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We consider the problem of monetising the extrinsic value of a natural gas storage. The storage is viewed as a basket of calendar spread options and is hedged dynamically using the delta-hedging approach. By way

of a real example, we go through the details of the daily corrections in the hedging. The methodology is tested over a six-month period using the Henry Hub daily forward curves and the implied volatilities. Among others, we test the effects of the models errors, parameter estimations and hedging frequencies. Our results show that there is a critical hedgeable correlation – significantly lower than the actual correlation – which results in a very high hedging efficiency and a reasonably high extrinsic value.

Theoretical backgroundIntrinsic value of a natural gas storage is the maximum value that can be attained by hedging the storage given current forward curves. This in itself defines an optimisation problem, over space, of all acceptable injection and withdrawal profiles. The target for the optimisation is to maximise the spread between the purchase costs and sale proceeds. Ratchets, carrying costs, injection and withdrawal fees, and fuel charges will define the restrictions.

In this article, we only consider the valuation of a storage before the beginning of its term that will have no inventory at the start or end of the contract term. In general, volumes and discounting factors need to be modified to account for initial inventories and/or existing inventories within the term of the contract. Also, the initial

inventories might be best hedged using futures rather than non-linear instruments. These aspects of storages are well-understood within the storage community and are not the focus of this article.

The following formula quantifies the intrinsic value:

IV t( ) = max max Fl t( )− Fm t( )− K , 0( )Wml e

−RTl

l=m+1

n∑

m=1

n∑⎛

⎝⎜⎞⎠⎟,

where Fm(t) is the value of the forward contract month m and Tm is its time to expiry in years at the time t, n is the term of the contract in months, R is the annual interest rate, K is the total cost of injections and withdrawals, including fuel and commodity charges and carrying costs, and Wml is the volume that will be injected in month m to be withdrawn in month l. Wml is positive for all values of m, and l and fulfils the following ratchet and capacity conditions:

Wml ≤ Im ,

l=m+1

n∑ Wml

l=1

m−1∑ ≤Wm , Wil

l=m

n∑

i=1

m−1∑ ≤ C, for all 1 ≤ m ≤ n,

where C is the storage capacity, Im and Wm are maximum injection and withdrawal monthly limits. The goal is to find an upper triangle matrix [Wml]m, l=1,..., n that maximises the value.

For the sake of simplicity, in this formula we assume a zero bid-offer spread in forward contracts. The factor max(Fl(t) –Fm(t)–K,0) ensures that injections and withdrawals happen only when forward spreads are greater than variable costs of the storage.

Natural gas storages are integral parts of gas distribution systems and play a key role in managing demand variations. Risk managers need to value storages on a daily basis, while traders face the challenge of effectively hedging storages. Ali Sadeghi presents a review of the basket-of-options approach for valuation and dynamically hedging the extrinsic value of a natural gas storage. The issue of market parameters and their impact on hedging efficiency is also discussed in detail

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HedgiNg THe exTRiNSic vAlueof A NATuRAl gAS SToRAgeNOT FOR

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Calculation of the intrinsic value is straightforward; there are numerous proprietary or public tools within the storage trading community that perform the task with reasonable accuracy. Given the liquidity and depth of the North American futures markets, intrinsic values are usually hedgeable, especially for the first five years of storage life. Therefore, many industry participants consider the intrinsic value as a ‘real’ number.

Gas storages are usually traded above their intrinsic values. In general, the market’s expectation is that over the remaining period of time – until the physical injection and withdrawal seasons – there will be better opportunities to hedge the storage. Using option theory’s terminology, in addition to the intrinsic value, the market deems an extrinsic value for the storage.

The expected maximum intrinsic value of a storage contract over its term, using the best possible exit strategy, is considered the storage value. Formally we can write:

V t( ) =

Et supτ

max max Fl τ( )− Fm τ( )− K , 0( )Wml e−R Tl−τ( )

l=m+1

n∑

m=1

n∑⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜

⎞

⎠⎟

under the same restrictions as in IV(t). The expectation is conditioned on the current forward curve.

V(t) as formulated above is defined on the probability space of all events in the time interval [t T ] endowed with the risk neutral measure. t is the set of all stopping times with values in [t T ]. Stopping times are well-defined exit strategies that determine a time to lock in the current intrinsic value. Fm(t) are forward values at time t corresponding to each event; from a computational standpoint, they represent the dynamic evolution of the forward curve.

Calculation of the extrinsic value using the above formula is not straightforward. In theory, Monte Carlo simulation can be employed. However, the optimisation process should stop only when the expected future value of the storage – given all possible permutations of the forward curve – is equal to the current intrinsic. This means that we not only need to simulate the forward curves, but also another simulation of the forward curves is needed for the time step of each simulated path. This exponentially adds to the complexity of the calculations.

Also, from a business standpoint, it is not possible to link back the resulting values of such simulations to market quotes, therefore rendering the results less ‘real’.

To avoid these problems, a lower bound for V(t) can be established by using Jensen’s inequality:

Et supτ

max max Fl τ( )− Fm τ( )−K , 0( )Wmll=m+1

n∑

m=1

n∑ e−R Tl−τ( )⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛

⎝⎜

⎞

⎠⎟ ≥

max Et supτmax Fl τ( )− Fm τ( )−K , 0( )⎛

⎝⎜⎞⎠⎟Wml

l=m+1

n∑

m=1

n∑ e−R Tl−τ( )⎛

⎝⎜⎞⎠⎟

Recall that

Aml = Et supτmax Fl τ( )− Fm τ( )− K , 0( ) e−R Tl−τ( )⎛

⎝⎜⎞⎠⎟

is by definition the value of an American call option on the spread of forward contract months l and m, so we have:

V t( ) ≥max Amll=m+1

n∑

m=1

n∑

Wml

⎛

⎝⎜

⎞

⎠⎟ .

This inequality constitutes the basis of the ‘basket-of-options’

approach for valuation and hedging of natural gas storages. If the strike is zero, then the two sides coincide. Except for the practically trivial case of zero strike, V(t) is always greater than the volume-weighted sum of the option values. Nevertheless, the option approach is promising for two reasons: it provides a tight lower bound for V(t) that is easier to calculate, and the resulting value can be hedged using calendar spread options. See Eydeland & Wolyniec (2003) for more details.

A less rigorous but more intuitive way of understanding this inequality is achieved by considering the fact that storages can be hedged by spread options (see the next section of this paper for more details). Therefore, storage value ought to be higher or equal to the sum of values of any baskets of options that satisfies physical and financial limitations.

In summary, the basket-of-options approach assumes that a natural gas storage will be hedged by calendar spread options, and the maximum volume-weighted value of the basket is the (approximate) value of the storage. The total costs of injections and withdrawals plus carrying costs define the strike price of the options. In the ‘Hedging strategies’ section, we will discuss the details of how to implement this approach while satisfying the storage constrains.

Note that any call option on the forward spread of months m and l with a strike K can be considered as a put option on the spread of months l and m with a strike –K.

As long as an American option is in-the-money, its holder can exercise it at any time to receive the intrinsic value. If the options market is liquid enough then the holder will short the option instead of exercising it, which will enable him to monetise the extrinsic value. In the absence of a liquid market, the owner might start a delta hedging programme to crystallise the theoretical extrinsic value of the option instead of exercising the option. In the same fashion, a storage owner can lock in 100% of the intrinsic value by entering into forward buys and sells at any time, as long as forward spreads are wide enough to cover the costs. The owner can also sell options and use the storage as their hedge, which will enable him to cash in the extrinsic value. Again, a delta hedging programme can be used to maintain the extrinsic value of the storage if no options market is available.

In a typical delta hedging situation, a seller of an option takes a long position on the underlying – the spread – to achieve a delta-neutral position. In the case of a storage owner, a long position in options (long volatility) has already been taken and, therefore, to achieve a delta-neutral position, short positions on the spread need to be taken. We will delve into the details of how this works within this article.

A minor difference between a physical storage and a basket-of-options is that, after forward hedging the injections/withdrawals, sometimes owners get a chance to unwind the existing hedges and enter into a second set of hedges that is more profitable. A parallel shift of the forward curve (upward or downward) changes the intrinsic value of the storage, but the mark-to-market value of hedges will offset the gains or losses completely. However, if the curve moves in a tilted way so that the order of the forward prices changes, then the optimal volume of injections and withdrawals will change,

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and therefore the change in the intrinsic value will not offset the mark-to-market change of hedges. In these cases, it makes sense to unwind the old and enter into new hedges for a gain. Unwinding the first set of hedges will entail some mark-to-market losses; therefore the second set of hedges will not take the owner to the same financial position as when they were used in the first place. This feature is referred to as rolling intrinsic. Although rolling intrinsic is not equivalent to a second chance of exercise, there is still some value in it and a trader will look at it as an opportunity.

Another minor difference is that, contrary to the case of options, when forward spreads are out-of-the-money there is still the possibility of making money out of a physical storage. During the injection period, every day that the forward prices for any withdrawal months are more than the spot price by an amount that is greater than total costs, the owner will inject the maximum daily volume and sell it immediately at the forward price to lock in a profit. This is called ‘spot-to-forward’ hedging. For our extrinsic valuation of the storage, we ignore spot-to-forward optimisation.

The above-mentioned theoretical reflections provide a natural framework for valuation of storages plus iterative methodologies for hedging not only the intrinsic but also the extrinsic values. To the best of our knowledge, there is no literature discussing the details of this approach, the effectiveness of the hedges, or the relationship of the results with the market and storage parameters. In most situations, market participants are confused about how to implement the delta-hedging approach to a storage asset. Accuracy and reliability of delta hedging is on the one hand a function of the mathematical tools used to value the options and its Greeks. On the other hand, the liquidity and depth of both physical and financial markets, plus the physical limitations of the storage, can greatly influence the results. The purpose of this article is to address these issues by looking at real examples.

Calendar spread optionsA calendar spread put option allows the buyer to assume a short position in the close month and a long position in the far month, while receiving a strike amount. This will imply the following as the pay-off function:

Pput = K − Sc − S f( ) .

where Sc and Sf are the close- and far-month future prices, and K is the strike price.

There is no exact formula for calculating the value and Greeks of a spread option.

There have been attempts in the literature to find approximate solutions in a closed-form that resembles the Black-Scholes formula. For instance, Bachelier’s formula is driven by the assumption that the spread is normally distributed.

Note that while the accuracy of the approximation directly impacts the quality of the results, the hedging methods proposed in this article are independent from the specific valuation methods used. A thorough discussion of spread options and their valuation methods may be found in Carmona & Durrleman (2003), Li et al (2008) and references therein.

Hedging strategiesWe will discuss further details of the basket-of-options and delta-hedging approaches using an example. Following New York Mercantile Exchange (Nymex) conventions, let us define the spread

between the future prices of two calendar months as the close month’s future less the far month’s future.

Throughout this paper we use a million British thermal units (mmBtu) as our standard unit for natural gas.

Consider a storage contract that allows for injections in July 2010 and withdrawals in January 2011. We assume the daily injection and withdrawal limits are 10,000 mmBtu per day for a total capacity of 310,000 mmBtu. The cost of borrowing is 6% annually and there is 1% fuel charge plus 2 cents per mmBtu charge on both injections and withdrawals. Assume it is January 4, 2010; current forward curves for July 2010 and January 2011 are $5.97 and $6.98, respectively. Therefore the current market spread is –$1.01.

The buyer of the contract has no choice other than injection in June and withdrawal in January. Therefore, there is no need for optimisation. The injection and withdrawal costs in this case are $0.08 and $0.09 per mmBtu. The injected natural gas ought to sit idle in the ground for five months; therefore there is a carrying cost of $0.15 per mmBtu. The total cost of the injection and withdrawals, as of January 4, is $0.32 per mmBtu.

Intrinsic valueThe intrinsic value of this contract if $0.69 per mmBtu. If the buyer decides to hedge the storage today, $213,900 profit will be locked in, and this is the minimum amount that the seller of the contract will consider.

Extrinsic valueNote that the buyer does not have to hedge the injections and withdrawals immediately. In fact, this contract can be considered as an American put option on the January/July spread at a strike that is currently –$0.32. Hedging the injections and withdrawals in this case is equivalent to exercising the option.

In this simple case, even though the spread might widen, there will never be another opportunity for the rolling intrinsic. This is because the injection and withdrawal volumes will never change and therefore any gains in the intrinsic value of the storage will be totally offset with the mark-to-market losses of the existing hedges.

Hedging with optionsAssume there is a quote in the market for an American-type put option at a strike price of –$0.32 for the July/January spread. The price of such an option has to be more than $0.69 since it would immediately pay off $0.69. Let us assume such an option is quoted at $1.09 per mmBtu. We ignore the bid-offer spreads for the sake of simplicity and assume this quote is liquid enough. Note that since the strike price is a function of market prices, it needs to be set on the day that options are purchased.

In this case, the total value of the storage is $1.09 per mmBtu, of which $0.69 is intrinsic and $0.40 is extrinsic. By selling this option the storage is completely hedged and the owner will realise the extrinsic value.

Next we explain in more detail how the hedge works.If at any time before settlement of the July contract the buyer

decides to exercise the option, then the storage owner will lock in the injections and withdrawals immediately. Therefore the payoff will be covered by the storage.

Otherwise, the contract for July 2010 settles in June. On the settlement day, the owner of the storage will know the payoff amount.

Assume the spread settles at –$2.00 and therefore the option pays

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off $1.68. On the same day, the storage owner locks in the injections and withdrawals, which after all the costs, will leave $1.68. The storage owner will pay this amount to the buyer of the option, therefore keeping the $1.09 premium.

As long as the spread settles anywhere below the strike price of –$0.32, exactly the same approach will keep the owner whole at the premium. Now assume the spread settles somewhere above the strike – that is, more than –$0.32. In this case, the option will not pay off. The owner of the storage contract will not inject either, since the spread is not wide enough to cover costs. Again, the resulting net profit is the $1.09 premium.

Note that in the case of the spread settling above strike, the owner still might use the cash-to-forward strategy to gain additional profit: during the month of July, every day that the January price is more than the spot price by at least $0.32, the owner will inject 10,000 mmBtu and sell it immediately at the January forward price for a profit.

European instead of American optionsThe spread options in Nymex are usually of the European type. The price of a European option is less than its American counterpart, but is still higher than the intrinsic value. Therefore, by using a European option, the hedging will work in exactly the same way, except the owner will not be able to realise 100% of the extrinsic value.

Options with other strikesHow would this approach work if options are quoted only for strikes that are different than the cost of $0.32? In this case, the owner can still sell the options to realise some part of the extrinsic value. However, the basket-of-options and storage do not cover each other perfectly.

For instance, let us assume the closest strike available is 0. This put option is worthier than a –$0.32 put, hence there will be an excess premium collected.

If the spread settles below –$0.32, then the owner will be short by $0.32 after locking in. But this amount will be offset, at least partially, by the excess premium.

If the spread settles above 0, then payoff is zero and the storage is not economical. As before, the storage owner might still use the storage for spot-to-forward optimisation. Plus, the whole premium is kept.

If the spread settles between –$0.32 and 0, then payoff will be positive while storage is not economical. The owner has to pay out of his own pocket. However, the amount of payout is limited to a maximum of $0.32, and again it will be partially offset by the excess premium.

In summary, in this case the option premium will cover more than the extrinsic value of the storage, but there is significant downside risk for a limited amount of uncovered payout. The maximum uncovered payout amount is the storage costs.

Next, let us assume the closest strike available is –$0.50. This put option is cheaper than a put with a strike of –$0.32.

If the spread settles below –$0.50, then the owner will gain $0.18 after settling the payoff.

If the spread settles above –$0.32, then payoff is zero and the storage is not economical. As before, the storage owner might still use the storage for spot to forward optimisation.

The main difference is when the spread settles between –$0.50 and –$0.32. In this case, payoff is zero and the storage owner has no

obligation, but the storage is still economical. The owner will lock in an additional amount between 0 and $0.18.

To summarise this case, the option value will not cover the whole extrinsic, but it provides some additional upside potential that might help the owner achieve the extrinsic. Note that the amount of upside is not unlimited. Also there are no downside risks.

Delta hedgingIn this section, we assume there aren’t any option markets available and therefore the above-mentioned hedging strategies are not applicable. We also assume that the underlying natural gas market at the storage location is liquid both physically and financially.

On January 4, 2010 our model value for the storage is $1.30. How do we extract $1.30 out of this storage?

The owner of the contract has a positive delta with respect to January 2011 contracts and a negative delta with respect to July 2010 contracts. The delta is zero for all other months. Therefore, to maintain a delta neutral position, a short position needs to be taken in January and a long position in July. The exact amount of the positions depends on the deltas. In our example, July has been at –62% and January was 71%. Therefore, the owner would hedge 62% of the injection volume and 71% of the withdrawal volume on January 4, 2010. These volumes will change gradually in a way that any profit and loss in the option value will be offset by the profit and losses made by the hedges. Therefore, no net change to the value of the portfolio occurs.

We used the Bachelier model and market-implied volatilities – from calls and puts on outright forwards – for this valuation.

We tested the results for various correlations including the historical correlation.

Figure 1 compares the daily development of the intrinsic, total (intrinsic plus extrinsic), and the delta hedged values. The latter includes the portfolio value of the storage contract plus delta-hedging positions. On the second axis, figure 1 also shows daily deltas for July and January. As we can see, the portfolio value has been relatively unchanged (from $0.98 to $0.83). At the same time, the unhedged storage value (the ‘total value’) has dropped from $0.98 to $0.59. Both deltas converge to 100% as time approaches

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Delta-hedged Total value Intrinsic July delta Jan delta

F1. Daily values & deltas of the storage DailyvaluesanddeltasofthestoragewithandwithouthedgingSource: Author

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the end of June. Volatilities and prices are actual market quotes and vary on a daily basis. Correlation in figure 1 is assumed to be 60%.

Timing factorIt is worth noting that the hedging could have started sooner or later than January 4. In both of these cases, the starting contract value would be different and the hedging would result in a completely different financial situation. As an example, let us assume that hedging started in the middle of May 2010 when, in hindsight, there has been a peak in the value of the storage, as seen in figure 2.

In this case, the storage value starts at $1.11 and, if delta hedged, will end up at $1.10. If unhedged, the contract will settle at $0.56. Bachelier is used with all parameters, in the same way as in figure 1.

Obviously the efficiency of the delta hedging is directly related to the time to expiry.

Modelling errorIn this section, we compare delta-hedging results driven from a Monte Carlo simulation with 250 iterations versus the Bachelier

formula. This is shown in figure 3. All parameters are the same and the correlation is assumed to be 60%.

The two methods show a relatively high difference in the valuation of the extrinsic at the beginning of the term. However, both methods approach towards the same end results after the delta hedging. This can be attributed to the fact that delta hedging uses the Greeks and not the option values itself. At the end of the term, all valuation formulas approach the intrinsic value, so the final net value is the intrinsic plus all the gains and losses from the hedges. Therefore, theoretical differences between different methods play a role only to the extent that their Greeks are different.

Parameter estimation error – hedgeable correlation Spread options are known to be very sensitive to market parameters, particularly correlations. Volatilities can be implied from the more liquid call and put markets, so there are no major issues. However, in the absence of market quotes for spread options, there is almost no way to drive a market-implied value for correlations.

It is a common practice to drive correlations from historical data. However, historical correlations are typically over 90% – in our case, actually 95% – which implies small extrinsic values. This, in most cases, does not match the market view of a storage’s extrinsic value. Therefore, market participants have to guess a reasonable correlation factor, possibly based on recent transactions in the marketplace.

This raises an important question about the integrity of delta hedging. If two traders have different views of the correlations, they are going to hedge two different extrinsic values. Does this mean they will settle two different values at the end? If so, then why not assume the lowest conceivable correlation and make the maximum amount of the extrinsic?

To answer this question, we must note that performance of each delta-hedging process is a direct result of how accurate the parameters are. A low value of the correlation will lead to a high original extrinsic value, but the delta-hedging approach will not be very efficient and most of the value will be lost as time goes by. Therefore, the high extrinsic value at the beginning is more illusion than reality. On the other hand, a trader who uses the correct parameters will achieve a very good efficiency in hedging without any initial illusions regarding the value.

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F2. Hedged and unhedged storage valuesPerunitoptionanalysis:Latestart(hedgingcommencedinMay)Source: Author

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F4. Effect of the correlationNetvalueofthestorageandhedgesusingdifferentestimatesofthecorrelationsSource: Author

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To illustrate the effects of correlations on both extrinsic value and hedging efficiency, we run our example using a variety of correlations. The details are shown in figure 4.

This example shows that there is a critical hedgeable correlation, in this case about 75%, which will yield the highest level of hedging performance. Using correlations higher than 75% will result in a lower realised value. On the other hand, a lower than 75% correlation will create an illusionary high initial extrinsic value, while the realised value will still be close to that of the 75% case.

Note that the hedgeable 75% correlation is significantly less than the actual historical correlations during January–June 2010. Our experience indicates that such inconsistencies are systematic, rather than specific to the data or year in our example. Also, we have observed the same behaviour using other valuations formulas, including the Monte Carlo simulation. Different reasons can be mentioned: the valuation methods are made for liquid markets while options are not liquid. Plus, implied volatilities do not match the historical volatility of that year.

For example, volitility for January–June 2010 for the contract month of January 2011 was 25%, while its implied volatility during that period has been in the 30–40% range. This means the implied volatility that we used is more a market view than a best estimate for future events, which then implies that correlations need to be adjusted correspondingly.

Hedging frequencyWe have thus far rebalanced the hedges on a daily basis to achieve a delta-neutral position. There might be reasons for a trader for not balancing the delta every day. For example, for a smaller storage, the delta-hedge volumes may not be large enough to be transacted in the market every day. The question which then arises is, what is the impact of such delays in the hedging results? The answer to this question of course depends on how volatile the forward markets are. For a smooth period of time, one would expect a daily correction to be unnecessary. Our results, as shown in figure 5, confirm that this is in fact the case.

SummaryThere is a spectrum of hedging strategies available to a storage operator: At one end, the storage can be left unhedged until physical injections start, in which case the owner is fully exposed to market volatilities. The other end is hedging 100% of the volume, in which case the intrinsic value of storage will be locked in and any exposure to market volatilities will be removed. The downside of this extreme end is that the owner forfeits the chances of wider seasonal spreads in the futures market.

To bridge the gap between the two extremes, we considered the basket-of-options approach. The underlying concept here is that storages behave the same way as financial options on the seasonal spreads, and can be hedged by short positions on such options. The value of the optimal basket of options provides a lower bound for the value of the storage. In the absence of an active options market, delta hedging enables the owner to cover risks while taking benefit from potentially wider spreads by changing the hedged volumes dynamically.

In this article, we considered a simple case of the storage with a term of two months, which consequently can be hedged by one spread option. In reality, storages are contracted for periods of at

least one year and require a portfolio of spread options for hedging. While this increases the complexity of the problem, storages can still be valued and hedged using spread options and their Greeks.

While the mathematical theory behind dynamic hedging is well understood for financially written options, there are ambiguities around the implementation of this theory to a physical asset with embedded optionalities. In the absence of a liquid options market, one has to rely on model valuations for hedging. Also, volatilities and correlations can be implied poorly, which makes the models less reliable and brings to light the issue of hedging efficiency. We showed that hedge efficiency is a direct result of the quality of the parameters involved. A high efficiency in hedging can be achieved for a certain range of correlations. Inaccurate correlations can lead to illusionary valuations that cannot be hedged. ■

References

Carmona R and Durrleman V, 2003Pricing and hedging spread optionsSIAM Review, Volume 45, Number 4, pp 627–685

Li M, Deng S and Zhou J, 2008Closed-form approximations for spread option prices and GreeksThe Journal of Derivatives, Spring 2008, Volume 15, Number 3, pp 58–80

Eydeland A and Wolyniec K, 2003Energy and power risk management: new development in modeling, pricing, and hedgingJohn Wiley & Sons

Wolyniec K, 2002Storage valuation: spread options and alternative approachesMirant Research notes, not publicly available to the best of our knowledge

Bjerksund P, Stensland G, Vagstad F, 2008Gas storage valuation: price modelling v. optimization methodsAvailable at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1288024&

Ali Sadeghi, associate partner, Resources2 Energy CanadaEmail: [email protected]

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0

Apr 12

, 201

0

May 5,

2010

May 28

, 201

0

June

23, 2

010

Daily Every other day Every week Every other week

$

F5. Hedging frequenciesNetvalueofthestorageandhedgesusingdifferenthedgingfrequenciesSource: Author


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