We provide an integral representation for the (implied) copulas of dependent random variables in terms of their moment generating functions. The proof uses ideas from Fourier methods for option pricing. This representation can be used for a large class of models from mathematical finance, including L\'evy and affine processes. As an application, we compute the implied copula of the NIG L\'evy process which exhibits notable time-dependence.
Copulas provide a complete characterization of the dependence structure between random variables and link in a very elegant way the joint distribution with the marginal distributions via Sklar's Theorem. However, they are a rather static concept and do not blend well with stochastic processes which can be used to describe the random evolution of dependent quantities, e.g. the evolution of several stock prices. Therefore other methods to create dependence in stochastic models have been developed. Multivariate stochastic processes spring immediately to mind, for example Lévy or affine processes (cf. e.g. Sato 1999, Duffie, Filipović, and Schachermayer 2003, Cuchiero, Filipović, Mayerhofer, and Teichmann 2011or Muhle-Karbe, Pfaffel, and Stelzer 2012), while in mathematical finance models using time-changes or linear mixture models have been developed; see e.g. Luciano and Schoutens (2006), Luciano and Semeraro (2010), Kawai (2009), Eberlein and Madan (2010) or Khanna and Madan (2009), to mention just a small part of the existing literature. In these approaches however the copula is typically not known explicitly. Another very interesting approach is due to Kallsen and Tankov (2006), who introduced Lévy copulas to characterize the dependence structure of Lévy processes.
In this note, we provide a new representation for the (implied) copula of a multidimensional random variable in terms of its moment generating function. The derivation of the main result borrows ideas from Fourier methods for option pricing, and the motivation stems from the knowledge of the moment generating function in most of the aforementioned models. This paper is organized as follows: in Section 2 we provide the representation of the copula in terms of the moment generating function; the results are proved for random variables for simplicity, while stochastic processes are considered as a corollary. In Section 3 we provide two examples to showcase how this method can be applied, for example, in performing sensitivity analysis of the copula with respect to the parameters of the model. Finally, Section 4 concludes with some remarks.
Let R n denote the n-dimensional Euclidean space, •, • the Euclidean scalar product and R n -the negative orthant, i.e. R n -= {x ∈ R n : x i < 0 ∀i}. We consider a random variable X = (X 1 , . . . , X n ) ∈ R n defined on a probability space (Ω, F, IP). We denote by F the cumulative distribution function (cdf) of X and by f its probability density function (pdf). Let C denote the copula of X and c its copula density function. Analogously, let F i and f i denote the cdf and pdf respectively of the marginal X i , for all i ∈ {1, . . . , n}. In addition, we denote by
We denote by M X the (extended) moment generating function of X:
for all u ∈ C n such that M X (u) exists. Let us also define the set
In the sequel, we will assume that the following condition is in force. Theorem 2.2. Let X be a random variable that satisfies Assumption (D). The copula of X is provided by
where u ∈ [0, 1] n and R ∈ R.
Proof. Assumption (D) implies that F 1 , . . . , F n are continuous and we know from Sklar’s Theorem that the copula of X is unique and provided by McNeil, Frey, and Embrechts (2005, Theorem 5.3) for a proof in this setting and Rüschendorf (2009) for an elegant proof in the general case.
We will evaluate the joint cdf F using the methodology of Fourier methods for option pricing. That is, we will think of the cdf as the ‘price’ of a digital option on several fictitious assets. Let us define the function
(2.4) and denote by g its Fourier transform. Then we have that
where we have applied Theorem 3.2 in Eberlein, Glau, and Papapantoleon (2010). The prerequisites of this theorem are satisfied due to Assumption (D) and because
Finally, the statement follows from (2.3) and (2.5) once we have computed the Fourier transform of g. We have for
which concludes the proof.
Remark 2.3. If the moment generating function of the marginals is known, the inverse function can be easily computed numerically. We have that
where the expectation can be computed using (2.5) again, while a root finding algorithm provides the infimum (using the continuity of F i ).
We can also compute the copula density function using Fourier methods, which resembles the computation of Greeks in option pricing.
Lemma 2.4. Let X be a random variable that satisfies Assumption (D) and assume further that the marginal distribution functions F 1 , . . . , F n are strictly increasing and continuously differentiable. Then, the copula density function c of X is provided by
where u ∈ (0, 1) n and R ∈ R.
Proof. The distribution functions F and F 1 , . . . , F n are absolutely continuous hence the copula density exists, cf. McNeil et al. (2005, p. 197). Let u ∈ (0, 1) n , then we have that x i = F -1 i (u i ) is finite for every i ∈ {1, . . . , n}, hence e -R,x is bounded. Using Assumption (D) we get that the function M X (R + iv)e -R+iv,x is integrable an
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