Publication Risk Journal

W
e 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 background
Intrinsic 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withdrawn in month l. Wml is positive for all values of m, and l and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 value
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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 sup
τ
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 [tT ] endowed with the risk neutral
measure. t is the set of all stopping times with values in [tT ].
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 sup
τ
max max Fl τ( )− Fm τ( )− K, 0( )Wml
l=m+1
n
∑
m=1
n
∑ e
−R Tl −τ( )⎛
⎝⎜
⎞
⎠⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟ ≥
max Et sup
τ
max Fl τ( )− Fm τ( )− K, 0( )⎛
⎝⎜
⎞
⎠⎟ Wml
l=m+1
n
∑
m=1
n
∑ e
−R Tl −τ( )⎛
⎝⎜
⎞
⎠⎟

Recall that

Aml
= Et sup
τ
max 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 Aml
l=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 options
A 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 − Sf( ).

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 strategies
We 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 value
The 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 value
Note 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 options
Assume 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 options
The 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 strikes
How 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 hedging
In 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
0.40
0.60
0.80
1.00
1.20
$ %
Jan
4,2010Jan
19,2010
Feb
2,2010Feb
17,2010
M
ar3,2010M
ar17,2010M
ar31,2010Apr14,2010
Apr28,2010M
ay
12,2010M
ay
26,2010Jun
10,2010Jun
24,2010
–1.0
–0.5
0
0.5
1.0
Delta-hedged Total value Intrinsic July delta Jan delta
F1. Daily values deltas of the storage
Daily values and deltas of the storage with and without hedging Source: 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 factor
It 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 error
In 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.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
M
ay 11, 2010
M
ay 18, 2010
M
ay 25, 2010
June 2, 2010
June 9, 2010
June 16, 2010
June 23, 2010
-1.5
-1
-0.5
0
0.5
1
1.5
Delta hedged Unhedged July delta Jan delta
$ %
F2. Hedged and unhedged storage values
Per unit option analysis: Late start (hedging commenced in May) Source: Author
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
Jan
4,2010
Jan
26,2010
Feb
17,2010
M
ar10,2010
M
ar31,2010
Apr21,2010
M
ay
12,2010
Jun
3,2010
Jun
24,2010
Monte Carlo valution Bachelier valuation
$
F3. The impact of different modelling approaches
Modelling error: Bachelier versus Monte Carlo simulation Source: Author
0.60
0.80
1.00
1.20
1.40
1.60
1.80
$
Jan
4,2010
Feb
2,2010
M
ar3,2010
M
ar31,2010
Apr28,2010
M
ay
26,2010
Jun
24,2010
–1.0 –0.5 0 0.25 0.5 0.6 0.75 0.9 0.95
F4. Effect of the correlation
Net value of the storage and hedges using different estimates of the correlations Source: 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 frequency
We 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.
Summary
There 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, 2003
Pricing and hedging spread options
SIAM Review, Volume 45, Number 4, pp 627–685
Li M, Deng S and Zhou J, 2008
Closed-form approximations for spread option prices and Greeks
The Journal of Derivatives, Spring 2008, Volume 15, Number 3, pp 58–80
Eydeland A and Wolyniec K, 2003
Energy and power risk management: new development in modeling,
pricing, and hedging
John Wiley Sons
Wolyniec K, 2002
Storage valuation: spread options and alternative approaches
Mirant Research notes, not publicly available to the best of our knowledge
Bjerksund P, Stensland G, Vagstad F, 2008
Gas storage valuation: price modelling v. optimization methods
Available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1288024
Ali Sadeghi, associate partner, Resources2 Energy Canada
Email: sadeghia@telus.net
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1.00
Jan
4,2010
Jan
28,2010
Feb
23,2010
M
ar18,2010
Apr12,2010
M
ay
5,2010
M
ay
28,2010
June
23,2010
Daily Every other day Every week Every other week
$
F5. Hedging frequencies
Net value of the storage and hedges using different hedging frequencies Source: Author
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