.


International Journal of Energy Economics and Policy | Vol 10 • Issue 1 • 2020 471

International Journal of Energy Economics and 
Policy

ISSN: 2146-4553

available at http: www.econjournals.com

International Journal of Energy Economics and Policy, 2020, 10(1), 471-480.

Crude Oil Option Market Parameters and Their Impact on the 
Cost of Hedging by Long Strap Strategy

Bartosz Łamasz*, Natalia Iwaszczuk

AGH University of Science and Technology, Poland. *Email: blamasz@zarz.agh.edu.pl

Received: 22 August 2019 Accepted: 02 November 2019 DOI: https://doi.org/10.32479/ijeep.8613

ABSTRACT

This study aims to examine the impact of selected market parameters of the European crude oil options on the hedging costs and break-even points 
(BEPs) in the long strap strategy. The paper analyses the impact of the following market parameters: Volatility and the future price of crude oil, the 
strike price and time to expiration. The theoretical aspect consisted in using the black model to calculate the value of the option price and the long strap 
strategy BEP in the condition of ever-changing market parameters. These calculations, by determining implied volatilities of the options, have been 
adapted to the actual data from the exchange market for the options on WTI futures contract. It was made possible owing to the quik strike platform 
made available by a CME group exchange. To obtain information about the impact of volatility, time and price of futures on the costs of hedging and 
BEPs in the long strap strategy, the authors calculated the Greeks (delta, gamma, vega and theta) for the crude oil options. Having done that, not only 
could they determine the direction but also the power of impact that the parameters had on the final results in the long strap strategy.

Keywords: Commodity Options, Crude Oil Price Risk, Long Strap Option Strategy 
JEL Classifications: G13, G32

1. INTRODUCTION

In the era of progressive globalisation, which, among others, 
results in faster information exchange, market risk, understood 
as price risk, plays an increasingly important role. The news of 
economic events is reflected in price fluctuations of financial and 
non-financial assets. Such an issue can be particularly seen in the 
commodity market and it has significant consequences for both 
producers and consumers.

Crude oil undoubtedly belongs to the group of raw materials that 
are of great importance for the global economy, as it is a raw 
material which has been an essential energy source in the world 
for many years. Currently, about 1/3 of primary energy is produced 
due to the process of oil crude refining. However, there are two 
basic restrictions regarding the increase in world oil consumption. 
First of all, it is non-renewable energy source, which means that 
its resources will run out in the future. Secondly, oil deposits are 

unevenly distributed around the world and their largest proven 
reserves are situated in such countries as: Saudi Arabia, Iran, Iraq, 
Kuwait, United Arab Emirates, Libya, Venezuela and Russia. 
Some of these countries cannot be classified as economically and 
politically stable. Moreover, the small number of oil suppliers 
(especially in the future) is the reason to consider this market 
oligopolistic.

Accordingly, there is high probability of disturbing the supply 
of this raw material, which is frequently reflected in significant 
price fluctuations. Price fluctuations are particularly important 
for oil producers, as they largely determine the state of the 
economy of these countries. A prominent example of a country 
that is dependent on oil crude market is Russia, where export of 
this raw material and its products constitutes more than a half of 
Russia’s total export (in 2017 it was 60% of total export). Hence, 
it is claimed that in the long term, fluctuations in world oil prices 
may have a destructive effect on the stability of the oil industry 

This Journal is licensed under a Creative Commons Attribution 4.0 International License



Łamasz and Iwaszczuk: Crude Oil Option Market Parameters and Their Impact on the Cost of Hedging by Long Strap Strategy

International Journal of Energy Economics and Policy | Vol 10 • Issue 1 • 2020472

in this country. Chikunov et al. (2019) emphasise that there is a 
strong need to develop scientific approaches which may lead to 
the evaluation and diagnosis of financial risk in Russian oil sector.

On the other hand, Arour et al. (2011) and Khamis et al. (2018) 
revealed that there is a significant impact of oil price fluctuations 
on the stock market at Gulf Cooperation Council Countries (Qatar, 
Kuwait, Oman, KSA, Bahrain, UAE). Tabash and Khan (2018) 
showed a strong interdependence between crude oil price volatility 
and the gross domestic product of UEA and Saudi Arabia. The 
impact of oil prices on the economy of Saudi Arabia was also 
studied by Foudeh (2017).

Also, oil prices fluctuations significantly influence the economic 
situation of countries that import large quantities of this raw 
material. These mostly include developing Asian countries, such 
as China and India. The consequences of oil price fluctuations 
can be noticed in some of the industries of the above-mentioned 
countries, as they determine the costs of production. For instance, 
this appears in the aviation sector. Kathiravan et al. (2019) showed 
that crude oil price fluctuations (WTI, Brent, Dubai) between 2007 
and 2018 had a significant impact on return rates on shares of 
companies involved in the aviation sector in India. Additionally, 
changes in crude oil price affect the economic activity of both, 
developing and developed countries (Cunado and Perez de Garcia, 
2005; Hamilton, 2003; Edelstein and Kilian, 2009).

Crude oil prices also affect such aspects of the economy as: the 
market value of crude oil companies, the inflation rate in oil 
importing countries, as well as the price of alternative energy 
sources (An et al., 2019). Following that, it is of crucial importance 
to search for the hedging methods that may help to avoid negative 
consequences of the changes of crude oil price on the economy of 
countries and individual enterprises, especially the refinery sector.

This paper focuses on the commodity (crude oil) options as well 
as on an option strategy structured with the derivatives. The long 
strap strategy discussed in the paper, when appropriate used, offers 
the opportunity of hedging effectively against the risk of crude oil 
price fluctuation. One of the objectives of the paper is to present 
the method for structuring the above-mentioned strategy as well as 
show some patterns for calculating the final result of its application. 
In addition, for the effective application of the long strap strategy, 
it is essential to have the skills sufficient to set out successfully 
the strategy break-even and stop-loss points. These numbers are 
strongly correlated with the price of the commodity options, which 
is why more light should be shed on the option pricing model. To 
this end, the authors decided to use the Black’s model.

All the calculations made in the paper are referred to the 
commodity options in which the price of WTI oil futures contracts 
(traded on the NYMEX) is the underlying instrument. To determine 
the implied volatility of options other than ATM (at-the-money), 
the quikstrike platform available on the CME group website was 
also used for calculations. The calculated option premiums (option 
prices) were used to calculate the costs of hedging and break-even 
points (BEPs) in the long strap strategy and analyse their response 
to changing values of selected market parameter, which was the 

purpose of this paper. A more precise determination of the power 
of impact of these parameters was possible through meticulous 
calculation and analysis of four Greeks: Delta, gamma, vega and 
theta. They provided some information about the change in the 
cost of hedging in the long strap strategy when changing a selected 
market parameter by a unit. Practical application of the Greeks 
is manifested in supporting decisions of price risk managers. By 
calculating the Greeks, they can adjust their option parameters and 
strategies to the expected directions of changes in the commodity 
prices.

The remainder of the paper is organized as follows. Section 2 
provides a literature review, in section 3 the construction of long 
strap strategy is presented. Section 4 discusses the used method 
and data of the study and final results is presented in sector 5. 
Finally, section 6 concludes the research paper.

2. LITERATURE REVIEW

Options, as derivative instruments with non-symmetric risk 
distribution, may be used by market participants in many different 
ways. Speculators trade options to profit from drops, rises or 
stagnation in prices of the underlying instrument. On the other side 
of the market there are hedgers, who consider options as tools to 
protect them against the risk of price fluctuation in financial assets 
(e.g., stock, bonds, currency exchange rates) or commodities (gold, 
oil, gas). However, each option market participant tends to focus on 
two key issues: the price of the base instrument on the last trading day 
(contract expiration day) and the option price. While the first one is 
unpredictable and may fluctuate freely in the future, the option price 
is already known on the date of taking a position in an option contract. 
This value is the key from the point of view of a success of followed 
option strategies created by short or long positions in different options.

Calculation of the value of an option (i.e. the option price or 
option’s premium) is the most complicated process, as one may 
see by comparing valuation methods applied for different types of 
derivatives. In a nutshell, a valuation of these derivatives is a search 
for answers to the question: how much should a buyer of an option 
pay for the option for the price to be fair1 for each party? The issue 
is rather complex as it requires setting the value of an option when 
bought (or sold). In turn, the value should counterbalance the payout 
to which the option buyer is entitled at a certain moment in the future 
i.e., on the last trading day. Consequently, many scholars tried to find 
the most effective options pricing model (2-1) and understand the 
relationship between market parameters and option’s premium (2-2).

2.1. Option Pricing
The first attempts at valuating options date to the turn of 19th and 
20th century. They are deemed modelled after Louis Bachelier’s 
doctoral thesis of 1900. The thesis focused also on modelling 
stock prices and, according to Bachelier, the prices were to move 
according to the arithmetic Brownian motion. Many years later, 
in 1960, several papers were published that pushed forward the 
search for option valuation models. They mostly applied to stock 

1 “Fair price” is a price which does not open the door to a potential arbitration 
on the market.



Łamasz and Iwaszczuk: Crude Oil Option Market Parameters and Their Impact on the Cost of Hedging by Long Strap Strategy

International Journal of Energy Economics and Policy | Vol 10 • Issue 1 • 2020 473

options. The most important papers devoted to the issue were 
written by J. Boness and P. Samuelson (Smithson, 1998).

However, the work by Black and Scholes published in 1973 is 
considered the breakthrough in the search for the model to estimate 
the option price. They presented a model valuating the European 
call option in which underlying asset was a dividend-free stock. The 
solution presented by Black and Scholes is based, as initially assumed, 
on the structure of a risk-free portfolio, using European options (Black 
and Scholes, 1973). The model has quickly gained popularity and 
the scientific circles made regular attempts at its improvement. In 
consequence, as early as 1973, Merton, using Black and Scholes’s line 
of thinking as the basis, developed a model valuating the European 
call option in which the underlying asset was a stock with a fixed 
dividend payable before the expiry date of the option (Merton, 1973).

Continued efforts to improve the option pricing model and expand 
it by adding more types of underlying assets have led to the 
discovery of a method for valuating of commodity derivatives. It 
happened as early as in 1976 and the discovery was made by Black. 
Also note that the solution presented by Black applied not only to 
valuation of European commodity options but also to futures and 
forwards for commodities (Black, 1976).

Apart from the European commodity options, valuation of 
American options remains important, as their holders have the right 
to exercise them on any day before their expiration days. Similarly, 
to the European options, the options are traded on commodity 
exchanges for raw materials and energy such as NYMEX or ICE 
and are the most popular among participants of the exchanges. The 
leading method used to valuate the American option is the analytical 
approximation method combining solutions proposed in numerical 
models (including Black’s model) and analytical models (such as 
binominal model (Cox, 1979)). Many research papers describing 
different approaches to the valuation of the options have been 
written. One of the first was the solution proposed by Barone-Adasi 
and Whaley (1987). In reference to Whaley’s concepts (1986), 
they developed both the spot price model to valuate commodity 
options as well as options for commodity future contracts. In their 
study, they used Black’s model for the European commodity option 
onto which the early exercise premium was added, arising from 
the optional early exercising the American option. Structured as 
described above, the model was discussed in papers published 
in subsequent years and the subject was further expanded by 
Bjerksund and Stensland (1993, 2002) as well as other analysts.

The literature offers examples of many other commodity option 
valuation models (both European and American), based on Black’s 
concepts; however, they have been modified at a rather large scale. 
Revisions and attempts at improving valuation of these options 
were made by Schwartz (1997), Shreve (2004) and Clark (2011). 
However, until now, the solutions proposed by Black come as the 
most popular valuation model for these options.

2.2. Black’s Model in Commodity Options and the 
Greeks
Black’s solutions dating back to 1976 is a fairly easy method of 
valuating European commodity prices. The model is based on 

the assumption that the underlying instrument prices move as 
particles in the Brownian movement, which is a particular case 
of a stochastic Wiener process. In turn, calculation of the option 
premium concentrates on searching for a value balancing the 
payout for the option buyer on the contract expiry date. Eventually, 
with the respective initial assumptions, prices of European 
commodity call and put options are expressed with the following 
formulas (Clark, 2014; Hull, 2012):

  V e f N d KN d
0

C r T

0 1 2

d

= ( )− ( ) 
−  (1)

  V e KN d f N d
0

P r T

2 0 1

d

= −( )− −( ) 
−  (2)

Where

   d =
ln
f

K
+
2
T

T
1

0

2







σ

σ
 (3)

   d =

f

K 2
T

T
2

0

2

ln






−

σ

σ
 (4)

And
V
0

C – Price of the commodity call option,
V
0

P – Price of the commodity put option,
f0– The future price of the underlying instrument,
K– Strike price of the option,
rd– Continuously compounded riskless interest rate,
T– Time left until the option expiration,
σ2– Yearly variance rate of return for the underlying instrument,
N– The standard normal cumulative distribution function.

The impact of some of the option parameters is analysed by using 
the Greeks. They provide information on how the option may 
change as a result of a changing a parameter by a unit. In this 
paper, four different Greeks were used: delta, gamma, vega and 
theta. Their mathematical interpretation and formulas which can 
be used to calculate the value based on Black model, are presented 
in Table 1.

The first of the presented coefficients, the delta, shows the option 
price response to a change in the future price of the underlying 
instrument (a commodity) by a unit. Delta is also identified with 
the proxy for the probability that an option will expire in the money 
(Hull, 2012). In turn, by calculating the gamma, one can learn 
about the impact of the future price of the underlying asset on the 
change in the delta. Mathematically, the delta is a partial second 
derivative of the option price against the price of the underlying 
asset. Another Greek, vega, is a partial first derivative of the option 
price calculated against price volatility of the underlying asset. Its 
value indicates how and by how much the option price is going 
to change given a 1% p.p. increase (or a decrease) in volatility of 
the underlying assets for which the option was sold. Theta offers 
some crucial information on the impact on the number of days to 
the last trading day on the option price. Theta shows a potential 
change in the value of the option when reducing the time to the 
last trading day by a time unit. the structure of the Black’s model 



Łamasz and Iwaszczuk: Crude Oil Option Market Parameters and Their Impact on the Cost of Hedging by Long Strap Strategy

International Journal of Energy Economics and Policy | Vol 10 • Issue 1 • 2020474

would indicate a year as the unit but the typically considered unit 
is a day instead. Such analysis provides therefore more precise 
information on the impact of theta on the option price (Bittman, 
2009; Hull, 2012).

Works by Taleb (1996), Hull (2012) or Węgrzyn (2013) offer some 
information on the Greeks covered in this paper and the method 
for their shaping depending on the type of the analysed options. 
In this paper, the Greeks have been used to analyse the power of 
the impact of some selected market parameters on the level of 
costs and BEPs in the long strap strategy. However, before they 
were determined, the method of structuring the strategy had been 
described briefly as well as equations helpful to determine the BEP 
points in the strategy were presented in more detail.

3. LONG STRAP STRATEGY – STRUCTURE 
AND BEP POINTS

The long strap strategy is formed by a combination of long 
positions in call and put options for the same underlying asset 
and the same time to expiration. Furthermore, the strike prices of 
these options are set at an identical level. It is also assumed that 
the number of positions taken in the call option is twice as high 
as the number of position in the put option. Therefore, in order 
to calculate the final result achieved in the long strap strategy, it 
is necessary to set out the final results in long positions of both 
put and call options with the right parameters and, subsequently, 
placing them in a manner appropriate for the strategy.

Assuming that K is the strike price of an option, c is a unit option 
premium for the call option and p is a unit option premium for 
the put option and FT is the future price of a commodity on the 
expiration day, the profit function of long position in n call options 
(C(K)) is,

 
( ) ( )

T

T T

nc                    if      F <K
C K

n F K c      if       F K
−

− ≥


=  −
 (5)

and the profit function of long position in m put options (P(K)) is

 
( ) ( )T T

T

m K F p     if     F <K
P K

mp                    if     F K
 −

=
≥

−

−

 (6)

Applying the symbols used in the equation (5) and (6) and taking 
into account the structuring of the long strap strategy (i.e. assuming 
that n=2m), its final result R(K) is described by the following 
equation:

 

( )
( )T T

T T

m K F p 2c           if      F <K 
R K 1

2m F K c p        if      F K
2

 − − −


=   
− − − 


≥

  
(7)

From the point of view of a successful application of each 
option strategy, the BEP is a very important component. 
A BEP is a price level of the underlying asset giving the final 
result equal to 0 in a strategy. Therefore, the BEP indicates the 
necessary change in the price to avoid losses on the strategy. 
In case of a two-sided hedges i.e. protecting both against 
price increases and price drops (with the long strap strategy 
as a good example of such hedging), there are two BEPs. The 
first one informs about the minimum price drop (against the 
expiration price). When reached, the BEP will compensate for 
the losses resulting from paid option premiums. In the paper, 
it is recorded as BEPD. The price of the underlying asset above 
which the long strap strategy (in case of an increase in prices) 
will be profitable is the second BEP, recorded as BEPU. Their 
calculation method is closely related to the equation (7) and 
described as follows:

  BEP K+
p

2
+c

U
=  (8)

  BEP K p 2c
D
= − −  (9)

Since in the long strap strategy the number of positions 
occupied in the call option is twice as high as in the put option, 
the difference (K–BEPD) will be always twice bigger than the 
(BEPU–K) difference. Accordingly, the strategy must be applied 
predominantly when expecting rises in the prices of the underlying 
asset. Since the long strap strategy also uses a long position in the 

Table 1: Mathematical interpretation and equations for selected Greeks
The Greeks The Greeks as a derivative Calculating the value of a coefficient based on Black’s model
Delta

δ =
∂
∂
V

f
θ δ ω ω= e N( d )

r T

1

d−

Gamma
γ

δ
=
∂
∂f γ σ

=
−
e N'(d )

f T

r T

1

d

Vega
v =

∂
∂
V

σ
v e fN (d ) T

r T

1

d

= − '

Theta
θ = −

∂
∂
V

T

 

θ ω ω ω
σ

= r e fN d r e KN( d )
e fN'(d )

2 T

d r T

1

d -r T

2

r T

1
d d

d

−
−

( )−




−

Source: own study based on (Węgrzyn, 2013), (Alexander, 2008), (Hull, 2012)

V–unit price of an option; ω=1 for the call option ω= –1 for the put option; N’(x) – standard normal probability density function, calculated from 

2x /2e
N'(x)

2 T

−

=
π

; other symbols and 

in the black’s model



Łamasz and Iwaszczuk: Crude Oil Option Market Parameters and Their Impact on the Cost of Hedging by Long Strap Strategy

International Journal of Energy Economics and Policy | Vol 10 • Issue 1 • 2020 475

put option, earnings are possible also in case of drops in the prices 
of the underlying assets. However, the price must fall below BEPD.

When analysing the equations (8) and (9), one may also notice 
that the BEPD distance from the option price K is equal to the total 
cost (TC) of hedging which is equal to the sum of paid option 
premiums. In turn, the distance between BEPD and the strike price 
K is twice shorter. In consequence, when TC stands for the TCs 
paid in the long strap strategy (TC = p + 2c), (8) and (9) equations 
may look as follows:

   BEP =K+
TC

2
U  (10)

   BEP K TCD = −  (11)

and when subtracted, give the dependency:

   BEP BEP
3

2
TC

U D
− =  (12)

The equation (12) shows that the distance between the BEPs is 
strictly connected with the costs borne in the long strap strategy 
and is always 1.5 times higher. Consequently, a change in the value 
of BEPU and BEPD will result from a change in prices of options 
(option premiums) used in the strategy. For this reason, further 
on, when analysing the impact of individual market parameters, 
the notion of the costs of the strategy and the distance between 
the BEPs will be substituted.

4. DATA AND METHODOLOGY

As argued above, the BEPs in the long strap strategy (which also 
represent the TCs) are also dependent on the set strike price of an 
option. To analyse the dependence, the authors decided to analyse 
different situations, depending on the value of market parameters 
necessary to calculate prices of commodity options.

Market parameters of commodity options are based on the 
following initial assumptions:
• A 2% annual risk-free interest rate
• The future price of a commodity upon taking a position in the 

option contract for the commodity was 66 monetary units
• Volatility of the future price of the underlying asset 

(a commodity) expressed for the year, reached one of the 
three levels: 25%, 35% or 45%

• The last trading day of an option is 30, 20 or 10 days
• The strike price equal to 64, 65, 66 (ATM options), 67 or 68 

monetary units.

The market parameters used in the equations have been selected 
carefully to offer the best reflection of the situation prevailing 
on the real market of WTI oil future option prices. The annual 
volatility of the commodity price typically oscillates within a 
0.2-0.5, which prompted the authors’ decision to use three different 
values from the range (0.25, 0.35 and 0.45) in the analyses. The 
future price of oil set at 66 USD/b is, in turn, a WTI price of June 
01, 2018 and refers to an option with the delivery date falling in 
July (its expiration date is June 15, 2018 and a potential sale of 
the commodity takes place in July). In addition, these analyses 

also took into account periods of 30, 20 and 10 days since the 
number of option transactions on NYMEX or ICE is clearly the 
highest in the last month of the option activity. Accordingly, it was 
concluded that such option (and, consequently, option strategies) 
were the most interesting from the point of view of an exchange 
trading participant.

From the perspective of matching a theoretical valuation of options 
with their real (market) parameters, problematic is the relationship 
between volatility and the strike price. In the literature, it is referred 
to as the volatility smile. The name comes from the shape (curve) 
of the graph of the function where the strike price is an argument 
and values are equal to the volatility implied in the option. In 
this case, the graph of the function is a U-shaped parabole whose 
minimum is around the ATM strike price.

Therefore, the occurrence of this phenomenon differentiates the 
level of volatility of the implied options for the same instrument, 
having the same strike prices. To determine volatility of the implied 
options issued for WTI oil with their strike price different that the 
ATM option strike price, the authors used quikstrike platform. The 
platform is available on the CME Group website and offers many 
embedded tools which can be used also (apart from its capability 
to determine the above-mentioned volatility) to valuate the option 
by using different models (using the Black models as well as other 
models, see: (CME Group, 2018a, b)) or checking the effectiveness 
of a strategy by simulating the future prices of the underlying asset.

5. RESULTS

The tools available on quickstrike platform helped to determine the 
implied volatility for the options which strikes price are different 
that the price of the ATM option (in this case option with K=66 
USD/b). The results of their application are presented in Figure 1. 
Values of other market parameters required for option valuation 
were assumed at the level identical to the level in section 4. The 
option strike price was changed every 1 USD and ranged between 
64 and 68 USD/b.

When calculated the implied volatility for options which are not 
ATM options (K ≠ 66), three cases were analysed. The first one 
assumed volatility of the ATM option at 0.25, the second one at 
0.35 and the third one at 0.45. According to Figure 1, a decrease 
in the strike price (K = 64 or K = 65) resulted in an increasing 
volatility of the option. For an option with the strike price K = 67 or 
K = 68, the volatilities were lower than the volatility of a respective 
ATM option. The number of days to expiration of an option did not 
have any major impact on the changes in the volatility parameter 
in the analysed options.

The value of implied volatilities obtained by using the quikstrike 
platform will allow to determine the prices of the European call 
and put options for the future WTI oil price. Black’s model was 
used to calculate the price of the option. Obtained values allowed 
to determine the level of BEPs and the TC in the long strap 
strategy. To analyse how the TCs respond to a variable strike price, 
percentage changes in the BEPs were calculated while increasing 
the strike price by a unit. All results are presented in Table 2.



Łamasz and Iwaszczuk: Crude Oil Option Market Parameters and Their Impact on the Cost of Hedging by Long Strap Strategy

International Journal of Energy Economics and Policy | Vol 10 • Issue 1 • 2020476

The calculations led to formulation of the following conclusions 
connecting the costs of hedging by the long strap strategy with the 
strike price and other market parameters of an option:
1. Increased volatility (with other option parameters remaining 

unchanged) increases the TCs in the long strap strategy. it is 
a consequence of a growth in the strike price both for the put 
and the call option

2. A reduction in the number of days to expiration (of the option 
strategy), with other option parameters unchanged, reduces the 
strike price both for the put and the call option. The dependence 
translates into a reduction of the TCs of the long strap strategy

3. By increasing the strike price, the put option price goes down 
and the call price goes up. Since in the long strap strategy twice 
as many positions are taken in call options than in put options, 
a higher strike price (with other parameters unchanged) limits 
the TCs of the long strap strategy.

These dependencies are derived directly from the Black’s model 
as well as from the method underlying the long strap strategy 

structuring. Furthermore, they also affect the rate of changing the 
total strategy costs in case of a growing strike price (Figure 2).

As presented above, an increase in the strike price would typically 
result in decreasing distance between BEPU and BEPD (TCs). The 
rate of the decreases is slower when setting increasingly higher 
strike prices. Therefore, a reduction in the TCs of the long strap 
strategy will become the most intensive when changing the strike 
price from 64 to 65 units and hardly perceptible (usually below 2%) 
when “moving” from 67 to 68 units. Taking the impact of other 
market parameters into account, note that the rate of reduction of 
distances between BEPs in the long strap strategy is going down 
when volatility of the underlying asset is increasing. On the other 
hand, limitation of the number of days to expiration increases the 
rate of the reduction, with the only exception being a change in the 
strike price from 67 to 68 units. In that case, as a result of a shorter 
time to expiration (with other parameters remaining unchanged), 
the reduction rate of the TCs remains unchanged. Taking the lowest 
of the volatilities used (about 0.25), an increase in the TCs as a 

Source: Own analysis

64 65 66 67 68
ATM (σ=0,25; T=30) 0.2661 0.2569 0.25 0.246 0.2419
ATM (σ=0,25; T=20) 0.2653 0.2572 0.25 0.2464 0.2423
ATM (σ=0,25; T=10) 0.2666 0.2567 0.25 0.2454 0.2436
ATM (σ=0,35; T=30) 0.3663 0.3575 0.35 0.3452 0.3425
ATM (σ=0,35; T=20) 0.3664 0.3561 0.35 0.3456 0.343
ATM (σ=0,35; T=10) 0.3667 0.3562 0.35 0.3459 0.3425
ATM (σ=0,45; T=30) 0.4655 0.4561 0.45 0.445 0.443
ATM (σ=0,45; T=20) 0.4654 0.4572 0.45 0.4454 0.4428
ATM (σ=0,45; T=10) 0.4649 0.456 0.45 0.4466 0.4433

0.2

0.25

0.3

0.35

0.4

0.45

0.5

V
ol

at
ili

ty

Strike price

Figure 1: Changes in volatility implied for the future price of WTI oil options at set market parameters of ATM options

Source: Own analysis

65
66

67
68

-25%

-20%

-15%

-10%

-5%

0%

5%

10 20 30 10 20 30 10 20 300,25 0,35
0,45

To
ta

l c
os

ts
 c

ha
ng

es

Volatility

Numbers of days to expiration

0%-5%

-5%-0%

-10%--5%

-15%--10%

-20%--15%

-25%--20%
Strike price

Figure 2: Percentage changes in total cost with a strike price changed by a unit



Łamasz and Iwaszczuk: Crude Oil Option Market Parameters and Their Impact on the Cost of Hedging by Long Strap Strategy

International Journal of Energy Economics and Policy | Vol 10 • Issue 1 • 2020 477

result of changing the strike price from 67 to 68 units may be also 
observed for a 10-day strategy.

The dependencies presented so far did not show the power of 
impact of individual market parameters onto costs in the long strap 
strategy. For this reason, the final stage of the analysis consisted 
in calculating Greek coefficients delta, gamma, vega and theta 
for options for the future prices of WTI oil. Such approach made 
it also possible to calculate the Greeks for strategies built on the 
basis of the above-mentioned options.

The results presented in the Table 3 show that, in the long strap 
strategy using options with strike prices set below the present 
future price of a commodity (66 USD/b), an increase in any 
parameter: the strike price, volatility or the days to expiration 

(while other parameters remain unchanged) reduces the delta. 
In case when the strategy structure plans to use options with 
strike prices above the ATM option strike price, the impact 
of volatility and time to expiration onto delta is reversed 
while the direction of the strike price impact does not change. 
Accordingly, the TC of the long strap strategy with a lower 
strike price (K = 64 or K = 65/b) will be growing quite rapidly 
in case of increasing future prices of WTI. This outcome will 
be particularly noticeable in case of 10 days to expiration strike 
days when a growth in the price of an oil barrel by 1 USD 
translates into an increase in the sum of paid option prices by 
approximately 1.3 USD (for K = 64 USD/b). However, the 
cost response to changes in the future price of the commodity 
at higher strike prices (K = 67 or K = 68 USD/b) will be 
significantly less intensive. Even more so, given short periods 

Table 2: TC, BEP and final results for long strap option strategy
Parameters Option premium BEP and TC BEP and TC changes (%)

σ T K Call Put D U TC D U TC
0,2661 30 64 3,13 1,13 56,6 67,7 7,40
0,2569 30 65 2,46 1,46 58,62 68,19 6,38 3,57 0,72 −13,78
0,25 30 66 1,88 1,88 60,35 68,83 5,65 2,95 0,94 −11,39
0,246 30 67 1,41 2,41 61,77 69,62 5,23 2,35 1,15 −7,43
0,2419 30 68 1,02 3,02 62,94 70,53 5,06 1,89 1,31 −3,31
0,2653 20 64 2,80 0,80 57,6 67,2 6,40
0,2572 20 65 2,12 1,12 59,64 67,68 5,36 3,54 0,71 −16,25
0,25 20 66 1,54 1,54 61,38 68,31 4,62 2,92 0,93 −13,81
0,2464 20 67 1,08 2,08 62,76 69,12 4,24 2,25 1,19 −8,23
0,2423 20 68 0,72 2,72 63,84 70,08 4,16 1,72 1,39 −1,89
0,2666 10 64 2,41 0,41 58,77 66,61 5,23
0,2567 10 65 1,68 0,68 60,96 67,02 4,04 3,73 0,62 −22,70
0,25 10 66 1,09 1,09 62,73 67,63 3,27 2,90 0,91 −19,14
0,2454 10 67 0,65 1,65 64,05 68,48 2,95 2,10 1,26 −9,59
0,2436 10 68 0,36 2,36 64,92 69,54 3,08 1,36 1,55 4,29
0,3663 30 64 3,83 1,83 54,51 68,75 9,49
0,3575 30 65 3,20 2,20 56,41 69,3 8,59 3,49 0,80 −9,48
0,35 30 66 2,64 2,64 58,09 69,95 7,91 2,98 0,94 −7,99
0,3452 30 67 2,15 3,15 59,54 70,73 7,46 2,50 1,12 −5,65
0,3425 30 68 1,74 3,74 60,78 71,61 7,22 2,08 1,24 −3,22
0,3664 20 64 3,36 1,36 55,92 68,04 8,08
0,3561 20 65 2,71 1,71 57,86 68,57 7,14 3,47 0,78 −11,63
0,35 20 66 2,15 2,15 59,54 69,23 6,46 2,90 0,96 −9,52
0,3456 20 67 1,68 2,68 60,96 70,02 6,04 2,38 1,14 −6,50
0,343 20 68 1,29 3,29 62,13 70,94 5,87 1,92 1,31 −2,76
0,3667 10 64 2,77 0,77 57,69 67,16 6,31
0,3562 10 65 2,09 1,09 59,73 67,64 5,27 3,54 0,71 −16,47
0,35 10 66 1,52 1,52 61,43 68,29 4,57 2,85 0,96 −13,27
0,3459 10 67 1,07 2,07 62,79 69,1 4,21 2,21 1,19 −8,02
0,3425 10 68 0,72 2,72 63,84 70,08 4,16 1,67 1,42 −1,11
0,4655 30 64 4,54 2,55 52,37 69,81 11,63
0,4561 30 65 3,93 2,93 54,21 70,4 10,79 3,51 0,85 −7,17
0,45 30 66 3,39 3,39 55,83 71,08 10,17 2,99 0,97 −5,81
0,445 30 67 2,90 3,90 57,3 71,85 9,70 2,63 1,08 −4,59
0,443 30 68 2,48 4,48 58,56 72,72 9,44 2,20 1,21 −2,68
0,4654 20 64 3,93 1,93 54,21 68,9 9,79
0,4572 20 65 3,32 2,32 56,04 69,48 8,96 3,38 0,84 −8,51
0,45 20 66 2,77 2,77 57,69 70,15 8,31 2,94 0,96 −7,29
0,4454 20 67 2,29 3,29 59,13 70,94 7,87 2,50 1,13 −5,22
0,4428 20 68 1,88 3,88 60,36 71,82 7,64 2,08 1,24 −2,96
0,4649 10 64 3,15 1,15 56,55 67,73 7,45
0,456 10 65 2,51 1,51 58,47 68,27 6,53 3,40 0,80 −12,34
0,45 10 66 1,96 1,96 60,12 68,94 5,88 2,82 0,98 −10,00
0,4466 10 67 1,50 2,50 61,5 69,75 7,40 2,30 1,17 −6,46
0,4433 10 68 1,12 3,12 62,64 70,68 6,38 1,85 1,33 −2,55
Source: Own analysis σ: Volatility, T: Number of day to expiration, K: Strike price, D=BEPD, U=BEPU, TC: Total cost



Łamasz and Iwaszczuk: Crude Oil Option Market Parameters and Their Impact on the Cost of Hedging by Long Strap Strategy

International Journal of Energy Economics and Policy | Vol 10 • Issue 1 • 2020478

of strategy delivery and low volatility (0.25), an increase in the 
future prices of oil may result in a reduced sum of the option 
prices. Volatility, a strike price and days to expiration do not have 
a major impact on the delta for the strategy with ATM options 
and typically oscillates between 0.54 and 0.58.

The value of the gamma which, in turn, informs about the response 
of the delta to fluctuations in the future price of a commodity 
reaches the highest value in long strap strategies with ATM options. 
By reducing the time to expiration, gamma is increased. In turn, 
higher volatility limits the level of the gamma.

A change in the strike price has a rather similar way of affecting the 
value of the two latter Greeks, vega and theta (its absolute value). 

Each reaches it highest level (assuming that other parameters 
are not volatile) for the ATM option. It means that the long strap 
strategies using such options are the most sensitive to volatility 
of the underlying instrument (as indicated by vega) and time to 
expiration (theta). Vega and theta also increase when influenced 
by volatility of options used. Strategies involving ATM options 
were an exception hereof as their vega is not dependent on 
volatility of the underlying instrument, observed on the market at 
the time. In turn, a reduction in the time to strike limits the vega 
and increases theta.

Accordingly, costs in the long strap strategies with ATM options, 
realised within 30 days from the moment of taking a position in 
such strategy, will be the most sensitive to shifts in volatility. The 

Table 3: The value of Greeks for single options and long strap strategy
Parameters Options Long Strap

σ T K Delta C Delta P Gamma Vega Theta C Theta P Delta Gamma Vega Theta
0,2661 30 64 0,67 −0,33 0,07 0,068 −0,03 −0,03 1,01 0,22 0,205 −0,09
0,2569 30 65 0,60 −0,40 0,08 0,073 −0,03 −0,03 0,79 0,24 0,219 −0,09
0,25 30 66 0,51 −0,49 0,08 0,075 −0,03 −0,03 0,54 0,25 0,226 −0,09
0,246 30 67 0,43 −0,57 0,08 0,074 −0,03 −0,03 0,29 0,25 0,223 −0,09
0,2419 30 68 0,35 −0,65 0,08 0,070 −0,03 −0,03 0,04 0,24 0,209 −0,08
0,2653 20 64 0,70 −0,30 0,08 0,054 −0,04 −0,04 1,10 0,25 0,161 −0,13
0,2572 20 65 0,61 −0,39 0,10 0,059 −0,05 −0,05 0,84 0,29 0,177 −0,14
0,25 20 66 0,51 −0,49 0,10 0,062 −0,05 −0,05 0,54 0,31 0,185 −0,14
0,2464 20 67 0,41 −0,59 0,10 0,060 −0,05 −0,04 0,22 0,31 0,180 −0,14
0,2423 20 68 0,31 −0,69 0,09 0,054 −0,04 −0,04 −0,07 0,28 0,163 −0,12
0,2666 10 64 0,76 −0,24 0,11 0,034 −0,07 −0,07 1,29 0,32 0,101 −0,20
0,2567 10 65 0,65 −0,35 0,13 0,041 −0,08 −0,08 0,94 0,40 0,122 −0,24
0,25 10 66 0,51 −0,49 0,15 0,044 −0,08 −0,08 0,52 0,44 0,131 −0,25
0,2454 10 67 0,36 −0,64 0,14 0,041 −0,08 −0,08 0,09 0,42 0,123 −0,23
0,2436 10 68 0,24 −0,76 0,12 0,034 −0,06 −0,06 −0,29 0,35 0,101 −0,19
0,3663 30 64 0,64 −0,36 0,05 0,071 −0,04 −0,04 0,91 0,16 0,213 −0,13
0,3575 30 65 0,58 −0,42 0,06 0,074 −0,04 −0,04 0,74 0,17 0,222 −0,13
0,35 30 66 0,52 −0,48 0,06 0,075 −0,04 −0,04 0,56 0,18 0,226 −0,13
0,3452 30 67 0,46 −0,54 0,06 0,075 −0,04 −0,04 0,38 0,18 0,225 −0,13
0,3425 30 68 0,40 −0,60 0,06 0,073 −0,04 −0,04 0,20 0,18 0,219 −0,12
0,3664 20 64 0,66 −0,34 0,06 0,057 −0,06 −0,06 0,97 0,19 0,170 −0,19
0,3561 20 65 0,59 −0,41 0,07 0,060 −0,06 −0,07 0,77 0,21 0,180 −0,20
0,35 20 66 0,52 −0,48 0,07 0,062 −0,07 −0,07 0,55 0,22 0,185 −0,20
0,3456 20 67 0,44 −0,56 0,07 0,061 −0,07 −0,06 0,33 0,22 0,183 −0,19
0,343 20 68 0,37 −0,63 0,07 0,058 −0,06 −0,06 0,11 0,21 0,175 −0,18
0,3667 10 64 0,70 −0,30 0,09 0,038 −0,10 −0,11 1,11 0,26 0,113 −0,31
0,3562 10 65 0,61 −0,39 0,10 0,042 −0,11 −0,11 0,84 0,29 0,125 −0,34
0,35 10 66 0,51 −0,49 0,10 0,044 −0,12 −0,12 0,53 0,31 0,131 −0,35
0,3459 10 67 0,41 −0,59 0,10 0,042 −0,11 −0,11 0,22 0,31 0,127 −0,34
0,3425 10 68 0,31 −0,69 0,09 0,038 −0,10 −0,10 −0,07 0,28 0,115 −0,30
0,4655 30 64 0,62 −0,38 0,04 0,072 −0,05 −0,06 0,85 0,13 0,216 -0,16
0,4561 30 65 0,57 −0,43 0,05 0,074 −0,05 −0,06 0,72 0,14 0,222 −0,17
0,45 30 66 0,53 −0,47 0,05 0,075 −0,06 −0,06 0,58 0,14 0,226 −0,17
0,445 30 67 0,48 −0,52 0,05 0,075 −0,06 −0,05 0,44 0,14 0,226 −0,17
0,443 30 68 0,43 −0,57 0,05 0,074 −0,05 −0,05 0,30 0,14 0,223 −0,16
0,4654 20 64 0,63 −0,37 0,05 0,058 −0,08 −0,08 0,90 0,16 0,175 −0,25
0,4572 20 65 0,58 −0,42 0,06 0,060 −0,08 −0,09 0,73 0,17 0,181 −0,25
0,45 20 66 0,52 −0,48 0,06 0,061 −0,08 −0,08 0,56 0,17 0,184 −0,25
0,4454 20 67 0,46 −0,54 0,06 0,061 −0,08 −0,08 0,39 0,17 0,184 −0,25
0,4428 20 68 0,41 −0,59 0,06 0,060 −0,08 −0,08 0,22 0,17 0,180 −0,24
0,4649 10 64 0,67 −0,33 0,07 0,040 −0,14 −0,14 1,01 0,21 0,119 −0,42
0,456 10 65 0,59 −0,41 0,08 0,042 −0,15 −0,15 0,78 0,23 0,127 −0,44
0,45 10 66 0,51 −0,49 0,08 0,044 −0,15 −0,15 0,54 0,24 0,131 −0,45
0,4466 10 67 0,43 −0,57 0,08 0,043 −0,15 −0,15 0,30 0,24 0,129 −0,44
0,4433 10 68 0,36 −0,64 0,08 0,041 −0,14 −0,14 0,07 0,23 0,122 −0,41
Source: Own analysis



Łamasz and Iwaszczuk: Crude Oil Option Market Parameters and Their Impact on the Cost of Hedging by Long Strap Strategy

International Journal of Energy Economics and Policy | Vol 10 • Issue 1 • 2020 479

impact of the change of time on the sum of the paid out option 
premiums will be then the most observable in strategies with the 
shortest strike date (10 days), the highest anticipated volatility 
(0.45) for which ATM options may be also used.

6. CONCLUSION

The method used to structure the long strap strategy and setting 
the BEPs presented in the paper showed that the strategy should 
be used largely in case of anticipated increases in the prices of 
the underlying asset. Apart from the long put option, the strategy 
also uses a long position in the call option and, as a result, it offers 
the opportunity of generating a profit also with dropping prices 
of the underlying asset. From the point of view of a successful 
application of the long strap strategy, the levels of the BEPs are 
important (BEPD, BEPU), they are directly related to the TCs 
i.e. with the sum of paid option premiums.

The key objective of the paper was to present the manner in 
which the costs will be changing in the long strap strategy when 
influenced by fluctuations of such parameters as the future price 
of a commodity and its volatility (non-elective parameters) as well 
as the strike price and the time to expiration (elective parameters). 
The analyses are based on the data from the WTI market for oil 
futures. For this reason, the results of the study are particularly 
valuable to entities actively participating in oil trading and exposed 
to the risk of oil price fluctuations.

Calculations underlying the conclusions showed that an increase 
in the strike prices reduces the costs. Their drop rate is particularly 
intensive when put option-based strategies are created which are 
effectively out-of-the-money options (in consequence, the call 
options used are in-the-money options, which results from the 
manner in which the long strap strategy is structured). Furthermore, 
note that an option with the strike price used in the structure of the 
strategy was different from the ATM strike price by not more than 
2 USD per barrel. Among the key conclusions from the analysis 
the way in which the implied volatility for such options is formed. 
The coefficient is higher for options with lower strike prices and 
lower for options with higher strike prices than the ATM option. 
The phenomenon contributes to some intensified reductions in 
the TCs while increasing the strike prices used in the long strap 
strategy. Also, a cost increase was also observed when increasing 
volatility and extending the time to expiration.

By calculating the Greeks for the long strap strategy, one could 
learn about the difference in the strike price when a market 
parameter changes by a unit. It turned out that an increase in the 
TC caused by a growth in the WTI oil future prices was particularly 
intensive when using options with lower strike prices. In turn, 
gamma, vega and theta were the highest for ATMs.

Consequently, the Greeks contributed some important knowledge 
about the dependence between the strike price and market 
parameters which affect the option. The coefficients may be then 
considered important tools supporting the decision-making process 
when selecting an option in the process of building strategies 
hedging against the price risk. These decisions may also concern 

a change in option positions when responding to some received 
values of the Greeks calculated for the strategy (a dynamic 
hedging). Therefore, one may conclude that the above-mentioned 
tools improve the decision-making flexibility and, consequently, 
give an opportunity for generating improved results by reducing 
the costs of hedging or obtaining a higher amount from an option 
payout function.

7. ACKNOWLEDGEMENT

This publication is financed by AGH University of Science and 
Technology within the subsidy for maintaining reserach potential.

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