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UNIVERSITÉ PARIS I – PANTHÉON-SORBONNE
MASTER MMMEF – Parcours Finance
Ramzi MAALOUF
PARTICULARITIES of the COMMODITIES MARKET
Résumé
Ce rapport de stage présente une vue globale du marché des matières premières. L’importance
de ce marché n’a pas cessé d’augmenter ces dernières années. En effet, l’année 2008 était une
année très intéressante, durant laquelle le marché a été témoin d’une envolée des prix du pétrole, de l’or, du cuivre, et des produits agricoles.
La première partie du document introduit le marché et met l’accent sur la courbe forward.
Ensuite, elle se concentre sur la « convenience yield », une notion particulière au monde des
commos. La deuxième partie élabore le sujet des options et volatilités « early expiry », qui est
aussi un concept spécifique au monde des commos. La troisième partie décrit une méthode
utilisée pour extraire la corrélation implicite entre deux sous-jacents à partir d’une option basket. La dernière partie souligne l’importance des options binaires dans le marché des
commos et l’impact qu’a le smile des volatilités sur les prix de ces options.
Rapport de stage présenté leLUNDI 15 SEPTEMBRE a 16H45
JuryGuillaume FOUCHERES BICH Philippe Natixis Université Paris 1
Commodity Derivatives
Lieu du Stage Natixis
47, Quai d'Austerlitz
75013 Paris
France
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Table of Contents1 Futures contracts in commodity markets............................................................................ 3
1.1 Different commodity asset classes ............................................................................. 31.2 Popularity of futures contracts in commodity markets .............................................. 3
1.3 Convenience yield ...................................................................................................... 4
1.4 Comparing futures and forwards................................................................................ 5
1.5 Shapes of the forward curve....................................................................................... 6
1.5.1 Contago .............................................................................................................. 71.5.2 Backwardation.................................................................................................... 8
1.5.3 Impact of the shape of forward curves on option prices .................................... 9
2 Early Expiry Options........................................................................................................ 10
2.1 Implied Volatilities................................................................................................... 10
2.2 Early expiry options: problem description ............................................................... 122.3 Methodology ............................................................................................................ 14
2.4 Numerical example................................................................................................... 15
3 Options on Baskets........................................................................................................... 17
3.1 Pricing options on baskets........................................................................................ 17
3.2 Moment matching..................................................................................................... 18
3.3 Determining implied correlation .............................................................................. 204 Digital options .................................................................................................................. 21
4.1 Popularity of digital options ..................................................................................... 21
4.2 Volatility’s impact on the price of a digital option .................................................. 23
4.3 Smile’s impact on the price of a digital option ........................................................ 25
5 References ........................................................................................................................ 27
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1 Futures contracts in commodity markets
1.1 Different commodity asset classes
The different commodity asset classes are the precious metals (gold, silver, platinum and
palladium), the base metals (copper, nickel, and aluminum), the energy commodities (crudeoil, natural gas …), the agricultural products , also called soft commodities, (wheat, soybean,
coffee …), the CO2 emissions permits and credits, and last but not least, electricity.
1.2 Popularity of futures contracts in commodity markets
The main players in commodity markets are:
1- producers and consumers looking at hedging the commodity’s price risk2- arbitragers looking for a riskless and profitable trading strategy in commodity markets3- investors and speculators looking for an exposure to commodity price moves
In contrast to the equity market, the typical underlying for commodity derivative products is
not the spot or physical commodity, but rather the futures contract on the desired commodity.
For example, in equity markets, a typical call option on a Microsoft share has the share itself
as underlying. In commodity markets on the other hand, a typical call option on Brent has the
Brent futures contract as underlying.
Thanks to the arbitragers (the 2nd class of players in commodity markets), the price of thefutures contract converges to the spot price at the futures maturity. Thus, the payoff of a
European derivative product on a spot commodity is equivalent to a similar product on the
commodity’s futures contract. Mathematically,
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So, theoretically, both call options are worth the same. This can be generalized to all types of
European payoffs. Thus, given the premiums are identical (almost identical in practice),
people tend to choose the most liquid asset as underlying. Consequently, in equity markets,stocks are preferred. In commodity markets however, futures are preferred for the followingreasons:
1- they avoid all risks associated with spot trading, namely transportation, storage, andcounterparty risk
2- their prices are readily available on exchanges, unlike spot prices which can only beknown by contacting a number of dealers
3- futures and futures options are generally traded on the same exchange, facilitatinghedging and speculation and reducing arbitrage
1.3 Convenience yield
In equity markets, the absence of arbitrage defines a strict relation between forward and spot
prices. Note that I have purposefully used the term forward instead of futures because I do not
want to consider daily margin calls.
This is not true for commodity forward contracts due to the following reasons:
1- the difficulty of short selling some spot commodities2- the illiquidity of some spot commodities3- the transportation costs4- the storage costs5- the benefits obtained by the ownership of the physical commodity
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Since in the commodities markets, the interest rates are not the main source of uncertainty,they are considered to be deterministic. Thus, futures and forwards are used interchangeably.
This is what I will be assuming for the rest of the document.
1.5 Shapes of the forward curve
We can see that in contrast to equity markets where the entire forward curve can be
determined from the spot price, a commodity’s forward curve is directly read from the market.
Different futures contracts for the same commodity are considered as distinct (though
correlated) underlying assets.
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Consequently, in practice, we work backwards from the forward curve to deduce the implied
convenience yield. Some forward curves may be complex enough requiring a non-constant
convenience yield to explain their shape.
The graph shows the price per barrel on the y-axis and the maturity of the futures contract on the x-axis
So, a deterministic function of time will do the trick.
The generalized formula becomes:
),(
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T t B
eS F
ds y
t T
t
s
T
t
Once the convenience yield is deduced from the available market quotes, it can be used to
determine the prices of futures that are not quoted on the exchanges.
The most common shapes of forward curves in the commodities markets are the next
subsections.
1.5.1 Contago
If we assume for simplicity that the interest rates and convenience yield are constants,
Contago is the case where (r – y) > 0.
Usually, it is the case where the current world inventories for the commodity in question are
relaxed, and the common feeling in the market is the fear of a possible supply crunch in the
future, stimulating the buying of long dated futures. We therefore see the long dated end of
the curve higher than the short dated one.
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Below is an example of contago:
The graph shows the price per barrel on the y-axis and the maturity of the futures contract on the x-axis
1.5.2 Backwardation
The forward curve is in backwardation when (r-y) < 0.
Usually, this is the case where the current world inventories are tight, and consumers prefer to
buying now what they will be using in the near future, fearing that inventories will keep ondecreasing. We therefore see the short dated end of the curve higher than the long dated one.
Below is an example of backwardation:
The graph shows the price per barrel on the y-axis and the maturity of the futures contract on the x-axis
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It is worth noting that although it occurs very rarely, a forward curve may change from
backwardation to contago (or vice versa). This was the case for the Brent and WTI crude oilfutures during May and June 2008. Indeed, consumers feared that oil fields and reserves willsoon be depleted because of the massive demand of emerging countries like China and India.
This pushed the forward curves from contago to backwardation.
1.5.3 Impact of the shape of forward curves on option prices
Speculators, consumers and producers usually buy at the money options. For them, the
important reference is the spot commodity price. However, they usually buy options on thefutures contract with maturity closest to that of the option. These are called standard options
(More on standard options later)
B&S formula for a call option on a future gives us:
.
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20
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Moreover, because of the absence of arbitrage, the sport price is generally closest to the
futures price with the shortest maturity, called the first nearby. This explains the results shownin the table below
Shape of forwardCurve
Future Compared to strike Call premium
Contango ATM meansT Nearby first f f S K 000
So, the option is actually ITM
Expensive
Backwardation ATM meansT Nearby first f f S K 000
So, the option is actually OTM
Cheap
Table 1-1
Obviously, the opposite is true for a put option.
As a conclusion, backwardation is suitable when investors have a bullish view, and contango
is suitable when they have a bearish view, since in these mentioned cases, the premium they
pay is cheap.
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2 Early Expiry Options
2.1 Implied Volatilities
Since the probability distribution of underlying assets is not lognormal as assumed in the B&S
world, a volatility smile (or skew) exists in the market for a given option maturity. This givesthe implied volatility for each strike. The implied volatility is the volatility that the market
uses to price options for given strikes. They are thus obtained by taking the option premium
from the market and inversing the B&S formula numerically.
Below is an example of a volatility smile:
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
4 0
4 5 .
5 1
5 6 .
6 2
6 7 .
7 3
7 8 .
8 4
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9 5
1 0 1
1 0 6
1 1 2
1 1 7
1 2 3
1 2 8
1 3 4
1 3 9
1 4 5
1 5 0
1 5 6
The graph shows the volatility smile of WTI crude oil options with 2 years to maturity. On the x-axis, we
have the strikes. On the y-axis, we have the implied volatilities.
If we consider several maturities, we obtain what is called a volatility surface. In order to have
consistent smiles, we should normalize the strikes by dividing them by the futures price. Weobtain what is called a moneyness ratio.
An example of a volatility surface is shown below:
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45%60%
75%90%
105%120%
135%
97
127
160
188
219
251
279309
337370
553735
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
Implied Vol
Moneyness
Maturity
The graph shows the volatility surface of Brent Crude Oil. The maturity is in days.
The volatility matrix corresponding to the above surface is shown below:
MoneynessMaturity in
days45% 60% 75% 90% 105% 120% 135%
97 49.59% 47.52% 44.15% 39.31% 34.92% 33.35% 34.18%127 48.10% 45.63% 43.69% 39.07% 34.79% 33.17% 33.60%160 47.69% 45.00% 42.40% 38.03% 34.31% 32.68% 32.76%188 46.15% 43.83% 41.53% 37.35% 33.84% 32.26% 32.19%219 44.93% 42.90% 40.68% 36.67% 33.35% 31.81% 31.64%251 44.02% 42.12% 39.99% 36.16% 32.95% 31.35% 31.05%
279 41.75% 40.12% 38.04% 34.92% 32.48% 31.38% 31.17%309 41.91% 39.92% 37.58% 34.47% 32.07% 30.96% 30.69%337 39.04% 37.85% 36.18% 33.72% 31.81% 30.85% 30.56%370 38.84% 37.50% 35.84% 33.47% 31.59% 30.66% 30.35%553 39.17% 38.07% 35.94% 32.80% 30.32% 28.86% 28.25%735 37.49% 36.80% 34.53% 31.44% 29.05% 27.56% 26.89%
Table 2-1
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2.2 Early expiry options: problem description
As mentioned in section 1.4.3, market participants tend to buy or sell standard options. Astandard option is an option whose underlying is the futures contract which matures slightly
after the expiry of the option. To clarify matters, we denote:
futureTn the maturity of the nth futures contract listed on the exchange
optionTn the maturity of the standard option on the nth futures contract described above
Note that the standard options are listed on the same exchange as their corresponding futurescontract, and they obviously mature a few days earlier than the futures contract maturity
(option
Tn <futureTn ). The reason for that is to avoid physical delivery of the commodity if
the option is exercised at maturity.
Since for the nth future, standard options with several strikes are listed on the exchange and
are generally liquid products, a volatility smile can be implied for that future. We call this
smile the standard volatility smile of future “n”.
So, for every futures contract (distinguished from other contracts by its maturity), a standard
volatility smile can be implied from the market. The result is a standard volatility matrix for
the commodity in question.
Standard volatility matrix
Moneyness
FutureFuturesMaturity
Options maturity(in days)
80% 90% ATM 110% 120%
Future 1 futureT1 option
1T = 6 49.12% 43.80% 37.84% 34.71% 36.32%
Future 2 futureT2 option
2T = 35 45.59% 41.38% 37.56% 35.28% 35.09%
Future 3 futureT3 option
3T = 66 44.47% 40.62% 37.09% 34.93% 34.29%
Future 4 futureT4 option
4T = 97 42.86% 39.31% 36.10% 34.09% 33.35%
Future 5 futureT5
option
5T = 127 42.45% 39.07% 35.97% 33.98% 33.17%
Future 6 futureT6 option
6T = 160 41.09% 38.03% 35.34% 33.56% 32.68%
Future 7 futureT7 option
7T = 188 40.23% 37.35% 34.83% 33.14% 32.26%
Table 2-2
Sometimes, however, participants may be interested in buying or selling options on futures
contract with longer dated maturities. To be more specific, they are interested in options
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expiring much earlier than their underlying futures contract, hence the name early expiry
options.
As an example, the WTI crude oil options with maturities beyond 1 year are only listed for thematurity months of June and December. So, if the current month is July 2008, and a consumer
wants to hedge against an increase in price of oil in September 2009, he/she will not find the
appropriate call option in the exchange. He/she must buy an Over the Counter early maturity
option. In order to price that option, we need to know the early expiry implied volatility.
So, the problem at hand is the following:
Given a standard volatility matrix (in moneyness format) of future contracts on a certain
commodity, we need to generate an array of early volatility matrices (one early volatility
matrix per future).
Now the question is:
Given the standard volatility matrix shown in the table above, we need to find the volatility of
the option with maturityoption
Tk
on the future “n” as underlying, for all k smaller than n. In
other words, the problem is the one of generating, for each future, the following early
volatility matrix:
Early volatility matrix of future “5”
MoneynessFuture
FuturesMaturity
Options maturity(in days)
80% 90% ATM 110% 120%
option
1T = 6 ? ? ? ? ? option
2T = 35 ? ? ? ? ? option
3T = 66 ? ? ? ? ?
option
4T = 97 ? ? ? ? ?
Future 5 futureT5
option
5T = 127 42.45% 39.07% 35.97% 33.98% 33.17%
Table 2-3
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2.3 Methodology
To tackle this problem, we take two main assumptions:
Assumption 1: For a given commodity, ATM instantaneous volatility is stationary
By stationary, we mean that the instantaneous volatility does not depend on time but rather on
the distance to maturity.
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In words, this means that seen from today, the volatility during the last month of the life of anoption is the same as the volatility of an option that expires in one month. Obviously, thisinterpretation applies to all durations other than one month.
Visually,
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Particularities of the Commodities Market
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.
Assumption 2: For a given futures contract, the standard volatility smile is replicated in itsearly volatility smiles. The reason for that is that traders believe it is the best estimate for the
early volatility smiles. They assume market participants will keep the same skewed
probability distribution for the underlying, independent of the option’s maturity.
Therefore, the methodology to generate an array of early volatility matrices is the following:For future “n”, we first start by computing the ATM early volatilities based on assumption 1.
Then, we compute early ITM and OTM volatilities based on the previously computed earlyATM volatilities and assumption 2.
2.4 Numerical example
The following example better illustrates the algorithm. Suppose our standard volatility matrix
is the one shown in table 2-2. Suppose, moreover, that we want to fill the early volatility
matrix of future “5”, shown in table 2-3.
Note that the last row is exactly the 5th
row of standard volatilities given in the standardvolatility matrix. The remaining volatilities are to be calculated. We show in details the
calculation of the shaded volatilities
We first calculate the 66days early ATM volatility
According to the notation described in the previous section, the first volatility in question is:127
66,0
We know that:
61)(66)(127)( 2127127,662127
66,0
2127
127,0 (1)
T1
T1 T2
T1 T2 T3
Same average ATM volatility
for the specified duration (T1)
Same average ATM volatility
for the specified duration (T2)
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By assumption 1, we have: 127127,66 =61
61,0
Formula (1) thus becomes: 61)(66)(127)( 26161,0212766,02127127,0
Moreover, from the standard volatility matrix, we can directly read the following volatilities:
%83.3466
612)6161,0
(1272)127127,0
(127
66,0,So
37.16%6161,0
:findwe
)3535,0
,(35and)6666,0
,(66couplese between thioninterpolatlinear byThus,
%56.373535,0
%09.376666,0
%97.35127127,0
Figure 1- Illustrating the assumption of stationary ATM volatilities
We now calculate the 66days early 80% volatility (the second volatility in question),
denoted by %80early
By assumption 2, we have: ATM ATM
early
Standard
%80Standard
early
%80
(2)
0 66 127
Early Volatility
(Unknown)
Volatility determined by interpolating standard
volatilities given by the market
(After assuming ATM volatilities are stationary)
Standard volatility given by the market
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The right hand side of the equation can directly be read from the standard volatility matrix.
ATM early , on the other hand, is calculated in the previous step
We therefore have:
%10.41%97.35
45.4283.34%80
early
In the same fashion, the rest of the early volatility matrix is computed. The result is shown inthe table below.
Early volatility matrix of future “5”
Moneyness
FutureFuturesMaturity
Options maturity(in days)
80% 90% ATM 110% 120%
option
1T = 6 41.84% 38.51% 35.45% 33.49% 32.70%option
2T = 35 41.54% 38.24% 35.21% 33.26% 32.47%option
3T = 66 41.10% 37.83% 34.83% 32.90% 32.12%
option
4T = 97 41.83% 38.50% 35.45% 33.48% 32.69%
Future 5 futureT5
option
5T = 127 42.45% 39.07% 35.97% 33.98% 33.17%
Table 2-4
3 Options on Baskets
3.1 Pricing options on baskets
Options on a basket of underlying assets are very common in commodity markets. We can
imagine a consumer who uses both natural gas (Nat Gas) and crude oil (WTI) for themanufacturing of his goods. This consumer would be interested in buying a basket call option
to hedge an increase in price rather than buying 2 separate call options, since it is cheaper to
do so. Certainly, this reduction in cost results from a weaker protection, since the basket calloption is only profitable when the average price exceeds the strike.
Mathematically, if we denote
2underlyingof at t pricespot:
1underlyingof at t pricespot:
Tatmaturing2contractfuturestheof at t pricethe:
Tatmaturing1contractfuturestheof at t pricethe:
2
1
,2
,1
t
t
T
t
T
t
S
S
F
F
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Particularities of the Commodities Market
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2
0
,2
1
0
,1
t
T
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nearby first i
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The two most common ways of pricing basket options are:
1. Monte Carlo2. Moment Matching
I will not address the first method since it is pretty straight forward. The next section will
describe the 2nd
method.
3.2 Moment matching
We know that (under specific conditions), the product of 2 lognormal stochastic process is a
lognormal stochastic process. Their sum however is not. Empirical studies have shown that itcan be approximated as such without loosing a lot of accuracy when calculating the price of
vanilla options. The advantage is the speed of calculation compared to Monte Carlo.
So, if we assume Yt is a lognormal stochastic process, then B&S can be applied to determinethe value of a call on an equally weighted basket.
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In fact, B&S gives:
Q)unermartingaleais pricefuturesthe becauseisThis()(21][
:Let
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Y S
F S
F Y E M
Y E
T d d T
T K
Y E
d
withd N K d N Y E T BC
T T
T Q
basket T
Q
basket
basket
basket T
Q
T
Qbasket
Now that we found the first unknown, let us try to find the second, basket :
To do so, we have to pass by an intermediate step:
ds
T W d
Qds
T
W d dsQT T
t
Q
T
t
QT
t
QT
t
T
t
Q
T
Q
T
s
T
s s
T
s
T
s s
T
s
e F e E e F
e E F F E
S
F E
S
F E
S S
F F E Y E M
.2,1
0
.2
0
.2,1
0
2
..2
1
0
2,1
0
2,1
0
22
0
2,2
0
21
0
2,1
0
2
0
1
0
,2,1
02
02
0
21
0
1
0
2
0
1
0
21
..
.})({
:havewe
tic,determinisisassumingandQ,undermartingalelexponentiaanis pricefuturesthesinceBut
)(
})({
)(
})({
.
}.{.2
4
1][
:Let
ds
T W d
Qds
T
W d dsQT T
t
Q
T
s
T
s s
T
s
T
s s
T
s
e F e E e F
e E F F E
.2,2
0
.2
0
.2,2
0
2
..2
1
0
2,2
0
2,2
0
0
22
0
2
0
2
0
2
0
22
..
.})({
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Particularities of the Commodities Market
20
)1(.
.
..2.)(.)({
4
1][
:getweconstant,isthatsimplicityforassumefurtherweIf
....
....
....}.{
2122
21
0
21
0
21
0
2
21
0
21
0
21
0
21
0
21
0
21
2
0
1
0
,2
0
,1
02
2
0
,2
02
1
0
,1
02
02
.,2
0
,1
0
.,2
0
,1
0
.2
1.
2
1.
2
1
,2
0
,1
0
).(
0
.2
1.
2
1
,2
0
,1
0
,2,1
0
T T T
T T
T T
T
Q
dsT T
dsT T
dsdsdsT T
W d Q
dsdsT T T
t
T
t
Q
e
S S
F F e
S
F e
S
F Y E M
e F F e F F
eee F F
e E ee F F F F E
T T
T
s s
T
s
T
s
T
s s s
T
s
T
s
Now that we found M2, we have to find a relation between M2 and basket .
For any exponential martingale, and for Yt in particular, we have:
)2(ln.1
.}{...
2
1
22
.2
1
..2
0
.2
02
.2
1.
0
222
M
M
T
e M e E eY M eY Y
basket
T W QT T W
T basket T basket basket
basket T basket
Now that we found all the parameters we need, B&S formula can readily be applied.
3.3 Determining implied correlation
Sometimes the problem faced by traders is inverted. This means, they observe in the market
(namely brokers, since basket options are not traded on exchanges), prices of products calleddispersions.
A dispersion product between WTI and Nat Gas is the difference between the price of the
separate calls and the price of a call on a basket.In other words:
Dispersion = 0.5 x ( Call(WTI) + Call(Nat Gas) ) – Call(basket)
The trader can easily price the separate calls. So, given the dispersion price, the price of the
call basket is determined.
Once the call basket premium is determined, the B&S formula can be numerically inverted to
find the implied basket volatility, basket .
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Particularities of the Commodities Market
21
Then, from equation (2) above, we findT
basket T
basket
eS
F
S
F e M M
T T 222
2
0
,2
0
1
0
,1
02
122
1.
T
S S
F F
eS
F e
S
F M
T T
T T
T T
21
2
0
1
0
,2
0
,1
0
2
2
0
,2
02
1
0
,1
02
2
}
.
.
2
1
].)(.)([4
1
ln{
:usgives(1)equationRearraging(1).equationfrom
ncorrelatioimpliedthedeterminetoneedweeverythinghavewe,MgdetermininAfter
22
21
In most cases, the implied correlation turns out to be different than the historical correlation, just like implied and historical volatilities are different. Implied correlation is interpreted as
the correlation the market is using to price basket options.
From equations (1) and (2), we can see that M2 is an increasing function of , and basket is
an increasing function of M2. We also know that the call value is an increasing function of
volatility. This shows that the price of a call basket is monotonically increasing with .
So, if the market is pricing a high implied correlation, it means there is a common sentiment
that both underlying assets are going to increase together. This creates a high demand on call
baskets for hedging purposes, pushing their price higher.
Once this implied correlation is determined, it can either be used to price other basket options,or it can be used to build a correlation matrix, which is in turn used to price basket options on
more than 2 underlying assets.
4 Digital options
4.1 Popularity of digital options
A European digital (or binary) option with maturity T and strike K on an underlying futures
contract F
T
t, is the option whose payoff at T is K F T T 1 .
Digital or binary options are very common in the commodities market. They are used in the
pricing of popular payoffs, like European barrier options (knock in and knock out options).The reason for that popularity is that the option structure disregards the highly improbable
events with very large payoffs, reducing the premium for the client.
An example of such a payoff is shown below:
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We can see that the investor is protected against the rise in the price of the underlying up tothe level of 175%, after which a rebate of 25% of the nominal is paid. Since this event is
considered as highly unlikely, giving away this payoff is not much of a problem for the
investor, and will reduce the overall cost of the structured product.
On the other hand, if the price drops, the client is not exposed to the downside loss unless thedrop is bigger than 25%. In the latter case, the investor experiences a 100% participation loss,
which is capped at 50% of the nominal.
The above payoff is easily structured using vanilla and digital calls and puts.
In fact, if we denote
N Nominal amount invested in the structure
C(K) Vanilla call option with strike K
P(K) Vanilla put option with strike K
DC(K) Digital call option with strike KDP(K) Digital put option with strike K
Payoff = P(50%) – P(75%) – 0.25 x N x DP(75%) + C(100%) – C(175%) – 0.5 x N x
DC(175%)
100755025 175150125
Percentage
Moneyness
Payoff
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4.2 Volatility’s impact on the price of a digital option
Unlike vanilla call options, the price of a digital call is not an increasing function of volatility.This can be explained intuitively.
In fact, two cases are to be considered.
Case 1: The digital call is in the money (ITM)
In this case, the intrinsic value of the call is 1, but the time value is negative, since with time,
we risk to fall back below the strike. So, when the option is ITM, we want the volatility to beas low as possible, ideally 0, to remain above the strike. Traders say they are short volatility.
Case 2: The digital call is out of the money (OTM)
In this case, the answer is not very obvious. It depends on how close we are to the strike price.If we are very far below, we are long volatility. If we are very close below, we are short
volatility. Some calculations have to be made to determine the exact level at which digital
call’s Vega changes signs.
Briefly speaking, the digital call option behaves differently than vanilla call options when itcomes to volatility. The reason is that the Digital’s upside potential is limited.
Let’s study the above results mathematically.
)().,0().,0(,
N(0,1)~ z where
.
2
1ln
2
1ln..
).,0(1.),0(
:getwe,QmeasureneutralforwardthetochangingBy
1.
220
2
20
2
0
.2
1
0
00
T
.
00
2
0
d N T Bd z QT B DC So
d T
T K
F
z T F
K W K e F K F
K F QT B E T B DC
e E DC
T
T
T T
W T T T
T
T
T T K F
Q
K F
dt r Q
T
T T
T
T T
T
t
Now, we will study the first derivative of d2 with respect to volatility, which will directly give
us the behavior of the price since N(x) is an increasing function of x.
T T
K
F
d
T
2
1
.
ln
2
0
2
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Particularities of the Commodities Market
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Case 1: The digital call is in the money (ITM)
00ln 20
d K F
T
So, the price of the digital call is always decreasing with volatility. (displays a -ve Vega)
ITM Digital Call Option (Moneyness = 150%)
0
0.2
0.4
0.6
0.8
1
1.2
1 %
7 %
1 3 %
1 9 %
2 5 %
3 1 %
3 7 %
4 3 %
4 9 %
5 5 %
6 1 %
6 7 %
7 3 %
7 9 %
8 5 %
9 1 %
9 7 %
1 0 3 %
1 0 9 %
1 1 5 %
1 2 1 %
1 2 7 %
1 3 3 %
1 3 9 %
1 4 5 %
Volatilities
D i g i t a l C a l l P r i c e
Case 2: The digital call is out of the money (OTM)
otherwised
T
K
F
whend
K
F
T
T
0
ln
200ln
2
0
20
So, the price of the digital call is increasing with volatility. (+ve Vega) up to a certain point,where more volatility will actually hinder the call’s value.
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OTM Digital Call Option (Moneyness = 50%)
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
1 %
9 %
1 7 %
2 5 %
3 3 %
4 1 %
4 9 %
5 7 %
6 5 %
7 3 %
8 1 %
8 9 %
9 7 %
1 0 5 %
1 1 3 %
1 2 1 %
1 2 9 %
1 3 7 %
1 4 5 %
1 5 3 %
1 6 1 %
1 6 9 %
1 7 7 %
1 8 5 %
1 9 3 %
Volatilities
D i g i t a l C a l l P r i c e
4.3 Smile’s impact on the price of a digital option
Traders hedge digital call options by selling the replicating portfolio. The market standard is
to approximately replicate a digital call option with strike K by a vanilla call spread, centered
at K with a very small difference between the 2 strikes K 1 and K 2.
K 1 K 2 K Spot Price
Payoff
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In practice, this replicating portfolio rarely has the same value as the formula for the price of adigital call derived in the previous section. This difference is clearly seen when the volatilitysmile is not flat.
Intuitively, when the volatility skew is upward sloping, the replicating portfolio consists of
buying a large number of calls with strike K1 and selling the same number of calls with strike
K2. Since C(K1) is priced with a volatility V1 a bit smaller than V2 ( which is used to priceC(K2) ), the replicating portfolio is worth less than the case where V1 = V2 (when the
volatility smile is flat). The opposite can be said when the volatility is downward sloping.
So, given the same parameters (Maturity, Strike, Volatility, and interest rates), and the
different volatility smiles shown below,
We can say, DC(Smile 1) > DC(Smile 2) > DC(Smile 3)
To show this result mathematically:
smilevolatilityof SlopeCall Vanillaof Vega
K
d N T B
K dK
K d r T K S C
K
r T K S C
K
r T K K S C r T K K S C r T K K S DC
.,,,,,,,,
,,,,
.2
,,,,lim,),(,,
)().,0(
0
2
First, we observe that when the smile is flat (B&S world), the slope is zero, and the equation
reduces to the formula obtained in the previous section.
Second, since the Vega of a vanilla call option is always positive, the price is only affected by
the sign of the slope parameter. In the case it is positive, the price decreases and vice versa.
Smile 1
Smile 2
Smile 3
Strikes
Volatilities
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5 References
1- Commodities and Commodity Derivatives: Modeling and Pricing for Agriculturals, Metals, and Energy by Helyette ,Prof. Geman
2- Options, Futures and Other Derivatives, by John C. Hull3- Commodity trade definitions and pricing, ECDA Analytics, Natixis
4- Interest Rate Models –Theory and Practice, by Damiano Brigo · Fabio Mercurio