CIMPA SCHOOL, 2007 Jump Processes and Applications to Finance Monique Jeanblanc

Transcription

1 CIMPA SCHOOL, 27 Jump Processes and Applications to Finance Monique Jeanblanc 1

2 Jump Processes I. Poisson Processes II. Lévy Processes III. Jump-Diffusion Processes IV. Point Processes 2

3 I. Poisson Processes 1. Counting Processes and Stochastic Integral 2. Standard Poisson Process 3. Inhomogeneous Poisson Processes 4. Stochastic Intensity Poisson Processes 5. Compound Poisson Processes 3

4 1 Counting Processes and Stochastic Integral Let T n be an increasing sequence of random times. n if t [T n,t n+1 [ N t = + otherwise or, equivalently N t = n 1 1 {Tn t} = n 1 n 1 {Tn t<t n+1 }. We denote by N t the left-limit of N s when s t, s < t and by N s = N s N s the jump process of N. 4

5 The stochastic integral is defined as (C N) t = C s dn s = C s dn s ],t] C s dn s = C Tn 1 {Tn t}. n=1 5

6 2 Standard Poisson Process 2.1 Definition The standard Poisson process is a counting process such that - for every s, t, N t+s N t is independent of Ft N, - for every s, t, ther.v.n t+s N t has the same law as N s. Then, λt (λt)k P (N t = k) =e k! 6

7 2.2 First properties E(N t )=λt, Var (N t )=λt for every x>, t>,u,α IR E(x N t )=e λt(x 1) ; E(e iun t )=e λt(eiu 1) ; E(e αn t )=e λt(eα 1). 7

8 2.3 Martingale properties For each α IR, for each bounded Borel function h, the following processes are F-martingales: (i) M t = N t λt, (ii) M 2 t λt =(N t λt) 2 λt, (iii) exp(αn t λt(e α 1)), (iv) exp [ h(s)dn s λ (e h(s) 1)ds ]. 8

9 For any β> 1, any bounded Borel function h, and any bounded Borel function ϕ valued in ] 1, [, the processes exp[ln(1 + β)n t λβt] =(1+β) N t e λβt, exp [ h(s)dm s + λ (1 + h(s) e h(s) )ds ], exp [ ln(1 + ϕ(s))dn s λ exp [ ln(1 + ϕ(s))dm s + λ are martingales. t ϕ(s)ds ], (ln(1 + ϕ(s)) ϕ(s)ds) ], 9

10 Let H be an F-predictable bounded process, then the following processes are martingales (H M) t := H s dm s = H s dn s λ ((H M) t ) 2 λ Hs 2 ds ( t ) exp H s dn s + λ (1 e H s )ds H s ds These properties does not extend to adapted processes H. For example, from (N s N s )dm s = N t, it follows that martingale. N s dm s is not a 1

11 Remark 2.1 Note that (i) and (iii) imply that the process (Mt 2 N t ; t ) is a martingale. The process λt is the predictable quadratic variation M, whereas the process (N t,t ) is the optional quadratic variation [M]. For any µ [, 1], the processes Mt 2 (µn t +(1 µ)λt) are also martingales. 11

12 2.4 Infinitesimal Generator The Poisson process is a Lévy process, hence a Markov process, its infinitesimal generator L is defined as L(f)(x) =λ[f(x +1) f(x)]. Therefore, for any bounded Borel function f, the process C f t = f(n t ) f() L(f)(N s )ds is a martingale. Furthermore, C f t = [f(n s +1) f(n s )]dm s. 12

13 Exercise: Extend the previous formula to functions f(t, x) defined on IR + IR and C 1 with respect to t, and prove that if L t = exp(log(1 + φ)n t λφt) then dl t = L t φdm t. 13

14 2.4.1 Watanabe s characterization Let N be a counting process and assume that there exists λ>such that M t = N t λt is a martingale. Then N is a Poisson process with intensity λ. 14

15 2.5 Change of Probability If N is a Poisson process, then, for β> 1, L t =(1+β) N t e λβt is a strictly positive martingale with expectation equal to 1. Let Q be the probability defined as dq dp F t = L t. The process N is a Q-Poisson process with intensity equal to (1 + β)λ. 15

16 2.6 Hitting Times Let T x =inf{t, N t x}. Then, for n x<n+1,t x isequaltothe time of the n th -jump of N, hence has a Gamma (n) law. 16

17 3 Inhomogeneous Poisson Processes 3.1 Definition Let λ be an IR + -valued function satisfying λ(u)du <, t. An inhomogeneous Poisson process N with intensity λ is a counting process with independent increments which satisfies for t>s Λ(s,t) (Λ(s, t))n P (N t N s = n) =e n! where Λ(s, t) =Λ(t) Λ(s) = s λ(u)du, andλ(t) = λ(u)du. 17

18 For any real numbers u and α, for any t E(e iun t ) = exp((e iu 1)Λ(t)) E(e αn t ) = exp((e α 1)Λ(t)). 18

19 3.2 Martingale Properties Let N be an inhomogeneous Poisson process with deterministic intensity λ. The process (M t = N t λ(s)ds, t ) is an F N -martingale, and the increasing function Λ(t) = called the compensator of N. λ(s)ds is Let φ be an F N - predictable process such that E( φ s λ(s)ds) < for every t. Then, the process ( φ sdm s,t ) is an F N -martingale. 19

20 In particular, E ( φ s dn s ) = E ( ) φ s λ(s)ds. Let H be an F N - predictable process. The following processes are martingales (H M) t = H s dm s = H s dn s ((H M) t ) 2 λ(s)hs 2 ds ( t ) exp H s dn s λ(s)(e H s 1)ds. λ(s)h s ds 2

21 3.3 Watanabe s Characterization Proposition 3.1 (Watanabe characterization.) Let N be a counting process and Λ an increasing, continuous function with zero value at time zero. Let us assume that M t = N t Λ(t) is a martingale. Then N is an inhomogeneous Poisson process with compensator Λ. 21

22 3.4 Stochastic calculus Integration by parts formula Let X t = x + predictable processes. x s dn s and Y t = y + y s dn s, where x and y are X t Y t = xy + s t (XY ) s = xy + s t Y s X s + s t X s Y s + s t X s Y s = xy + X s dy s + Y s dx s +[X, Y ] t where [X, Y ] t = s t X s Y s = s t x s y s N s = x s y s dn s. 22

23 If dx t = µ t dt + x t dn t and dy t = ν t dt + y t dn t, one gets X t Y t = xy + Y s dx s + Y s dx s +[X, Y ] t where [X, Y ] t = x s y s dn s. If dx t = x t dm t and dy t = y t dm t, the process X t Y t [X, Y ] t is a martingale. 23

24 3.4.2 Itô s Formula Let N be a Poisson process and f a bounded Borel function. The decomposition f(n t )=f(n )+ [f(n s ) f(n s )] <s t is trivial and corresponds to Itô s formula for a Poisson process. [f(n s ) f(n s )] = [f(n s +1) f(n s )] N s <s t = <s t [f(n s +1) f(n s )]dn s. 24

25 Let X t = x + x s dn s = x + T n t x Tn, with x a predictable process. The trivial equality F (X t )=F (X )+ s t (F (X s ) F (X s )), holds for any bounded function F. 25

26 Let dx t = µ t dt + x t dm t =(µ t x t λ(t))dt + x t dn t and F C 1,1 (IR + IR). Then F (t, X t ) = F (,X )+ t F (s, X s )ds + x F (s, X s )(µ s x s λ(s))ds + s t F (s, X s ) F (s, X s ) (3.1) = F (,X )+ t F (s, X s )ds + x F (s, X s )dx s + s t [F (s, X s ) F (s, X s ) x F (s, X s )x s N s ]. 26

27 The formula (3.1) can be written as F (t, X t ) F (,X )= = = + t F (s, X s )ds + + t F (s, X s )ds + + t F (s, X s )ds + [F (s, X s ) F (s, X s )]dn s x F (s, X s )dx s x F (s, X s )(µ s x s λ(s))ds [F (s, X s ) F (s, X s ) x F (s, X s )x s ]dn s x F (s, X s )dx s [F (s, X s + x s ) F (s, X s ) x F (s, X s )x s ]dn s. 27

28 3.5 Predictable Representation Property Proposition 3.2 Let F N be the completion of the canonical filtration of the Poisson process N and H L 2 (F ), N a square integrable random variable. Then, there exists a predictable process h such that and E( h 2 sds) <. H = E(H)+ h s dm s 28

29 3.6 Independent Poisson Processes Definition 3.3 A process (N 1,,N d ) is a d-dimensional F-Poisson process if each N j is a right-continuous adapted process such that N j =and if there exists constants λ j such that for every t s P [ d j=1(n j t N j s = n j ) F s ] = d j=1 e λ j(t s) (λ j(t s)) n j n j!. Proposition 3.4 An F-adapted process N is a d-dimensional F-Poisson process if and only if (i) each N j is an F-Poisson process (ii) no two N j s jump simultaneously. 29

30 4 Stochastic intensity processes 4.1 Definition Definition 4.1 Let N be a counting process, F-adapted and (λ t,t ) a non-negative F- progressively measurable process such that for every t, Λ t = λ sds < P a.s.. The process N is an inhomogeneous Poisson process with stochastic intensity λ if for every non-negative F-predictable process (φ t,t ) the following equality is satisfied ( ) ( ) E φ s dn s = E φ s λ s ds. Therefore (M t = N t Λ t,t ) is an F-local martingale. If φ is a predictable process such that t, E( φ s λ s ds) <, then ( φ sdm s,t ) is an F-martingale. 3

31 An inhomogeneous Poisson process N with stochastic intensity λ t can be viewed as time changed of Ñ, a standard Poisson process N t = ÑΛ t. 31

32 4.2 Itô s formula The formula obtained in Section 3.4 generalizes to inhomogeneous Poisson process with stochastic intensity. 32

33 4.3 Exponential Martingales Proposition 4.2 Let N be an inhomogeneous Poisson process with stochastic intensity (λ t,t ), and(µ t,t ) a predictable process such that µ s λ s ds <. Then, the process L defined by exp( L t = µ sλ s ds) if t<t 1 n,t n t (1 + µ T n )exp( (4.1) t µ sλ s ds) if t T 1 is a local martingale, solution of dl t = L t µ t dm t, L =1. (4.2) Moreover, if µ is such that s, µ s > 1, [ L t =exp µ s λ s ds + ln(1 + µ s ) dn s ]. 33

34 The local martingale L is denoted by E(µ M) and named the Doléans-Dade exponential of the process µ M. This process can also be written L t = <s t (1 + µ s N s )exp [ ] µ s λ s ds. Moreover, if t, µ t > 1, then L is a non-negative local martingale, therefore it is a supermartingale and L t = exp = exp [ µ s λ s ds + s t µ s λ s ds + ln(1 + µ s ) N s ln(1 + µ s ) dn s ] [ = exp [ln(1 + µ s ) µ s ] λ s ds + ln(1 + µ s ) dm s ]. 34

35 The process L is a martingale if t, E(L t )=1. If µ is not greater than 1, then the process L defined in (4.1) is still a local martingale which satisfies dl t = L t µ t dm t. However it may be negative. 35

36 4.4 Change of Probability Let µ be a predictable process such that µ> 1 and λ s µ s ds <. Let L be the positive exponential local martingale solution of dl t = L t µ t dm t. Assume that L is a martingale and let Q be the probability measure equivalent to P defined on F t by Q Ft = L t P Ft. Under Q, the process (M µ t def = M t µ s λ s ds = N t (µ s +1)λ s ds, t ) is a local martingale. 36

37 5 Compound Poisson Processes 5.1 Definition and Properties Let λ be a positive number and F (dy) be a probability law on IR. A (λ, F ) compound Poisson process is a process X =(X t,t ) of the form X t = N t Y k k=1 where N is a Poisson process with intensity λ>andthe(y k,k IN) are i.i.d. square integrable random variables with law F (dy) =P (Y 1 dy), independent of N. 37

38 Proposition 5.1 A compound Poisson process has stationary and independent increments; the cumulative function of X t is P (X t x) =e λt n (λt) n F n (x). n! If E( Y 1 ) <, the process (Z t = X t tλe(y 1 ),t ) is a martingale and in particular, E(X t )=λte(y 1 )=λt yf(dy). If E(Y 2 1 ) <, the process (Z 2 t tλe(y 2 1 ),t ) is a martingale and Var (X t )=λte(y 2 1 ). 38

39 Introducing the random measure µ = δ Tn,Y n on IR + IR and n denoting by (f µ) t the integral f(x)µ(ω, ds, dx), we obtain that IR M f t =(f µ) t tν(f) = f(x)(µ(ω, ds, dx) dsν(dx)) is a martingale. IR 39

40 Corollary 5.2 Let X be a (λ, F ) compound Poisson process independent of W.Let Then, In particular, if 1+Y 1 >,a.s. ds t = S t (µdt + dx t ). N t S t = S e µt (1 + Y k ) S t = S exp(µt + k=1 N t k=1 ln(1 + Y k )). The process (S t e rt,t ) is a martingale if and only if µ + λe(y 1 )=r. 4

41 5.2 Martingales We now denote by ν the measure ν(dy) =λf (dy),a(λ, F ) compound Poisson process will be called a (λ, ν) compound Poisson process. Proposition 5.3 If X is a (λ, ν) compound Poisson process, for any α such that e αu ν(du) <, the process Z (α) t =exp ( ( )) αx t + t (1 e αu )ν(du) is a martingale and E(e αx t )=exp ( ( )) t (1 e αu )ν(du). 41

42 Proposition 5.4 Let X be a (λ, ν) compound Poisson process, and f a bounded Borel function. Then, the process ( Nt ) exp f(y k )+t (1 e f(x) )ν(dx) k=1 is a martingale. In particular ( ( Nt )) E exp f(y k ) =exp k=1 ( ) t (1 e f(x) )ν(dx) 42

43 For any bounded Borel function f, we denote by ν(f) = f(x)ν(dx) the product λe(f(y 1 )). Proposition 5.5 Let X be a (λ, ν) compound Poisson process. The process M f t = s t f( X s ) 1 { Xs } tν(f) is a martingale. Conversely, suppose that X is a pure jump process and that there exists a finite positive measure σ such that f( X s ) 1 { Xs } tσ(f) s t is a martingale for any f, thenx is a (σ(1),σ) compound Poisson process. 43

44 5.3 Hitting Times Let X t = bt + N t k=1 Y k. Assume that the support of F is included in ], ]. The process (exp(ux t tψ(u)),t ) is a martingale, with ψ(u) =bu λ (1 e uy )F (dy). Let T x =inf{t : X t >x}. Since the process X has no positive jumps, X Tx = x. Hence E(e ux t Tx (t Tx)ψ(u) )=1andwhent goes to infinity, one obtains E(e ux Txψ(u) 1 {Tx < }) =1. 44

45 If ψ admits an inverse ψ, one gets if T x is finite E(e λt x )=e xψ (λ). Setting Z k = Y k, the random variables Z k can be interpreted as losses N t for insurance companies. The process z + bt Z k is called the Cramer-Lundberg risk process. The time τ =inf{t : X t } is the bankruptcy time for the company. k=1 45

46 One sided exponential law. If F (dy) =θe θy 1 {y<} dy, one obtains ψ(u) =bu λu, hence inverting ψ, θ + u E(e κt x 1 {Tx < }) =e xψ (κ), with ψ (κ) = λ + κ θb + (λ + κ θb) 2 +4θb 2b. 46

47 5.4 Change of Measure Let X be a (λ, ν) compound Poisson process, λ >and F a probability measure on IR, absolutely continuous w.r.t. F and ν(dx) = λ F (dx). Let L t =exp t(λ λ)+ ( ) d ν ln ( X s ). dν s t Set dq Ft = L t dp Ft. Proposition 5.6 Under Q, the process X is a ( λ, ν) compound Poisson process. 47

48 6 An Elementary Model of Prices including Jumps Suppose that S is a stochastic process with dynamics given by ds t = S t (b(t)dt + φ(t)dm t ), (6.1) where M is the compensated compensated martingale associated with a Poisson process and where b, φ are deterministic continuous functions, such that φ> 1. The solution of (6.1) is S t = S exp = S exp [ φ(s)λ(s)ds + [ ] [ b(s)ds exp ] b(s)ds (1 + φ(s) N s ) s t ln(1 + φ(s))dn s ] φ(s)λ(s)ds. 48

49 ( ) Hence S t exp b(s)ds is a strictly positive martingale. We denote by r the deterministic interest rate and by ( R t =exp ) t r(s)ds the discounted factor. Any strictly positive martingale L can be written as dl t = L t µ t dm t with µ> 1. 49

50 Let (Y t = R t S t L t,t ). Itô s calculus yields to dy t = Y t ((b(t) r(t))dt + φ(t)µ t d[m] t ) = Y t (b(t) r(t)+φ(t)µ t λ(t)) dt. Hence, Y is a local martingale if and only if µ t λ(t) = b(t) r(t). φ(t) Assume that µ> 1 andq Ft = L t P Ft. Under Q, N is a Poisson process with intensity ((µ(s) + 1)λ(s),s ) and ds t = S t (r(t)dt + φ(t)dm µ t ) where (M µ (t) =N t (µ(s)+1)λ(s) ds, t ) is the compensated Q-martingale. Hence Q is the unique equivalent martingale measure. 5

51 6.1 Price Process Let ds t =(αs t + β) dt +(γs t + δ)dx t (6.2) where X is a (λ, ν) compound Poisson process. The solution of (6.2) is a Markov process with infinitesimal generator L(f)(x) = + [f(x + γxy + δy) f(x)] ν(dy)+(αx + β)f (x). Let S be the solution of (6.2). The process e rt S t is a martingale if and only if γ yν(dy)+α = r, δ yν(dy)+β =. 51

52 Let F be a probability measure absolutely continuous with respect to F and L t =exp t(λ λ)+ ( λ d ln F ) ( X s ). λ df s t Let dq Ft = L t dp Ft. The process (S t e rt,t ) is a Q-martingale if and only if λγ y F (dy)+α = r, λδ y F (dy)+β = Hence, there are an infinite number of e.m.m.. One can change the intensity of the Poisson process, and/or the law of the jumps. 52