poisson jump propro-cess driven by a double Exponential distribution. Thus, as in the MRVG model, the presence of the parameter α is the primary driver of the storage value. The param-eters relating to the jump-diffusion are σ , the diffusion volatility, λ , the jump arrival rate, and µ which controls the size of both positive and negative jumps. The jump diffusion specification is what allows us to replicate the volatility smile using this model, in a similar manner to the MRVG model.
The final model, the MRVG-3x presented in Section 3.1, is a multifactor extension of the first. The inclusion of the second factor allows more flexibility in modelling the covariance structure of the for-ward curve and therefore should produce storage values more representative of the underlying dynamics.
Recall that the dynamics of the log spot price under the MRVG-3x model are given by
dx(t) =∂ f (0, t)
∂ t + dy(1)(t) + dy(2)(t)
where
dy(1)(t, T ) =
−κj(1)(b exp (−α (T − t)) σ )
dt+ b exp (−α (T − t)) σ dX (t) dy(2)(t, T ) = −1
2(exp (−ε (T − t)) c1σ )2dt+ (exp (−ε (T − t)) c1) σ dW (t) (4.2.2) and dW (t) is a standard Brownian Motion.
The model is composed of two factors. The first, which accounts for the majority of the forward curve variability is a Mean Reverting Variance Gamma process. As with the MRVG model, the main parameters for this factor can be calibrated to the options market. The parameter b represents the proportion of total variance attributed to the first factor and would need to be estimated from the historical data. The second factor is specified such that it approximates the typical shape of the sensitivity, which we refer to as the volatility function, of the forward curve to the second principal component of the forward curve returns covariance matrix. An example of this is given below by Figure 4.2.1
Figure 4.2.1: MRVG-3x 2nd Factor Volatility Function
The fit between the model and historically estimated second volatility function for the MRVG-3x model.
As outlined in Chapter 3, the parameters relating to the second factor can be estimated directly from the eigenvector values. The parameter c1 determines the value of the function as it approaches zero time to maturity and ε controls the decay of the function as maturity increases. The shape of the volatility function, which is determined by these two parameters, will have a direct impact on the covariance of different maturities along the forward curve. A sharply decaying curve will decrease the covariance be-tween prompt forward prices and the back of the curve, which will lead to greater time-spread variance and thus higher storage value. This model, as it encompasses the first, will generally attribute more value to a storage asset through this additional decorrelation of the forward curve returns. One obvious con-cern, which we hope to address, is the extent to which this additional value should be discounted on the basis of the parameter estimation risk inherent in the model estimation.
We will again focus on the storage deal valued in previous chapters, a simple 20in/20out deal. This mean the asset takes 20 days to inject to full capacity from empty (In) and 20 days to empty once full (Out).
The deal commences immediately on the options quote date and lasts 1 year. As before, the option data was sourced from Bloomberg with a quote date of 19th December 2012. The data set used to estimate the historical covariance matrix covers the 3 years ending 19th December 2012, where each day in the sample contains the day-ahead and month ahead quotes spanning 11 months.
The procedure for estimating calibration risk outlined by Bannör and Scherer (2013b) begins by dis-cretizing the parameter space and then evaluating the pricing error to benchmark instruments for each parameter combination in the space. Thus they reduce the multidimensional parameter constellation to
a vector of error terms. Given that a large majority of these errors would fall outside of what would be deemed a reasonable calibration, the authors propose discarding parameter combinations which return an error greater than some cut-off point. In their numerical examples the authors use a cut-off for the error term, given by the root mean square error, of 2% above the minimum error term. Obviously the choice of cut-off will depend upon the average bid-offer percentage spread observed in the options market. For this example, we have chosen a cut-off of 3% in order to reflect the relatively lower liquidity of natural gas options.
There are two main sources of risk associated with a given parameter combination which yields a RMSE within this cut-off level.
1. Risk due to the option pricing model, emanating from the assumed dynamics of the model, and any numerical error inherent in the pricing algorithm. Further, as outlined in Cont and Tankov (2003) the calibration error landscape in exponential Lévy models may have flat regions in which the error is insensitive to variations in the parameters, thus the behaviour of the objective function minimization algorithm within these areas creates an additional source of risk. Combined these risks are characterized by an acceptance that the true minimum lies within a narrow region con-taining the calibrated values. Whilst the variability in parameters may be low within this region the potential effect on the pricing of complex derivatives means the risk can still be quite substantial.
In collectively identifying these risks we will follow the convention of Deryabin (2012) who refers to the risk due to small variations in the parameters as “local” calibration risk.
2. Risk due to the presence of multiple local minima. Generally speaking, the calibration problem is ill-posed and the solution may have a strong dependency on the starting point of the search algorithm. This problem is made worse as one widens the acceptable cut-off error, which may lead to valid solutions at many distant points in the parameter space. Cont and Tankov (2003) demonstrate the dependency of the Merton jump-diffusion model on the choice of initial parameter values when calibrating to equity options and conclude that the number of local minima depends less on the number of model parameters relative to option prices and more on the convexity of the objective function. Obviously, this issue is more severe than the first as it could potentially destroy all confidence in the calibrated parameters and lead to a very high level of model risk. We will refer to this risk as “multiple minima” calibration risk. Although some authors suggest overcoming this issue through regularization of the objective function, Avellaneda et al. (1997); Crépey (2003), we tend to agree with Cont and Ben Hamida (2005) that the presence of multiple minima contains information on the level of model risk which may be lost through regularization.
For our purposes, one could in theory, search over a dense and wide ranging discretization of the parame-ter space in order to identify regions containing local minima. However, there is a computational demand when proceeding in this manner which necessitates the need for approximation. This is particularly true
when calibrating to non-standard option contracts, as is the case here for monthly swaptions contracts.
We therefore suggest, first searching for local minima by selecting the initial parameter values in ones search function from a wide and relatively sparse discretization of the parameter space. Once each of these local minima have been identified one can construct a region encompassing each in order to iden-tify the local calibration risk.
For each of the models specified above the calibration to our options data using the simplex method of Lagarias et al. (1998) did produce multiple local minima dependent upon the choice of initial parameters.
However, in each case the local minimum either yielded a RMSE far in excess of our acceptable cut-off or returned parameter values similar to our optimal values and thus could be considered “local” calibration risk as defined above. Therefore we will focus only on a reduced parameters space encompassing our initial parameter estimates.
Example 1: Market Based Calibration Risk
In this first example we will focus on the MRVG and MRJD models as they require estimation from market data alone. Following the example of Bannör and Scherer (2013b), we choose hm() to be the normal transformation given by Equation 4.1.3. For the scaling parameter, λ , we have used the sample variance of the calibrated errors centered on the minimum value and chosen c such that the density values sum to unity. For the MRVG model, we see from Equation 4.2.1 the parameter set of interest is given by θm= {α, σ , ν}. The initial calibrated parameters are given in Table 4.2.1. The associated root mean square error (RMSE) is 1.07%.
α σ ν RMSE
MRVG 0.2162 0.201 0.2560 1.07%
Table 4.2.1: MRVG Model Parameters.
The calibrated model parameters for the MRVG model. ’RMSE’ refers to the root mean square error between the model and market option prices.
To investigate the local risk of our initial calibration, we select α values ranging from 70% to 130% in steps of 5% of the calibrated α. For the σ and ν values we use a much tighter range of 90% to 110% in steps of 1.5%, which reflects the greater sensitivity of the RMSE to changes in these parameters. In total, this yielded 2548 distinct parameter combinations. We next iterate over this parameter space and discard any parameter combinations which exceed the minimum RMSE by the 3% limit. The resulting parameter space consisted of 807 parameter combinations. Using the error terms associated with these parameters we derive an error density as per Equation 4.1.3 and then evaluate the storage value at each point in
this reduced parameter space. The resulting calibration risk induced storage value density is displayed graphically in Figure 4.2.29 and the valuation results are given in Table 4.2.2. As is evident from the graph, the density appears almost uniform over the acceptable calibration range. The apparent presence of columns of points in Figure 4.2.2 correspond to storage values at each α point in the parameter space.
This behaviour can be rationalized by observing that changes in α have little effect on the calibration error relative to changes in σ and ν due to the sensitivity of the implied volatility smile to changes in model volatility and kurtosis. This means that the a much wider range of α values fall within the acceptable parameter space. Further, due to the sensitivity of storage values to the model covariance structure, the choice of α will impact the value much more than both σ and ν.
Figure 4.2.2: MRVG Storage Value Push Forward Density
The empirical calibration risk induced storage value probability density for the MRVG model. The red line corresponds to the storage value with the smallest calibration error.
Value
Modus 11.2087
Expected Value 11.2116 Coefficient of Variation 0.38%
Skewness -0.034
Minimum RMSE 1.05%
Table 4.2.2: MRVG Storage Value Push Forward Distribution
Summary statistics of the calibration risk induced storage value distribution under the MRVG model.
9Values were obtained using the FFT-m based algorithm detailed in Chapter 2 with a grid size of 211.
For the MRJD model, the parameter set is θm= {α, σ , λ , µ}. The initial calibrated parameters are given below in Table 4.2.3, the RMSE is slightly higher then the MRVG model at 1.10%.
α σ λ µ RMSE
MRJD 0.2099 0.0334 8.7966 0.047 1.10%
Table 4.2.3: MRJD Model Parameters
The calibrated model parameters for the MRJD model. ’RMSE’ refers to the root mean square error between the model and market option prices.
We have again selected the α range to be 70% to 130% in steps of 5% of the calibrated α. The values for σ , λ and µ were selected to range from 90% to 110% in steps of 4% of the initial calibrated values, again this tighter range is reflective of the greater sensitivity of the RMSE to changes in these parameters.
In total this gave 2808 distinct parameter combinations, of which 795 were within the cut-off threshold.
The resulting push forward storage density is displayed in Figure 4.2.3 and summary statistics are given in Table 4.2.4.
Figure 4.2.3: MRJD Storage Value Push Forward Density
The empirical calibration risk induced storage value probability density for the MRJD model. The red line corresponds to the storage value with the smallest calibration error.
Value
Modus 11.2127
Expected Value 11.2039 Coefficient of Variation 0.36%
Skewness 0.021
Minimum RMSE 1.08%
Table 4.2.4: MRJD Storage Value Push Forward Distribution
Summary statistics of the calibration risk induced storage value distribution under the MRJD model.
As we can see visually, and also from the summary statistics, there is not much difference between the push forward densities under each model. The MRVG model returns a higher expected value than the MRJD model but also has a slightly higher level of variability. It would appear on the basis of this analysis that both models carry a roughly equivalent level of calibration risk. Later, we will use these results to construct bid-offer levels which may shed some additional light on the relative riskiness of these models.
Example 2: Joint Calibration and Parameter Estimation Risk
As discussed above the MRVG-3x model requires calibration to market data and also parameter es-timation from historical data. We see from Equation 4.2.2 that the full parameter set in this case is θ = {α , σ , ν , b, c1, ε}. The market calibrated parameter set is θm= {α, σ , ν}. Recall that these parame-ters determine the shape of the volatility term structure of the first factor of the model. The historically estimated parameters, θh= {b, c1, ε}, determine the shape of the volatility term structure of the second factor of the model. The initial market calibrated and historically estimated model parameters are given in Table 4.2.5.
α σ ν b c1 ε RMSE
MRVG-3x 0.1148 0.2518 0.1675 0.7511 0.6254 12.734 1.86%
Table 4.2.5: MRVG-3x Model Parameters
The calibrated and historically estimated model parameters for the MRVG-3x model. ’RMSE’ refers to the root mean square error between the model and market option prices.
Recall from Equation 4.1.4 that the calibration risk induced density for this model shall be decomposed into the conditional market calibration error density hm(ε (θ ) |θh) and the parameter estimation risk den-sity corresponding to the historically estimated parameters p(θh). We shall proceed by first estimating the former. We have chosen the same parameter space discretization as in the MRVG case, that is α
values ranging from 70% to 130% in steps of 5% of the calibrated α and σ ,ν values ranging from 90%
to 110% of their initial calibrated values in steps of 1.5%. This returned 2548 parameter combinations, of which 838 were within the threshold of 3% of the minimum calibration error of 1.35%10. The push
to 110% of their initial calibrated values in steps of 1.5%. This returned 2548 parameter combinations, of which 838 were within the threshold of 3% of the minimum calibration error of 1.35%10. The push