Lecture 1: Lévy processes

Transcription

1 Lecture 1: Lévy processes A. E. Kyprianou Department of Mathematical Sciences, University of Bath 1/ 22

2 Lévy processes 2/ 22

3 Lévy processes A process X = {X t : t 0} defined on a probability space (Ω, F, P) is said to be a (one dimensional) Lévy process if it possesses the following properties: (i) The paths of X are P-almost surely right continuous with left limits. (ii) P(X 0 = 0) = 1. (iii) For 0 s t, X t X s is equal in distribution to X t s. (iv) For 0 s t, X t X s is independent of {X u : u s}. 2/ 22

4 Lévy processes A process X = {X t : t 0} defined on a probability space (Ω, F, P) is said to be a (one dimensional) Lévy process if it possesses the following properties: (i) The paths of X are P-almost surely right continuous with left limits. (ii) P(X 0 = 0) = 1. (iii) For 0 s t, X t X s is equal in distribution to X t s. (iv) For 0 s t, X t X s is independent of {X u : u s}. Some familiar examples (i) Linear Brownian motion σb t at, t 0, σ, a R. (ii) Poisson process with λ, N = {N t : t 0}. (iii) Compound Poisson processes with drift N t ξ i + ct, t 0, i=1 where {ξ i : i 1} are i.i.d. and c R. 2/ 22

5 Lévy processes Note that in the last case of a compound Poisson process with drift, if we assume that E( ξ 1 ) = x F (dx) < and choose c = λ xf (dx), R R then the centred compound Poisson process N t i=1 ξ i λt xf (dx), t 0, R is both a Lévy process and a martingale. 3/ 22

6 Lévy processes Note that in the last case of a compound Poisson process with drift, if we assume that E( ξ 1 ) = x F (dx) < and choose c = λ xf (dx), R R then the centred compound Poisson process N t i=1 ξ i λt xf (dx), t 0, R is both a Lévy process and a martingale. Any linear combination of independent Lévy processes is a Lévy process. 3/ 22

7 The Lévy-Khintchine formula 4/ 22

8 The Lévy-Khintchine formula As a consequence of stationary and independent increments it can be shown that any Lévy process X = {X t : t 0} has the property that, for all t 0 and θ, E(e iθx t ) = e Ψ(θ)t where Ψ(θ) = log E(e iθx 1 ) is called the characteristic exponent. 4/ 22

9 The Lévy-Khintchine formula As a consequence of stationary and independent increments it can be shown that any Lévy process X = {X t : t 0} has the property that, for all t 0 and θ, E(e iθx t ) = e Ψ(θ)t where Ψ(θ) = log E(e iθx 1 ) is called the characteristic exponent. Theorem. The function Ψ : R C is the characteristic of a Lévy process if and only if Ψ(θ) = iaθ σ2 θ 2 + (1 e iθx + iθx1 ( x <1) )Π(dx). R where σ R, a R and Π is a measure concentrated on R\{0} which respects the integrability condition (1 x 2 )Π(dx) <. R 4/ 22

10 Key examples of L-K formula 5/ 22

11 Key examples of L-K formula For the case of σb t at, Ψ(θ) = iaθ σ2 θ 2. 5/ 22

12 Key examples of L-K formula For the case of σb t at, Ψ(θ) = iaθ σ2 θ 2. For the case of a compound Poisson process N t i=1 ξi, where the the i.i.d. variables {ξ i : i 1} have common distribution F and the Poisson process of jumps has rate λ, Ψ(θ) = (1 e iθx )λf (dx) R 5/ 22

13 Key examples of L-K formula For the case of σb t at, Ψ(θ) = iaθ σ2 θ 2. For the case of a compound Poisson process N t i=1 ξi, where the the i.i.d. variables {ξ i : i 1} have common distribution F and the Poisson process of jumps has rate λ, Ψ(θ) = (1 e iθx )λf (dx) R For the case of independent linear combinations, let X t = σb t + at + N t i=1 ξi λ x F (dx)t, where N has rate λ and R {ξ i : i 1} have common distribution F satisfying x F (dx) < R Ψ(θ) = iaθ σ2 θ 2 + (1 e iθx + iθx)λf (dx) R 5/ 22

14 The Lévy-Itô decomposition Ψ(θ) = iaθ σ2 θ 2 + (1 e iθx + iθx)λf (dx) R Ψ(θ) = iaθ σ2 θ 2 + (1 e iθx + iθx1 ( x <1) )Π(dx). R 6/ 22

15 The Lévy-Itô decomposition Ψ(θ) = iaθ σ2 θ 2 + (1 e iθx + iθx)λf (dx) R Ψ(θ) = iaθ σ2 θ 2 + (1 e iθx + iθx1 ( x <1) )Π(dx). R Ψ(θ) = {iaθ + 12 } { } σ2 θ 2 + (1 e iθx )λ 0F 0(dx) x 1 + { } (1 e iθx + iθx)λ nf n(dx) n 0 2 (n+1) x <2 n where λ 0 = Π(R\( 1, 1)) and λ n = Π({x : 2 (n+1) x < 2 n }) F 0(dx) = λ 1 0 Π(dx) { x 1} and F n(dx) = λ 1 n Π(dx) {x:2 (n+1) x <2 n } 7/ 22

16 The Lévy-Itô decomposition Ψ(θ) = iaθ σ2 θ 2 + (1 e iθx + iθx)λf (dx) R Ψ(θ) = iaθ σ2 θ 2 + (1 e iθx + iθx1 ( x <1) )Π(dx). R Suggestive that for any permitted triple (a, σ, Π) the associated Lévy processes can be written as the independent sum N t 0 N t n X t = at + σb t + ξi 0 + ξi n xλ nf n(dx) 2 (n+1) x <2 n i=1 n=1 i=1 where {ξi n : i 0} are families of i.i.d. random variables with respective distributions F n and N n are Poisson processes with respective arrival rates λ n The condition R (1 x2 )Π(dx) < ensures that all these processes "add up". 8/ 22

17 Brownian motion / 22

18 Compound Poisson process / 22

19 Brownian motion + compound Poisson process / 22

20 Unbounded variation paths / 22

21 Bounded variation paths / 22

22 Bounded vs unbounded variation paths 14/ 22

23 Bounded vs unbounded variation paths Paths of a Lévy processes are either almost surely of bounded variation over all finite time horizons or almost surely of unbounded variation over all finite time horizons. 14/ 22

24 Bounded vs unbounded variation paths Paths of a Lévy processes are either almost surely of bounded variation over all finite time horizons or almost surely of unbounded variation over all finite time horizons. Distinguishing the two cases can be identified from the Lévy-Itô decomposition: N t 0 N t n X t = at + σb t + ξi 0 + ξi n xπ(dx) 2 (n+1) x <2 n i=1 n=1 i=1 14/ 22

25 Bounded vs unbounded variation paths Paths of a Lévy processes are either almost surely of bounded variation over all finite time horizons or almost surely of unbounded variation over all finite time horizons. Distinguishing the two cases can be identified from the Lévy-Itô decomposition: N t 0 N t n X t = at + σb t + ξi 0 + ξi n xπ(dx) 2 (n+1) x <2 n i=1 n=1 Bounded variation if and only if σ = 0 and i=1 ( 1,1) x Π(dx) < 14/ 22

26 Infinite divisibility 15/ 22

27 Infinite divisibility Suppose that X is an R-valued random variable on (Ω, F, P), then X is infinitely divisible if for each n = 1, 2, 3,... X d = X (1,n) + + X (n,n) where {X (i,n) : i = 1,..., n} are independent and identically distributed and the equality is in distribution. 15/ 22

28 Infinite divisibility Suppose that X is an R-valued random variable on (Ω, F, P), then X is infinitely divisible if for each n = 1, 2, 3,... X d = X (1,n) + + X (n,n) where {X (i,n) : i = 1,..., n} are independent and identically distributed and the equality is in distribution. Said another way, if µ is the characteristic function of X then for each n = 1, 2, 3,... we have that µ = (µ n) n where µ n is the the characteristic function of some R-valued random variable. 15/ 22

29 Infinite divisibility Suppose that X is an R-valued random variable on (Ω, F, P), then X is infinitely divisible if for each n = 1, 2, 3,... X d = X (1,n) + + X (n,n) where {X (i,n) : i = 1,..., n} are independent and identically distributed and the equality is in distribution. Said another way, if µ is the characteristic function of X then for each n = 1, 2, 3,... we have that µ = (µ n) n where µ n is the the characteristic function of some R-valued random variable. For any Lévy process: X t = (X t X (n 1) t ) + (X (n 1) n t n X (n 2) t ) + + (X t 1 X n n 0) from which stationary and independent increments implies infinite divisibility. 15/ 22

30 Infinite divisibility Suppose that X is an R-valued random variable on (Ω, F, P), then X is infinitely divisible if for each n = 1, 2, 3,... X d = X (1,n) + + X (n,n) where {X (i,n) : i = 1,..., n} are independent and identically distributed and the equality is in distribution. Said another way, if µ is the characteristic function of X then for each n = 1, 2, 3,... we have that µ = (µ n) n where µ n is the the characteristic function of some R-valued random variable. For any Lévy process: X t = (X t X (n 1) t ) + (X (n 1) n t n X (n 2) t ) + + (X t 1 X n n 0) from which stationary and independent increments implies infinite divisibility. This goes part way to explaining why E(e iθx t ) = e Ψ(θ)t = [E(e iθx 1 )] t 15/ 22

31 Lévy processes in finance and insurance 16/ 22

32 Financial modelling: Share value, a day and a year 17/ 22

33 Financial modelling 18/ 22

34 Financial modelling Black Scholes model of a risky asset: S t = exp{x + σb t at}, t 0 (Initial value S 0 = e x ). Criticised at many levels. 18/ 22

35 Financial modelling Black Scholes model of a risky asset: S t = exp{x + σb t at}, t 0 (Initial value S 0 = e x ). Criticised at many levels. Lévy model of a risky asset: S t = exp{x + X t}, t 0. Does better than Black-Scholes on some issues (eg infinitely divisibility instead of Gaussian increments) but still generally viewed as a statistically poor fit with real data. 18/ 22

36 Financial modelling Black Scholes model of a risky asset: S t = exp{x + σb t at}, t 0 (Initial value S 0 = e x ). Criticised at many levels. Lévy model of a risky asset: S t = exp{x + X t}, t 0. Does better than Black-Scholes on some issues (eg infinitely divisibility instead of Gaussian increments) but still generally viewed as a statistically poor fit with real data. Modelling with a Lévy process means choosing the triplet (a, σ, Π). 18/ 22

37 Financial modelling Black Scholes model of a risky asset: S t = exp{x + σb t at}, t 0 (Initial value S 0 = e x ). Criticised at many levels. Lévy model of a risky asset: S t = exp{x + X t}, t 0. Does better than Black-Scholes on some issues (eg infinitely divisibility instead of Gaussian increments) but still generally viewed as a statistically poor fit with real data. Modelling with a Lévy process means choosing the triplet (a, σ, Π). The inclusion of σ is a choice of the inclusion of noise and the choice of Π models jump structure and a can be used to deal with so-called risk neutrality: The existence of a measure P under which X is a Lévy process satisfying E(e X T ) = e qt in other words Ψ( i) = q. 18/ 22

38 Some favourite Lévy processes in finance 19/ 22

39 Some favourite Lévy processes in finance The Kou model: η 1, η 2, λ > 0, p (0, 1), σ 2 0 and Π(dx) = λpη 1e η 1x 1 (x>0) dx + λ(1 p)η 2e η 2 x 1 (x<0) dx Ψ(θ) = iaθ σ2 θ 2 λpiθ λ(1 p)iθ + η 1 iθ η 2 + iθ 19/ 22

40 Some favourite Lévy processes in finance The Kou model: η 1, η 2, λ > 0, p (0, 1), σ 2 0 and Π(dx) = λpη 1e η 1x 1 (x>0) dx + λ(1 p)η 2e η 2 x 1 (x<0) dx Ψ(θ) = iaθ σ2 θ 2 The KoBoL/CGMY model: λpiθ λ(1 p)iθ + η 1 iθ η 2 + iθ σ 2 0 and Π(dx) = C e Mx x 1+Y 1 (x>0)dx + C e G x x 1+Y 1 (x<0)dx Ψ(θ) = iaθ σ2 θ 2 + CΓ( Y ){(M iθ) Y M Y + (G + iθ) Y G Y } 19/ 22

41 Some favourite Lévy processes in finance The Kou model: η 1, η 2, λ > 0, p (0, 1), σ 2 0 and Π(dx) = λpη 1e η 1x 1 (x>0) dx + λ(1 p)η 2e η 2 x 1 (x<0) dx Ψ(θ) = iaθ σ2 θ 2 The KoBoL/CGMY model: λpiθ λ(1 p)iθ + η 1 iθ η 2 + iθ σ 2 0 and Π(dx) = C e Mx x 1+Y 1 (x>0)dx + C e G x x 1+Y 1 (x<0)dx Ψ(θ) = iaθ σ2 θ 2 + CΓ( Y ){(M iθ) Y M Y + (G + iθ) Y G Y } Spectrally negative Lévy processes: σ 2 0 and Π(0, ) = 0. 19/ 22

42 Some favourite Lévy processes in finance The Kou model: η 1, η 2, λ > 0, p (0, 1), σ 2 0 and Π(dx) = λpη 1e η 1x 1 (x>0) dx + λ(1 p)η 2e η 2 x 1 (x<0) dx Ψ(θ) = iaθ σ2 θ 2 The KoBoL/CGMY model: λpiθ λ(1 p)iθ + η 1 iθ η 2 + iθ σ 2 0 and Π(dx) = C e Mx x 1+Y 1 (x>0)dx + C e G x x 1+Y 1 (x<0)dx Ψ(θ) = iaθ σ2 θ 2 + CΓ( Y ){(M iθ) Y M Y + (G + iθ) Y G Y } Spectrally negative Lévy processes: σ 2 0 and Π(0, ) = 0....and others... Variance Gamma, Meixner, Hyperbolic Lévy processes, β-lévy processes, θ-lévy processes, Hypergeometric Lévy processes,... 19/ 22

43 Need to know for option pricing 20/ 22

44 Need to know for option pricing Note: X is a Markov process. Can work with P x to mean P( X 0 = x). Standard theory dictates that the price of a European-type option over time horizon [0, T ] as a function of the initial value of the stock where q 0 is the discounting rate. E x(e qt (f(e X T )) 20/ 22

45 Need to know for option pricing Note: X is a Markov process. Can work with P x to mean P( X 0 = x). Standard theory dictates that the price of a European-type option over time horizon [0, T ] as a function of the initial value of the stock E x(e qt (f(e X T )) where q 0 is the discounting rate. More exotic financial derivatives require an understanding of first passage problems. 20/ 22

46 Need to know for option pricing Note: X is a Markov process. Can work with P x to mean P( X 0 = x). Standard theory dictates that the price of a European-type option over time horizon [0, T ] as a function of the initial value of the stock E x(e qt (f(e X T )) where q 0 is the discounting rate. More exotic financial derivatives require an understanding of first passage problems. The value of an American put with strike K > 0: E x(e qτ a (K e X τ a ) + ) where τ a = inf{t > 0 : X t < a} for some a R. 20/ 22

47 Need to know for option pricing Note: X is a Markov process. Can work with P x to mean P( X 0 = x). Standard theory dictates that the price of a European-type option over time horizon [0, T ] as a function of the initial value of the stock E x(e qt (f(e X T )) where q 0 is the discounting rate. More exotic financial derivatives require an understanding of first passage problems. The value of an American put with strike K > 0: E x(e qτ a (K e X τ a ) + ) where τ a = inf{t > 0 : X t < a} for some a R. Barrier option (up and out call with strike K > 0): for some b > log K. E x(e qt (e X t K) + 1 (sups t X s b)) 20/ 22

48 Need to know for option pricing Note: X is a Markov process. Can work with P x to mean P( X 0 = x). Standard theory dictates that the price of a European-type option over time horizon [0, T ] as a function of the initial value of the stock E x(e qt (f(e X T )) where q 0 is the discounting rate. More exotic financial derivatives require an understanding of first passage problems. The value of an American put with strike K > 0: E x(e qτ a (K e X τ a ) + ) where τ a = inf{t > 0 : X t < a} for some a R. Barrier option (up and out call with strike K > 0): E x(e qt (e X t K) + 1 (sups t X s b)) for some b > log K. More generally, complex instruments such as credit-default swaps and convertible contingencies are built upon the key mathematical ingredient P x(inf Xs > 0) s t 20/ 22

49 Insurance mathematics The first passage problem also occurs in the related setting of insurance mathematics. 21/ 22

50 Insurance mathematics The first passage problem also occurs in the related setting of insurance mathematics. The classical risk insurance ruin problem sees the wealth of an insurance problem modelled by the so-called Cramér-Lundberg process: N t X t := x + ct ξ i, with the understanding that x is the initial wealth, c is the rate at which premiums are collected and {N t : t 0} is a Poisson process describing the arrival of the i.i.d. claims {ξ i : i 0}. i=1 21/ 22

51 Insurance mathematics The first passage problem also occurs in the related setting of insurance mathematics. The classical risk insurance ruin problem sees the wealth of an insurance problem modelled by the so-called Cramér-Lundberg process: N t X t := x + ct ξ i, with the understanding that x is the initial wealth, c is the rate at which premiums are collected and {N t : t 0} is a Poisson process describing the arrival of the i.i.d. claims {ξ i : i 0}. i=1 This is nothing but a spectrally negative Lévy process. 21/ 22

52 Insurance mathematics The first passage problem also occurs in the related setting of insurance mathematics. The classical risk insurance ruin problem sees the wealth of an insurance problem modelled by the so-called Cramér-Lundberg process: N t X t := x + ct ξ i, with the understanding that x is the initial wealth, c is the rate at which premiums are collected and {N t : t 0} is a Poisson process describing the arrival of the i.i.d. claims {ξ i : i 0}. i=1 This is nothing but a spectrally negative Lévy process. A classical field of study, so called Gerber-Shiu, theory, concerns the study of the joint law of τ 0, X τ and X 0 τ, 0 the time of ruin, the deficit at ruin and the wealth prior to ruin. 21/ 22

53 Ruin x v u We are interested in E x(e qτ 0 ; X τ 0 du, X τ 0 dv). 22/ 22