jump-type levy processesartax.karlin.mff.cuni.cz/~adaml5am/seminar/1011z/vorisek.pdf ·...

Jump-type Levy Processes


Ernst Eberlein

Handbook of Financial Time Series


Outline

Table of contentsProbabilistic Structure of Levy Processes

Levy processLevy-Ito decompositionJump part

Probabilistic Structure of Levy ProcessesThe distribution of a Levy processLevy-Khintchine formulaIntegrability propertiesProperties of the process

Financial ModelingClassical modelExponential Levy modelPricing of derivativesModels for interest rates

Levy Processes with Jumps


Probabilistic Structure of Levy Processes

Levy process

Levy process

I X = (Xt)t≥0 is a process with stationary and independentincrements

I underlying it is a filtered probability space (Ω,F , (Ft)t≥0,P)

I filtration (Ft)t≥0 is complete and right continuous

I proces Xt is Ft-adapted

I Levy process has version with cadlag path (Theorem 30 inProtter (2004)), i.e. right-continuous with limits from the left

? every Levy process is a semimartingale



Levy process

One-dimensional Levy process

Xt = X0 + bt +√

cWt + Zt +∑s≤t

∆XsI|∆Xs |>1 (1)

I b, c ≥ 0 are real numbers

I (Wt)t≥0 is standard Brownian motion

I (Zt)t≥0 is purely discontinuous martingale

I (Wt)t≥0 and (Zt)t≥0 are independent

I ∆Xs := Xs − Xs− denotes the jump at time s

? in case where c = 0, the process is purely discontinuous

? (1) is the so-called canonical representation forsemimartingales



Levy-Ito decomposition

Levy-Ito decomposition

I for semimartingale Y = (Yt)t≥0:

Yt −∑s≤t

∆YsI|∆Xs |>1

is a special semimartingale (see I.4.21 and I.4.24 in Jacod,Shiryaev (1987)), which admits unique decomposition intolocal martingale M = (Mt)t≥0 and a predictable process withfinite variation V = (Vt)t≥0, which for Levy processes is the(deterministic) linear function of time bt

I any local martingale M with (M0 = 0) admits a uniquedecomposition (see I.4.18 in Jacod, Shiryaev (1987)):

M = Mc + Md =√

cW + Z

where the second equality holds for Levy processes



Jump part

Jump part

I cadlag paths over finite intervals => any path has only finitenumber of jumps with absolute jump size larger than ε > 0(i.e. sum of jumps bigger than ε is a finite sum for each path)

I the sum of small jumps:∑s≤t

∆XsI|∆Xs |≤1 (2)

does not converge in general (infinitely many small jums), butone can force this sum to converge by compensating it, i.e. bysubtracting the corresponding average increase of the process

I the average increase can be expressed by the intensity F (dx)with which the jumps arrive



Jump part

Compensation

limε→0

∑s≤t

∆XsIε≤|∆Xs |≤1 − t

∫xIε≤|x |≤1F (dx)

(3)

I this limit exists in the sense of convergence in probability

I the sum represents the (finitely many) jumps

I the integral is the average increase of the process

I one cannot separate the difference, because neither of the twoexpressions has a finite limit as ε→ 0

? one can express the (3) using the random measure of jumpsof the process X denoted by µX



Jump part

Random measure of jumps

µX (ω; dt, dx) =∑s≤0

I|∆Xs |6=0δ(s,∆Xs(ω))(dt, dx)

I if a path of the process X given by ω has a jump of size∆Xs(ω) = x at time s, then the random measure µX (ω; ., .)places unit mass δ(s,x) at the point (s, x) ∈ R+ × R

I consequently for a time interval [0, t] and a set A ⊂ R,µX (ω; [0, t]× A) counts jumps of size within A:

µX (ω; [0, t]× A) = |(s, x) ∈ [0, t]× A|∆Xs(ω) = x|

I average number of jumps expressed by an intensity measure:

E[µX (.; [0, t]× A)

]= tF (A)



Jump part

Another expression

I the sum of the big jumps:∫ t

0

∫R

xI|x |>1µX (ds, dt)

I (Zt), the martingale of compensated small jumps:∫ t

0

∫R

xI|x |≤1

(µX (ds, dt)− dsF (dx)

)(4)

? µX (ω; dt, dx) is a random measure, i.e. it depends on ω

? dsF (dx) is a product measure on R+ × R? again these mesures cannot be separated in general



The distribution of a Levy process


I the distribution of a Levy process X = (Xt)t>0 is completelydetermined by any of its marginal distributions L(Xt)

I lets consider L(X1) and arbitrary natural number n:

X1 = X1/n + (X2/n − X1/n) + . . .+ (Xn/n − Xn−1/n)

I by stationarity and independence of the increments, L(X1) isthe n-fold convolution of L(X1/n):

L(X1) = L(X1/n) ∗ · · · ∗ L(X1/n)

I consequently L(X1) and analogously any L(Xt) are infinitelydivisible distributions




Infinitely divisible distributions

I conversely any infinitely divisible distribution ν generates in anatural way a Levy process X = (Xt)t>0 which is uniquelydetermined by setting L(X1) = ν

I if for n > 0, νn is the probability measure such thatν = νn ∗ · · · ∗ νn then one gets immediately for rational timepoints t = k/n L(Xt) as a k-fold convolution of νn

I for irrational time points t, L(Xt) is determined by acontinuity argument (see Chapter 14 in Breiman (1968))

I the process to be constructed has independent increments =>it is sufficient to know the one-dimensional distribution

I if a specific infinitely divisible distribution is chatacterizedby a few parameters the same hold for the correspondingLevy process (crucial for estimation of parameters)



Levy-Khintchine formula


I the Fourier transform E [exp(iuX1)] of a Levy process given aninfinitely divisible distribution ν = L(X1) is:

exp

[iub − 1

2u2c +

∫R

(eiux − 1− iuxI|x |≤1

)F (dx)

](5)

I the b, c,F determine the L(X1), and thus the process X

I (b,c,F) is called the Levy-Khintchine triplet or insemimartingale terminology the triplet of local characteristics

I the truncation function h(x) = xI|x |≤1 used in (5) could bereplaced by other versions of truncation functions (e.g.smooth one: x (identity) near the origin, else goes to zero)

? changing h results in a different drift parameter b, whereasthe diffusion coeficient c ≥ 0 and the Levy measure F remain




RemarksI Levy measure F does not have mass on 0 and satisfies:∫

Rmin(1, x2)F (dx) <∞

I conversely any measure on R with these two properties plusb ∈ R and c ≥ 0 defines via (5) an infinitely divisibledistribution and thus a Levy process.

I Levy-Khintchine formula (5) with ψ(u) (characteristicexponent):

E [exp(iuX1)] = exp(ψ(u))

I again by independence and stationarity of the increments:

E [exp(iuXt)] = exp(tψ(u))

? important for computation of derivative value E [f (XT )],parameters of X estimated as the parameters of L(X1)



Integrability properties

Integrability properties of Levy measure F

I finiteness of of moments of the process depends only onfrequency of large jumps since it is related to integration by Fover |x | > 1

Proposition (1)

Let X = (Xt)t≥0 be a Levy process with Levy measure F .

1. Xt has finite p-th moment for p ∈ R+, i.e. E [|Xt |p] <∞, ifand only if

∫|x |>1 |x |

pF (dx) <∞.

2. Xt has finite p-th exponential moment for p ∈ R+, i.e.E [exp(pXt)] <∞, if and only if

∫|x |>1 exp(px)F (dx) <∞.

Proof: see Theorem 25.3 in Sato (1999)




Finite expectation of L(X1)I =>

∫|x |>1 xF (dx) <∞ => we can add

−∫

iuxI|x |>1F (dx) to the (5) and get:

E [exp(iuX1)] = exp

[iub − 1

2u2c +

∫R

(eiux − 1− iux

)F (dx)

]I =>

∫ t0

∫R xI|x |>1dsF (dx) exists => to the (4) we can add:∫ t

0

∫R

xI|x |>1

(µX (ds, dt)− dsF (dx)

)?∫ t

0

∫R xI|x |>1µ

X (ds, dt), always exists for every pathI as a result we get (1) in simpler representation:

Xt = X0+b∗t+√

cWt+Zt+

∫ t

0

∫R

x(µX (ds, dt)− dsF (dx)

)(6)




Finite expectation of L(X1) II

I the drift coefficient b∗ = E [X1] because the W and Z aremartingales

I Levy processes used in finance have finite first moments, thanwe get the (6) representation

I the existence of moments is determined by the frequency ofthe big jumps

I the fine structure of the path is related to the frequency ofthe small jumps

? process has finite activity if almost all path have only finitenumber of jumps along any time interval of finite length

? process has infinite activity if almost all path have infinitelymany jumps along any time interval of finite length



Properties of the process

Activity of the process

Proposition (2)


1. X has finite activity if F (R) <∞.

2. X has infinite activity if F (R) =∞.

I by definituon a Levy measure satisfies∫

R I|x |>1F (dx) <∞=> the assumption F (R) <∞ (or F (R) =∞) is equivalenttu assumption of finitenes (or infinitenes) of

∫R I|x |≤1F (dx)

I the path of Brownian motion have infinite variation

I whether the purely discontinuous component (Zt) in (1) orintegral in (6) has path with infinite variation depends on thefrequency of the small jumps



Properties of the process

Variation of the process

Proposition (3)


1. Almost all path of X have finite variation if c = 0 and∫|x |≤1 |x |F (dx) <∞.

2. Almost all path of X have infinite variation if c 6= 0 or∫|x |≤1 |x |F (dx) =∞.

Proof: see Theorem 21.9 in Sato (1999)I the integrability of F (

∫|x≤1 |x |F (dx) <∞) guarantees also

that the sum of the small jumps (2) converges (a.s.)? in this case one can separate the measures µX and F in (6):

∫ t

0

∫R

x(µX (ds, dt)− dsF (dx)

)=

∫ t

0

∫R

xµX (ds, dt)−t

∫R

xF (dx)


Ito formula for jump process

Ito formula for jump process

f (Xt) = f (X0) +

∫ t

0f ′(Xs)dX c

s +1

2

∫ t

0f ′′(Xs)dX c

s dX cs +

+∑

0<s≤t

[f (Xs)− f (Xs−)]

X cs = bt +

√cWt


Financial Modeling

Classical model

Classical model

dSt = µStdt + σStdWt

I the geometric Brownian motion goes back to Samuelson(1965)

St = S0 exp(σWt + (µ− σ2/2)t)

I the exponent of this price process is Levy process

I given (1): b = µ− σ2/2,√

c = σ, Zt = 0 and no big jumps

I log returns with time step 1 normally distributed

N(µ− σ2/2, σ2)


Financial Modeling

Exponential Levy model


I one can identify a parametric distribution ν by fitting anempirical distribution

I considering the Levy process X = (Xt)t≥0 such thatL(X1) = ν, the model

St = S0 exp(Xt) (7)

produces log returns exactly equal to ν (infinitely divisible)

I the stochastic differential equation describing the process

dSt = St−(dXt + (c/2)dt + e∆Xt − 1−∆Xt)

I the distribution of the log returns of this process is not knownin general


Financial Modeling


Exponential Levy model II

I a Levy version of differential equation

dSt = St−dXt

has the solution the stochastic exponential

St = S0 exp(Xt − ct/2)∏s≤t

(1 + ∆Xs) exp(−∆Xs)

I one can directly see from (1 + ∆Xs) that this model canproduce negative prices as soon as the driving Levy process Xhas negative jumps larger than 1

I the Levy measures of distributions used in finance have strictlypositive densities on the whole negative half line (negativejumps of arbitrary size)


Financial Modeling

Pricing of derivatives


I price process given by (7) has to be a martingale

? pricing is done by taking expectations under risk neutral(martingale) measure

I for (St)t≥0 to be a martingale, expectation has to be finite

? candidates for the driving process are Levy processes X with afinite first exponential moment E [exp(Xt)] <∞

I Proposition 1 characterizes these processes in terms of theirLevy measure

? the necessary assumption of finiteness of the first exponentialmoment a priori excludes stable processes


Financial Modeling


Xt = X0+bt+√

cWt +Zt +∫ t

0

∫R x(µX (ds, dt)− dsF (dx)

)

I let X be given in the representation (6) then St = S0 exp(Xt)is a martingale if

b = −c

2−∫

R(ex − 1− x)F (dx) (8)

I this can be seen by applying Ito formula to St = S0 exp(Xt),where (8) guarantees that the drift component is 0


Financial Modeling

Models for interest rates

Forward rate approach

I the dynamics of the instantaneous forward rate with matutityT at time t

f (t,T ) = f (0,T ) +

∫ t

0α(s,T )ds −

∫ t

0σ(s,T )dXs

I the coefficients α(s,T ) and σ(s,T ) can be deterministic orrandom

I one gets zero-coupon bond prices in a form comparable to thestock price (7)

B(t,T ) = B(0,T ) exp

(∫ t

0(r(s)− A(s,T ))ds +

∫ t

0Σ(s,T )dXs

),

? where r(s) = f (s, s) is the short rate and A(s,T ) and Σ(s,T )are derived from α(s,T ) and σ(s,T ) by integration


Financial Modeling

Models for interest rates

Levy LIBOR market modelI the forward LIBOR rates L(t,Tj) for the time points Tj

(0 ≤ j ≤ N) are chosen as the basic ratesI as a result of bacward induction one gets for each j the rate

L(t,Tj) = L(0,Tj) exp

(∫ t

0(λ(s,Tj)dX

Tj+1s

),

? where λ(s,Tj) is a volatility structure, X Tj+1 = (XTj+1t )t≥0 is

the process derived from an initial Levy processX TN = (X TN

t )t≥0 and the L(t,Tj) is considered under PTj+1

(the forward martingale measure)

I closely related to the LIBOR model is the forward processmodel, where forward processes

F (t,Tj ,Tj+1) = B(t,Tj)/B(t,Tj+1)

are chosen as the basic quantities



Levy Jump Diffusion

Poisson process

I the simplest Levy measure is ε1 (a point mass in 1)

I by adding an intensity parameter λ > 0, one gets F = λε1

I this Levy measure generates a process X = (Xt)t≥0 withjumps of size 1 which occur with an average rate of λ in aunit time interval

I X is called a Poisson process with intensity λ

I the drift parameter b in Fourier transform is E [X1], which isλ, therefore it takes the form:

E [exp(iuXt)] = exp[λt(e iu − 1)]

I any variable Xt has a Poisson distribution with parameter λt

P[Xt = k] = exp(−λt)(λt)k

k!



Levy Jump Diffusion

Exponentially distributed waiting times

I one can show that the successive waiting times from one jumpto the next are independent exponentially distributed randomvariables with parameter λ

I starting with a sequence (τi )i≥1 of independent exponentially(λ) distributed r.v. and setting Tn =

∑ni=1 τi , the associated

counting process

Nt =∑n≥1

ITn≤t

is a Poisson process with intensity λ



Levy Jump Diffusion

Compound Poisson processI a natural extension of the Poisson process is a process where

the jump size is randomI let Y = (Yt)t≤0 be a sequence of iid. r. v., L(Y1) = ν

Xt =Nt∑i=1

Yi ,

? where (Nt)t≥0 is a Poisson process (λ > 0) independent of(Yi )i≥0, defines a compound Poisson process X = (Xt)t≥0

with intensity λ and jump size distribution νI Fourier transform is given by

E [exp(iuXt)] = exp

[λt

∫R

(e iux − 1)ν(dx)

]I the Levy measure is given by F (A) = λν(A) for measurable

sets A in R



Levy Jump Diffusion

Levy jump diffusion

I a Levy jump diffusion is a Levy process where the jumpcomponent is given by a compound Poisson process

Xt = bt +√

cWt +Nt∑i=1

Yi ,

? where b ∈ R, c > 0, (Wt)t≥0 is a standard Brownian motion,(Nt)t≥0 is a Poisson process with intensity λ > 0 and (Yi )i≥0

is a sequence of iid. r.v. independent of (Nt)t≥0

I one can use e.g. normally or double-exponentially distributedjump sizes Yi

I any other distribution could be considered, but the question isif one can control explicitly the quantities one is interested in(for example, L(Xt))



Hyperbolic Levy porcesses

Hyperbolic Levy processes

I hyperbolic distributions which generate hyperbolic Levyprocesses X = (Xt)t≥0 - also called hyperbolic Levy motions-constitute a four-parameter class of distributions withLebesgue density

dH(x) =

√α2 − β2

2α δ K1(δ√α2 − β2)

exp

(−α√δ2 + (x − µ)2 + β(x − µ)

),

? where Kj denotes the modified Bessel function of the thirdkind

I α determines the shape, β with 0 ≤ |β| < α the skewness,µ ∈ R the location and δ > 0 is a scaling parameter

I taking the logarithm of dH , one gets a hyperbola



Hyperbolic Levy porcesses

The Fourier transform of a hyperbolic distribution

φH(u) = exp(iuµ)

(α2 − β2

α2 − (β + iu)2

)1/2K1(δ

√α2 − (β + iu)2)

K1(δ√α2 − β2)

I moments of all order exists and

E [X1] = µ+βδ√α2 − β2

K2(δ√α2 − β2)

K1(δ√α2 − β2)

I in case of hyperbolic distributions c = 0, which means thathyperbolic Levy motios are purely discontinuous processes



Generalized hyperbolic Levy processes

Generalized hyperbolic distributions

I hyperbolic distributions are a subclass of a more powerfullfive-parameter class, the generalized hyperbolic distributions(Barndorff-Nielsen (1978))

I the additional class parameter λ ∈ R has the value 1 forhyperbolic distributions

dGH(x) = a(λ, α, β, δ)(δ2 + (x − µ)2)(λ− 12

)/2·

·Kλ−1/2

(α√δ2 + (x − µ)2

)exp (β(x − µ)) ,

? where the normalizing constant is given by

a(λ, α, β, δ) =(α2 − β2)λ/2

√2παλ−1/2δλKλ(δ

√α2 − β2)




Generalized inverse Gaussian distributions

I generalized hyperbolic distribution can be represented as anormal mean-variance mixtures

dGH(x) =

∫ ∞0

dN(µ+βy)(x) · dGIG (x ;λ, δ,√α2 − β2)dy

? where the mixing distribution is generalized inverse Gaussianwith density

dGIG (x ;λ, δ, γ) =(γδ

)λ xλ−1

2Kλ(δγ)exp

(− 1

2x(δ2 + γ2x2)

)for x > 0





I the moment generating function MGH(u) for |u + β| < α:

MGH(u) = exp(µu)

(α2 − β2

α2 − (β + iu)2

)λ/2Kλ(δ

√α2 − (β + u)2)

Kλ(δ√α2 − β2)

I as a consequence, exponential moments are finite, which iscrucial fact for pricing of derivatives under martingalemeasures

I the Fourier transform φGH is obtained from the relationφGH(u) = MGH(iu)

I again c = 0, so generalized hyperbolic Levy motions arepurely discontinuous processes

I the Levy measure F has a closed-form density




Normal inverse Gaussian distributions

I setting λ = −1/2 we get the normal inverse Gaussiandistributions

dNIG (x) =αδK1(α

√δ2 + (x − µ)2)

π√δ2 + (x − µ)2

exp(β(x−µ)+δ√α2 − β2)

I their Fourier transform is simple becauseK−1/2(z) = K1/2(z) =

√π/(2z)e−z :

φNIG (u) = exp(iuµ) exp(δ√α2 − β2

)exp

(−δ√α2 − (β + iu)2

)I one can see that NIG are closed under convolution in

parameters δ and µ



α-Stable Levy processes


I stable distributions constitute a four-parameter class ofdistributions with Fourier transform given by

φst(u) =

exp

(iuµ− |σu|α

(1− iβsign(u) tan πα

2

))ifα 6= 1,

exp(iuµ− |σu|

(1 + iβsign(u) 2

π ln |u|))

ifα = 1

I the parameter space is 0 < α ≤ 2, σ ≥ 0, −1 ≤ β ≤ 1 andµ ∈ R

I for α = 2 one gets the Gaussian distribution with mean µ andvariance 2σ2

I for α < 2 there is no Gaussian part, which means the paths ofan α-stable Levy motion are purely discontinuous




Special cases of α-Stable distributions

I explicit densities are known in three cases only:

? Gaussian distribution, α = 2, β = 0

? Cauchy distribution, α = 1, β = 0

? Levy distribution, α = 1/2, β = 1

I usefulness in particular as a pricing model is limited for α 6= 2by the fact that finiteness of the first exponential moment innot satisfied



Meixner Levy processes

Meixner Levy processes

I the Fourier transform of Meixner distributions is given by

φM(u) =

(cos(β/2)

cosh((αu − iβ)/2)

)2δ

? for α > 0, |β| < π, δ > 0

I the corresponding Levy processes are purely discontinuouswith paths of infinite variation

I the density of Levy measure F is

gM(x) =δ

x

exp(βx/α)

sinh(πx/α)



CGMY and variance gamma Levy processes

CGMY and variance gamma Levy processes

I the class of CGMY distributions (infinitely divisible) extendsthe variance gamma model

I CGMY Levy processes have purely discontinuous paths andthe density of Levy mesure is given by

gCGMY (x) =

C exp(−G |x |)

|x |1+Y x < 0,

C exp(−Mx)x1+Y x > 0

? with parameter space C ,G ,M > 0 and Y ∈ (−∞, 2)

I the process has infinite activity iff Y ∈ [0, 2)

I the paths have infinite variation iff Y ∈ [1, 2)

I for Y = 0 one gets the three-parameter variance gammadistributions which are a subclass of the generalizedhyperbolic distributions


References

Barndorff-Nielsen, O.E. (1978): Hyperbolic distributions anddistributions on hyperbolae. Scandinavian Journal of Statistics5, 151-157.

Breiman, L (1968): Probability. Addison-Wesley, Reading.

Jacod, J., Shiryaev, A.N. (1987): Limit Theorems forStochastic Processes. Springer, New York.

Protter, P.E. (2004): Stochastic Integration and DifferentialEquations. (2nd ed.) Volume 21 of Applications ofMathematics. Springer, New York.

Samuelson, P. (1965): Rational theory of warrant pricing.Industrial Management Review 6:13-32.

Sato, K.-I. (1999): Levy Processes and Infinitely DivisibleDistributions. Cambridge University Press, Cambridge.

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