Introduction - Paul Carlisle KettlerClayton-Lévy: C(u;v) = u + v 1 (4.5) ; >0...

LÉVY-COPULA-DRIVEN FINANCIAL PROCESSES

PAUL C. KETTLER

Abstract. This paper proposes a general non-Gaussian Ornstein-Uhlenbeck model for ajoint financial process based on marginal Lévy measures joined by a Lévy copula. Simulatedprocesses then result from choices of marginal measures and Lévy copulas, with resultingstatistics and inferences. Selected for analysis are the 3/2-stable and Gamma marginal Lévymeasures, along with Clayton, Gumbel, and Complementary Gumbel Lévy versions of ordi-nary [probability] copulas, with the last two being here introduced. A relationship betweenthe original coupled subordinated processes and the terminal dependency relationship be-tween the simulated variables is observed and calibrated. Normal inverse Gaussian andtempered stable measures are also noted, as are additional Lévy copulas constructed fromthe Gumbel and Frank ordinary copulas, with some analysis and suggestion for using themin future research.

1. Introduction

A recent work of Fred Espen Benth with the author (Benth and Kettler 2011) investigatedthe relationship between electricity and gas prices by estimating marginal distributions anda theoretical copula joining them. That study simulated the model process, concluding withoption prices for the spark spread, the difference of these two prices.

This paper proposes a general non-Gaussian Ornstein-Uhlenbeck subordinated model for ajoint financial process. The model is founded not on process laws and corresponding marginaldistributions with an ordinary [probability] copula, but rather on marginal Lévy measuresjoined by a Lévy copula. Simulated processes then result from choices for these measuresand copula. Statistical analysis produces summary results, and a section on theory probesthe relationship between an originating subordinator and the terminal relationship of thesimulated variables.

The principal inferences and conclusions of this study are that the choice of Lévy copulais not material in differentiating the character or statistics of the price series, and that theterminal ordinary copula of the logarithmic price relatives is nearly the independent copula,regardless of choice of subordinating Lévy processes. These findings imply that financial pro-cesses modeled in this fashion are robust across functional forms and parameter settings. Aswell, the resulting logarithmic price relatives exhibit marked departure from normal distribu-tions, an anticipated result, given the character of the marginal driving Lévy measures. The

Date: 02 November 2012.2010 Mathematics Subject Classification. Primary: 91B24, 91B70. Secondary: 62M10, 62M20.1991 Journal of Economic Literature Subject Classification. C51, G13.Key words and phrases. Lévy copula, finance, stochastic processes, model construction, simulation, time

series.The author wishes to thank Fred Espen Benth and Frank Proske for valuable insights.The R Foundation for Statistical Computing made available the statistical packages for this study (R

Development Core Team 2005; Würtz et al. 2005; Genz, Bretz, and Hothorn (R port) 2005).1

LÉVY-COPULA-DRIVEN FINANCIAL PROCESSES 2

calibrations are interesting, as evidenced in various summary statistics such as the Anderson-Darling test for normality.

Appendices A and B provide background information respectively on Lévy copulas andOrnstein-Uhlenbeck processes.

2. A general subordinated model

The paper is a report on research into the joint behavior of stock prices when they are definedin a geometric process with dependence on subordinated pure jump Ornstein-Uhlenbeck Lévyprocess. Within the subordinated process one joins marginal Lévy measures by a specifiedLévy copula to produce stochastic variables then introduced into the geometric process. Thisstructure of subordination is a Background Driving Lévy Process (BDLP) in the manner ofBarndorff-Nielsen and Shephard. See, e.g., (Barndorff-Nielsen and Shephard 2001, Section 1.1,pp. 167–169).

Here is the setup, beginning with the coupled Ornstein-Uhlenbeck process in the two di-mensional case.

dY 1t = −λ1Y

1t dt+ dL1

t , Y1

0 = 0

dY 2t = −λ2Y

2t dt+ dL2

t , Y2

0 = 0,(2.1)

where L1t and L2

t are the subordinators. The variables (Y 1t , Y

2t ) then enter the geometric

equations as follows.

d logS1t = (µ1 + β1Y

1t ) dt+

√Y 1t dB1

t , logS10 = 0

d logS2t = (µ2 + β2Y

2t ) dt+

√Y 2t dB2

t , logS20 = 0,

(2.2)

where B1t ⊥⊥ B2

t are Brownian motions.The experimental design then calls for simulation of the joint Ornstein-Uhlenbeck process

of Equations (2.1) with a Lévy copula, followed by simulation of the joint geometric process ofEquations (2.2). The study begins by examining the relationship of the subordinators througha Lévy copula, by example, and continues through analysis of the simulated joint stock priceseries, with accompanying tables and charts.

3. Random selection from a Lévy copula

Recall that a Lévy copula is like an ordinary copula in that it is a function which retains all ofthe dependence information of a Lévy measure, while leaving all of the remaining informationin the marginal Lévy measures. Let ν( dx dy) be such a bivariate Lévy measure. Tail integralsof this measure, which are the analogues of distribution functions, are defined as follows. Firstfor the joint measure, in this study supported on R2

+,

U(a, b) :=∞∫b

∞∫a

ν( dx dy)


and for the marginal measures,

U1(a) :=∞∫

0

∞∫a

ν( dx dy)

U2(b) :=∞∫b

∞∫0

ν( dx dy)

The Lévy copula CL(u, v), then, defined on the same domain, is this.

CL(u, v) := U(U−1

1 (u), U−12 (v)

),

or equivalently,

CL(U1(a), U2(b)

):= U(a, b),

assuming all inverses are defined in the generalized sense.

Remark. One may think of a Lévy copula as itself a joint Lévy measure with uniform margins,much in the same sense that an ordinary copula is a joint probability distribution with uniformmargins. In the case of the Lévy copula the uniform margins extend from [0,∞], whereaswith the ordinary copula they extend from [0, 1]. In the sense of information, the Lévy copulaprovides only the dependency, and nothing of the margins, again in the same sense as anordinary copula with its margins. Together, the Lévy copula and its margins provide theentire information set of the joint Lévy measure.

For illustration consider a Clayton-Lévy copula subordinator model with common α-stablemarginal Lévy measures. This is one of the six pairwise choices of copula and marginal measurefor the later simulations. At the heart of selecting a jump pair is the choice of point in thecopular domain. A presentation on this process appears here (Tankov 2003, Example 5.1,p. 20), and follows this plan. Let C(u, v) be a Clayton-Lévy copula as such.

C(u, v) :=(u−θ + v−θ

)− 1θ, (u, v) ∈ [0,∞)2, θ > 0

To simulate a joint α-stable subordinator on the chosen unit time interval one generatesprocesses Xs and Ys given the common marginal tail integral of

U(x) = x−α,(3.1)

for which the inverse is

U−1(y) = y−1α(3.2)

The α-stable subordinator has finite activity if α < 1 because |x| integrates the measure ofthe small jumps. In the simulations, however, the choice is α = 3/2 to be more representativeof what is observed in the financial markets. Specifically in the present context, applying theFundamental Theorem of the Calculus to U(x),

(3.3) −1∫

0

xU ′(x) =α

1− α> 0


Call Γi the ith jump time of a Poisson process with intensity λ, and select a pair (Wi,1,Wi,2)of independent uniform variates on [0, 1]. Then,

X(1)s =

∞∑i=1

U−1(Γi)1{[0,s]}(Wi,1)

X(2)s =

∞∑i=1

U−1(F−1(Wi,2|Γi)

)1{[0,s]}(Wi,1),

(3.4)

Further, given the conditional distribution on the copula as

F (v|u) =∂Fθ(u, v)

∂u=

[1 +

(uv

)θ]−(1+ 1θ ),(3.5)

it follows that

F−1(Wi,2|u) = u ·(W− θ

1+θ

i,2 − 1

)− 1θ

(3.6)

Equations (3.4) appear also in (Cont and Tankov 2004, Chapter 6, Section 5, p. 195).The Clayton-Lévy copula is the only one of the copulas chosen for simulation which admits

a closed-form expression for the inverse of the conditional copula distribution. The othersrequire numeric inversion procedures for their conditional distributions.

For the present modeling purposes one wishes to simulate the BDLP by selecting jumptimes {Γi}, 1 ≤ i ≤ Nλ(T ), from a standard Poisson process over a revised time interval [0, T ],and then to calculate paired jumps

{x

(1)i , x

(2)i

}at these times.

As the distribution of a waiting time ∆i := Γi − Γi−1, with Γ0 = 0, conventionally, is

Φ(∆i) = 1− exp(−λ∆i) = Wi,1,

so

∆i = Φ−1(Wi,1)

One then constructs the {Γi} iteratively as

Γi = Γi−1 + ∆i,

continuing until determining ΓNλ(T ).Next, with a view to the discrete simulation, construct an inventory of Nλ(T ) paired jumps

in the manner of Equations (3.4), of which this is the ith.

x(1)i = U−1(Γi)

x(2)i = U−1

(F−1(Wi,2|Γi)

)(3.7)

These jumps shall appear in order of the {Wi,1}, as well, in harmony with Equations (3.4).


Remark. This diagram shows the sequence of calculations to produce the ith pair of jumpcomponents

(x

(1)i , x

(2)i

).

Γi = uF−1(Wi,2|·)−−−−−−−→ v

U−1

y U−1

yx

(1)i x

(2)i

4. Models

One may address models other than the Clayton-Lévy subordinator model with α-stablemargins by allowing either other copulas or other margins, or both. Further, one may considerbidirectional copulas and margins, meaning those non-trivially supported on Rn \{0}, with orwithout subordination. Among the marginal choices are the Gamma, normal inverse Gaussian(NIG), and the tempered stable processes (including as a limit the variance gamma,) andthe bidirectional α-stable process, among others. Copula choices include the Gumbel-Lévy,herein defined, and Complementary Gumbel-Lévy, called complementary because its generatoris the inverse of the Gumbel-Lévy generator.1 The study now proceeds to examine somecombinations of these seriatim.

For the α-stable and Gamma processes tail integrals and their inverses exist in closed form.For the α-stable processes one has Equations (3.1) and (3.2). For the Gamma processes onehas these.

UG(x) = νe−%x, ν > 0(4.1)

for which the inverse is (UG)−1

(y) = max

{0,−1

%log(yν

)}(4.2)

See (Barndorff-Nielsen and Shephard 2001, Section 2.3.4, p. 175).The Lévy measure νNIG(x) on R \ {0} of the NIG(α, β, µ, δ) process is this, with notation

of (Barndorff-Nielsen 1998, p. 47, Equation 2.9). K1(·) is the modified Bessel function of thirdorder and index 1. As well, δ > 0 and 0 ≤ |β| ≤ α.

(4.3) νNIG(x) =δα

π |x|K1(α |x|)eβx

The Lévy measure νTS(x) on R\{0} of the tempered stable processes is this, with notationof (Cont and Tankov 2004, Chapter 4, Section 5, p. 119, Equation 4.26). As well, c−, c+, λ−,and λ+ are positive coefficients, and α > 0.

(4.4) νTS(x) =c−

|x|1+α e−λ−|x|1{x<0} +c+

x1+αe−λ+x1{x>0}

The limiting case for α = 0, and c := c− = c+ is the Lévy measure of the variance gammaprocess. See (Tankov 2006, p. 3, Equation 2.4).

1Your author has chosen these names in honor of the late Professor Emil Julius Gumbel, founder of extremevalue theory and Nazi antagonist. As there are many ways to chose Lévy copulas inspired by ordinary copulas,these are only two such choices.


The inverse tail integrals of the NIG and tempered stable processes are only known bynumerical approximation. Though these processes are of interest to financial economists andmathematicians, these ideas are left for future study. For pertinent reading on the relationshipbetween process probability and Lévy densities, including that of the Gamma distribution,see (Barndorff-Nielsen 2000).

Both the NIG and tempered stable processes have infinite activity, for the measures do notintegrate |x| near {0}, cf. Equation(3.3).

Following are the functional representations of the named Lévy copulas, including a bidirec-tional Clayton-Lévy version (Tankov 2006, p. 6, Equation 3.1), adapted from ordinary copulasof the same names. See (Cherubini, Luciano, and Vecchiato 2004, p. 124) for a presentation onordinary copulas. Included for comparison are the Product-Lévy (Independent) and Fréchet-Lévy upper limit copula C↑(u, v); no analogous Lévy version exists for the Fréchet-Lévy lowerlimit copula.

Clayton-Lévy:

C(u, v) =(u−θ + v−θ

)− 1θ, θ > 0(4.5)

Clayton-Lévy, bidirectional:

CB(u, v) =(|u|−θ + |v|−θ

)− 1θ (η1{uv≥0} − (1− η)1{uv<0}

), θ > 0(4.6)

Gumbel-Lévy:

CG(u, v) = exp

{[(log(u+ 1)

)−θ+(

log(v + 1))−θ]− 1

θ

}− 1, θ > 0(4.7)

Complementary Gumbel-Lévy:

CG(u, v) ={

log[exp

(u−θ

)+ exp

(v−θ)− 1]}− 1

θ, θ > 0(4.8)

Product-Lévy (Independent) for marginal Lévy measures ν1, ν2 ∈ [0,∞]:

C⊥(u, v) =

u : (u, v) ∈ [0, ν1]× [ν2]

v : (u, v) ∈ [0, ν2]× [ν1]

u+ v : (u, v) = (ν1, ν2)

0 : elsewhere

(4.9)

Fréchet-Lévy Upper:

C↑(u, v) = min(u, v)(4.10)

The following functions generate, respectively, the Clayton-Lévy, Gumbel-Lévy, and Com-plementary Gumbel-Lévy copulas in Archimedean analogy to their corresponding ordinary


copulas. In each case φ : [0,∞]→ [0,∞].

φC(x) := x−θ

φG(x) := [log(x+ 1)]−θ

φG(x) := exp(x−θ

)− 1

For a discussion of Lévy copula generation see (Kallsen and Tankov 2004, Section 6, pp. 21–23) and (Tankov 2003, Proposition 4.5, pp. 15–16). Note that φG(·) and φG(·) are inverses ofeach other (after re-parameterizing θ to 1/θ in either formulation.)

Figures 1 and 2 display a Clayton-Lévy copula and its level curves; Figures 3 and 4 displaya Gumbel-Lévy copula and its level curves; Figures 5 and 6 display a Complementary Gumbel-Lévy copula and its level curves. In each case θ = 1.

Observe the vertical scale of these. C(20, 20) = 10.0000 for the Clayton-Lévy copula;CG(20, 20) = 3.5826 for the Gumbel-Lévy copula; CG(20, 20) = 10.2439 for the Complemen-tary Gumbel-Lévy copula. Compare these values with C(20,∞) = C(∞, 20) = 20, as for theother (and all) Lévy copulas.

Alternative generation of Lévy copulas comes from reference to an ordinary copula by wayof a generator ψ : [0, 1]→ [0,∞]. Such procedures extend the possibilities for creating usefulcopulas in empirical research. For instance, one can begin with ordinary copulas such as theGumbel and Frank, respectively.

CG(u, v) = exp{−[(

(− log u)θ + (− log v)θ) 1θ

]}θ ∈ [1,∞),with Product copula for θ = 1

CF (u, v) = −1

θlog

[1 +

(exp(−θu)− 1

)(exp(−θv)− 1

)exp(−θ)− 1

]θ ∈ (−∞,∞) \ {0},with Product copula for θ = 0±

For a discussion of Lévy copula generation in this form also see (Kallsen and Tankov 2004,loc. cit.) and (Tankov 2003, loc. cit.). An example of such a generator, as proffered in (Tankov2004, Theorem 5.1, pp. 167–169) is ψ(x) = x/(1− x); another is ψ(x) = − log(1− x).

5. Simulation

The simulation proceeds in two phases, the first to develop the subordinated process, asdisplayed in Equations (2.1), the second to develop the geometric process, as displayed inEquations (2.2). Six models are selected, taking 3/2-stable subordinators or the Gamma sub-ordinators, and coupling them by a Clayton-Lévy, Gumbel-Lévy, or Complementary Gumbel-Lévy copula, with chosen parameters. The calculations include charts in the Clayton-Lévycopula choice to illustrate the findings.

5.1. The subordinated process. The way is clear now to devise an algorithm for generatingsequences of jumps joined by a Lévy copula. This algorithm generalizes mutatis mutandis tomarginal processes other than the α-stable and to copulas other than the Clayton-Lévy, asthis paper explores in the sequel.

Consider now that U(·) is the tail integral of an arbitrary Lévy measure.


(1) Select λ and T , then create a series of jump times {Γi}, 1 ≤ i ≤ Nλ(T ), by exponentialdelay. Note that if ε : = U−1(T ), then jumps smaller than ε, defined now as smalljumps, in the

{x

(1)i

}series will be eliminated, for

x(1)1 ≥ x(1)

2 ≥ . . . ≥ x(1)Nλ(T ) ≥ ε,

owing to the monotonicity of U(·).(2) Calculate an inventory of incremental jump component pairs

{x

(1)i , x

(2)i

}.

(3) Calculate(Y 1t , Y

2t

)iteratively as the accumulation of these jumps following inter-

jump exponential declines. Select the jumps for inclusion at time Γj on the order ofthe {Wi,1}, now indexing the BDLP by the jump times, as follows.

Y 1j = e−λ1∆jY 1

j−1 +

Nλ(T )∑i=1

x(1)i 1{Γj−1/T<Wi,1≤Γj/T}, Y

10 = 0

Y 2j = e−λ2∆jY 1

j−1 +

Nλ(T )∑i=1

x(2)i 1{Γj−1/T<Wi,1≤Γj/T}, Y

20 = 0

(5.1)

The first terms on the right of Equations (5.1) represent the inter-jump exponential declines ofthe Ornstein-Uhlenbeck process, whereas the second terms represent the accumulated jumpsoccurring between times Γj−1 and Γj . The jumps are indicated (literally) for inclusion bythe {Wi,1}, but actually occur when the next jump time Γj appears. By this means thesubordinator remains stationary in that the expected size of the accumulated jumps at ajump time is proportional to the waiting time.

Remark. Jumps catalogued by this algorithm in the{x

(1)i

}series are defined large jumps, to

complement the small jumps. Observe that ε is such that

(5.2) U(ε) = U(U−1(T )

)= T

Thus the Lévy measure of the large jumps, and therefore the intensity of the compoundPoisson process they represent, is T , independent of U(·). The small jumps, and a methodfor including them in the study, is the subject of Section 5.4.

5.2. Finite sample bias. In selecting pairs of jumps, the first coordinate jump, computed asin the first of Equations (3.7), is limited to a lower bound of ε, as reported in Subsection 5.1.The second coordinate jump, computed as in the second of Equations (3.7), is not so limited.In consequence, a bias exists in jump selection leading to expected lower values in the secondjump. The phenomenon is most pronounced for the Clayton-Lévy copula, so the correctionproposed is only implemented in that case.

To counteract the observed bias the simulations also restrict the second coordinate jump toa lower bound of ε. This selection arrives in a direct manner by choosing the uniform randomvariable Wi,2 not on the interval [0, 1], but rather on the interval [0, r̄i], with r̄i = F (T |Γi)chosen by the following reasoning. The revised requirement is that

x(2)i = U−1

(F−1(Wi,2|Γi)

)≥ ε(5.3)

So

U(x

(2)i

)= F−1(Wi,2|Γi) ≤ U(ε) = T


by Equation (5.2) and because U(·) is monotone decreasing. Therefore,

F(U(x

(2)i

) ∣∣Γi) = Wi,2 ≤ F (U(ε)|Γi) = F (T |Γi) = : r̄,(5.4)

independent of U(·), as F (·|Γi) is monotone increasing.An alternative plan would be to require

E[x

(2)i

]= x

(1)i

This scheme, while better in some ways, would make r̄i dependent on U(·), as revealed byEquation (5.3).

5.3. The geometric process. Herein one simply takes the{Y 1j , Y

2j

}terms developed by

simulating the subordinated Ornstein-Uhlenbeck process, inserting them into the discretetime version of the geometric process, cf. Equations (2.2), as so. This is implementation ofEuler’s Method (first order) on the deterministic part.

logS1j = logS1

j−1 + (µ1 + β1Y1j−1)∆j +

√Y 1j−1B̂

1∆j, logS1

0 = 0

logS2j = logS2

j−1 + (µ2 + β2Y2j−1)∆j +

√Y 2j−1B̂

2∆j, logS2

0 = 0,(5.5)

where B̂1∆j⊥⊥ B̂2

∆jare Brownian motions. Exponentiating the

{logS1

j , logS2j

}series allows

the recovery of the{S1j , S

2j

}series.

5.4. Amussen-Rosiński modification. The processes articulated in Section 4 are necessar-ily approximate in that small jumps, those below the threshold of ε such as those computedin the α-stable and Gamma processes in Equations (3.2) and (4.2), are ignored. One canimprove on this methodology by employing a method articulated by Amussen and Rosiński toapproximate the small jumps by a Brownian motion. The primary reference is (Amussen andRosiński 2001), with additional presentations in (Rosiński 2006; Prause 1999; Rosiński 1991).

The essence of the argument, with results incorporated in the simulations of this study, isthat one can approximate the small jumps of a Lévy process of infinite measure frequently,but not always, by a Brownian motion with drift. Therein, the authors provide a necessaryand sufficient condition that the normal approximation, as this capability is called, does nothold for any process with finite Lévy measure, such as the compound Poisson process, norfor the Gamma process, but does hold for the α-stable process for the entire admissible set{α∣∣ 0 < α ≤ 2}. See Equation (5.8) below. For the NIG process see (Amussen and Rosiński

2001, Theorem 2.1 and Proposition 2.1, and Examples 2.1–2.5, pp. 484–486).For the simulations using Gamma Lévy margins, ν is set to T so that εG :=

(UG)−1

(T ) = 0,reflecting the state of the Gamma process as having no small jumps. Insofar as UG(0) = ν <∞, the Gamma process has finite variation, and thus is a compound Poisson process.

Figure 7 displays conditional copula distribution functions in the manner of Equation (3.5),which appears for the Clayton-Lévy copula along with similar formulations for the Gumbel-Lévy and Complementary Gumbel-Lévy copulas. In each case the point of conditional eval-uation is u = 2. The rank of vertical scaling described for the copulas is evident in thesemeasures also for evaluations at (u, v) = (2, 5), at the right hand boundary of this chart.

Figure 8 displays the marginal Lévy measure for the 3/2-stable subordinate process withparameter θ = 1. For the Gamma subordinate process (not shown) the parameter choices


are ν = T = 20 and % = 1. The formulations to determine drift a(ε) and variance s2(ε) ofthe relevant Brownian motion for the α-stable process, adapted to the present circumstances,follow.

a(ε) = −1∫ε

x ν(dx)(5.6)

s2(ε) =

ε∫0

x2 ν(dx)(5.7)

The specific condition for the normal approximation to apply, with the α-stable process con-forming, is this for Lévy measures without atoms in (0, ε).

limε→0

s(ε)

ε=∞

Substituting −U ′(x) dx for ν(dx) by the Fundamental Theorem of the Calculus, and recallingthat ε = U−1(T ), one has that

a(ε) = α1−α

(1− ε1−α) = α

1−α

(1− T−

1−αα

)> 0, 0 < α ≤ 2(5.8)

s2(ε) = α2−α

(ε2−α) = α

2−α

(T−

2−αα

)> 0, 0 < α < 2(5.9)

Simple calculations show that

a(ε)|α→1 = log T

a(ε)|α=2 = −2(

1− T 1/2)

s2(ε)|α→2 =∞Note that the cases, α = 1 and α = 2 are the Cauchy and normal processes, respectively. Aswell, observe that by the assumption T = 20 the threshold for small jumps ε = 0.1357. In the3/2-stable process the marginal Lévy measure ν(ε) = 221.04.

To include the normal approximation to the original formulation of the subordinated processis straightforward. Simply amend Equations (2.1) as follows.

dY 1t =

(−λ1Y

1t + a(1)(ε)

)dt+ s(1)(ε) dB̂1

t + dL1t , Y

10 = 0

dY 2t =

(−λ2Y

2t + a(2)(ε)

)dt+ s(2)(ε) dB̂2

t + dL2t , Y

20 = 0,

(5.10)

where the new terms are underlined, and where B̂1t ⊥⊥ B̂2

t are Brownian motions.These inclusions also translate to the realm of the simulation by modification to Equa-

tions (5.1), as here.

Y 1j = e−λ1∆jY 1

j−1 + a(1)(ε)∆j + s(1)(ε)B̂1∆j

+

Nλ(T )∑i=1

x(1)i 1{Γj−1/T<Wi,1≤Γj/T}, Y

10 = 0

Y 2j = e−λ2∆jY 2

j−1 + a(2)(ε)∆j + s(2)(ε)B̂2∆j

+

Nλ(T )∑i=1

x(2)i 1{Γj−1/T<Wi,1≤Γj/T}, Y

20 = 0,

(5.11)

where the new terms are underlined, and where B̂1∆j⊥⊥ B̂2

∆jare Brownian motions.


Observe that the sum of the underlined terms in either of Equations (5.11) could be negativewith the random choice of a sufficiently negative Brownian path over the interval ∆j . In theseinstances the sum of such terms is forced to zero to preserve the non-negative incrementalcharacteristic of a subordinator. By the Doob Martingale Inequality the probability of theseinstances decreases exponentially with time, and therefore becomes insignificant.

5.5. Stability. At some level the simulated subordinated Ornstein-Uhlenbeck process is inequilibrium. In Equation (5.1) this is where either Y 1

j or Y 2j is such that the expected infin-

itesimal decline from the exponential term is matched by the expected infinitesimal advancefrom the pure jump term. Letting k ∈ {1, 2}, then at time Γj these respective rates areλkY

kj and ν(ε,∞) = U−1(ε) = T = λ for a generic U(·). Thus for Y k

j = λ/λk the Gammaprocess has conditional expectation of Y k

j , and therefore is a local martingale. It is desirable,consequently, to start each process at Y k

0 = λ/λk to ensure stability from the onset.In the alternate 3/2-stable process with the Amussen-Rosiński modification as in Equa-

tion (5.11) the corresponding starting point is Y k0 = λ/

(λk − a(k)(ε)

)to allow for the com-

pensating drift of the Brownian approximation.Accordingly, Equations (5.1) and (5.11) are reset to these starting values.

5.6. Statistics. The study examined six models. For margins the choice was either a 3/2-stable process, identical in each variable, or a Gamma process, also identical in each variable.For copulas the choice was either Clayton-Lévy, Gumbel-Lévy, or Complementary Gumbel-Lévy. Chosen parameters for the Gamma process were and ν = T = 20, as noted, and % = 1.In each copula θ = 1. In the geometric processes µ = 0.001 and β = 0.10 in each variable. Bythese choices the models were symmetric in all aspects, except for small residual biases in thechoices of jump pairs owing to finite sample biases.

Statistics and tabular results are reported across the three copular models and the twomarginal measures. Pseudo-random numbers used to generate sequences were the same forboth the 3/2-stable and Gamma processes so that the generated paths are directly comparable.

Accompanying the text is a pair of charts for the 3/2-stable marginal choice for the Clayton-Lévy copula, illustrating the empirical copula which resulted from simulations of 500 paths.Additionally appear four pairs of charts illustrating features of a single random path fromthese selections.

Charts for the other copulas and for the Gamma margin are not shown in the interest ofeconomy, as those charts are qualitatively similar to the ones which appear. A conclusion ofthis study is that the results of simulation are robust over the various choices, an idea to berevisited.

The first two charts of these 10 for the Clayton-Lévy copula show results of computing theempirical ordinary copula at the terminal prices. Figures 9 and 10 exhibit these respectiveaxial views. Note that a copula for prices is the same as a copula for logarithmic prices,because the logarithm is an increasing function. See (Schweizer and Wolff 1981, Theorem 3,p. 881).

An exercise in fitting a Clayton ordinary copula to the empirical copula in each model gavethe results appearing in Table 1. The model took Cγ(u, v) as the empirical copula, with

Cθ = (u−θ + u−θ − 1)−1θ ,


Copula Margin 3/2-stable GammaClayton θ 0.0703 0.0835

objective 0.0121 0.0168r2 0.7235 0.7711

Gumbel θ 0.0197 0.0102objective 0.0162 0.0219

r2 0.6908 0.7939Comple- θ 0.0000 0.0000mentary objective 0.0951 0.0914Gumbel r2 0.9176 0.9084

Table 1. Statistics fitting empirical copula

the ordinary Clayton copula, and evaluated

minθ

∑u,v

(Cγ − Cθ)2

Note this is the same as

(5.12) minθ

∑u,v

[(Cγ(u, v)− C⊥(u, v)

)−(Cθ(u, v)− C⊥(u, v)

)]2wherein the interpretation is that of comparing the differences of copulas to the independentcopula.

The coefficient of determination r2 was calculated by comparing the variances of the re-spective differences of the empirical copula and the Clayton copula as fit, to the independentcopula C⊥(u, v) = uv, for each of the models, as here.

r2 = 1− var(Cθ(u, v)− C⊥(u, v)

)/var

(Cγ(u, v)− C⊥(u, v)

)This result follows the formulation of Equation (5.12).

Other methods to fit, including by maximum likelihood, are described here (Frees andValdez 1998, Sec. 4, pp. 12–18).

Figures 11 and 12 show histograms of the logarithmic price relative series for the samplepaths for the Clayton-Lévy copula with 3/2-stable margins. Anderson-Darling tests for nor-mality cause rejection of the null hypothesis in each instance, as is evident from the histograms.Some statistics for the three copulas, and for both the 3/2-stable and Gamma margins, appearin Table 2.

Figures 13 and 14 show Q-Q probability plots of the logarithmic price relative series for thesample paths for the Clayton-Lévy copula with 3/2-stable margins. Figures 15 and 16 showsubordinating pure jump processes for the sample paths for this combination. Figures 17 and18 show prices for the sample paths, again for the same combination.

5.7. Inferences. Three principal inferences are discernible from the course of this study.(1) The terminal logarithmic price relative empirical copulas are immaterially different

from the independent copula, over all models. This fact is apparent from the entriesof Table 1. The Clayton ordinary copula does provide a good fit, but the optimizedparameter θ is close to zero in all cases (being a flat zero for the Complementary


Margin 3/2-stable GammaCopula Variable First Second First Second

A 1.8801 1.6225 1.4643 1.8964Clayton P 0.0001 0.0003 0.0008 0.0001

Skewness -1.4218 0.5737 -1.2053 0.7135Kurtosis 4.6838 0.6699 3.7792 1.6030

A 1.7602 2.2113 1.7031 2.2424Gumbel P 0.0002 0.0000 0.0002 0.0000

Skewness -0.1250 0.9393 -0.0310 0.7047Kurtosis 2.6468 4.2463 3.5151 3.7896

Comple- A 6.0889 1.2741 6.1703 1.0986mentary P 0.0000 0.0025 0.0000 0.0067Gumbel Skewness 4.5044 0.2667 4.1547 0.2061

Kurtosis 30.4387 1.4195 26.3826 1.3461

Table 2. Anderson-Darling statistics of the logarithmic price relative seriesfor the sample paths

Gumbel-Lévy copula,) the independent limit of the Clayton family. Further, the pro-jected views of the empirical copulas, as appearing in Figures 9 and 10 for the Clayton-Lévy copula with 3/2-stable margins show only patterns which are attributable to theaccumulation of computational errors; specifically they exhibit low amplitude wavepatterns typical of truncation errors in evaluating transcendental functions by seriesmethods.

(2) The logarithmic price relative series are distinctly not normal, exhibiting significantskewness and kurtosis, as revealed by all the Anderson-Darling and related statisticsappearing in Table 2. This is an expected result, given the nature of the driving3/2-stable and Gamma Lévy marginal subordinating processes.

(3) The choice of copula is not important in determining the quality of the inferences inthe two items above.

6. Conclusions

This study established that the proposed model provides a computationally reasonablescheme for generating financial processes. The model incorporates the freedom to describe thedependency relationship between variables with the generality of a Lévy copula, while alsopermitting flexible jump processes as often required.

Financial process modeling of the fashion proposed by this study appears to be robustacross choices of marginal Lévy measures and Lévy copulas. Subtle distinctions are evident,but in general all of the developed processes are remarkably similar.

Planned future research includes delving into the theory of Lévy-copula-driven financialprocesses by establishing a set of first principles, thus enabling informed prediction of terminalprocesses and copulas from the subordinators, ex ante.


Appendices

A. Lévy copulas

A Lévy copula is much like an ordinary [probability] copula in that it distills the depen-dence information from a joint measure – in this instance a joint Lévy measure – leaving theinformation from the marginal measures distinct. The main difference to an ordinary copulais that one integrates the Lévy measures in the upper tails rather than in the lower tails.This convention obtains for frequently the Lévy measures of interest are supported on thepositive real line, as with subordinators, and Lévy measures become uniformly unbounded atthe origin. For background reading on ordinary copulas see (Nelsen 1998), for Lévy copulassee (Cont and Tankov 2004, Sections 5.5–5.7, pp. 145–165).

For illustration, let µ be a Lévy measure on R2+. If

A := [x,∞]× R+

B := R+ × [y,∞]

then the tail integral of the measure (analogous to a probability distribution function) is

L(x, y) =

∫A∩B

dµ,

and the marginal tail integrals are

L1(x) =

∫A

dµ

and

L2(y) =

∫B

dµ

The marginal measures are independent if and only if they are supported respectively on theaxes.

The Lévy copula K(u, v) : R2+ → R+. Specifically,

K(u, v) = L(L−1

1 (u), L−12 (v)

),

or equivalently

K(L1(x), L2(y)

)= L(x, y)

Some properties of the Lévy copula are easy to establish. First, the copula is grounded withuniform margins.

K(0, v) = K(u, 0) = 0

and

K1(u) = K(u,∞) = u

K2(v) = K(∞, v) = v


Further, if µ has a density l(x, y), with marginal densities l1(x) and l2(y), then they relateto the copular density k(u, v) as follows.

k(u, v) =∂2

∂u∂vK(u, v) =

∂2

∂x∂yL(x, y)

/(∂L1

∂x

∂L2

∂y

)=

l(x, y)

l1(x)l2(y)

Two special Lévy copulas are worthy of note, the independent copula K⊥(u, v) and thecompletely dependent copula K‖(u, v), as follows.

K⊥(u, v) = u · 1{v=∞} + v · 1{u=∞}

K‖(u, v) = min(u, v)

B. Ornstein-Uhlenbeck processes

B.1. Introduction. This appendix sets forth principles for the analysis of dependency amongspecified kinds of Lévy processes. These are ones of the Ornstein-Uhlenbeck description, withsubordination. That is, each has a component of a pure jump process with non-negative jumpsincluded with the usual Brownian motion.

The study of these processes dates to a seminal paper by George Eugene Uhlenbeck andLeonard Salomon Ornstein, two highly respected Dutch mathematical physicists (Uhlenbeckand Ornstein 1930). Some say Laplace gave the process its start in 1810 (Jacobsen 1996), butalmost all who observe say the subject is correctly named.

These two researchers were contemporaries and collaborators with some of the great namesof physics, including Neils Bohr, Satyendra Bose, Paul Dirac, Paul Ehrenfest, Albert Einstein,Josiah Gibbs, Samuel Goudsmit, Werner Heisenberg, Oskar Klein, Wolfgang Pauli, Dirk JanStruik, and others. Among the discoveries of these persons are phenomena of high levelcurrent interest. Dirac invented the word “boson” for a particle conforming to the Bose-Einstein statistics, and today physicists are searching for the elusive Higgs boson (after PeterHiggs of the University of Edinburgh) to extend the Standard Model. This is not even tomention the Ornstein-Uhlenbeck process itself, still and increasingly, a subject of intensemodern investigation.

That a physical process was at the beginning of this concept is evident simply by readingthe cited paper. This was a study of the behavior of a molecule of a gas responding toa systematic force — friction — and to a random influence — fluctuation. This was notthe study of the mathematical formulation we now call the Wiener process (of course, afterNorbert Wiener, another luminary of the time.) Nonetheless, Ornstein’s and Uhlenbeck’sformulation was almost exactly what one sees today in modern mathematical terminologywith drift coefficients and Brownian differentials. This is remarkable prescience.

The present study offers a contribution in the realm of dependency analysis. Motivation ismanifold. Jump processes drive some of the most important events in business and commerce,economics and finance, in the natural and physical sciences, in biological systems and the oc-currence of natural disasters, in subatomic physics and theories of the universe, in humanbehavior itself. Evident among these phenomena is the suffusion of dependency relationships.Subatomic events have consequences in space and time; earthquakes occur not in isolation;galaxies influence one another, and, of course, people interact in many ways. On closer exam-ination these dependencies rarely present themselves neatly in the form, say, of multinormalcovariance matrices. Research questions abound, with significant positive consequences forunderstanding and explaining. This appendix examines the O-U process, as all affectionately


now call it, with the added ingredient of special kinds of jumps, offering a few results on theway.

This appendix begins by examining a time-scaled version of the standard O-U process,advancing to the multi-dimensional version, and then continuing with an analysis of jumpprocesses, first without, and then with an accompanying Brownian motion. Descriptive resultsand tools for measurement follow. The appendix concludes with examples to illustrate thetheory.

Now it is time to state the case.

B.2. The basic one dimensional O-U process. The problem is to solve

dY (t) = −λY (t) dt+ dB(κt), λ > 0, κ > 0,

where B(κt) is a Brownian motion. In this problem, as in those to follow, time is scaled bythe constant κ to provide a slight generalization from the usual formulation. The implicationof the special cases κ = 1 and κ = λ follows in Subsubsection B.2.2.

B.2.1. The solution. By Brownian scaling,

dY (t) = −λY (t) dt+ κ12 dB̂(t),

where B̂(t) is a Brownian motion. Apply factor eλt. Then,

eλt dY (t) = eλt[−λY (t) dt+ κ

12 dB̂(t)

](B.1)

By Itô’s formula (there being no second order term,)

d[eλtY (t)

]= eλt [λY (t) dt+ dY (t)](B.2)

Add Equations (B.1) and (B.2) above to get

d[eλtY (t)

]= eλtκ

12 dB̂(t) = eλt dB(κt)

So

eλtY (t) = eλ·0Y (0) +

t∫0

eλs dB(κs),

or

Y (t) = e−λtY (0) +

t∫0

e−λ(t−s) dB(κs)(B.3)


B.2.2. The moments. The mean µ(t) and variance σ2(t) are as follows.

µ(t) = E [Y (t)] = e−λtY (0) −−−→t→∞

0,

since the expectation of an Itô integral which is a martingale is zero.For σ2(t), recast the solution in the alternate process B̂ for a direct application of the Itô

Isometry (Øksendal 2003, Corollary 3.1.7, p. 29).

σ2(t) = E[Y (t)− E [Y (t)]

]2= E

κ 1

2

t∫0

e−λ(t−s) dB̂(s)

2

= κ

t∫0

e−2λ(t−s) ds

= κe−2λt

t∫0

e2λs ds

= κe−2λt

[1

2λe2λs

]t0

=κ

2λe−2λt

[e2λs

]t0

=κ

2λe−2λt

[e2λt − 1

]=

κ

2λ

[1− e−2λt

]−−−→t→∞

κ

2λ

Note that two cases of this result have special interest, the case where κ = 1, and the casewhere κ = λ. These are the cases where the Brownian motion, respectively, either is unscaledor is scaled by the decay parameter λ. In the former the limit of σ2(t) is 1/(2λ), whereas inthe latter the limit is 1/2.

B.2.3. Autocorrelation of the process. Let 0 ≤ s ≤ t < ∞. The coefficient of determinationr2s,t of the autocorrelated process is one, less the fraction of variance at time t contributed bya process starting at time s. As the process has stationary increments this contribution is

κ

2λ

[1− e−2λ(t−s)

]Since the total variance at time t is

κ

2λ

[1− e−2λt

],

then

r2s,t = 1−

[1− e−2λ(t−s)][1− e−2λt]

=e2λs − 1

e2λt − 1


The coefficient of correlation rs,t, therefore, is√r2s,t, positive in this instance. Thus,

(B.4) rs,t =

√e2λs − 1

e2λt − 1

Note also,r0,t = 0, rt,t = 1, and lim

t→∞rs,t = 0

Further, r2s,t is majorized in the interval 0 ≤ s ≤ t by s/t, so rs,t is majorized by

√s/t, the

square root being monotone. This is evident because s/t as a function of s is linear, and r2s,t

is convex, for∂2r2

s,t

∂s2=

4λ2e2λs

e2λt − 1> 0,

and the two functions coincide at 0 and t. Further, this majorant is the most constrained, for

limλ→0

rs,t =

√s

t,

as a first order Taylor’s series expansion of the numerator and denominator in Equation (B.4)above reveal. As well, rs,t has an inflection point at s = (log 2)/(2λ), independent of t, wherethe function changes from concave to convex with increasing s. The value of rs,t at this pointis 1/

√e2λt − 1.

B.2.4. The characteristic function. The characteristic function ϕ(t)Y (ζ) of the normal variate

Y (t), given µ(t) and σ2(t) is as follows.

ϕ(t)Y (ζ) = exp

[−1

2σ2(t)ζ2 + iµ(t)ζ

]−−−→t→∞

exp

[−1

4ζ2

]B.3. The basic multi-dimensional O-U process. First consider two parallel processesalong the design of the one solved in Subsubsection B.2.1. These are

dY1(t) = −λ1Y1(t) dt+ dB1(κ1t)

and

dY2(t) = −λ2Y2(t) dt+ dB2(κ2t)

The independent solutions, respectively, are

Y1(t) = e−λ1tY1(0) +

t∫0

e−λ1(t−s) dB1(κ1s)

and

Y2(t) = e−λ2tY2(0) +

t∫0

e−λ2(t−s) dB2(κ2s)


B.3.1. The moments. Moments of these processes, means µ1(t) and µ2(t), and variances σ21(t)

and σ22(t), in similarity to the computation of Subsubsection B.2.2, are

µ1(t) = e−λ1tY1(0) −−−→t→∞

0,

and

µ2(t) = e−λ2tY2(0) −−−→t→∞

0

Also,

σ21(t) =

κ1

2λ1

[1− e−2λ1t

]−−−→t→∞

κ1

2λ1,

and

σ22(t) =

κ2

2λ2

[1− e−2λ2t

]−−−→t→∞

κ2

2λ2

If the two processes correlate, then they do so in the manner of their Brownian motions.Specifically, we recognize that

dB̂1(t) dB̂2(t) = ρ dt,

where ρ is the ordinary coefficient of correlation, −1 ≤ ρ ≤ +1.This observation stimulates a computation of covariance, as before, by the Itô Isometry,

this time in its multivariate form (Karatzas and Shreve 1991, Proposition 2.10, p. 139).

cov(t) = E[[Y1(t)− E [Y1(t)]

][Y2(t)− E [Y2(t)]

]]= E

κ 121

t∫0

e−λ1(t−s) dB̂1(s)

κ 122

t∫0

e−λ2(t−s) dB̂2(s)

= ρκ

121 κ

122 e−(λ1+λ2)t

[1

λ1 + λ2e(λ1+λ2)s

]t0

= ρκ

121 κ

122

λ1 + λ2e−(λ1+λ2)t

[e(λ1+λ2)s

]t0

= ρκ

121 κ

122

λ1 + λ2

[1− e−(λ1+λ2)t

]−−−→t→∞

ρκ

121 κ

122

λ1 + λ2

Generalizing these results to n dimensions, with the natural indexing, and with the vectorof processes now Y = [Y1, Y2, . . . , Yn]T , gives the covariance matrix [cj,k(t)] as follows.

[cj,k(t)] =

ρj,k κ12j κ

12k

λj + λk

[1− e−(λj+λk)t

] −−−→t→∞

ρj,k κ12j κ

12k

λj + λk


B.3.2. The characteristic function. This formulation for covariance, along with that of thevector of means µ = [µ1, µ2, . . . , µn]T , provides the characteristic function:

ϕ(t)Y (ζ1, ζ2, . . . , ζn) = exp

−1

2

n∑k=1

n∑j=1

ζjcj,k(t)ζk + in∑j=1

µj(t)ζj

−−−→t→∞

exp

−1

2

n∑k=1

n∑j=1

ζj

ρj,k κ12j κ

12k

λj + λk

ζk

B.4. The subordinated Ornstein-Uhlenbeck pure jump Lévy process. Now turn at-tention to the process

(B.5) dY (t) = −λY (t) dt dz(κt), λ > 0, κ > 0,

where z(κt) is a pure non-negative jump Lévy process, called a subordinator. In this role, aspart of a further defined process like the O-U, a subordinator becomes a background drivingLévy process (BDLP.) This subordinated O-U process, or rather a slight modification of it withκ = λ, is a subject of investigation here (Barndorff-Nielsen and Shephard 2005, Subsection1.5). An implication of their Theorem 1.12 in the cited Subsection is that Equation (B.5)above has the solution

(B.6) Y (t) = e−λtY (0) +

t∫0

e−λ(t−s) dz(κs)

Note that this equation is of the same form as the solution to the basic one dimensional O-Uprocess represented by Equation (B.3) above, with the Lévy process replacing the Brownianmotion.

In this instance, however, the BDLP is not fully specified until one selects a Lévy measure.Several popular choices suggest themselves, being reminded that only non-negative jumps arepossible in this formulation, so any such measure must have support on the positive real line.This is also the reason that the Laplace transform, rather than the Fourier transform (such asthe characteristic function for the basic O-U processes) is the transform of choice. Candidatesfor such measure include the compound Poisson, gamma, and normal inverse Gaussian (NIG.)The relationship between the Lévy measure and the probability measure of the process revealsitself in the Lévy-Hinc̆in2 representation. For a description of this representation and also adiscussion of subordinators see here (Rogers and Williams 1994, Subsection I.28, pp. 73–80).

The authors continue with a careful extended analysis to derive the explicit form of theBDLP in such special cases. Herein consider the NIG, for illustration, with parameters(α, β, µ = 0, δ), with β ≥ 0. For a full discussion of this distribution, see the work on itby the same authors (Barndorff-Nielsen 1998).

Following their reasoning, with only minor changes to accommodate differences in notationand the inclusion here of the κ time scaling factor, they conclude (Barndorff-Nielsen andShephard 2005, Example 1.40, pp. 65–68) that the BDLP has the form

z(κt) = y(κt) + p(κt) + q(κt),

where2The late Prof. Hinc̆in’s surname is variously Westernized to Khinchin or Khintchine (Fr.)


(1) y(κt) is an [ordinary, not O-U] NIG Lévy process with parameters(α, β, 0, (1− ρ)δ

),

and

ρ =α

β> 0

As the authors state for reference (Barndorff-Nielsen and Shephard 2005, Example1.23, p. 35), again adapted to the notation herein, the logarithm of the Laplacetransform of y(κt), also known at the kumulant (sic) function K{θ ‡ ·}, is

K{θ ‡ y(κt)} = logL(θ, y(κt)

)= log E

[exp

(θy(κt)

)]= κt(1− ρ)δ

[{α2 − β2

}1/2 −{α2 − (β + θ)2

}1/2]

+ κtµθ

(2)

p(κt) =1

2α−1

(1− ρ2

)−1/2N(κt)∑i=1

(u2i − u′2i )

),

where N(κt) denotes a Poisson process with rate

1

δα

(1 + ρ

1− ρ

) 12

and the ui and u′i((i = 0, 1, 2, . . . , N(κt)

)are independent standard normally dis-

tributed variables, independent also of the process N(κt),

(3) and the kumulant of q(κt)

K{θ ‡ q(κt)} = logL(θ, q(κt)

)= log E

[exp

(θq(κt)

)]= κtρδ

[β {(α− β)/(α+ β)}1/2

− (θ + β) {(α− θ − β)/(α+ θ + β)}1/2]

From this formulation follow readily the density and moments.


List of Figures

1 Clayton-Lévy Copula 232 Clayton-Lévy Copula, Level Curves 233 Gumbel-Lévy Copula 244 Gumbel-Lévy Copula, Level Curves 245 Complementary Gumbel-Lévy Copula 256 Complementary Gumbel-Lévy Copula, Level Curves 257 Conditional Copula Distributions 268 Marginal Lévy Measures 269 Clayton: Copula, log price 1, 3/2-stable 2710 Clayton: Copula, log price 2, 3/2-stable 2711 Clayton: Histogram, log relative 1, 3/2-stable 2812 Clayton: Histogram, log relative 2, 3/2-stable 2813 Clayton: QQ – normal plot, log relative 1, 3/2-stable 2914 Clayton: QQ – normal plot, log relative 2, 3/2-stable 2915 Clayton: Subordinator, variable 1, 3/2-stable 3016 Clayton: Subordinator, variable 2, 3/2-stable 3017 Clayton: Prices, variable 1, 3/2-stable 3118 Clayton: Prices, variable 2, 3/2-stable 31


0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.00.0

4.0

8.0

12.0

16.0

20.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

copula values

u values

v values

Clayton-Lévy Copula

Figure 1. Clayton-Lévy Copula

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

copula values

u values

v values

Clayton-Lévy CopulaLevel Curves

Figure 2. Clayton-Lévy Copula, Level Curves


0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.00.0

2.04.0

6.08.0

10.012.0

14.016.0

18.020.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

copula values

u values

v values

Gumbel-Lévy Copula

Figure 3. Gumbel-Lévy Copula

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

copula values

u values

v values

Gumbel-Lévy CopulaLevel Curves

Figure 4. Gumbel-Lévy Copula, Level Curves


0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.00.0

2.04.0

6.08.0

10.012.0

14.016.0

18.020.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

copula values

u values

v values

Complementary Gumbel-Lévy Copula

Figure 5. Complementary Gumbel-Lévy Copula

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

copula values

u values

v values

Complementary Gumbel-Lévy CopulaLevel Curves

Figure 6. Complementary Gumbel-Lévy Copula, Level Curves


Conditional Copula Distributions� = 1, u = 2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

v values

cum

ulat

ive

prob

abili

ty Clayton

ComplementaryGumbel

Gumbel

Figure 7. Conditional Copula Distributions

Marginal Lévy Measures3/2-stable: � = 1.5

Gamma: � = 20, � = 1

0.0

2.0

4.0

6.0

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24.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

copula axis

mar

gina

l mea

sure

3/2-stable

Gamma

Values are insignificantly different from zero above 5.0.

Figure 8. Marginal Lévy Measures


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Empirical Copula Difference 3/2−stable

log price_1Fractile

Diff

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ce

Figure 9. Clayton: Copula, log price 1, 3/2-stable

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Empirical Copula Difference 3/2−stable

log price_2Fractile

Diff

eren

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Figure 10. Clayton: Copula, log price 2, 3/2-stable


LP1 3/2−stable

First variable bins

Fre

quen

cy

−3 −2 −1 0 1 2

05

1015

2025

Figure 11. Clayton: Histogram, log relative 1, 3/2-stable

LP2 3/2−stable

Second variable bins

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quen

cy

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05

1015

20

Figure 12. Clayton: Histogram, log relative 2, 3/2-stable


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Theoretical Quantiles −− LP_1

Sam

ple

Qua

ntile

s

Figure 13. Clayton: QQ – normal plot, log relative 1, 3/2-stable

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Theoretical Quantiles −− LP_2

Sam

ple

Qua

ntile

s

Figure 14. Clayton: QQ – normal plot, log relative 2, 3/2-stable


0 5 10 15 20

05

1015

2025

30

Subordinator_1 −− 3/2 stable

Poisson ProcessJump Times

Pro

cess

Val

ue

Figure 15. Clayton: Subordinator, variable 1, 3/2-stable

0 5 10 15 20

05

1015

2025

3035

Subordinator_2 −− 3/2 stable


Pro

cess

Val

ue

Figure 16. Clayton: Subordinator, variable 2, 3/2-stable


0 5 10 15 20

01

23

4

Prices_1 −− 3/2 stable


Pric

e

Figure 17. Clayton: Prices, variable 1, 3/2-stable

0 5 10 15 20

010

2030

4050

60

Prices_2 −− 3/2 stable


Pric

e

Figure 18. Clayton: Prices, variable 2, 3/2-stable

References 32

References

Amussen, S. and J. Rosiński (2001, Jun.). Approximation of small jumps of Lévyprocesses with a view towards simulation. J. Appl. Probab. 38 (2), 482–493.

Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. FinanceStochastics 2, 41–68.

Barndorff-Nielsen, O. E. (2000, July). Probablilty densities and Lévy densities. WorkingPaper Series No. 66, Centre for Analytical Finance, University of Aarhus.

Barndorff-Nielsen, O. E. and N. Shephard (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. Roy. Statist. Soc.Ser. B 63 (2), 167–241. With discussion.

Barndorff-Nielsen, O. E. and N. Shephard (2005, May). Continuous Time Approach toFinancial Volatility. Cambridge: Cambridge University Press.

Benth, F. E. and P. C. Kettler (2011, Mar.). Dynamic copula models for the spark spread.Quant. Finance 11 (3), 407–421.

Cherubini, U., E. Luciano, and W. Vecchiato (2004). Copula Methods in Finance.Chichester: Wiley.

Cont, R. and P. Tankov (2004). Financial Modelling with Jump Processes. Boca Raton:Chapman & Hall/CRC.

Frees, E. W. and E. A. Valdez (1998, Jan.). Understanding relationships using copulas. N.Amer. Actuarial J. 2 (1), 1–25. Discussions Vol. 2, No. 3, pp. 143–149, and Vol. 3,No. 1, pp. 137–141.

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Jacobsen, M. (1996). Laplace and the origin of the Ornstein-Uhlenbeck process.Bernoulli 2 (3), 271–286.

Kallsen, J. and P. Tankov (2004). Characterization of dependence of multidimensionalLévy processes using Lévy copulas. Preprint, Technische Universität München.

Karatzas, I. and S. E. Shreve (1991). Brownian Motion and Stochastic Calculus (2nd ed.).New York: Springer-Verlag.

Nelsen, R. B. (1998). An Introduction to Copulas. New York: Springer-Verlag.Øksendal, B. K. (2003). Stochastic Differential Equations (6th ed.). Berlin:

Springer-Verlag.Prause, K. (1999, Oct.). The generalized hyperbolic model: estimation, financial

derivatives, and risk measures. Ph. D. thesis, University of Freiburg, Germany.Advisor, Prof. Dr. Wolfgang Soergel.

R Development Core Team (2005). R: A language and environment for statisticalcomputing. Vienna, Austria: R Foundation for Statistical Computing. ISBN3-900051-07-0.

Rogers, L. and D. Williams (1994). Diffusions, Markov Processes, and Martingales (2nded.), Volume 1, Foundations. Chichester: Wiley.

Rosiński, J. (1991). On a class of infinitely divisible processes represented as mixtures ofGaussian processes. In S. Cambanis, G. Samorodnitsky, and M. S. Taqqu (Eds.), StableProcesses and Related Topics: A Selection of Papers from the Mathematical SciencesInstitute Workshop, Volume 25 of Progress in Probability, pp. 27–41. Birkhäuser, Basle.Meetings January 9–13, 1990.

References 33

Rosiński, J. (2006). Tempering stable processes. To appear in Stoch. Proc. Appl.Schweizer, B. and E. F. Wolff (1981, Jul.). On nonparametric measures of dependence for

random variables. Ann. Statist. 9 (4), 879–885.Tankov, P. (2003, Dec.). Dependence structure of spectrally positive multidimensional

Lévy processes. Applied Probability Trust, Centre de Mathématiques Appliquées,École Polytechnique, Palaiseau, France.

Tankov, P. (2004, Sep.). Lévy Processes in Finance: Inverse Problems and DependenceModelling. Ph. D. thesis, Centre de Mathématiques Appliquées, École Polytechnique,Route de Saclay, 91128 Palaiseau cedex, Paris. Rama Cont, advisor.

Tankov, P. (2006). Simulation and option pricing in Lévy copula models. To appear in: M.Avellaneda and R. Cont (eds.), Mathematical Modelling of Financial Derivatives, IMAVolumes in Mathematics and Applications, Springer.

Uhlenbeck, G. E. and L. S. Ornstein (1930). On the theory of Brownian motion. Phys.Rev. 36, 823–841.

Würtz, D. et al. (2005). Financial Software Collection — fBasics. R Foundation forStatistical Computing. R package version 220.10063.

(Paul C. Kettler)Centre of Mathematics for ApplicationsDepartment of MathematicsUniversity of OsloP.O. Box 1053, BlindernN–0316 OsloNorway

E-mail address: [email protected]: http://www.paulcarlislekettler.net/

Introduction - Paul Carlisle KettlerClayton-Lévy: C(u;v) = u + v 1 (4.5) ; >0...

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