A Dynamic Lévy Copula Model for the Spark Spread

Abstract

We present a model for the spark spread on energy markets which is implied by a two-dimensional model for the electricity and gas spot prices. The marginal price processes are supposed to follow sums of (not necessarily Gaussian) Ornstein-Uhlenbeck components and the main focus of this paper is on the two-dimensional dependence modeling via Lévy copulas. We will introduce a specific class of skewed Lévy copulas and estimate the complete model on data from UK markets. Further, due to the arithmetic structure of the model, we are able to employ Fourier transform techniques to derive semi-analytic expressions for option prices.

Appendix

Lévy Copulas

We will shortly review the main definitions and results needed for the concept of two-dimensional Lévy copulas for Lévy measures with positive support. For further informations on Lévy copulas we refer to [7, 9]. We first need a multidimensional extension of the notion of increasing functions.

Definition 9.1 (F-volume). 

Let $F: S \rightarrow \bar{ \mathbb{R}}$ for some subset $S \subset \bar{ {\mathbb{R}}}^{d}$. Then for ${u}^{1} = (u_{1}^{1},\ldots,u_{d}^{1})$, ${u}^{2} = (u_{1}^{2},\ldots,u_{d}^{2}) \in S$ with ${u}^{1} \leq {u}^{2}$ and $[{u}^{1},{u}^{2}] \subset S$, the F -volume is defined by

$$\displaystyle{V _{F}([{u}^{1},{u}^{2}]):=\sum \limits _{ j_{1}=1}^{2}\ldots \sum \limits _{ j_{d}=1}^{2}{(-1)}^{j_{1}+\ldots +j_{d} }F(u_{1}^{j_{1} },\ldots,u_{d}^{j_{d} })}$$

Definition 9.2 (d-increasing). 

A function $F: S \rightarrow \bar{ \mathbb{R}}$ for some subset $S \subset \bar{ {\mathbb{R}}}^{d}$ is called d -increasing if V _F ([u, v]) ≥ 0 for all u, v ∈ S with u ≤ v and [u, v] ⊂ S. 

Definition 9.3 (Grounded functions). 

A function $F: {[0,\infty ]}^{d} \rightarrow [0,\infty ]$ is grounded if $F(x) = 0,\ x = (x_{1},\ldots,x_{d})$ as soon as any of $x_{1},\ldots,x_{d}$ equals 0.

Note that in general the notion of d-increasing functions does not coincide with functions that increase in each margin. However, in the following lemma some connection is made in the two-dimensional and positive case.

Lemma 9.1.

Let $F: {[0,\infty ]}^{2} \rightarrow [0,\infty ]$ be grounded and in C ^1,1 then F is 2-increasing, if $\frac{{\partial }^{2}} {\partial u\partial v}F(u,v) > 0,\forall (u,v) \in {[0,\infty ]}^{2}$

Next, we need the definition of the tail integral of a positive Lévy measure.

Definition 9.4 (Tail integral). 

A d-dimensional tail integral is a function $U: {[0,\infty ]}^{d} \rightarrow [0,\infty ]$ such that

• 1. ( − 1)^d U is a d-increasing function.

• 2. U is equal to zero if one of its arguments is equal to ∞.

• 3. U is finite everywhere except at zero and $U(0,\ldots,0) = \infty$.

The margins $U_{k},\ k = 1,\ldots,d$ of a tail integral U are defined by

$$\displaystyle{U_{k}(x) = U(0,\ldots,0,\mathop{x}\limits_{\mathop{ \uparrow }\limits_{ k\text{-th position}}},0,\ldots,0).}$$

The name “tail integral” makes more sense, as soon as the following link between Lévy measures and tail integrals is made:

Definition 9.5 (Tail integral of a positive Lévy measure). 

The tail integral U of a Lévy measure ν on [0, ∞)^d for $x = (x_{1},\ldots,x_{d})$ is defined by

• $U(x) = 0\text{, if }x_{k} = \infty \text{ for any }k = 1,\ldots,d$

• $U(x) =\nu ([x_{1},\infty ) \times \ldots \times [x_{d},\infty ))\text{ for }x \in [0,\infty {)}^{d}\setminus \{0\}$

• $U(0,\ldots,0) = \infty$

We are now ready to introduce Lévy copulas for two-dimensional Lévy measures with positive support. We remark that the general framework would need some small modification.

Definition 9.6 (Lévy copula). 

A two-dimensional Lévy copula for Lévy processes with positive Lévy jumps is a function $F: {[0,\infty ]}^{2} \rightarrow [0,\infty ]$, which

• Is grounded

• Two-increasing

• Has uniform margins, that is, $F(x,\infty ) = F(\infty,x) = x,\ \forall x \in [0,\infty ]$

In analogy to Sklar’s Theorem for copulas, the following theorem forms the main result in the theory of Lévy copulas.

Theorem 9.1 (Sklar’s theorem for Lévy copulas). 

Let $(X_{t},Y _{t})$ be a two-dimensional Lévy process with positive jumps having tail integral U and marginal tail integrals U ₁ and U ₂ . There exists a positive Lévy copula F such that

$$\displaystyle\begin{array}{rcl} U(x_{1},x_{2}) = F(U_{1}(x_{1}),U_{2}(x_{2})),\ \forall x_{1},\ x_{2} \in [0,\infty ].& &{}\end{array}$$

(9.16)

It is unique on $\mathrm{Ran\ }U_{1} \times \mathrm{ Ran\ }U_{2}$ , the product of the ranges of one-dimensional tail integrals.Conversely, if F is a positive Lévy copula and (X _t ), (Y _t ) are two one-dimensional Lévy processes with tail integrals U ₁ ,U ₂ , then there exists a two-dimensional Lévy process such that its tail integral is given by (9.16).

Finally, we recall the major representatives of Lévy copulas applied in practice.

Proposition 9.3 (Archimedean Lévy copula). 

Let $\phi: [0,\infty ] \rightarrow [0,\infty ]$ be a strictly decreasing convex function such that ϕ(0) = ∞ and ϕ(∞) = 0. Then

$$\displaystyle{F(x,y) {=\phi }^{-1}(\phi (x) +\phi (y))}$$

defines a two-dimensional Lévy copula.

In analogy to ordinary copulas, the Lévy copulas constructed this way are called Archimedean Lévy copulas and ϕ is called the generator.

Example 9.1.

A few examples of Archimedean Lévy copulas, each with respect to a parameter θ > 0, following the naming in [10] wherever given, are:

• 1. The Clayton-Lévy copula generated by $\phi _{\mathrm{C}}(u) = {u}^{-\theta }$:

  $$\displaystyle{F_{\theta }(x,y) ={ \left ({x}^{-\theta } + {y}^{-\theta }\right )}^{-1/\theta }}$$

  In this case a greater parameter θ means higher dependence of jumps. This includes F _⊥ for θ → 0 and $F_{\uparrow \uparrow }$ for θ → ∞.

• 2. The Gumbel-Lévy copula generated by $\phi _{\mathrm{G}}(u) ={ \left [\log (u + 1)\right ]}^{-\theta }$

  $$\displaystyle{F_{\theta }(x,y) =\exp \left \{{\left [{\left (\log (x + 1)\right )}^{-\theta } + {(\log (y + 1))}^{-\theta }\right ]}^{-1/\theta }\right \} - 1.}$$

• 3. The complementary Gumbel-Lévy copula generated by the inverted generator: $\phi _{\bar{\mathrm{G}}}(u) =\exp ({u}^{-\theta }) - 1$

  $$\displaystyle{F_{\theta }(x,y) ={ \left \{\log \left [\exp ({x}^{-\theta }) +\exp ({y}^{-\theta }) - 1\right ]\right \}}^{-1/\theta }.}$$

• 4. Generated by $\phi _{\exp }(u) = \frac{1} {{e}^{\theta u}-1}$

  $$\displaystyle{F_{\theta }(x,y) = \frac{1} {\theta } \log \left [{\left ( \frac{1} {{e}^{\theta x} - 1} + \frac{1} {{e}^{\theta y} - 1}\right )}^{-1} + 1\right ].}$$

Remark 9.2.

Even though unnamed in literature, the last example should be included in any relevant list. It could play an important role as an alternative to the Clayton-Lévy, since, in the above list, these are the only two cases where the inverse of the partial derivative can be given in closed form. The relevance of this feature comes with the simulation algorithm of Lévy copulas.