Crack spread option pricing with copulas

Abstract

A copula-based 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 copula-based 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)) mis-price 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.

Appendices

Appendix A

1.1 Survey of Frank, Clayton, and Gumbel copulas

Three one parameter bivariate Archimedean copulas are given in Table 4. Nelsen (1999: 94–97) tabulates 22 one-parameter 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.

Table 4 Frank, Clayton, and Gumbel Families

1.2 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 (non-parametric) version. The steps are:

• Step 1 Estimate from the data the Kendall’s correlation using the non-parametric or distribution free measure:

$$ {\tau_E} = {\left( {\begin{array}{*{20}{c}} n \\ 2 \\ \end{array} } \right)^{{ - 1}}}\sum\limits_{{i < j}} {{\text{Sign}}\left[ {\left( {{X_i} - {X_j}} \right){{\left( {{Y_i} - Y} \right)}_i}} \right]} $$

• 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:

$$ {Z_i} = \frac{{{\text{number}}\left\{ {\left( {{X_i},{Y_j}} \right){\text{ such that }}{X_j} < {X_i}{\text{ and }}{Y_j} < {Y_i}{ }} \right\}}}{{n - 1}} $$

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.

1.3 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:

1.3.1 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.

1.3.2 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

2.1 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

$$ C\left( {{S_T}} \right) = \max \left( {{S_1} - {S_2} - K,0} \right) $$

(B1)

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 closed-form solution for spread options with correlated asset prices and a non-zero strike price (see Boyle et al. (2004) and Carmona and Durrleman (2003) for discussions of pricing spread options with non-zero strike prices).

2.2 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 European-style 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 risk-free rate r. The Kirk and Aron (1995) formula to price the crack spread is as follows:

$$ C = \left( {{S_2} + K} \right)\left[ {{e^{{ - rT}}}\left( {{S_A}N\left( {{d_1}} \right) - N\left( {{d_2}} \right)} \right)} \right] $$

(B2)

where

$$ \begin{gathered} \quad \quad \quad \quad \quad {S_A} = \frac{{{S_1}}}{{{S_2} + K}} \hfill \\ \sigma = \sqrt {{\sigma_1^2 + {{\left[ {{\sigma_2}\frac{{{S_2}}}{{{S_1} + K}}} \right]}^2} - 2\rho {\sigma_1}{\sigma_2}\frac{{{S_2}}}{{{S_1} + K}}}} \hfill \\ \end{gathered} $$

${d_1} = \frac{{\ln \left( {{S_A}} \right) + \left( {r + {\sigma^2}/2} \right)\left( {T - t} \right)}}{{\sigma \sqrt {{T - t}} }}$ and ${d_2} = {d_1} - \sigma \sqrt {{T - t}}$

The Kirk and Aron (1995) formula is an approximate model since there is no closed-form solution to a spread option on correlated assets with a non-zero exercise price.

2.3 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 risk-free rate. Define the instantaneous linear correlation between two assets as ρ. In the two assets binomial tree, each node has four branches and the risk-neutral probabilities for each branch is given by:

$$ {p_1} = \frac{1}{4}\left[ {1 + \sqrt {{\Delta t}} \left( {\frac{{{\mu_1}}}{{{\sigma_1}}} + \frac{{{\mu_2}}}{{{\sigma_2}}}} \right) + \rho } \right] $$

(B3.a)

$$ {p_2} = \frac{1}{4}\left[ {1 + \sqrt {{\Delta t}} \left( {\frac{{{\mu_1}}}{{{\sigma_1}}} - \frac{{{\mu_2}}}{{{\sigma_2}}}} \right) - \rho } \right] $$

(B3.b)

$$ {p_3} = \frac{1}{4}\left[ {1 - \sqrt {{\Delta t}} \left( {\frac{{{\mu_1}}}{{{\sigma_1}}} + \frac{{{\mu_2}}}{{{\sigma_2}}}} \right) + \rho } \right] $$

(B3.c)

$$ {p_4} = \frac{1}{4}\left[ {1 - \sqrt {{\Delta t}} \left( {\frac{{{\mu_1}}}{{{\sigma_1}}} - \frac{{{\mu_2}}}{{{\sigma_2}}}} \right) - \rho } \right] $$

(B3.d)

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 ${\widehat{d}_i} = {e^{{ - {\sigma_i}\Delta t}}}$.

2.4 Non-copula 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 log-returns 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 log-returns are used). In this case, payoff values can be simulated from the terminal distribution. For a European-type spread option, it is not necessary to simulate the entire price path. The non-copula Monte Carlo simulation approach, however, is also based on the same two weak assumptions (i.e., normal distribution of log-returns and linear dependence) as the Kirk and Aron (1995) model and the binomial approach.