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1. Building Levy process

1.1. Building new Levy processes from known ones.

1.1.1. Linear transformation. To construct new Levy processes, we us three basictypes of transformations, under which the class of Levy processes is invariant:linear transformation, subordination (time changing a Levy process with anotherincreasing Levy processes) and exponential tilting of the Levy measure.

Theorem 1.1. Let (Xt)t≥0 be a Levy process on Rd with characteristic triplet(A, v, γ) and let M be an n× d matrix. Then Yt = MXt is a Levy process on Rn

with characteristic triplet (AY , vY , γY ) where

AY = MAM t (1)

vY (B) = v(x : Mx ∈ B), ∀B ∈ B(Rn) (2)

γY = Mγ +

∫Rdy(1|y|≤1(y)− 1− S1(y)vY (dy) (3)

S1 is the image by M of a unit ball in Rd : S1 = Mx : |x| ≤ 1

Proof. Proof see Cont and Tankov (2004), pp. 106.

Example 1.2. Sums of independent Levy processesLet (XT )t≥0 and (Yt)t≥0 be two independent Levy processes with characteristic

triplets (A1, v1, γ1) and (A2, v2, γ2). By using Theorem 1.1 with M = (1 1) andProposition 5.3 from Chapter 5 in Cont Tankov we obtain that Xt + Yt is a Levyprocess with characteristic triplet (A, v, γ) where

A = A1 + A2,

v(B) = v1(B) + v2(B) ∀B ∈ B(R)

γ = γ1 + γ2 −∫

[−√

2,−1]∪[1,√

2]

For Levy processes of infinite variation, Theorem 1.1 can be simplified. Namely,let (Xt)t≥0 be a Levy process of finite variation on Rd with characteristic function

E[eiz.Xt ] = exp t

ib.z +

∫R(eiz.x − 1)v(dx)

(4)

1

2 TANKOV-CONT

and let M be an n×d matrix. Then Yt = MXt is a Levy process on Rn with Levymeasure vY (B) = v(x : Mx ∈ B) and by = Mb.

1.2. Subordination. Let (St)t≥0 be a subordinator, that is, a Levy process sat-isfying one of the equivalent conditions of Proposition 3.10 which means in par-ticular that its trajectories are almost surely increasing. Since St is a positiverandom variable for all t, we describe it by using Laplace transform rather thanFourier transform. Let the characteristic triplet of S be (0, ρ, b). then the momentgenerating function of St is

E[euSt ] = et l(u) ∀u ≤ 0, where l(u) = bu+

∫ ∞0

(eux − 1)ρ(dx) (5)

We call l(u) the Laplace exponent of S. Since process S is increasing it can beinterpreted as ”time deformation” and used to ”time change” other Levy processesas shown in the following theorem.

Theorem 1.3. Subordination of a Levy process Fix a probability space (Ω,F ,P).Let (Xt)t≥0 be a Levy process on Rd with characteristic exponent Ψ(u) and triplet(A, v, γ) and let (St)t≥0 be a subordinator with Laplace exponent l(u) and triplet(0, ρ, b). Then the process (Yt)t≥0 defined for each ω ∈ Ω by Y (t, ω) by Y (t, ω) =X(S(t, ω), ω) is a Levy process. Its characteristic function is

E[eiuYt ] = etl(Ψ(u)), (6)

i.e. the characteristic exponent of Y is obtained by composition of the Laplaceexponent of S with the characteristic exponent of X. the triplet (AY , vY , γY ) of Yis given by

AY = bA

vY (B) = bv(B) +

∫ ∞0

pXs (B)ρ(ds), ∀B ∈ B(Rd), (7)

γY = bγ +

∫ ∞0

ρ(ds)

∫|x|≤1

xpXs (dx) (8)

where pXt is the probability distribution of Xt.

(Yt)t≥0 is said to be subordinate to the process (Xt)t≥0.

Proof. Let us first prove that Y is a Levy process. Denote by FSt the filtration of(St)t≥0 with FS = F∞. For every sequence of times t0 < t1 < · · · < tn we obtain,using the independent increments property of X, Levy-Khinchin formula for X

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and the independent increments property of S:

E[Πni=1e

iui(X(Sti )−X(Sti−1))]

= EE[Πni=1e

iui(X(Sti )−X(Sti−1)) | FS]

= E

Πni=1E

[eiui(X(Sti )−X(Sti−1)) | FS

]= E

Πni=1e

(Sti−Sti−1)Ψ(ui)

= Πni=1E

e(Sti−Sti−1)Ψ(ui)

= Πn

i=1Eeiui(Sti−Sti−1)

Therefore, Y has independent increments. The stationarity of increments ca beshown in the same way. To show that Y is continuous in probability, first ob-serve that every Levy process is uniformly continuous in probability, due to thestatinarity of its increments. Further, for every ε > 0 and δ > 0, one can write:

P | X(Ss)−X(St) |> ε≤ P | X(Ss)−X(St) |> ε || Ss − St < δ+ P | Ss − St ≥ δ

The first term can be made arbitrarily small simultaneously for all values of sand t by changing δ, because X is uniformly continuous in probability. as for thesecond term,it limit as s→ t is always zero, because S is continuous in probability.Hence, P| X(Ss)−X(St) |> ε → 0 as s→ t. The formula (6) is easily obtainedby conditioning on FS:

E[eiuX(St)

]= E

E[eiuX(St) | FS

]= EeStΨ(u) = etl(Ψ(u).

For the detailed proof see Sato (1999), Theorem 30.1.We explain what’s going on in a simple example. Suppose S is a compound

Poisson subordinator with characteristic triplet (0, ρ, 0). Then Y is again a com-pound Poisson process with the same intensity, because it moves only by jumpsand its jumps occur at the same time as those of S. Therefore the drift and Gauss-ian component are equal to zero. To compute its jump measure, suppose that Shas a jump at t. Conditionally on St − S− = s, the size of jump in Y has thedistribution pXs . Integrating with respect to jump measure of S, we obtain formula(7). Finally, rewriting the characteristic triplet of Y with respect to the truncationfunction 1|x|≤1, we obtain formula (8) for γY .

Example 1.4. A stable subordinator is an α-stable process with α ∈ (0, 1), Levymeasure concentrated on the positive half-axis and a non-negative drift (such aprocess is a subordinator because it satisfies the last condition of Proposition 3.10tankov. Let (St)t≥0 be a stable subordinator with zero drift. Its Laplace exponentis

4 TANKOV-CONT

l(u) = c1

∫ ∞0

eux − 1

x1+αdx = −c1Γ(1− α)

α(−u)α (9)

for some constant c1 > 0. Let (Xt)t≥0 be a symmetric β-stable process on R withcharacteristic exponent Ψ(u) = −c2|u|β for some constant c2 > 0. Then the process(YT )t≥0 subordinate to X by S has characteristic exponent l(Ψ(u)) = −c|u|βα,where c = c1c2Γ(1 − α)/α, that is, Y is symmetric stable process with index ofstability βα. In particular, when X is a Brownian motion, the subordinate processis 2α−stable.

1.2.1. Tilting and tempering the Levy measure. One way to specify a Levy processis by giving an admissible Levy triplet: in particular, the Levy measure must verifythe constraints: ∫

|x|≤1

|x|2v(dx) <∞∫|x|≥1

v(dx) <∞

Any transformation of the Levy measure, respecting the integrability constraintabove, will lead to a new Levy process. Examples of such transformations areobtained by multiplying v(·) by an exponential function. If there exists θ ∈ Rd

such that∫|x|≥1

eθ.xv(dx) <∞ the the measure v defined by

v(dx) = eθ.xv(dx) (10)

is a Levy measure. Then for any Levy process (Xt)t≥0 on Rd with characteris-tic triplet (A, v, γ), the process with characteristic triplet (A, v, γ) is also a Levyprocess, called the Esscher transform of X. The transform given by (10) is calledexponential tilting of the Levy measure. Esscher transforms are discussed in chap-ter 9 Cont and Tankov (2004).

When d = 1 we can consider a asymmetric version of the transformation: if v isaLevy measure on R then

v(dx) = v(dx)(1x>0e−λ+x + 1x<0e

−λ−|x|,

where λ+ and λ− are positive parameters, is also a Levy measure and defines aLevy process whose large jumps are ”tempered” i.i. the tail of the Levy measureare exponentially damped.

1.3. Models of jump-diffusion type. A Levy process of jump-diffusion typehas the following form:

Xt = γt+ σWt +Nt∑i=1

Yi (11)

where (Nt)t≥0 is the Poisson process counting the jumps of X and Yi are jumpsizes (i.i.d. variables). To define the parametric model completely, we must nowspecify the distribution of jump sizes v0(x). Is is especially important to specifythe tail behaviour of v0 correctly depending on one’s belief about behaviour of

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extremal events, because as we have seen, the tail behaviour of th jump measuredetermiedn to a large extent the tail behaviour of probability density of the process( Propositions 3.13 and 3.14 in Cont Tankov)

In the Merton model, jumps in the log-price Xt are assumed to have a Gaussiandistribution: Yi ∼ N(µ, δ2). This allows to obtain the probability density of Xt asquickly converging series. Indeed,

PXt ∈ A =∞∑k=0

PXt ∈ A|Nt = kPNt = k

which entails that the probability density of Xt satisfies

pt(x) = e−λt∞∑k=0

(λt)k exp− (x−γt−kµ)2

2(σ2t+kδ2

k!√

2π(σ2t+ kδ2)(12)

Prices of european options in the Merton model can be obtained by a series whereeach term involves a Black-Scholes formula.

In the Kou model (Rosinski, 2001), the distribution of jump sizes is an asym-metric exponential with density of the form

v0(dx) = [pλ+e−λ+x1x>0 + (1− p)λ−e−λ−|x|1x<0]dx (13)

with λ+ > 0, λ− > 0 governing the decay of the tails for the distribution of positivean negative jump sizes and p ∈ [0, 1] representing the probability of upward jump.The probability distribution of returns in this model has semi-heavey (exponential)tails. The advantage of this model compared to the previous one is that due tothe memoryless property of exponential random variables, analytical expressionsfor expectations involving first passage times may be obtained.

1.4. Building Levy processes by Brownian subordination.

1.4.1. General results. Let (St)t≥0 be a subordinator with Laplace exponent l(u)and let (Wt)t≥0 be a Brownian motion independent from S. Subordinating Brow-nian motion with drift µ by the process S, we obtain a new Levy process Xt =σW (St) + µst. This process is a Brownian motion if it is observed on a new timescale, that is, the stochastic time scale given by S.

This time scale as the financial interpretation of business time, that is the in-tegrated rate of information arrival. This interpretation makes models based onsubordinated Brownian motion easier to understand than general Levy models.Equation (6) entails that X has characteristic exponent

Ψ(u) = l(−u2σ2/2 + iµu).

This allows to compute cumulants of Xt from those of St. Consider the symmetriccase µ = 0. Then Xt is symmetric, therefore has zero mean and skewness, an one

6 TANKOV-CONT

can easily compute variance and excess kurtosis:

V arXt = σ2E[St],

k(Xt) =3V ar StE[St]2

Therefore, Xt is leptokurtic if the subordinator is not a deterministic process.The representation of Brownian subordination can make the model easier to un-

derstand and adds tractability, it imposes some important limitations on the formof Levy measure. The following theorem characterises Levy measure of processesthat can be represented as subordinated Browniani motion with drift.

Recall that a function f : [a, b] → R is called completely monotonic if all its

derivatives exists and (−1)k dkf(u)duk

for all k ≥ 1.

Theorem 1.5. Let v be a Levy measure on R and µ ∈ R. There exists a Levyprocess (Xt)t≥0 with Levy measure v such that Xt = W (Zt) + µZt for some sub-ordinator (Zt)t≥0 and some Brownian motion (Wt)t≥0 independent from Z if andonly if the following conditions are satisfied:

(1) v is absolutely continuous with density v(x)(2) v(x)eµx = v(−x)eµx for all x.(3) v(

√u)e−µ

√u is a completely monotonic function on (0,∞).

This theorem allows to describe the jump structure of a process, that can berepresented as time changed Brownian motion with drift. For example the Levymeasure of such process has an exponentially tilted version that is symmetric onR. since the exponential tilting mainly affects the big jumps, this means that smalljumps of such process will always be symmetric.

Let v be a Levy measure on Rd. It can be the Levy measure of a subordinatedBrownian motion (with our drift) off it is symmetric and v(

√u) is completely

monotonic function on (0,∞). Furthermore, consider a subordinator with zerodrift and Levy measure ρ.

Formula (7) entail that a Brownian motion with drift µ time changed by thissubordinator will have Levy density v(x) given by

v(x) =

∫ ∞0

e−(x−µ)2

2tρ(dt)√

2πt(14)

We can symbolically denote this operation by BSµ(ρ) = v, where BS stands forBrownian subordination. The inverse transform is denoted by

BS−1µ (v) = ρ.

Then (14) allows to write

BS−1µ (v) = eµ

2t/2BS−10 (ve−µx) (15)

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Subordinator Gamma process Inverse Gaussian process

Levy density ρ(x) = ceλx

x1x > 0 ρ(x) = ceλx

x3/21x > 0

Laplace Transform E[euSt ] = (1− u/λ)−ct E[euSt ] = e−2ct√π(√λ−u−

√λ

Probability pt(x) = λct

Γ(ct)xct−1eλx pt(x) = ct

x3/2e2ct√πλe−λx−πc

2t2/x

density for x > 0 for x > 0

Hence, we can deduce that the time changed Brownian motion representation foran exponentially tilted Levy measure from the representation for its symmetricmodification.

Proof. of Theorem 1.5 The only if part. The absolute continuity of v is a directconsequence of (7), because the Gaussian probability distribution is absolutelycontinuous. Omitting the constant factor, the formula (14) can be rewritten as

v(x)eµx =

∫ ∞0

e−x2

2t e−µ2t2 t−2ρ(dt)

which shows that v(x)e−µx must be symmetric. Further, by making the variablechange u = x2/2 and s1/t we obtain (simplify notation and assume that ρ hasdensity) ∫ ∞

0

e−use−µ2

2s s−3/2ρ(1/s)ds = v(√

2u)e−µ√

2u (16)

Proof of the only if part see Cont and Tankov (2004) pp. 114.

1.4.2. Subordinating processes. Let us consider the tempered stable subordinator, that s, an exponentially tempered version of the stable subordinator, discussedin Example 1.4. It is a three-parameter process with Levy measure

ρ(x) =ceλx

xα+11x > 0, (17)

where c and λ are positive constants and 1 > α ≥ 0. For greater generality weinclude the case α = 0 (the gamma process) although it cannot be obtained fromthe stable subordinator via exponential tilting.

A tempered stable subordinator is, of course, a tempered stable process in thesense of formula (add this reference coming later26). The parameter c alters theintensity of jumps of all sizes simultaneously; in other worlds, it changes the timescale of the process, λ fixes the decay rate of big jumps and α determines therelative importance of small jumps in the path process. The Laplace exponent oftempered stable subordinator in the general case (α 6= 0) is

l(u) = cΓ(−α)(λ− u)α − λα (18)

and l(u) = −c log(1 − u/λ) if α = 0. The probability density of temperedstable subordinator is only known explicit form for α = 1/2 (inverse Gaussian

8 TANKOV-CONT

subordinator) and α = 0 (gamma subordinator). These two cases are comparedin Table 1.4.2.

The tempered stable subordinator possesses the following scaling property. Let(St(α, λ, c))t≥0 be a tempered stable subordinator with parameters α, λ and c.Then every r > 0, rSt(α, λ, c) has the same law as Srαt(α, λ/r, c). Bacause ofthis scaling property and the scaling property of Brownian motion (rW : t hasthe same law as Wr2t), in subordinated models it is sufficient to consider onlytempered stable subordinators with E[St] = t, which in this case form a two-parameter family. for all computations related to characteristic function, momentsand cumulants it is convenient to use the parametrization

ρ(x) =1

Γ(1− α)

(1− αk

)1−αe−(1−α)x/k

x1+α(19)

where α is the index of stability. In this new parametrization k is equal to thevariance of subordinator at time 1. Since the expectation of the subordinator attie 1 is equal to 1, k actually determines how random the time change is, and thecase k = 0 corresponds to a deterministic function. in the variance gamma casethis formula simplifies to

ρ(x) =1

k

e−x/k

x(20)

and in the inverse Gaussian case

ρ(x) =1√2πk

e−x2k

x3/2(21)

1.4.3. Models based on subordinated Brownian motion. By time changing an in-dependent Browniani motion (with volatility σ and drift θ) by a tempered stylesubordinator , we obtain the so called normal tempered stable process (processderived from Brownian subordinators are ”normal”). Its characteristic exponentis

Ψ(u) =1− αkα

1−

(1 +

k(u2σ2/2− iθu)

1− α

)α(22)

in the general case and

Ψ(u) = −1

klog1 +

u2σ2k

2− iθku (23)

in the variance gamma case (α = 0). The Levy measure of a normal temperedstable prices can be computed using equation (7). It has density v(x) given by

JUMP PROCESSES 9

v(x) =2c

σ√

2π

∫ ∞0

e−(xθ)2

2tσ2−λt dt

tα + 3/2

=2c

σ√

2π

(√θ2 + 2λσ2

|x|

)α+1/2

Kα+1/2

(|x|√θ2 + 2λσ2

σ2

)

=C(α, k, σ, θ

|x|α+1/2eθx/σ

2

Kα+1/2

|x|√θ2 + 2

kσ2(1− α)

σ2

,

(24)

where

C(α, k, σ, θ) =2

Γ(1− α)σ√

2π

(1− αk

)1−α

(θ2 +2

kσ2(1− α))α/2+1/4

and K is the modified Bessel function of the second kind (See Appendix A.1 Contand Tankov, 2004). In accordance to Theorem 1.5, this measure is an exponentialtilt of a symmetric measure. Introducing tail decays rates

λ+ =1

σ2

(√θ2 +

2

kσ2(1− α)− θ

)and

λ− =1

σ2

(√θ2 +

2

kσ2(1− α) + θ

)we can rewrite this measure in the following form:

v(x) =C

|x|α+1/2ex(λ−−λ+)/2Kα+1/2(|x|(λ− + λ+)/2). (25)

From the asymptotic behaviour formulae for K, we deduce that

v(x) ∼ 1

|x|2α+1, when x→ 0,

v(x) ∼ 1

|x|2α+1e−λ+x when x→∞,

v(x) ∼ 1

|x|2α+1e−λ−|x| when x→ −∞,

That is, the Levy measure has a stable like behaviour near zero and exponentialdecay with decay rates λ+ and λ− at the tails.

Because the probability density of tempered stable subordinator is known inclosed form for α = 1/2 and α = 0, the corresponding subordinated processes arealso more mathematically tractable and easier to simulate and therefore they have

10 TANKOV-CONT

been widely used in the literature. Namely, the variance gamma process has beenused as a model for the log of stock prices and the normal inverse Gaussian process(NIG) has been used for financial modelling.

1.5. Tempered stable process. The tempered stable process is obtained bytaking one-dimensional stable process and multiplying the Levy measure with de-creasing exponential on each half of the real axis. after this exponential softening,the small jumps keep their initial stable-like behaviour whereas the large jumpsbecome much less violent. A tempered stable process is thus a Levy process on Rwith no Gaussian component and a Levy density of the form

v(x) =c−|x|1+α

e−λ−|x|1x<0 +c+

x1+αe−λ+x1x>0, (26)

where the parameter satisfy c− > 0, c+ > 0, λ− > 0, λ+ > 0 and α < 2. Thismodel was used in Cont, Bouchud, Potters (1997).

Remark 1.6. Generalized tempered stable process Unlike the stable pro-cesses, which can only be defined for α > 0, in the tempered stable case there isno natural lower bound on α and the expression (26) yields a Levy measure for allα < 2. In fact, taking negative values of α we obtain compound Poisson modelswith rush structure. It may also be interesting to allow for different values of α onthe two sides of real axis. To include these cases into our treatment, we use thename ”tempered stable” for the process with Levy measure of the form (26) withα > 0 (because only in this case the small jump have stable-like behaviour) and weuse the term generalised tempered stable model for the process with Levy measure

v(x) =c

|x|1+α−e−λ−|x|1x < 0 +

c

x1+α+e−λ+x1x>0 (27)

with α+ < 2 and α− < 2. all formulae of this section will be given for generalisedtempered stable model.

The following proposition shows that the tempered stable model allows for richerstructures than the subordinated Brownian motion models that we have treadedin the preceding section.

Proposition 1.7. Time changed Brownian motion representation fortempered stable process A generalised tempered stable process (27) can berepresented as a time changed Brownian motion (with drift) iff c− = c+ andα− = α+ = α ≥ −1.

Remark 1.8. The subordinator in this representation can be expressed via specialfunctions: the representation is given by mu = (lambda− − λ+)/2 and

ρ(t) =c

tα/2+1etµ

2/2−tλ2/4D−α(λ√t) (28)

where λ = (λ− + λ+)/2, c is a constant and D−α(z) denotes the Whittaker’sparabolic cylinder function.

JUMP PROCESSES 11

Remark 1.9. The condition on the coefficients means that the small jumps mustbe symmetric whereas decay rates for big jumps may be different. In other words theclass of tempered stable processes which are representable as time changed Brow-nian motion coincides with the models discussed by Carr et al under the nameCGMY. Hence in the class of tempered stable processes one has a greater modellingfreedom than with models based on Brownian subordination because tempered sta-ble models allow for asymmetric large jumps, the CGMY subclass is probably asflexible as the whole tempered stable process.

From Propositon SEE HERE 3.8, 3.9, 3.10 we deduce that a generalised tem-pered stable process

(1) is of compound Poisson type if alpha+ < 0 and alpha− < 0,(2) has trajectories of finite variation if α+ < 1 and α− < 1,(3) is a subordinator (positive Levy process) if c− = 0, α+ < 1 and the drift

para,ever is positive.

The limiting case α− = α+ = 0 corresponds to an infinite activity process. If inaddition c+ = c−, we recognise the variance gamma model of the previous section.Excluding the deterministic drift parameter we see the the generalised temperedstable prices is a parametric model with six parameters. We will discuss their rolea little later after computing the characteristic function of the process.

Working with tempered stable process becomes more convenient if we use theversion of Levy-Khinchin formula without truncation of big jumps, namely wewrite

E[eiuXt ] = exp t

iuγc +

∫ ∞−∞

(eiux − 1− iux)v(x)dx

(29)

this form can be used because of exponential decay of the tails of Levy measure.in this case E[Xt] = γct. To compute the characteristic function, we first considerthe positive half of the Levy measure and suppose that α± = 1 and α± 6= 0.∫ ∞

0

(eiux − 1− iux)e−λx

x1+αdx =

∞∑n=2

(iu)n

n!

∫ ∞0

xn−1−αe−λxdx

=∞∑n=2

(iu)n

n!λα−nΓ(n− α)

= λαΓ(2− α) 1

2!

(iu

λ

)2

+2− α

3!

(iu

λ

)3

+ · · ·

· · ·+ (2α)(3− α)

4!

(iu

λ

)4

+ · · ·

The expression in braces resembles to the power series

(1 + x)µ = 1 + µx+ µ(µ− 1)x2

2!+ · · ·

12 TANKOV-CONT

Comparing the two series we conclude that

∫ ∞0

(eiux − 1− iux)e−λx

x1+αdx = λαΓ(−α)

(1− iu

λ

)α− 1 +

iuα

λ

(30)

The interchange of sum and integral and the convergence of power series thatwe used to obtain this expression can be formally justified if |u| < λ but theresulting formula can be extend via analytic continuation to other values of u suchthat Zu > −λ. To compute the power in (30) we choose a branch of zα thatis continuous in the upper help plane and maps positive half-line into positivehalf-line.

A similar computation in the case α = 1, yields that∫ ∞0

(eiux − 1− iux)e−λx

x2= (λ− iu)log

(1− iu

λ

)+ iu (31)

and if α = 0, ∫ ∞0

(eiux − 1− iux)e−λx

xdx =

u

iλ+ log

iλ

u+ iλ(32)

Assembling together both parts of the Levy measure, we obtain the characteristicfunction of the generalized tempered stable process.

Proposition 1.10. Let (Xt)t≥0 be a generalized tempered stable process. In thegeneral case (α± 6= 1 and α± 6= 0) its characteristic exponent Ψ(u) = t−1 log E[eiuX ]is

Ψ(u) = iuγc + Γ(−α+)λα+

+ c+

(1− iu

λ+

)α+

− 1 +iuα

λ+

+ Γ(−α−)λ

α−− c−

(1 +

iu

λ−

)α−

− 1− iuα−λ−

.

and if α+ = α− = 1,

Ψ(u) = iu(γc + c+ − c−) + c+(λ+ − iu) log

(1− iu

λ+

)+ c−(λ− + iu) log

(1 +

iu

λ−

),

and if α+ = α− = 0,

JUMP PROCESSES 13

Ψ(u) = iuγc − c+

iu

λ+

+ log

(1− iu

λ+

)− c−

− iuλ−

+ log

(1 +

iu

λ−

)The other cases (when only one of the α− s is equal to 0 or 1) can be obtained ina similar fashion.

Proposition 3.13 (SEE HERE !!!!!!!!!!) allows to compute the first cumulants ofthe tempered stable process. Taking derivatives of the characteristic exponent, wefind in the general case:

K1 = E[Xt] = tγc,

K2 = V arXt = tγ(2− α+)c+λα+−2+ + tΓ(2− α−)c−λ

α−−2− ,

K3 = tΓ(3− α+)c+λα+−3+ − tΓ(3− α−)c−λ

α−−3− ,

K4 = tΓ(4− α+)c+λα+−4+ + tΓ(4− α−)c−λ

α−−4− .

These expression do not clarify completely the role of different parameters. Forexample, suppose that the process id symmetric. Then the excess kurtosis of thedistribution of Xt is

k =K4

(V arXt)2=

(2− α)(3− α)

ctλα

which shows that we can decrease the excess kurtosis by either increasing λ (thejumps become smaller and the process becomes closer to a continuous one) or in-creasing c (the jumps become more frequent and therefore the central limit theoremworks better).

The expression does not allow for distinguishing the effect of c and λ. To fullyunderstand the role of different parameters we must pass to the dynamical view-point and look at the Levy measure rather than at the moments of the distribution.Then it becomes clear that λ− and λ+ determine the tail behavior of the Levy mea-sure, they tell us how far the process may jump, and from the point of view ofa risk manager this corresponds to the amount of money that we can lose (orearn) during a short period of time. c+ and c− determine the overall and relativefrequency upward and downward jumps; of course the total frequency is infinite,but if we are interested only in jumps larger than a given value, the these twoparameters, tell us, how often we should expect such events. Finally, α+ and α−determine the local behavior of the process (how the price evolves between bigjumps). When α+ and α− are close to 2, the process behaves much like a Brown-ian motion, with many small oscillations between big jump. On the other hand,

14 TANKOV-CONT

if α+ and α− is small, most of price changes are due to big jumps with periods ofrelative tranquillity between them.

1.6. Generalized hyperbolic model. This shows the modeling approach byspecifying the probability density directly. Let us first come back to the inverseGaussian subordinator with probability density at some fixed time

p(x) = c(X , ζ)x−3/2e−12

(Xx−ζ/x1x>0, (33)

Introducing an additional parameter into this distribution, we obtain the so-calledgeneralized inverse Gaussian law (GIG):

p(x) = c(λ,X , ζ)xλ−1e−12

(Xx−ζ/x)1x>0. (34)

This distribution was proven to be infinitely divisible (see Halgreen (1979)) andcan generate a Levy process (a subordinator). However, GIG laws do not form aconvolution closed class, which means that distributions of this process at othertimes will not be GIG. Let S be a GIG random variable and W be an independentstandard normal random variable. Then the law of

√SW + µS, where µ is a con-

stant, is called normal variance-mean mixture with mixing distribution GIG. Thiscan be seen as a static analog of Brownian subordination (indeed, the distributionof the subordinated process at time t is a variance-mean mixture with the mixingdistribution being that of the subordinator at time t).

Normal variance-mean mixtures of GIG laws are called generalized hyperbolicdistributions (GH) and they are also infinitely divisible. The one-dimensional GHlaw is a five-parameter family that is usually defined via its Lebesgue density:

p(x;λ, α, β, δ, µ) = C(δ2 + (x− µ)2)λ2− 1

4Kλ− 12(α√δ2 + (x− µ)2eβ(x−µ)

C =(α2 − β2)λ/2

√2παλ−1/2δλKλ(δ

√α2 − β2)

, (35)

where K is the modified Bessel function of the second kind. The characteristicfunction of this law has the following form:

Φ(u)eiµu(

α2 − β2

α2 − (β + iu)2

)λ/2Kλ(δ

√λ2 − (β + iu)2)

Kλ(δ√α2 − β2)

(36)

The main disadvantage of GH laws is that hey are not closed under convolution:The sum of two independent GH random variable is not a GH random variable.This fact makes GH laws inconvenient for working with data on different scales.For example it is difficult to calibrate ea GH model to a price sheet with optionof several maturities. In this case one has to choose one maturity at which thestock price distribution is supposed to be generalized hyperbolic, and distributionat other maturities must be computed as convolution powers of this one. On thecontrary, it is relatively easy to sample from a GH distribution and to estimate

JUMP PROCESSES 15

its parameters when all data are on the same time scale (e.g. one disposes of anequally spaced price series).

Because the GH law is infinitely divisible, one can construct a generalized hyper-bolic Levy process whose distributions at fixed times have characteristic functionsΦ(u)t. The Levy measure of this process is difficult to compute and work with butsome of its properties can be read directly from the characteristic function, usingthe following lemma.

Lemma 1.11. Let (Xt) be a Levy process with characteristic exponent Ψ(u) =1tlogE[eiuXt ] and let Ψ(u) = Ψ(u)+Ψ(−u)

2be the symmetric characteristic exponent.

(1) If X is a compound Poisson process then Ψ(u) is bounded.

(2) If X is a finite variation process then Ψu→ 0 as u→∞

(3) If X has no Gaussian part then Ψu2→ 0 as u→∞

Proof. Proof see Cont and Tankov pp. 126

For the GH distribution we find, using asymptotic properties of Bessel functions,that Ψ(u) ∼ −δ | u | when u→∞. This relation is valid in the general case but itdoes not hold in some particular case (e.g., when δ = 0). This means that exceptin some particular cases generalized hyperbolic Levy process is an infinite variationprocess without Gaussian part.

The principal advantage of the GH law is its rich structure and great variety ofshapes. Indeed, many of the well-known probability distributions are subclassesof GH family. For example,

(1) The normal distribution is the limiting case of GH distribution when δ →∞ and δ/α→ σ2.

(2) The case λ = 1 corresponds to the hyperbolic distribution with density

p(x;α, β, δ, µ) = a(α, β, δ)exp(−α√δ2 + (x− µ)2 + β(x− µ))