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**Krzysztof Wolyniec presents a simple factor model to manage the uncertainty associated with volumetric risk in energy and commodity markets.**

Commodity demand is uncertain, driven by consumption, which is an exogenous variable. As such, commodities, and in particular energy markets, contain significant levels of volumetric risk.

In financial markets, trade volumes do not have physical constraints and can match the demand as long as the risk-carrying capacity of the participants is sufficient. The risk profile of a financial transaction can be described by a single variable: the price. Volume is a free parameter that can be chosen at will.

The uncertainty over future volumes and prices in commodity markets has led to the development of risk management tools that can shift the cashflow risk away from end-users and back to market-makers and investors.

For example, electric utilities and other load serving entities (LSEs) enter contracts with suppliers to secure future unknown demand for power at fixed or banded prices. Similarly, natural gas utilities contract their future suppliers to cover unforeseeable heating demand fluctuations.

For suppliers, a variety of interruptible, non-dispatchable generation sources, such as solar, wind and hydro, are also characterised by uncertain generation volumes. The owners often contract with energy merchants for guaranteed off-take agreements, whereby all volumetric and price risk is shifted to the marketer.

Ultimately, whoever bears the cashflow risk in the commodities world cannot avoid exposure. The question is how should it be measured, managed and priced.

## 1 The cashflow risk management problem

The fundamental analytical unit is the underlying cashflow:

$$\mathrm{CF}=\sum _{i=1}^{N}{P}_{i}{L}_{i}$$ | (1) |

where both price ($P$) and volume ($L$) are stochastic variables.

Equation (1) assumes a market and a price for every demand (volumetric) subperiod. But that is not often the case, which has led to the growth of storage industries and demand-reduction techniques.

It is worth noting the distinction between market and physical demand periods is not always useful, since different markets operate at different horizons and resolutions. In both the power and natural gas markets, agreements can be annual, quarterly, monthly, weekly, daily, hourly or based on peak blocks, depending on demand. The depth of the respective markets also varies significantly at different contract resolutions.

### 1.1 Example: power pool markets.

In deregulated power pool markets, demand/supply settlements take place on an hourly basis. Demand itself is effectively continuous, with physical system balancing performed at 15-minute intervals.

Since basic forward contracting takes place on a monthly basis, we can express the cashflows as follows:

$$\mathrm{CF}=\sum _{m}\sum _{d}\sum _{h}{P}_{m,d,h}^{\mathrm{Off}}{L}_{m,d,h}^{\mathrm{Off}}+{P}_{m,d,h}^{\mathrm{On}}{L}_{m,d,h}^{\mathrm{On}}$$ | (2) |

where the $m$, $d$ and $h$ indices sum monthly, daily and hourly periods within a block, respectively.

The power market is the most instructive example, as it typically shows high granularity. In other markets, the smallest demand time horizon unit could be a day (natural gas) or a week (crude products).

The flexibility (balancing) services may be provided by system operators or independent entities. As such, they may have their own markets. They may also be accounted for via use charges by system operators, or LSEs may acquire their own physical flexibility assets (ie, storage or generation) to smooth out demand fluctuations. For example, explicit balancing markets are typical for power pool markets.

Clearly, the cashflow formula is focused on spot (cash) exposures. In general, we might have to introduce additional pricing terms into our cashflow formula to account for the additional cost of the balancing services or assets. The following analysis would not change significantly if we accounted for those additional terms. In practice, however, the balancing market might not exist at all, or it might be less liquid and transparent than the spot/forward market, making explicitly accounting for the associated risks more difficult.

From here on in, this article will not discuss balancing terms. For further information, we refer the reader to Eydeland & Wolyniec (2016).

## 2 Risk management in commodity markets

In general, risk management involves forming optimal portfolios of underlying exposures and risk management tools, such as forwards or options.

To obtain the optimal portfolio (see Eydeland & Wolyniec 2016), we need to minimise the tradeable (liquid) risk and optimise the residual (non-traded) risk.

We can describe the risk management problem for a specific contractual month by the following risk minimisation problem:

$$\underset{{\beta}_{t},{\alpha}_{t},{\gamma}_{t}}{\mathrm{arg}\mathrm{min}}\left\{\mathrm{Risk}\left({\mathrm{CF}}_{m}-{\int}_{{t}_{0}}^{{T}_{m}}{\alpha}_{t}d{F}_{t}^{m}-{\int}_{{t}_{0}}^{{T}_{m}}{\beta}_{t}d{C}_{t}^{m}-{\int}_{{t}_{0}}^{{T}_{m}}{\gamma}_{t}d{D}_{t}^{m}\right)\right\}$$ | (3) |

where ‘$\mathrm{Risk}$’ is a risk function such as expected variance, $\mathrm{Risk}(\cdot )={E}^{P}[{(\cdot )}^{2}]$, and $F$, $C$ and $D$ are the liquid tradeable assets. In power markets, for example, the major trading instruments are forwards, monthly options and daily options, respectively.

## 3 Solving for the optimal portfolio

The solution to (3) can be described by the risk-neutral pricing functional (see Cont & Tankov 2003):

$${E}_{t}^{\ast}[{\mathrm{CF}}_{m}]$$ | (4) |

where the star refers to the appropriate risk-neutral density. It is important to underscore the formal expectation in (4) is potentially very different to the expectation we use in the definition of the risk function in (3). The latter is the physical (statistical) distribution of the underlying risk, while the former is only an analytical tool.

We can now impose some structure on the optimal solution. Let us assume the risk functional is simply the local variance. This allows us to obtain a very general characterisation of the pricing measure and, consequently, the optimal portfolio.

The resulting pricing measure is the so-called minimal martingale measure. It does not price the residual risk and only accounts for market tradeable risk (see Eydeland & Wolyniec 2016). Formally, the measure does not price risks that are orthogonal (in terms of covariance) to the market tradeable ones. This assumption is operationally equivalent to simply hedging the market risk and accounting for residual risk separately, eg, by adding a charge.

With the minimal martingale structure in place, we can now construct the pricing measure explicitly for our problem.

## 4 Constructing market measure

We will not attempt the full characterisation of the pricing measure for all possible problems. Rather, our construction will be parsimonious and easy to estimate.

In order to construct the market measure, first we need to know what market tradeables will impact the valuation of our cashflow. As an example, let us consider the power (pool) market, as the analysis translates easily to other energy markets with a few simplifications.

The most liquid traded contract is the forward one with monthly resolution. We can also find monthly and daily options trading with fluctuating liquidity (see Eydeland & Wolyniec 2016). Next, we investigate the relationship between the pricing functional and the traded contracts.

It is worth noting the standard modelling strategy is to estimate the underlying data-generating process (measure), adjust for risk and finally obtain the relationship between the pricing functional and the tradeables. However, this procedure is grossly inefficient, as it requires an enormous amount of information that is simply not available from the small samples we often find in commodity markets. It can result in a very noisy estimate of information that is often misleading.

Constructing the pricing measure directly has the advantage of parsimony, as it shows exactly what information is required for the optimal portfolio. It allows us to characterise the minimal information set of the optimisation problem in question. The resulting estimation problem is normally more efficient than any of the extant modelling strategies.

Returning to our power pool example, we need to evaluate the following expectation:

$${E}^{\ast}[\mathrm{CF}]={E}^{\ast}\left[\sum _{m}\sum _{d}\sum _{h}{P}_{m,d,h}^{\mathrm{Off}}{L}_{m,d,h}^{\mathrm{Off}}+{P}_{m,d,h}^{\mathrm{On}}{L}_{m,d,h}^{\mathrm{On}}\right]$$ | (5) |

Given the resolution of the most liquid forward contracts is monthly, we would like to express the cashflow in terms of the forward contract at settlement. To simplify notation, we only analyse the on-peak block for a specific month. The full formula would just be a sum of the subproblems.

We are interested in evaluating the following risk-neutral expectations for every month and block:

$${E}_{t}^{\ast}[{\mathrm{CF}}_{m}^{\mathrm{On}}]={E}_{t}^{\ast}[{E}_{T}^{\ast}[{\mathrm{CF}}^{m}]]={E}_{t}^{\ast}\left[{E}_{T}^{\ast}\left[\sum _{d,h}{P}_{d,h}^{\mathrm{On}}{L}_{d,h}^{\mathrm{On}}\right]\right]$$ | (6) |

where $t$ refers to current evaluation time; $T$ refers to the forward expiry; and $d$ and $h$ are indices summing over the contractual period $T$ through $T+30$ days.

We start by looking at the conditional expectation of the cashflow at forward expiry $T$.

### 4.1 Cash covariance

We can rewrite the inner expectation in (6) as follows:

${E}_{T}^{\ast}\left[\sum _{d,h}{P}_{d,h}^{\mathrm{On}}{L}_{d,h}^{\mathrm{On}}\right]$ | $=\sum _{d,h}{E}_{T}^{\ast}[{P}_{d,h}^{\mathrm{On}}{L}_{d,h}^{\mathrm{On}}]$ | |||

$=\sum _{d,h}{E}_{T}^{\ast}[{P}_{d,h}^{\mathrm{On}}]{E}_{T}^{\ast}[{L}_{d,h}^{\mathrm{On}}]+{cov}_{T}^{\ast}({P}_{d,h}^{\mathrm{On}}{L}_{d,h}^{\mathrm{On}})$ | (7) |

which is the standard way of analysing the problem. It requires an estimate of all the individual covariances. The typical practice is even worse than that, as it involves the decomposition of the covariance into individual correlations and variances, which increases the complexity of the estimation.

Instead, using (5), (6) can be rewritten as:

${E}_{T}^{\ast}[{\mathrm{CF}}_{m}^{\mathrm{On}}]$ | $={E}_{T}^{\ast}\left[\sum _{d,h}{P}_{d,h}^{\mathrm{On}}{L}_{d,h}^{\mathrm{On}}\right]$ | |||

$={E}_{T}^{\ast}\left[\sum _{d,h}{P}_{d,h}^{\mathrm{On}}\sum _{{d}^{\prime},{h}^{\prime}}{L}_{{d}^{\prime},{h}^{\prime}}^{\mathrm{On}}\frac{{\sum}_{d,h}{P}_{d,h}^{\mathrm{On}}{L}_{d,h}^{\mathrm{On}}}{{\sum}_{d,h}{P}_{d,h}^{\mathrm{On}}{\sum}_{{d}^{\prime},{h}^{\prime}}{L}_{{d}^{\prime},{h}^{\prime}}^{\mathrm{On}}}\right]$ | (8) |

Here, a simple transformation is introduced. We separate the total monthly block cashflow into a monthly realised price and load averages multiplied by the price-weighted realised load shape. The reason for this transformation will become clear shortly. Next, we express the cashflow in terms of functions of tradeable contracts. If such a construction is possible, then the construction of the measure – and, hence, the optimal portfolio – would be greatly simplified.

We introduce the following notation:

$$\begin{array}{cc}\frac{{\sum}_{d,h}{P}_{d,h}^{\mathrm{On}}{L}_{d,h}^{\mathrm{On}}}{{\sum}_{d,h}{P}_{d,h}^{\mathrm{On}}{\sum}_{{d}^{\prime},{h}^{\prime}}{L}_{{d}^{\prime},{h}^{\prime}}^{\mathrm{On}}}& \doteq {\overline{S}}^{m}\\ \sum _{d,h}{P}_{d,h}^{\mathrm{On}}& \doteq {\overline{P}}^{m}\\ \sum _{{d}^{\prime},{h}^{\prime}}{L}_{{d}^{\prime},{h}^{\prime}}^{\mathrm{On}}& \doteq {\overline{L}}^{m}\end{array}\}$$ | (9) |

With those in place, we can rewrite the conditional expectations as follows:

$${E}_{T}^{\ast}[{\overline{P}}^{m}{\overline{L}}^{m}{\overline{S}}^{m}]={E}_{T}^{\ast}[{\overline{P}}^{m}{\overline{L}}^{m}]{E}_{T}^{\ast}[{\overline{S}}^{m}]+{cov}_{T}^{\ast}({\overline{P}}^{m}{\overline{L}}^{m},{\overline{S}}^{m})$$ | (10) |

The splitting of the product into the product of expectations and covariance is purely formal. It assumes no underlying model.

At this stage, we focus on the load and price averages, ignoring the shape coefficient for now.

We have:

$${E}_{T}^{\ast}[{\overline{P}}^{m}{\overline{L}}^{m}]={E}_{T}^{\ast}[{\overline{P}}^{m}]{E}_{T}^{\ast}[{\overline{L}}^{m}]+{cov}_{T}^{\ast}({\overline{P}}^{m},{\overline{L}}^{m})$$ | (11) |

Given that, by construction:

$${E}_{T}^{\ast}[{\overline{P}}^{m}]=N{F}_{T}^{m}$$ | (12) |

where $F$ is the forward/futures price at expiry, and $N$ is the number of hours in the block. We also have:

${E}_{T}^{\ast}[{\overline{P}}^{m}{\overline{L}}^{m}]$ | $=N{F}_{T}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}]+{cov}_{T}^{\ast}({\overline{P}}^{m}{\overline{L}}^{m})$ | |||

$=N{F}_{T}^{m}\left({E}_{T}^{\ast}[{\overline{L}}^{m}]+\frac{{cov}_{T}^{\ast}({\overline{P}}^{m},{\overline{L}}^{m})}{N{F}_{T}^{m}}\right)$ | (13) |

The first term represents the product of the forward block price at expiry, multiplied by the expected (at forward expiry) average load. The second term is the covariance of the average realised spot price and average realised load scaled by the forward price at expiry. In total, the term in parentheses tends to depend only weakly on the forward price level.

The major issue now is how to model the second term in (13), since the evaluation of the first term involves only straightforward load forecasting. The complexity of (cash) volumetric risk management comes down to modelling the (cash) covariance term. To be more specific, since we are constructing a pricing measure, our task is constructing the relationship between the covariance term and the available tradeables.

Here, we consider three major trading products: futures, monthly options and daily/index options. The first two expire at the beginning of the month ($T$). Consequently, only daily options would have any impact on the covariance term. In turn, as discussed in Wolyniec (2015), the proper sufficient statistic for describing the behaviour of a portfolio of daily options after forward expiry is the cash volatility (quadratic variation). Consequently, the central problem of modelling the cash covariance term is finding its relationship with cash volatility. Again, the conclusion is model independent. If no relationship exists, then the covariance would describe not market risk but residual risk, without any possibility of forming any hedges.

However, the expected size of the cash covariance term does matter, irrespective of its relationship to traded volatility, since it affects both the expected cost and the size of forward hedges.

Returning to the general expression, (10) can be rewritten as follows:

${E}_{T}^{\ast}[{\overline{P}}^{m}{\overline{L}}^{m}{\overline{S}}^{m}]$ | $=N{F}_{T}^{m}\left({E}_{T}^{\ast}[{\overline{L}}^{m}]+\frac{{cov}_{T}^{\ast}({\overline{P}}^{m},{\overline{L}}^{m})}{N{F}_{T}^{m}}\right){E}_{T}^{\ast}[{\overline{S}}^{m}]+{cov}_{T}^{\ast}({\overline{P}}^{m}{\overline{L}}^{m},{\overline{S}}^{m})$ | |||

$=N{F}_{T}^{m}\left[\left({E}_{T}^{\ast}[{\overline{L}}^{m}]+\frac{{cov}_{T}^{\ast}({\overline{P}}^{m},{\overline{L}}^{m})}{N{F}_{T}^{m}}\right){E}_{T}^{\ast}[{\overline{S}}^{m}]+\frac{{cov}_{T}^{\ast}({\overline{P}}^{m}{\overline{L}}^{m},{\overline{S}}^{m})}{N{F}_{T}^{m}}\right]$ | ||||

$=N{F}_{T}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}]\left[\left(1+\frac{{cov}_{T}^{\ast}({\overline{P}}^{m},{\overline{L}}^{m})}{N{E}_{T}^{\ast}[{\overline{L}}^{m}]{F}_{T}^{m}}\right){E}_{T}^{\ast}[{\overline{S}}^{m}]+\frac{{cov}_{T}^{\ast}({\overline{P}}^{m}{\overline{L}}^{m},{\overline{S}}^{m})}{N{E}_{T}^{\ast}[{\overline{L}}^{m}]{F}_{T}^{m}}\right]$ | ||||

$=N{F}_{T}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}]{\Phi}_{T}^{m}$ | (14) |

where the scaled covariance/shape term:

$${\Phi}_{T}^{m}\doteq \left[\left(1+\frac{{cov}_{T}^{\ast}({\overline{P}}^{m},{\overline{L}}^{m})}{N{E}_{T}^{\ast}[{\overline{L}}^{m}]{F}_{T}^{m}}\right){E}_{T}^{\ast}[{\overline{S}}^{m}]+\frac{{cov}_{T}^{\ast}({\overline{P}}^{m}{\overline{L}}^{m},{\overline{S}}^{m})}{N{E}_{T}^{\ast}[{\overline{L}}^{m}]{F}_{T}^{m}}\right]$$ |

collects all terms not expected to depend on the forward price.

As we can see, the expectation is composed of the product of the forward price and expected load under the risk-neutral measure (which is equal to the physical one under our minimal martingale measure) and the shape/covariance factor, which can only depend on cash volatility.

### 4.2 Estimation of shape/covariance factor

Significant research has been conducted on modelling the shape/covariance factor by building models of expected load shape and correlations of hourly loads and prices. However, this modelling strategy is frequently ineffective, as a very large parameterisation space is needed. Consequently, enormous estimation problems result. This effort is also misdirected, since only the overall shape/covariance factor matters and not its individual components.

One solution is to estimate the shape/covariance factor directly, as this guarantees much more efficient estimates.

Those considerations are non-trivial. It can be shown that for the typical sample sizes available in energy markets, direct estimation is normally more efficient and the gains in efficiency are large.

This consideration is relevant, as we actually require something more important than just an expectation of the shape/covariance factor. We need to obtain the relationship between this factor and the appropriate cash volatility:

$${\Phi}_{T}^{m}={\Phi}_{T}^{m}({\sigma}_{C}^{m})$$ |

In principle, the shape factor may also depend on forward price (average price level) and forward volatility. However, in practice, those dependencies are very weak.

### 4.3 Example: estimation of the factor

In our example of a power load, the pricing location is PSEG hub in PJM. We consider the behaviour of a certain wholesale load, which is an interesting mixture of industrial, commercial and residential retail load. In figure 1, we can see monthly on-peak averages of both the day-ahead locational marginal pricing for PSEG hub as well as the realised load in a period of large variation in price and volatility patterns. As expected, there is a clear seasonal pattern in power loads.

**Figure 1:** PSEG monthly on-peak prices and loads

Now, the ultimate goal of the modelling effort is to find out how much extra cost (hedging or otherwise) to serve the load accrues due to the covariance and shape contribution on a cash basis (ie, conditional on the expiry of the forward contract).

Cost-to-serve | $?-\text{?}$ | Price $ | Price QV | |
---|---|---|---|---|

Average | 237,815 | 3.3% | $48 | 23% |

Load | Load vol | Block correlation | Covariance co-eff | |

Average | 4,642 | 9% | 69% | 7% |

In figure 2, we can see a monthly excess cost of $S-1$. Again, the point of modelling and hedging is to understand the behaviour of ${S}^{m}-1$. As discussed, the only hedging instruments available intra-month are daily or index options. Consequently, we want to find a model explaining $S-1$ in terms of appropriate volatilities. Figure 3 shows the realised price quadratic variations for individual months. As discussed in Wolyniec (2015), this differs substantially from the return price volatility, but it is the appropriate measure for the hedging instrument in question. Now, we want to find a model that will explain the $S-1$ of figure 2 in terms of the price volatility (QV) of figure 3.

**Figure 2: **Excess block cost (

**Figure 3: **Block statistics

We can follow the prescription in (7) and calculate the two contributions of the expected shape and hourly covariances. We first estimated the average relative price and load shapes over the whole sample. We can see the shapes in figure 2. Obviously, estimation over the whole sample hides seasonal variation, but using seasonal subestimates does not meaningfully change the final results; hence, we skip over the details in the presentation here.

**Figure 4:** Price and load shapes

Using the average shapes from figure 4, we can calculate the contribution of the coincident shape to the excess cost $S-1$ corresponding to the first term in (7). The average additional contribution, expressed as a percentage of the flat block, is 0.7%.

Our next step is to look at the contribution of hourly covariances. To perform the correlation analysis, we cannot form one sample with all the data, since the presence of seasonality would induce spurious (de)correlation and volatility. Consequently, we estimate correlations for individual months. This has the undesirable result of greatly limiting individual sample sizes and dramatically expanding estimation standard errors. As discussed in Eydeland & Wolyniec (2016), correlation estimates with fewer than 200 observations are very susceptible to noise. A partial solution would be to attempt seasonal estimation, although that will not be pursued here.

For ease of exposition, we do not estimate hourly correlations and volatilities. Instead, we estimated block price-load correlation, which can be seen in figure 3 for every month. We treat the residual contribution of hourly correlation beyond block one as a fixed correction to the overall estimate of $S-1$. In simple terms, we add an additional fixed term to the shape contribution calculated above. Together, the fixed term is 1.7%.

Now, using the block correlation estimate, we can attempt to see how accurately the estimated model explains $S-1$ in-sample. We take the average of the product of correlation and load volatility, which, as can be seen from the last column of table A, is equal to 7%. Now, for every month, we multiply the average product by the realised price QV in order to mimic an actual application of the model, where we have a quoted daily (index) volatility market. We add to the resulting covariance contribution the fixed shape/hourly contribution calculated above to arrive at the final modelled projection of $S-1$. In figure 5, we show the absolute residual for every month between the projected and realised quantities of $S-1$.

**Figure 5:** Model residuals

The average absolute error for this modelling strategy is 1.8%, with a maximum error as large as 8%. In fact, the percentage absolute errors understate the poor performance of the model, since the large residuals are typically negative and happen when prices and/or volatilities are high. The model underperforms when we need to hedge the most.

This underperformance is not due to the inefficient estimation, since our analysis is in-sample. Rather, it may be due to the assumptions on the relationship between hourly and block correlations and volatilities. It may also be explained by the fact that the residual is large.

However, it is easy to check this assumption. We ran a regression of the excess cost $S-1$ directly on price QV. The absolute error of that model is shown in figure 5 as well. The average error is substantially lower at 1.2%, and the maximum error tops out at 4.5%. On top of that, the maximum error is positive. The model ends up overhedging when prices are high.

In principle, the estimation of the missing hourly structure can improve the model performance. However, it cannot improve on the direct in-sample estimation by construction. In addition, the estimate of the individual hourly covariances would increase the confidence intervals’ width of the parameter by a factor of as much as 16 (it could be effectively less due to cross-correlations).

Direct estimation, however, exploits all the implicit dependencies by construction. We do not have to think through the interaction of the various components at all, and Stein’s paradox works for us.

The block example already shows that exploiting only part of the underlying structure leads to a suboptimal estimation strategy. We can converge to the equivalent of the direct strategy by modelling the underlying structure exactly. However, this is impossible, as we do not know all the details of the underlying process. In other words, we do not know the cross-correlations (or rather cross-concordances). Even if we did, the resulting estimate would have much wider confidence intervals due to the independent estimation of model components (parameterisation).

Note the direct estimation is exactly the modelling strategy described by (14).

In some applications, such as power or natural gas demand, the shape factor is strongly dependent on weather. However, given the rescaled nature of the factor, the driving statistic is weather volatility, a function of weather levels themselves. Consequently, extra hedging efficiency can be gained by considering (semi) static weather hedges for certain locations, where the liquidity and cost are sufficiently attractive to consider (see Eydeland & Wolyniec 2016).

CME offers several monthly and seasonal (quarterly) cooling degree day (CDD) and heating degree day (HDD) futures and options, although the liquidity is quite spotty and, hence, price discovery is not likely to be very efficient. In addition, for a specific need, several banks, hedge funds and (re)insurance companies offer structured cross-price/weather-structured products.

## 5 Forward covariance

So far, we have looked at the cash contribution to the pricing measure. In other words, we have looked at the conditional risk-neutral expectation at forward expiry, which primarily depends on cash volatility. We now return to the overall expression to consider the contribution of the forward volatility (in the form of a quadratic variation).

By substituting (14) in (6), we obtain the following:

$${E}_{t}^{\ast}[{\mathrm{CF}}^{m}]={E}_{t}^{\ast}[{E}_{T}^{\ast}[{\mathrm{CF}}^{m}]]=N{E}_{t}^{\ast}[{F}_{T}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}]{\Phi}_{T}^{m}({\sigma}_{C}^{m})]$$ | (15) |

where the dependence of the cash shape factor on the cash volatility is explicit.

We can again expand the terms to properly collect the various contributions of the forward volatility:

${E}_{t}^{\ast}[{F}_{T}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}]{\Phi}_{T}^{m}({\sigma}_{C}^{m})]$ | $={E}_{t}^{\ast}[{F}_{T}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}]]{E}_{t}^{\ast}[{\Phi}_{T}^{m}({\sigma}_{C}^{m})]+{cov}_{t}^{\ast}({F}_{T}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}],{\Phi}_{T}^{m}({\sigma}_{C}^{m}))$ | (16) |

we have already seen the covariance term in the definition of the shape factor, however, it is important to underscore the cash covariance (ie, conditioned on forward expiry) ${cov}_{T}^{\ast}(\cdot )={E}^{\ast}[(\cdot )\mid {F}_{T}^{m}]$ is, in general, very different from the forward covariance ${cov}_{t}^{\ast}(\cdot )={E}^{\ast}[(\cdot )\mid {F}_{t}^{m}]$, despite formal similarities.

### 5.1 Forward volatility and forward price/demand covariance

As discussed in Grzywacz & Wolyniec (2011), energy prices have several different volatility (information) scales. Different factors drive price formation at different horizons.

For example, in power demand, short (cash) horizon is driven by weather, while longer (forward) horizon might be driven by competitive migration between various LSEs. This might result in relatively small cash covariance with a positive sign: customer load correlates with the system load, which in turn induces higher prices and results in positive correlation between price, load and shape on short (cash) time scales. However, the presence of migration might induce very significant positive but ‘one-sided’ covariance between price and demand: customers migrate away (and demand drops accordingly) when forward prices decline significantly, making the switch to a more competitive provider. Nevertheless, changes in demand are small when forward prices increase, since customers cannot migrate into the original contractual arrangements. Clearly, this makes the forward volatility contribution path-dependent.

Those examples of power loads indicate the typical mechanism that induces forward covariance dependence on forward price volatility. Consequently, the next step in building a proper risk optimisation is developing the relationship between the covariance term and forward volatility. Analogous to the cash case, we can roll all terms unlikely to depend on forward price directly into the forward shape/covariance factor in order to obtain the final pricing formula.

## 6 Pricing formula

With a rescaling analogous to that applied to the cash expectations, we obtain the pricing formula:

$${E}_{t}^{\ast}[{F}_{T}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}]{\Phi}_{T}^{m}({\sigma}_{C}^{m})]={F}_{t}^{m}{E}_{t}^{\ast}[{\overline{L}}^{m}]\left[\left(1+\frac{{cov}_{t}^{\ast}({F}_{T}^{m},{E}_{T}^{\ast}[{\overline{L}}^{m}])}{N{F}_{t}^{m}{E}_{t}^{\ast}[{\overline{L}}^{m}]}\right){E}_{t}^{\ast}[{\Phi}_{T}^{m}({\sigma}_{C}^{m})]+\frac{{cov}_{t}^{\ast}({F}_{T}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}],{\Phi}_{T}^{m}({\sigma}_{C}^{m}))}{N{F}_{t}^{m}{E}_{T}^{\ast}[{\overline{L}}^{m}]}\right]$$ | (17) |

where ${F}_{t}^{m}={E}_{t}^{\ast}[{F}_{T}^{m}]$.

In practical demand applications such as natural gas or power load serving contracting, the contribution of the last term – the forward covariance between the shape and average load and price – is typically small on time scales exceeding four to six months. The reason for the limited time scale is conditional weather information flows only within four to six months before expiry, and weather is the main factor driving the shape.

There are exceptions, though. For example, for wholesale power loads composed from a mixture of the three main types of retail demand (industrial, commercial and residential), significant forward price changes cause the migration of the largest customer first (industrial), which has the least sensitivity to weather. The remaining customers will exhibit higher sensitivity to weather because of the changes in the composition of the mixture. Structurally, this will manifest itself in a potentially significant (negative) contribution from the last term in (17): forward volatility increases the cash shape factor.

In supply applications such as hydro run-of-river or intermittent non-dispatchable generation (ie, wind and solar), the contribution might matter depending on the supply stack structure.

On the other hand, in the California Independent System Operator (Caiso), high forward prices might induce more solar panel adoption, exacerbating the infamous hourly duck curve, which induces a high contribution of the cash shape factor.

## 7 Structure of pricing formula

Those special cases aside, we can drop the shape forward covariance term and rewrite our pricing formula as follows:

$${E}_{t}^{\ast}[{\mathrm{CF}}^{m}]\approx N{F}_{t}^{m}{E}_{t}^{\ast}[{\overline{L}}^{m}]\left(1+\frac{{cov}_{t}^{\ast}({F}_{T}^{m},{E}_{T}^{\ast}[{\overline{L}}^{m}])}{N{F}_{t}^{m}{E}_{t}^{\ast}[{\overline{L}}^{m}]}\right){E}_{t}^{\ast}[{\Phi}_{T}^{m}({\sigma}_{C}^{m})]$$ | (18) |

The first term is the current forward price. The second is the expected load under a risk-neutral measure, which in this case is the same as the expectations under the physical measure, since we are not pricing residual untraded risk under the minimal martingale measure.^{1}^{1}Although we do price in the component of the load risk, which is

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