Jump-type Levy Processes

Transcription

1 Jump-type Levy Processes Ernst Eberlein Handbook of Financial Time Series

2 Outline Table of contents Probabilistic Structure of Levy Processes Levy process Levy-Ito decomposition Jump part Probabilistic Structure of Levy Processes The distribution of a Levy process Levy-Khintchine formula Integrability properties Properties of the process Financial Modeling Classical model Exponential Levy model Pricing of derivatives Models for interest rates Levy Processes with Jumps

3 Probabilistic Structure of Levy Processes Levy process Levy process X = (X t ) t 0 is a process with stationary and independent increments underlying it is a filtered probability space (Ω, F, (F t ) t 0, P) filtration (F t ) t 0 is complete and right continuous proces X t is F t -adapted Levy process has version with cadlag path (Theorem 30 in Protter (2004)), i.e. right-continuous with limits from the left every Levy process is a semimartingale

4 Probabilistic Structure of Levy Processes Levy process One-dimensional Levy process X t = X 0 + bt + cw t + Z t + s t X s I { Xs >1} (1) b, c 0 are real numbers (W t ) t 0 is standard Brownian motion (Z t ) t 0 is purely discontinuous martingale (W t ) t 0 and (Z t ) t 0 are independent X s := X s X s denotes the jump at time s in case where c = 0, the process is purely discontinuous (1) is the so-called canonical representation for semimartingales

5 Probabilistic Structure of Levy Processes Levy-Ito decomposition Levy-Ito decomposition for semimartingale Y = (Y t ) t 0 : Y t s t Y s I { Xs >1} is a special semimartingale (see I.4.21 and I.4.24 in Jacod, Shiryaev (1987)), which admits unique decomposition into local martingale M = (M t ) t 0 and a predictable process with finite variation V = (V t ) t 0, which for Levy processes is the (deterministic) linear function of time bt any local martingale M with (M 0 = 0) admits a unique decomposition (see I.4.18 in Jacod, Shiryaev (1987)): M = M c + M d = cw + Z where the second equality holds for Levy processes

6 Probabilistic Structure of Levy Processes Jump part Jump part cadlag paths over finite intervals => any path has only finite number of jumps with absolute jump size larger than ɛ > 0 (i.e. sum of jumps bigger than ɛ is a finite sum for each path) the sum of small jumps: X s I { Xs 1} (2) s t does not converge in general (infinitely many small jums), but one can force this sum to converge by compensating it, i.e. by subtracting the corresponding average increase of the process the average increase can be expressed by the intensity F (dx) with which the jumps arrive

7 Probabilistic Structure of Levy Processes Jump part Compensation lim X s I {ɛ Xs 1} t s t ɛ 0 xi {ɛ x 1} F (dx) (3) this limit exists in the sense of convergence in probability the sum represents the (finitely many) jumps the integral is the average increase of the process one cannot separate the difference, because neither of the two expressions has a finite limit as ɛ 0 one can express the (3) using the random measure of jumps of the process X denoted by µ X

8 Probabilistic Structure of Levy Processes Jump part Random measure of jumps µ X (ω; dt, dx) = I { Xs =0}δ (s, Xs(ω))(dt, dx) s 0 if a path of the process X given by ω has a jump of size X s (ω) = x at time s, then the random measure µ X (ω;.,.) places unit mass δ (s,x) at the point (s, x) R + R consequently for a time interval [0, t] and a set A R, µ X (ω; [0, t] A) counts jumps of size within A: µ X (ω; [0, t] A) = {(s, x) [0, t] A X s (ω) = x} average number of jumps expressed by an intensity measure: [ ] E µ X (.; [0, t] A) = tf (A)

9 Probabilistic Structure of Levy Processes Jump part Another expression the sum of the big jumps: t xi { x >1} µ X (ds, dt) 0 R (Z t ), the martingale of compensated small jumps: t 0 R ) xi { x 1} (µ X (ds, dt) dsf (dx) (4) µ X (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

10 Probabilistic Structure of Levy Processes The distribution of a Levy process The distribution of a Levy process the distribution of a Levy process X = (X t ) t>0 is completely determined by any of its marginal distributions L(X t ) lets consider L(X 1 ) and arbitrary natural number n: X 1 = X 1/n + (X 2/n X 1/n ) (X n/n X n 1/n ) by stationarity and independence of the increments, L(X 1 ) is the n-fold convolution of L(X 1/n ): L(X 1 ) = L(X 1/n ) L(X 1/n ) consequently L(X 1 ) and analogously any L(X t ) are infinitely divisible distributions

11 Probabilistic Structure of Levy Processes The distribution of a Levy process Infinitely divisible distributions conversely any infinitely divisible distribution ν generates in a natural way a Levy process X = (X t ) t>0 which is uniquely determined by setting L(X 1 ) = ν if for n > 0, ν n is the probability measure such that ν = ν n ν n then one gets immediately for rational time points t = k/n L(X t ) as a k-fold convolution of ν n for irrational time points t, L(X t ) is determined by a continuity argument (see Chapter 14 in Breiman (1968)) the process to be constructed has independent increments => it is sufficient to know the one-dimensional distribution if a specific infinitely divisible distribution is chatacterized by a few parameters the same hold for the corresponding Levy process (crucial for estimation of parameters)

12 Probabilistic Structure of Levy Processes Levy-Khintchine formula Levy-Khintchine formula the Fourier transform E[exp(iuX 1 )] of a Levy process given an infinitely divisible distribution ν = L(X 1 ) is: [ exp iub 1 ] ( 2 u2 c + e iux ) 1 iuxi { x 1} F (dx) R (5) the b, c, F determine the L(X 1 ), and thus the process X (b,c,f) is called the Levy-Khintchine triplet or in semimartingale terminology the triplet of local characteristics the truncation function h(x) = xi { x 1} used in (5) could be replaced 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, whereas the diffusion coeficient c 0 and the Levy measure F remain

13 Probabilistic Structure of Levy Processes Levy-Khintchine formula Remarks Levy measure F does not have mass on 0 and satisfies: min(1, x 2 )F (dx) < R conversely any measure on R with these two properties plus b R and c 0 defines via (5) an infinitely divisible distribution and thus a Levy process. Levy-Khintchine formula (5) with ψ(u) (characteristic exponent): E[exp(iuX 1 )] = exp(ψ(u)) again by independence and stationarity of the increments: E[exp(iuX t )] = exp(tψ(u)) important for computation of derivative value E[f (X T )], parameters of X estimated as the parameters of L(X 1 )

14 Probabilistic Structure of Levy Processes Integrability properties Integrability properties of Levy measure F finiteness of of moments of the process depends only on frequency of large jumps since it is related to integration by F over { x > 1} Proposition (1) Let X = (X t ) t 0 be a Levy process with Levy measure F. 1. X t has finite p-th moment for p R +, i.e. E[ X t p ] <, if and only if { x >1} x p F (dx) <. 2. X t has finite p-th exponential moment for p R +, i.e. E[exp(pX t )] <, if and only if { x >1} exp(px)f (dx) <. Proof: see Theorem 25.3 in Sato (1999)

15 Probabilistic Structure of Levy Processes Integrability properties Finite expectation of L(X 1 ) => { x >1} xf (dx) < => we can add iuxi { x >1} F (dx) to the (5) and get: [ E[exp(iuX 1 )] = exp iub 1 2 u2 c + R ( e iux 1 iux ) ] F (dx) => t 0 R xi { x >1}dsF (dx) exists => to the (4) we can add: t ) xi { x >1} (µ X (ds, dt) dsf (dx) 0 R t 0 R xi { x >1}µ X (ds, dt), always exists for every path as a result we get (1) in simpler representation: X t = X 0 +b t+ t ( ) cw t +Z t + x µ X (ds, dt) dsf (dx) 0 R (6)

16 Probabilistic Structure of Levy Processes Integrability properties Finite expectation of L(X 1 ) II the drift coefficient b = E[X 1 ] because the W and Z are martingales Levy processes used in finance have finite first moments, than we get the (6) representation the existence of moments is determined by the frequency of the big jumps the fine structure of the path is related to the frequency of the small jumps process has finite activity if almost all path have only finite number of jumps along any time interval of finite length process has infinite activity if almost all path have infinitely many jumps along any time interval of finite length

17 Probabilistic Structure of Levy Processes Properties of the process Activity of the process Proposition (2) Let X = (X t ) t 0 be a Levy process with Levy measure F. 1. X has finite activity if F (R) <. 2. X has infinite activity if F (R) =. by definituon a Levy measure satisfies R I { x >1}F (dx) < => the assumption F (R) < (or F (R) = ) is equivalent tu assumption of finitenes (or infinitenes) of R I { x 1}F (dx) the path of Brownian motion have infinite variation whether the purely discontinuous component (Z t ) in (1) or integral in (6) has path with infinite variation depends on the frequency of the small jumps

18 Probabilistic Structure of Levy Processes Properties of the process Variation of the process Proposition (3) Let X = (X t ) t 0 be a Levy process with Levy measure F. 1. Almost all path of X have finite variation if c = 0 and x F (dx) <. { x 1} 2. Almost all path of X have infinite variation if c 0 or x F (dx) =. { x 1} Proof: see Theorem 21.9 in Sato (1999) 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 µ X and F in (6): t 0 R x ( ) t µ X (ds, dt) dsf (dx) = xµ X (ds, dt) t xf (dx) 0 R R

19 Ito formula for jump process Ito formula for jump process t f (X t ) = f (X 0 ) + f (X s )dxs c + 1 t f (X s )dxs c dxs c [f (X s ) f (X s )] 0<s t X c s = bt + cw t

20 Financial Modeling Classical model Classical model ds t = µs t dt + σs t dw t the geometric Brownian motion goes back to Samuelson (1965) S t = S 0 exp(σw t + (µ σ 2 /2)t) the exponent of this price process is Levy process given (1): b = µ σ 2 /2, c = σ, Z t = 0 and no big jumps log returns with time step 1 normally distributed N(µ σ 2 /2, σ 2 )

21 Financial Modeling Exponential Levy model Exponential Levy model one can identify a parametric distribution ν by fitting an empirical distribution considering the Levy process X = (X t ) t 0 such that L(X 1 ) = ν, the model S t = S 0 exp(x t ) (7) produces log returns exactly equal to ν (infinitely divisible) the stochastic differential equation describing the process ds t = S t (dx t + (c/2)dt + e Xt 1 X t ) the distribution of the log returns of this process is not known in general

22 Financial Modeling Exponential Levy model Exponential Levy model II a Levy version of differential equation ds t = S t dx t has the solution the stochastic exponential S t = S 0 exp(x t ct/2) s t(1 + X s ) exp( X s ) one can directly see from (1 + X s ) that this model can produce negative prices as soon as the driving Levy process X has negative jumps larger than 1 the Levy measures of distributions used in finance have strictly positive densities on the whole negative half line (negative jumps of arbitrary size)

23 Financial Modeling Pricing of derivatives Pricing of derivatives price process given by (7) has to be a martingale pricing is done by taking expectations under risk neutral (martingale) measure for (S t ) t 0 to be a martingale, expectation has to be finite candidates for the driving process are Levy processes X with a finite first exponential moment E[exp(X t )] < Proposition 1 characterizes these processes in terms of their Levy measure the necessary assumption of finiteness of the first exponential moment a priori excludes stable processes

24 Financial Modeling Pricing of derivatives X t = X 0 +bt + cw t +Z t + t 0 R x ( µ X (ds, dt) dsf (dx) ) let X be given in the representation (6) then S t = S 0 exp(x t ) is a martingale if b = c 2 (e x 1 x)f (dx) (8) R this can be seen by applying Ito formula to S t = S 0 exp(x t ), where (8) guarantees that the drift component is 0

25 Financial Modeling Models for interest rates Forward rate approach the dynamics of the instantaneous forward rate with matutity T at time t t t f (t, T ) = f (0, T ) + α(s, T )ds σ(s, T )dx s 0 0 the coefficients α(s, T ) and σ(s, T ) can be deterministic or random one gets zero-coupon bond prices in a form comparable to the stock price (7) ( t t ) B(t, T ) = B(0, T ) exp (r(s) A(s, T ))ds + Σ(s, T )dx s, 0 0 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

26 Financial Modeling Models for interest rates Levy LIBOR market model the forward LIBOR rates L(t, T j ) for the time points T j (0 j N) are chosen as the basic rates as a result of bacward induction one gets for each j the rate ( t L(t, T j ) = L(0, T j ) exp 0 (λ(s, T j )dx T j+1 s ), where λ(s, T j ) is a volatility structure, X T j+1 = (X T j+1 t ) t 0 is the process derived from an initial Levy process X T N = (X T N t ) t 0 and the L(t, T j ) is considered under P T j+1 (the forward martingale measure) closely related to the LIBOR model is the forward process model, where forward processes F (t, T j, T j+1 ) = B(t, T j )/B(t, T j+1 ) are chosen as the basic quantities

27 Levy Processes with Jumps Levy Jump Diffusion Poisson process the simplest Levy measure is ɛ 1 (a point mass in 1) by adding an intensity parameter λ > 0, one gets F = λɛ 1 this Levy measure generates a process X = (X t ) t 0 with jumps of size 1 which occur with an average rate of λ in a unit time interval X is called a Poisson process with intensity λ the drift parameter b in Fourier transform is E[X 1 ], which is λ, therefore it takes the form: E[exp(iuX t )] = exp[λt(e iu 1)] any variable X t has a Poisson distribution with parameter λt P[X t = k] = exp( λt) (λt)k k!

28 Levy Processes with Jumps Levy Jump Diffusion Exponentially distributed waiting times one can show that the successive waiting times from one jump to the next are independent exponentially distributed random variables with parameter λ starting with a sequence (τ i ) i 1 of independent exponentially (λ) distributed r.v. and setting T n = n i=1 τ i, the associated counting process N t = I {Tn t} n 1 is a Poisson process with intensity λ

29 Levy Processes with Jumps Levy Jump Diffusion Compound Poisson process a natural extension of the Poisson process is a process where the jump size is random let Y = (Y t ) t 0 be a sequence of iid. r. v., L(Y 1 ) = ν N t X t = Y i, where (N t ) t 0 is a Poisson process (λ > 0) independent of (Y i ) i 0, defines a compound Poisson process X = (X t ) t 0 with intensity λ and jump size distribution ν Fourier transform is given by [ ] E[exp(iuX t )] = exp λt (e iux 1)ν(dx) R the Levy measure is given by F (A) = λν(a) for measurable sets A in R i=1

30 Levy Processes with Jumps Levy Jump Diffusion Levy jump diffusion a Levy jump diffusion is a Levy process where the jump component is given by a compound Poisson process X t = bt + N t cw t + Y i, where b R, c > 0, (W t ) t 0 is a standard Brownian motion, (N t ) t 0 is a Poisson process with intensity λ > 0 and (Y i ) i 0 is a sequence of iid. r.v. independent of (N t ) t 0 one can use e.g. normally or double-exponentially distributed jump sizes Y i any other distribution could be considered, but the question is if one can control explicitly the quantities one is interested in (for example, L(X t )) i=1

31 Levy Processes with Jumps Hyperbolic Levy porcesses Hyperbolic Levy processes hyperbolic distributions which generate hyperbolic Levy processes X = (X t ) t 0 - also called hyperbolic Levy motionsconstitute a four-parameter class of distributions with Lebesgue density d H (x) = α 2 β 2 ( ) 2α δ K 1 (δ α 2 β 2 ) exp α δ 2 + (x µ) 2 + β(x µ), where K j denotes the modified Bessel function of the third kind α determines the shape, β with 0 β < α the skewness, µ R the location and δ > 0 is a scaling parameter taking the logarithm of d H, one gets a hyperbola

32 Levy Processes with Jumps Hyperbolic Levy porcesses The Fourier transform of a hyperbolic distribution ( α 2 β 2 ) 1/2 K1 (δ α φ H (u) = exp(iuµ) 2 (β + iu) 2 ) α 2 (β + iu) 2 K 1 (δ α 2 β 2 ) moments of all order exists and E[X 1 ] = µ + βδ K 2 (δ α 2 β 2 ) α 2 β 2 K 1 (δ α 2 β 2 ) in case of hyperbolic distributions c = 0, which means that hyperbolic Levy motios are purely discontinuous processes

33 Levy Processes with Jumps Generalized hyperbolic Levy processes Generalized hyperbolic distributions hyperbolic distributions are a subclass of a more powerfull five-parameter class, the generalized hyperbolic distributions (Barndorff-Nielsen (1978)) the additional class parameter λ R has the value 1 for hyperbolic distributions d GH (x) = a(λ, α, β, δ)(δ 2 + (x µ) 2 ) (λ 1 )/2 2 ) K λ 1/2 (α δ 2 + (x µ) 2 exp (β(x µ)), where the normalizing constant is given by a(λ, α, β, δ) = (α 2 β 2 ) λ/2 2πα λ 1/2 δ λ K λ (δ α 2 β 2 )

34 Levy Processes with Jumps Generalized hyperbolic Levy processes Generalized inverse Gaussian distributions generalized hyperbolic distribution can be represented as a normal mean-variance mixtures d GH (x) = 0 d N(µ+βy) (x) d GIG (x; λ, δ, α 2 β 2 )dy where the mixing distribution is generalized inverse Gaussian with density ( γ ) ( λ x λ 1 d GIG (x; λ, δ, γ) = δ 2K λ (δγ) exp 1 ) 2x (δ2 + γ 2 x 2 ) for x > 0

35 Levy Processes with Jumps Generalized hyperbolic Levy processes Generalized hyperbolic Levy processes the moment generating function M GH (u) for u + β < α: ( α 2 β 2 ) λ/2 Kλ (δ α M GH (u) = exp(µu) 2 (β + u) 2 ) α 2 (β + iu) 2 K λ (δ α 2 β 2 ) as a consequence, exponential moments are finite, which is crucial fact for pricing of derivatives under martingale measures the Fourier transform φ GH is obtained from the relation φ GH (u) = M GH (iu) again c = 0, so generalized hyperbolic Levy motions are purely discontinuous processes the Levy measure F has a closed-form density

36 Levy Processes with Jumps Generalized hyperbolic Levy processes Normal inverse Gaussian distributions setting λ = 1/2 we get the normal inverse Gaussian distributions d NIG (x) = αδk 1(α δ 2 + (x µ) 2 ) π δ 2 + (x µ) 2 exp(β(x µ)+δ α 2 β 2 ) their Fourier transform is simple because K 1/2 (z) = K 1/2 (z) = π/(2z)e z : φ NIG (u) = exp(iuµ) exp (δ ) ( ) α 2 β 2 exp δ α 2 (β + iu) 2 one can see that NIG are closed under convolution in parameters δ and µ

37 Levy Processes with Jumps α-stable Levy processes α-stable Levy processes stable distributions constitute a four-parameter class of distributions with Fourier transform given by { ( exp iuµ σu α ( 1 iβsign(u) tan πα )) φ st (u) = 2 ifα 1, exp ( iuµ σu ( 1 + iβsign(u) 2 π ln u )) ifα = 1 the parameter space is 0 < α 2, σ 0, 1 β 1 and µ R for α = 2 one gets the Gaussian distribution with mean µ and variance 2σ 2 for α < 2 there is no Gaussian part, which means the paths of an α-stable Levy motion are purely discontinuous

38 Levy Processes with Jumps α-stable Levy processes Special cases of α-stable distributions 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 usefulness in particular as a pricing model is limited for α 2 by the fact that finiteness of the first exponential moment in not satisfied

39 Levy Processes with Jumps Meixner Levy processes Meixner Levy processes the Fourier transform of Meixner distributions is given by for α > 0, β < π, δ > 0 ( cos(β/2) φ M (u) = cosh((αu iβ)/2) ) 2δ the corresponding Levy processes are purely discontinuous with paths of infinite variation the density of Levy measure F is g M (x) = δ x exp(βx/α) sinh(πx/α)

40 Levy Processes with Jumps CGMY and variance gamma Levy processes CGMY and variance gamma Levy processes the class of CGMY distributions (infinitely divisible) extends the variance gamma model CGMY Levy processes have purely discontinuous paths and the density of Levy mesure is given by { C exp( G x ) x < 0, g CGMY (x) = x 1+Y C exp( Mx) x > 0 x 1+Y with parameter space C, G, M > 0 and Y (, 2) the process has infinite activity iff Y [0, 2) the paths have infinite variation iff Y [1, 2) for Y = 0 one gets the three-parameter variance gamma distributions which are a subclass of the generalized hyperbolic distributions

41 References Barndorff-Nielsen, O.E. (1978): Hyperbolic distributions and distributions on hyperbolae. Scandinavian Journal of Statistics 5, Breiman, L (1968): Probability. Addison-Wesley, Reading. Jacod, J., Shiryaev, A.N. (1987): Limit Theorems for Stochastic Processes. Springer, New York. Protter, P.E. (2004): Stochastic Integration and Differential Equations. (2nd ed.) Volume 21 of Applications of Mathematics. Springer, New York. Samuelson, P. (1965): Rational theory of warrant pricing. Industrial Management Review 6: Sato, K.-I. (1999): Levy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.