Lévy Processes. Antonis Papapantoleon. TU Berlin. Computational Methods in Finance MSc course, NTUA, Winter semester 2011/2012

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1 Lévy Processes Antonis Papapantoleon TU Berlin Computational Methods in Finance MSc course, NTUA, Winter semester 2011/2012 Antonis Papapantoleon (TU Berlin) Lévy processes 1 / 41

2 Overview of the course 1 Motivation 2 Lévy processes Lévy processes: definition Lévy processes and infinite divisibility Lévy-Itô decomposition Properties of Lévy processes Lévy-driven SDEs 3 Affine processes 4 Bibliography 1 / 42

3 Motivation Empirical facts from finance: asset prices do not evolve continuously, they exhibit jumps or spikes! USD/JPY Oct 1997 Oct 1998 Oct 1999 Oct 2000 Oct 2001 Oct 2002 Oct 2003 Oct 2004 USD/JPY daily exchange rate, October 1997 October / 42

4 Motivation Asset log-returns are not normally distributed, they are fat-tailed and skewed! Empirical distribution of daily log-returns on the GBP/USD rate and fitted Normal. 3 / 42

5 Motivation Implied volatilities are constant neither across strike, nor across maturity! implied vol (%) delta (%) or strike maturity Implied volatilities of vanilla options on the EUR/USD rate, 5 November / 42

6 Motivation During the recent crisis... The Normal copula model required implied correlations up to 120% to match market prices. (Wim Schoutens GOCPS 2008) Before the collapse, Carnegie Mellon s alumni in the industry were telling me that the level of complexity in the mortgage-backed securities market had exceeded the limitations of their models. (Steven Shreve Don t Blame The forbes.com) Dependence, and tail dependence, risk where completely underestimated. Remark Naturally, jump processes appear in many other applications or sciences: in Physics, in Biology, in the Actuarial science, in Telecommunications, etc. 5 / 42

7 Motivation Lévy processes in finance and other fields Lévy processes provide a convenient framework to model the empirical phenomena from finance, since 1 the sample paths can have jumps 2 the generating distributions can be fat-tailed and skewed 3 the implied volatilities can have a smile shape. Lévy processes in: Hilbert and Banach spaces, LCA and Lie groups Quantum Mechanics, Free Probability (non-commutative) Lévy-type processes and pseudo-differential operators Branching processes and fragmentation theory,... References: Barndorff-Nielsen et al. (2001), Kyprianou (2005), Franz and Schürmann (2006),... 6 / 42

8 Lévy processes: definition Definition Let (Ω, F, (F t ) t 0, P) be a complete stochastic basis. Definition A càdlàg, adapted, real valued stochastic process L = (L t ) t 0 with L 0 = 0 a.s. is called a Lévy process if the following conditions are satisfied: (L1): L has independent increments, i.e. L t L s is independent of F s for any 0 s < t T. (L2): L has stationary increments, i.e. for any s, t 0 the distribution of L t+s L t does not depend on t. (L3): L is stochastically continuous, i.e. for every t 0 and ɛ > 0: lim s t P( L t L s > ɛ) = 0. 7 / 42

9 Lévy processes: definition Example 1: linear drift L t = bt, ϕ Lt (u) = exp ( iubt ) / 42

10 Lévy processes: definition Example 2: Brownian motion L t = σw t, ( ϕ Lt (u) = exp u2 σ 2 ) 2 t / 42

11 Lévy processes: definition Example 3: Poisson process L t = N t k=1 J k, J k 1 ϕ Lt (u) = exp ( ) tλ(e iu 1) / 42

12 Lévy processes: definition Example 4: compensated Poisson process (martingale!) L t = N t k=1 J k tλ = N t λt, ϕ Lt (u) = exp ( ) tλ(e iu 1 iu) / 42

13 Lévy processes: definition Example 5: compound Poisson process L t = N t k=1 J k, ϕ Lt (u) = exp ( ) tλ(e[e iuj 1]) / 42

14 Lévy processes: definition Example 6: Lévy jump-diffusion L t = bt + σw t + N t k=1 J k λte[j] / 42

15 Lévy processes: definition Characteristic function of a Lévy jump-diffusion E [ e ] iult = exp [ iubt ] [ E exp ( ) ] [ iuσw t E exp ( iu = exp [ iubt ] [ exp = exp [ iubt ] exp 1 ] 2 u2 σ 2 t [ 1 2 u2 σ 2 t N t k=1 J k iutλe[j] )] exp [λt ( E[e iuj 1] iue[j] )] ] exp [ ( λt e iux 1 iux ) F (dx)] R )] [ = exp t (iub u2 σ 2 + (e iux 1 iux)λf (dx) 2 R 1 Time and space factorize. Drift, diffusion and jumps separate. 2 Jumps of the form λ F. Question Are these observations always true? Answer: yes, yes, no!. 14 / 42

16 Lévy processes: definition Aim: the connection between... 1 Lévy processes 2 infinitely divisible laws 3 the Lévy triplet C (L t ) t 0 L(L t ) LI LI C (b, c, ν) LK Commutative diagram of the relationship between a Lévy process (L t ) t 0, the law of the infinitely divisible random variable L(L t ) and the Lévy triplet (b, c, ν), demonstrating the role of the Lévy Khintchine formula and the Lévy Itô decomposition. 15 / 42

17 Lévy processes and infinite divisibility Infinitely divisible distributions Definition The law P X of a random variable X is infinitely divisible, if for all n N there exist i.i.d. random variables X (1/n) 1,..., X n (1/n) such that X = d X (1/n) X n (1/n) (1) P X = P X (1/n)... P X (1/n). }{{} (2) n times Equivalently, ϕ X (u) = ( ϕ X (1/n)(u)) n. (3) 16 / 42

18 Lévy processes and infinite divisibility where X (1/n) Poisson( λ n ). 17 / 42 Infinitely divisible distributions: examples Example (Normal) The Normal distribution is infinitely divisible, because ( [ ϕ X (u) = exp iuµ 1 2 u2 σ 2] = exp [iu µ n 1 ] ) n ( σ2 n, u2 = ϕ 2 n X (1/n)(u)) where X (1/n) Normal( µ n, σ2 n ). Example (Poisson) Similarly, the Poisson distribution is infinitely divisible, since ( [ ] [ λ ] ) n ( n, ϕ X (u) = exp λ(e iu 1) = exp n (eiu 1) = ϕ X (1/n)(u))

19 Lévy processes and infinite divisibility The Lévy-Khintchine formula Theorem The law of a random variable X is infinitely divisible if and only if there exists a triplet (b, c, ν), with b R, c R 0 and a measure satisfying ν({0}) = 0 and R (1 x 2 )ν(dx) <, such that E [ e iux ] ( = exp ibu u2 c 2 + ( e iux ) ) 1 iux1 { x <1} ν(dx). (4) R The triplet (b, c, ν) is called the Lévy triplet and the exponent in (4) κ(u) = iub u2 c 2 + ( e iux ) 1 iux1 { x <1} ν(dx) (5) R is called the Lévy exponent. Moreover, b R is called the drift term, c R 0 the Gaussian coefficient and ν the Lévy measure. 18 / 42

20 Lévy processes and infinite divisibility Lemma P k infinitely divisible and P k P then P is infinitely divisible. Sketch of Proof: if part. Let (ε n) n N be a sequence in R, monotonic and decreasing to zero. Define [ ( ) ϕ Xn (u) = exp iu b xν(dx) u2 c ] ε n< x <1 2 + (e iux 1)ν(dx). x >ε n Each ϕ Xn is the convolution of a normal and a compound Poisson distribution, hence ϕ Xn is the characteristic function of an infinitely divisible probability measure P Xn. Then lim ϕ X n n (u) = ϕ X (u), and by Lévy s continuity theorem and the above Lemma, ϕ X is the characteristic function (given continuity at u = 0) of an infinitely divisible law. Corollary Every infinitely divisible law is the limit of a sequence of compound Poisson laws important for simulations! 19 / 42

21 Lévy processes and infinite divisibility Lévy processes have infinitely divisible laws Consider a Lévy process L = (L t ) t 0 ; then, for any n N and any t > 0, L t = L t + (L 2t n n L t ) (L t L (n 1)t ). (6) n n The stationarity and independence of the increments yield that (L tk L t(k 1) ) k 0 is an i.i.d. sequence of random variables, hence the n n random variable L t is infinitely divisible. Theorem For every Lévy process L, we have E[e iult ] = e tκ(u) = exp [ t ( ibu u2 c 2 + R (e iux 1 iux1 { x <1} )ν(dx) )] (7) where κ(u) is the characteristic exponent of L 1 := X, a random variable with an infinitely divisible distribution. 20 / 42

22 Lévy processes and infinite divisibility Lévy processes have infinitely divisible laws Sketch of Proof. Define the function φ u (t) = E[e iult ], then we have φ u (t + s) = E[e iult+s ] = E[e iu(lt+s Ls) e iuls ] = E[e iu(lt+s Ls) ]E[e iuls ] = φ u (t)φ u (s). Now, φ u (0) = 1 and the map t φ u (t) is continuous (by stochastic continuity). However, the unique continuous solution of the Cauchy functional equation (8) is (8) φ u (t) = e tκ(u), where κ : R C. (9) Since, L 1 is an infinitely divisible random variable, the statement follows. 21 / 42

23 Lévy processes and infinite divisibility Lévy processes and infinite divisibility: Recap Remark Every Lévy process can be associated to an infinitely divisible distribution. Question Is the opposite also true? i.e. given a random variable X with an infinitely divisible law, can we construct a Lévy process L = (L t ) t 0, such that L(L 1 ) = L(X )? 22 / 42

24 Lévy-Itô decomposition The Lévy-Itô decomposition Theorem Consider a triplet (b, c, ν) where b R, c R 0 and ν is a measure satisfying ν({0}) = 0 and R (1 x 2 )ν(dx) <. Then, there exists a probability space (Ω, F, P) on which four independent Lévy processes exist, L (1), L (2), L (3) and L (4), where L (1) is a constant drift, L (2) is a Brownian motion, L (3) is a compound Poisson process and L (4) is a square integrable pure jump martingale with an a.s. countable number of jumps of magnitude less than 1 on each finite time interval. Taking L = L (1) + L (2) + L (3) + L (4), we have that there exists a probability space on which a Lévy process L = (L t ) t 0 with characteristic exponent κ(u) = iub u2 c 2 + (e iux 1 iux1 { x <1} )ν(dx), (10) R for all u R, is defined. 23 / 42

25 Lévy-Itô decomposition Sketch of proof I. Split the Lévy exponent (10) into four parts κ = κ (1) + κ (2) + κ (3) + κ (4) where κ (1) (u) = iub, κ (2) (u) = u2 c 2, κ(3) (u) = (e iux 1)ν(dx), x 1 κ (4) (u) = (e iux 1 iux)ν(dx). x <1 The first part corresponds to a linear drift with parameter b, the second one to a Brownian motion with coefficient c and the third part to a compound Poisson process with arrival rate λ := ν(r \ ( 1, 1)) and jump magnitude F (dx) := ν(dx) ν(r\( 1,1)) 1 { x 1}. 24 / 42

26 Lévy-Itô decomposition Sketch of proof II. Let L (4) denote the jumps of the Lévy process L (4) and µ L(4) denote the random measure counting the jumps of L (4). Construct a compensated compound Poisson process L (4,ɛ) t = = 0 s t t 0 L (4) s 1 (4) {1> L s 1> x >ɛ ( ) >ɛ} t xν(dx) 1> x >ɛ ( ) xµ L(4) (dx, ds) t xν(dx) 1> x >ɛ then, the jumps of L (4,ɛ) form a Poisson point process; using results for Poisson point processes, we get that the characteristic function of L (4,ɛ) is κ (4,ɛ) (u) = (e iux 1 iux)ν(dx). ɛ< x <1 Then, there exists a Lévy process L (4) which is a square integrable martingale such that L (4,ɛ) L (4) uniformly on [0, T ] as ɛ 0+. Clearly, the Lévy exponent of the latter Lévy process is κ (4). 25 / 42

27 Lévy-Itô decomposition Sketch of proof III. Therefore, we can decompose any Lévy process into four independent Lévy processes L = L (1) + L (2) + L (3) + L (4), i.e. L t = bt + t t cw t + xµ L (ds, dx) + x (µ L ν L )(ds, dx) 0 x 1 0 x <1 where ν L = ν Leb. Here L (1) is a constant drift, L (2) is a Brownian motion, L (3) a compound Poisson process and L (4) is a pure jump martingale. Remark If E[ L t ] <, the Lévy-Itô decomposition resumes the form L t = b t + t cw t + x (µ L ν L )(ds, dx). 0 R 26 / 42

28 Properties of Lévy processes The Lévy measure The Lévy measure is a measure on R that satisfies ν({0}) = 0 and (1 x 2 )ν(dx) <. The Lévy measure describes the expected number of jumps of a certain height in a time interval of length 1 :: ν(a) = E[ s 1 1 A( L s )]. If ν is a finite measure, i.e. ν(r) = λ <, then F (dx) := ν(dx) λ is a probability measure. Thus, λ is the expected number of jumps and F (dx) the distribution of the jump size x. If ν(r) =, then an infinite number of (small) jumps is expected. Several important properties about the path of a Lévy process, e.g. activity and variation, can be read from its Lévy measure. R 27 / 42

29 Properties of Lévy processes Examples of Lévy measures The distribution function of the Lévy measure of the standard Poisson process (left). The density of the Lévy measure of a compound Poisson process with double-exponential jumps The density of the Lévy measure of the normal inverse Gaussian (left) and the α-stable process. 28 / 42

30 Properties of Lévy processes Activity and variation Proposition Let L be a Lévy process with triplet (b, c, ν). 1 If ν(r) < then almost all paths of L have a finite number of jumps on every compact interval. In that case, the Lévy process has finite activity. 2 If ν(r) = then almost all paths of L have an infinite number of jumps on every compact interval. In that case, the Lévy process has infinite activity. Proposition Let L be a Lévy process with triplet (b, c, ν). 1 If c = 0 and x ν(dx) < then almost all paths of L have finite x 1 variation. 2 If c 0 or x ν(dx) = then almost all paths of L have infinite x 1 variation. 29 / 42

31 Properties of Lévy processes Examples of Lévy processes I Paths of finite activity (left) and infinite activity finite variation Lévy process. 30 / 42

32 Properties of Lévy processes Examples of Lévy processes II Paths of continuous and pure jumps infinite variation Lévy process. 31 / 42

33 Properties of Lévy processes Moments Proposition Let L be a Lévy process with triplet (b, c, ν) and α R. Then E [ e ] αlt < if and only if e αx ν(dx) <. x 1 Sketch of Proof (sufficiency). The third term in the Lévy-Itô decomposition is a compound Poisson process with arrival rate ν(dx) λ := ν(r \ ( 1, 1)) and jump magnitude F (dx) := ν(r\( 1,1)) 1 { x 1}. Now, finiteness of E[exp(αL t)] implies finiteness of E[exp(αL (3) t )], where E [ ] e αl(3) t = e λt (λt) k ( ) k e αx F (dx) = e λt t k ( ) k e αx 1 { x 1} ν(dx) k! k 0 R k! k 0 R hence, for k = 1, e λt e αx 1 { x 1} ν(dx) < = e αx ν(dx) <. R x 1 32 / 42

34 Properties of Lévy processes Lévy martingales I Proposition Let L = (L t ) t 0 be a Lévy process with Lévy triplet (b, c, ν) and assume that E L 1 < ( x >1 x ν(dx) < ). 1 L is a martingale if and only if b + x1 { x >1} ν = 0; 2 (L t E[L t ]) t 0 is a martingale. Proof. Follow directly from the canonical decomposition, since E[L t ] = ( b + (x h(x)) ν ) t. 33 / 42

35 Properties of Lévy processes Lévy martingales II Proposition Let L = (L t ) t 0 be a Lévy process with cumulant κ and assume that E[e αlt ] <, α R. The process M = (M t ) t 0 is a martingale, where Proof. M t = eαlt e tκ(α). Applying Proposition 3, we get that E[e αlt ] = e tκ(α) <, for all 0 t T. For 0 s t, we can re-write M as M t = eαls e α(lt Ls ) e sκ(α) e (t s)κ(α) = e α(lt Ls ) Ms e (t s)κ(α). Using the fact that a Lévy process has stationary and independent increments, we get E [ ] [ M Fs e α(l t L s ) ] t = MsE Fs e (t s)κ(α) = M se (t s)κ(α) e (t s)κ(α) = M s. 34 / 42

36 Lévy-driven SDEs Lévy-driven SDEs Consider a Lévy process L and a Lipschitz function σ : R 0 R R 1 σ(t, x) σ(t, y) k x y, t R 0 2 t σ(t, x) is right continuous with left limits, x R. Let X 0 be finite and F 0 measurable. Then, the stochastic differential equation t X t = X 0 + σ(s, X s )dl s (11) 0 admits a unique solution. The solution is a semimartingale. 35 / 42

37 Affine processes Affine processes I Setup: 1 X = (X t ) t 0 a time-homogeneous Markov process 2 X takes values in D = R m + R n R d 3 X is affine, if the moment generating function satisfies: [ E x exp u, Xt ] = exp ( φ t (u) + ψ t (u), x ) (12) 4 where φ t (u) and ψ t (u) are functions taking values in R and R d respectively, 5 and x D. 36 / 42

38 Affine processes Affine processes II Lemma (Flow property) The functions φ and ψ satisfy the semi-flow equations: φ t+s (u) = φ t (u) + φ s (ψ t (u)) ψ t+s (u) = ψ s (ψ t (u)) (13) with initial condition for all suitable 0 t + s T and u U. φ 0 (u) = 0 and ψ 0 (u) = u, (14) Hence, φ and ψ satisfy (generalized) Riccati differential equations. 37 / 42

39 Affine processes Affine processes III We can show that F (u) := t t=0+ φ t (u) and R(u) := t t=0+ ψ t (u) (15) exist for all u U and are continuous in u. Moreover, F and R satisfy Lévy Khintchine-type equations: ( F (u) = b, u + e ξ,u 1 ) m(dξ) (16) and R i (u) = β i, u + D αi 2 u, u ( + e ξ,u 1 u, h i (ξ) ) µ i (dξ), (17) D where (b, m, α i, β i, µ i ) 1 i d are admissible parameters. 38 / 42

40 Affine processes Affine processes IV 1 Affine processes on R: the admissibility conditions yield F (u) = c + bu + a ( 2 u2 + e zu 1 uh(z) ) m(dz) R(u) = βu, for a, c R 0 and b, β R. Every affine process on R is an Ornstein Uhlenbeck (OU) process. 2 Affine processes on R 0 : the admissibility conditions yield ( F (u) = c + bu + e zu 1 ) m(dz) D R(u) = γ + βu + α ( 2 u2 + e zu 1 uh(z) ) µ(dz), D for b, c, α, γ R 0 and β R. There exist affine process on R 0 which are not OU process. R 39 / 42

41 Affine processes Affine processes V Admissible parameters for R 2 0-valued affine processes: b = ( ) +, β + 1 = a = 0, α 1 = ( ), β + 2 = ( ) +, α 0 2 = ( ) + ( ) 0 + m, µ 1, µ 2 are Lévy measures on R 2 0 µ 1, µ 2 can have infinite variation 40 / 42

42 Bibliography D. Applebaum Lévy processes and stochastic calculus. Cambridge University Press, R. Cont and P. Tankov Financial Modelling with Jump Processes. Chapman & Hall/CRC, D. Duffie, D. Filipovic and W. Schachermayer Affine processes and applications in finance. Annals of Applied Probability 13, , J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes (2nd ed.). Springer, A. E. Kyprianou Introductory lectures on fluctuations of Lévy processes with applications. Springer, A. Papapantoleon An introduction to Lévy processes with applications in finance. Lecture Notes, TU Vienna, 2008 (arxiv/ ). 41 / 42

43 Bibliography P. Protter Stochastic integration and differential equations (3rd ed.). Springer, K. I. Sato Lévy processes and infinitely divisible distributions. Cambridge University Press, A. N. Shiryaev Essentials of stochastic finance: facts, models, theory. World Scientific, / 42