Abstract
A copulabased approach for pricing crack spread options is described. Crack spread options are currently priced assuming joint normal distributions of returns and linear dependence. Statistical evidence indicates that these assumptions are at odds with the empirical data. Furthermore, the unique features of energy commodities, such as mean reversion and seasonality, are ignored in standard models. We develop two copulabased crack spread option models using a simulation approach that address these gaps. Our results indicate that the Gumbel copula and standard models (binomial, and Kirk and Aron (1995)) misprice a crack spread option and that the Clayton model is more appropriate. We contribute to the energy derivatives literature by illustrating the application of copula models to the pricing of a heating oil–crude oil “crack” spread option.
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Notes
Throughout the article we use the generic term “crack spread option” although the model specifically refers to a heating oil–crude oil crack spread option (NYMEX trading symbol CH). The copula approach is similar for pricing a gasoline oil–crude oil “crack” spread option (NYMEX trading symbol CG).
We would like to acknowledge Bob Biolsi of the CME Group for providing us with insight into how the settlement prices are determined.
More generally, the marginal distributions and joint distribution of the random variables must be elliptical distributions (such as multivariate normal, multivariate tdistribution, logistic distribution, and Laplace distributions).
We wish to sincerely thank the anonymous referees for suggesting the Laurence and Wang (2009) article which has helped improve an earlier version of this article considerably.
In the daily NYMEX crude oil and heating oil futures price data (519 parallel observations) over the period February 3, 2003, to March 3, 2005, we find no evidence of seasonality for both price series. The chisquare test statistic values are 6.84 (pvalue = 0.81) for crude oil and 14.52 (pvalue = 0.21) for heating oil respectively.
Dornier and Queruel (2000) include an additional term in order to consider seasonality in a meanreverting model.
While we only need the payoff at maturity to price the Europeantype crack spread option discussed in this paper, we generate the full random paths for the assets as an illustration since the copula simulation technique can be used for pricing other path dependent options (e.g., American type) in the energy sector.
The options data include the following items: “symbol,” “trading date,” “contract month,” “open interest,” “Call (c)/Put (p),” “strike price,” “High,” “Low,” “Last,” “Settlement Price (Actual Price),” and “Total Volume.”
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Acknowledgements
We also wish to acknowledge valuable comments from participants at the Institute of Industrial Engineers Research Conference, Nashville, Tennessee, May, 2007 and Institute of Industrial Engineers Research Conference, Orlando, Florida, May, 2006. Dr. Herath acknowledges research funding from the Social Sciences and Humanities Research Council (SSHRC) of Canada.
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Appendices
Appendix A
Survey of Frank, Clayton, and Gumbel copulas
Three one parameter bivariate Archimedean copulas are given in Table 4. Nelsen (1999: 94–97) tabulates 22 oneparameter Archimedean copula families. The parameter θ in each case measures the degree of dependence and controls the dependence between the two variables. For instance, when θ→0 there is no dependence and if θ→∞ there is perfect dependence. The dependence parameter θ which characterizes each family of Archimedean copulas can be related to Kendall’s Tau. This property is used to empirically determine the applicable copula form.
Empirically identifying a copula form
The first step in modeling and simulation is identifying the appropriate copula form. Genest and Rivest (1993) provide the following procedure (fit test) to identify an Archimedean copula. The method assumes that a random sample of bivariate data (X _{ i }, Y _{ i }) for i = 1,2.....,n is available. Assume that the joint distribution function H has an associated Archimedean copula \( {C_{\theta }} \), and then the fit allows us to select the appropriate generator φ. The procedure involves verifying how closely different copulas fit the data by comparing the closeness of the copula (parametric version) with the empirical (nonparametric) version. The steps are:

Step 1
Estimate from the data the Kendall’s correlation using the nonparametric or distribution free measure:

Step 2
Identify an intermediate variable Z _{ i } = F (X _{ i }, Y _{ i }) with a distribution function \( K(z) = \Pr \left( {{Z_i} \leqslant z} \right) \). Construct an empirical (non parametric) estimate of this distribution as follows:
The empirical version of the distribution function K(z) is K _{ E }(z) = proportion of Z _{ i } ≤ z

Step 3
Construct the parametric estimate of K (z). The relationship between this distribution function and the generator is given by \( K(z) = z  \frac{{\varphi (z)}}{{\varphi \prime (z)}} \), where φ′(z) is the derivative of the generator and 0 ≤ z ≤ 1. We provide below the specific form of K(z) for the three Archimedean copulas surveyed in this paper.

(i)
$$ {\text{Frank Copula}}\quad \quad \quad \quad K(z) = \frac{{\theta z  \left[ {1  \exp \left( {\theta z} \right)} \right]\ln \left[ {\frac{{\exp \left( {  \theta z} \right)  1}}{{\exp \left( {  \theta } \right)}}} \right]}}{\theta } $$(A1)

(ii)
$$ {\text{Clayton Copula}}\quad \quad \quad \quad K(z) = \frac{{z\left( {1 + \theta  {z^{\theta }}} \right)}}{\theta } $$(A2)

(iii)
$$ {\text{Gumbel Copula}}\quad \quad \quad \quad K(z) = \frac{{z\left( {\theta  \ln z} \right)}}{\theta } $$(A3)
Repeat Step 3 for several different families of copulas, i.e., several choices of generator functions, φ(.). By visually examining the graph of K(z) versus K _{ E } (z) or using statistical measures such as minimum square error analysis, one can choose the best copula. This copula can be used in modeling dependence and simulation.
Copula simulation procedures for the Clayton and Gumbel families
In order to simulate outcomes from the bivariate distribution of asset prices, we use the procedures developed by Genest (1987) and Lee (1993) for the Clayton copula:
Algorithm A.3.1 Generating bivariate outcomes from the Clayton family

Step 1
Generate independent (0, 1) uniform random numbers u _{1} and u _{2}

Step 2
Set \( {X_1} = {F^{{  1}}}\left( {u{}_1} \right) \)

Step 3
Compute \( u_2^{*} = {\left[ {1 + u_1^{{  \theta }}\left( {u_2^{{  \theta /1 + \theta }}  1} \right)} \right]^{{  1/\theta }}} \)

Step 4
\( {X_2} = {G^{{  1}}}\left( {u_2^{*}} \right) \)
The above algorithm is computationally expensive for simulating bivariate data from the Gumbel distribution (see discussion in Frees and Valdez 1998). An alternate algorithm suggested by Marshall and Olkin (1988) for the construction of compound copulas can be used instead. The algorithm is as follows:
For the Gumbel copula, the generator and Laplace transform are given in Equation (c) Column C, Appendix Table 4. The inverse generator is equal to the Laplace transform of a positive stable variate \( \gamma \sim {\text{St}}\left( {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }},1,\Theta, 0} \right) \) where \( \Theta = {\left( {\cos \left( {\frac{\prod }{{2\theta }}} \right)} \right)^{\theta }} \) and θ > 0.
Algorithm A.3.2 Generating bivariate outcomes from the Gumbel family

Step 1
Simulate a positive Stable variate \( \gamma \sim {\text{St}}\left( {\widehat{\alpha },1,\Theta, 0} \right) \)

Step 2
Simulate two independent uniform [0,1] random numbers u _{1} and u _{2}

Step 3
Set \( {X_1} = {F^{{  1}}}\left( {u_1^{*}} \right) \) and \( {X_2} = {G^{{  1}}}\left( {u_2^{*}} \right) \) where \( u_i^{*} = \phi \left( {\frac{1}{\gamma }\ln {u_i}} \right) \) and \( \varphi (t) = \exp \left( {  {t^{{\frac{1}{\theta }}}}} \right) \) for \( i \in \left[ {1,2} \right] \)
Cherubini et al. (2004) suggest the following procedure to simulate a positive random variable \( \gamma \sim {\text{St}}\left( {\widehat{\alpha },1,\Theta, \delta } \right) \):

Step 1 (a)
Simulate a uniform random variable \( \upsilon = U\left( {\frac{{  \prod }}{2},\frac{\prod }{2}} \right) \)

Step 1 (b)
Independently draw an exponential random variable (ε) with mean = 1

Step 1 (c)
\( {\theta_0} = \arctan \left( {\tan \left( {\frac{{\prod {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }}}}{2}} \right)} \right)/{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }} \) and compute
$$ z = \frac{{\sin {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }}\left( {{\theta_0} + \upsilon } \right)}}{{{{\left( {\cos {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }}{\theta_0}\cos \upsilon } \right)}^{{\frac{1}{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }}}}}}}}}{\left[ {\frac{{\cos \left( {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }}{\theta_0} + \left( {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }}  1} \right)\upsilon } \right)}}{\varepsilon }} \right]^{{\frac{{\left( {1  {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }}} \right)}}{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }}}}}}} $$ 
Step 1 (d)
\( \gamma = \Theta z + \delta \)
Appendix B
Current approaches to pricing crack spread options
Spread options are options whose payoffs depend on the difference between two or even three assets. The payoff of a two asset spread option at maturity for a call is
where K is the strike price of the option, S _{ i } is the price of the underlying asset i. When K = 0, the spread option is equivalent to an exchange option (see Margrabe 1978). There is no closedform solution for spread options with correlated asset prices and a nonzero strike price (see Boyle et al. (2004) and Carmona and Durrleman (2003) for discussions of pricing spread options with nonzero strike prices).
Kirk and Aron (1995) model
The Kirk and Aron (1995) crack spread option model is based on the Black (1976) futures contract model. It is a Europeanstyle spread option on futures contracts modified by equating the futures price to a general asset price S _{ i }, with strike price K, volatility of the respective assets σ _{ i }, the correlation (linear dependence) between the two assets ρ and the riskfree rate r. The Kirk and Aron (1995) formula to price the crack spread is as follows:
where
\( {d_1} = \frac{{\ln \left( {{S_A}} \right) + \left( {r + {\sigma^2}/2} \right)\left( {T  t} \right)}}{{\sigma \sqrt {{T  t}} }} \) and \( {d_2} = {d_1}  \sigma \sqrt {{T  t}} \)
The Kirk and Aron (1995) formula is an approximate model since there is no closedform solution to a spread option on correlated assets with a nonzero exercise price.
Binomial lattice approach
To price a crack spread option, we can use the binomial lattice approach for two correlated assets (see Kamrad and Ritchken 1991; Boyle 1988; Pickles and Smith 1993). Define the asset pair \( \left\{ {{S_1}(t),{S_2}(t)} \right\} \)over time t as having a bivariate lognormal density. Let the drift rate be μ _{ i } where σ _{ i } is the instantaneous standard deviation of the i ^{th} asset and r is the riskfree rate. Define the instantaneous linear correlation between two assets as ρ. In the two assets binomial tree, each node has four branches and the riskneutral probabilities for each branch is given by:
The value for the up and down movements for each asset i = 1,2 is given by \( {\widehat{u}_i} = {e^{{{\sigma_i}\Delta t}}} \) and \( {\widehat{d}_i} = {e^{{  {\sigma_i}\Delta t}}} \).
Noncopula Monte Carlo simulation approach
Monte Carlo simulation can be used to price spread options including crack spread options. If the volatility of the assets and their correlation are assumed to be deterministic, then one could estimate the joint distribution of the underlying assets which can be used directly in the simulation. For example, if the marginal distribution of the logreturns of each asset is assumed to be normal, then in the simulation, one can sample directly from a joint normal distribution (or joint lognormal if returns instead of logreturns are used). In this case, payoff values can be simulated from the terminal distribution. For a Europeantype spread option, it is not necessary to simulate the entire price path. The noncopula Monte Carlo simulation approach, however, is also based on the same two weak assumptions (i.e., normal distribution of logreturns and linear dependence) as the Kirk and Aron (1995) model and the binomial approach.
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Herath, H.S.B., Kumar, P. & Amershi, A.H. Crack spread option pricing with copulas. J Econ Finan 37, 100–121 (2013). https://doi.org/10.1007/s1219701191711
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DOI: https://doi.org/10.1007/s1219701191711
Keywords
 Copulas
 Nonlinear Dependence
 Crack Spread Options
 Energy Derivatives
 Oil and Gas
 Risk Management
JEL Classification
 C: Mathematical and Quantitative Methods