6. Stochastic calculus with jump processes

Transcription

1 A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio corresponding o follow his sraegy is given by d V φ = φ k S k k=1 = φ. S We will denoe by = T, T 1,, T n+1 = T he ransiion daes: beween 2 ransiion daes, he porfolio is unchanged, so ha we can denoe is composiion beween T i and T i+1 as φ i, and wrie: φ = φ I = + Discussion: why ]T i, T i+1 ]? n i= φ i I ]Ti,T i+1 ]() 1

2 A) Trading sraegies (2/3) A rading sraegy is hence modelled as a caglad process and no cadlag (conrarily o marke prices) Definiion: simple predicable process Process (φ ) [,T] ha can be represened by φ = φ I = + n i= φ i I ]Ti,T i+1 ] where T = < T 1 < T 2 < < T n < T n+1 = T are sopping imes and φ i are bounded random variables wih φ i F Ti (i.e. he value of φ i is revealed in T i ) The idea is o approach any caglad rading sraegy by simple sraegies 2

3 A) Trading sraegies (3/3) Example of simple predicable process 3

4 B) Sochasic inegral and gain process for simple rading sraegies (1/2) If φ is a simple rading sraegy, he gain realized beween T i and T i+1 is φ i (S Ti+1 S Ti ) and he oal accumulaed wealh of an invesor beginning wih a quaniy φ a is (for T j T j+1 ): j 1 G φ = φ. S + φ i. (S Ti+1 S Ti ) + φ j. S S Tj i= n = φ. S + φ i S Ti+1 S Ti i= Definiion: G φ is called he sochasic inegral of he predicable process φ wih respec o he price process (S ). We will denoe i by: φ u ds(u) 4

5 B) Sochasic inegral and gain process for simple rading sraegies (2/2) We will only consider self-financing sraegies, i.e. he value of he porfolio a following he sraegy is equal o he oal accumulaed wealh: V φ = φ. S = G (φ) Maringale preserving propery If S is a maringale, hen for any simple predicable process φ, he sochasic inegral/gain process G = φds maringale is also a 5

6 C) Consrucion of he sochasic inegral w.r. Brownian moion Iô inegral, i.e. case of S = W (Brownian moion): 1. definiion of he Iô inegral for simple processes, 2. proof of he Iô isomery and hence coninuiy of he Io inegral for he L 2 norm 3. densiy of he se of simple processes in he se of adaped square inegrable processes (wih he L² norm) and 4. exension by densiy of he Io inegral on he se of adaped square inegrable processes 6

7 D) Exension of he sochasic inegral o semi-maringales (1/8) If S is no a Brownian moion, i is no eviden a priori ha he sochasic inegral sill saisfies any sabiliy propery We would like o be able o pass o he limi in some sense: if φ n φ in he se of simple predicable processes (e.g. uniform convergence), hen φ n ds φds probabiliy) we need a sabiliy condiion in some sense (e.g. in This is also a desirable propery for models for coninuous ime rading: a small change in a sraegy mus lead o a small change in he gain process, unless he whole modelling process is useless Sabiliy is no reached for any adaped process S inroducion of he concep of semi-maringale 7

8 D) Exension of he sochasic inegral o semi-maringales (2/8) Semi-maringale: Definiion: An adaped cadlag process (S ) is a semimaringale if he sochasic inegral wih respec o S of simple predicable processes φ, i.e. φ = φ I = + n i= φ i T I ]Ti,T i+1 ] φds = verifies he following coninuiy (sabiliy) propery: for all φ (n), φ simple predicable sraegies, if n φ. S + φ i. (S Ti+1 S Ti ) i= sup,ω,t Ω φ n ω φ ω hen T φ (n) ds P T φds 8

9 D) Exension of he sochasic inegral o semi-maringales (3/8) Semi-maringales If he modelling process S does no verify his propery, i means ha a small error in he composiion of he sraegy can bring a large change in he porfolio value beer use semi-maringales for modelling asses in financial modelling On can show ha he above convergence acually occurs uniformly on [,T] (when considering he sochasic inegral process): sup,t φ n ds φds P Uniformly on compac ses in probabiliy : «ucp» convergence 9

10 D) Exension of he sochasic inegral o semi-maringales (4/8) One ofen defines he noion of semi-maringale slighly differenly: Semi-maringale (Definiion 2): A semi-maringale is a cadlag adaped process ha can be wrien as : S = S + M + A where M = A = a. s., M is a maringale and A is of finie variaion Acually his second definiion is sronger han he firs one: Any finie variaion process is a semi-maringale (in he sense of definiion 1) Any square inegrable maringale is a semi-maringale (in he sense of definiion 1) 1

11 D) Exension of he sochasic inegral o semi-maringales (5/8) Semi-maringales: Examples Brownian moion (squared inegrable maringale) (Compound) Poisson process (finie variaion) Any Levy process (by he Lévy Io decomposiion) Counerexample: Fracional Brownian moion B H (for H.5) Sochasic Inegral for a general caglad process Any caglad process can be approximaed uniformly by a sequence of simple predicable processes using he coninuiy/sabiliy propery of he sochasic inegral, we can define T φds as he limi of T φ n ds. Acually, one can even choose he usual Riemann sums as discree approximaion 11

12 illusraion fracional B.M. 12

13 D) Exension of he sochasic inegral o semi-maringales (6/8) This leads o he following resul: Sochasic Inegral via Riemann sums Le S be a semi-maringale, φ a caglad process and π (n) = (T n = < T 1 n < < T n n+1 = T) a sequence of sochasic pariions of [,T] such ha hen T n φ. S + = sup k n i= T k n n T k 1 φ Ti (S Ti+1 S Ti ) a. s. for n P φ u ds u uniformly in in [,T] ( ucp convergence) 13

14 D) Exension of he sochasic inegral o semi-maringales (7/8) One can show ha he following properies of he sochasic inegral are sill saisfied: If S is a semi-maringale, hen for any adaped caglad process (σ ) on (Ω, F, F, P): Semi-maringale preserving propery: X = σ u ds(u) is also a semi-maringale Associaiviy: if (φ ) is anoher adaped caglad process, hen φdx = φσds Maringale preserving propery: if X is a square-inegrable maringale and (φ ) is bounded, hen he sochasic inegral M = φdx is a square-inegrable maringale 14

15 D) Exension of he sochasic inegral o semi-maringales (8/8) Link wih he Io inegral: One can see ha we recover he Io inegral if S is a Brownian moion Sochasic inegral wih respec o a Poisson process: If S is a (compound) Poisson process, i easily follows from he definiion ha: φ s ds s = φ s ΔS s s ΔS s 15

16 E) Sochasic inegral wih respec o a Poisson random measure* For a given Levy process, he jump measure J X couns he number of jumps: J X, A = # {jumps occurring beween and and of size in A} I is a Poisson random measure defined on, T R\{}, wih inensiy measure ν, he Lévy measure We have seen ha J X is a sum of Dirac measures, locaed a he jump insans and sizes: J X = δ (,ΔX ) [,T] ΔX 16

17 E) Sochasic inegral wih respec o a Poisson random measure* We can consider inegrals wih respec o J X of measurable funcions defined on, T R\{}: φ:, T R R T φ( R, y)j X d dy = φ, ΔX [,T] ΔX This is a new random variable as he measure is sochasic We can also consider funcions φ ha are sochasic as well: φ: Ω, T R R 17

18 E) Sochasic inegral wih respec o a Poisson random measure* We consider a simple sraegy of he following form: n m φ, y = φ ij I ]Ti,T i+1 ] I Aj y i= j=1 where T i are sopping imes, φ ij is bounded and F Ti -measurable, and A j are disjoin Borel subses of R wih ν, T A j <. Then he sochasic inegral wih respec o he jump measure is defined as: T φ( R n,m, y)j X d dy = φ ij J X ( ]T i, T i+1 ] A j ) i,j=1 18

19 E) Sochasic inegral wih respec o a Poisson random measure* Paricular case In he special case where T i are he jump imes of he pure jump process X, and φ, y = ψ y where ψ = ψ i I ]Ti,T i+1 ], i.e. ψ is consan beween wo jumps, hen he inegral wih respec o J X corresponds o he sochasic inegral wih respec o X iself: T φ( R T, y)j X d dy = ψ y J X d dy = R [,T] ΔX φ, ΔX = ψ ΔX [,T] ΔX T = ψ dx 19

20 E) Sochasic inegral wih respec o a Poisson random measure* Paricular case This correspondence is only valid in his special case: If φ is general, he inegral wih respec o he jump measure canno be expressed as a sochasic inegral wih respec o he underlying process (X ). So in general, inegraion wih respec o J X and o X are wo differen conceps 2

21 E) Sochasic inegral wih respec o a Poisson random measure* Maringale preserving propery: For all simple predicable φ: Ω, T R R, he process defined by X = φ s, y M(ds dy) R is a squared inegrable maringale, and we have he isomery: E X 2 = E φ s, y 2 μ ds dy R where M is a Poisson random measure, and M denoes he compensaed measure associaed o M: M A = M A μ A for A, T R Thanks o his propery, one can exend he compensaed inegral o any L 2 funcion (same consrucion as for he Iô inegral) 21

22 E) Sochasic inegral wih respec o a Poisson random measure* In summary: 2 definiions, one defined as an L² limi, he oher as a limi in probabiliy. In he case of a predicable process φ consruced from a sraegy consising o rebalance he porfolio a jump insans (φ = yψ), and when considering he jump measure associaed o pure jump processes (no diffusion) boh definiions yield he same quaniy 22

23 F) Quadraic variaion: inroducion We consider a process X observed on a ime grid (pariion) π = < 1 < < n < n+1 = T π We can consider is realised variance: V X π = X i+1 X i 2 i π Now, each erm of his sum can be wrien as: 2 X i+1 X i = 2 Xi+1 X 2 i 2X i (X i+1 X i ) So ha he realised variance can be wrien as: V X π = X T 2 X 2 2 X i X i+1 X i i π Looks like a Riemann sum, converging under condiions o a sochasic inegral wih respec o X() 23

24 F) Quadraic variaion: inroducion If (X ) is a semi-maringale, hen i is cadlag (and no caglad) Now, if we consider X_ X [,T] hen his new process is caglad, and if we consider he Riemann sum as above, i will converge in probabiliy o he sochasic inegral: T X u dx u 24

25 F) Quadraic variaion: inroducion Hence, if (X ) is a semi-maringale wih moreover X =, hen he realised variance V X π converges in probabiliy o: X, X T = X T 2 2 T X u dx u which is a random variable. By definiion, his will be called he quadraic variaion of X on [,T] Repeaing his consrucion for any [, T], we ge he quadraic variaion process X, X 25

26 F) Quadraic variaion: Definiion Definiion: If (X ) is a semi-maringale, hen is quadraic variaion is he adaped cadlag process defined by X, X = X 2 2 X u dx u Resul: if π (n) (n) (n) (n) (n) = = < 1 < < n < n+1 sequence of pariions wih sup k (n) (n) k k 1 i < 2 Realized variance = X i+1 X i i π n for n, where he convergence is uniform in [, T] = T is a if n, hen P X, X 26

27 F) Quadraic variaion: Properies The Q.V., X, X is an increasing process The jumps of X, X are direcly relaed o he jumps of X: Δ X, X = ΔX 2 In paricular, X, X has coninuous sample pahs if and only if X does. So we can wrie: wih X, X c = X c, X c X, X = X, X c + ΔX s 2 Coninuous par of X, X s ΔX s 27

28 F) Quadraic variaion: Properies If X is coninuous and has finie variaion pahs, hen is quadraic variaion is zero: X, X T = If X is a maringale and X, X T =, hen X is a consan a.s.: X = X a. s. In paricular, a coninuous maringale wih pahs of finie variaion has a null quadraic variaion and is hence consan: Maringales Coninuous processes of finie variaion = Consans 28

29 F) Quadraic variaion: Properies Maringales Coninuous processes of finie variaion = Consans We can disinguish 2 broad classes of processes: maringales (e.g. BM or compensaed Poisson process) which are ypical examples of noise processes, and coninuous processes of finie variaion, ha can be inerpreed as drifs/rends. There is no nonrivial process who belongs o boh caegories 29

30 F) Quadraic variaion: Examples (exercises) Quadraic variaion of a Brownian moion: if B = σw, hen: B, B = σ 2 Quadraic variaion of a (compound) Poisson process: X, X = N i=1 Y i 2 = ΔX s 2 s ΔX s (acually, he same holds for any pure jump process) 3

31 F) Quadraic variaion: Examples Quadraic variaion of a Levy process wih riple (σ 2, ν, γ) X, X = σ 2 + ΔX s 2 s ΔX s = σ 2 + x 2 J X (ds dx), R = X, X c + ΔX s 2 s ΔX s 31

32 F) Quadraic variaion: Examples Quadraic variaion of a Brownian sochasic inegral X = σ dw where (σ ) is caglad (and in L 2 ) We have hen: X, X = σ 2 s ds 32

33 G) Io formula If f: R R, g:, T R are C 1 deerminisic funcions, hen f g f g = f g s g s ds In paricular if f x = x 2, his leads o: g 2 g 2 = 2 Now, if (X ) is a semi-maringale, hen X 2 X 2 = 2 X s dx s g s dg s + X, X The sandard chain rule does no work anymore for sochasic inegral wih respec o semi-maringales 33

34 G) Io formula: case of a Brownian sochasic inegral (Io Lemma): s if X s = σ u dw u i.e. dx = σ dw, hen: f X = f X + f X s σ s dw s σ 2 s f X s ds (if f and σ safisfy some regulariy condiions so ha everyhing above is well defined ) We will generalize his o he case of semi-maringales 34

35 G) Io formula: Deerminisic case: Le x:, T R a deerminisic funcion wih a finie number of discouninuiy poins T i (piecewise coninuous), and C 1 elsewhere. We can wrie x() as: x = b s ds where b(s) is coninuous (b=x ) + Ti Δx i where Δx i = x T i x(t i ) We will suppose ha x() is cadlag (x T i = x(t i + )) Le f be a C 1 funcion. Then on each open inerval ]T i, T i+1 [ we have he usual formula: f x T i+1 T f x T i = i+1 f x x T d = i+1 f x b d T i A each discouninuiy poin, we have: f x T i+1 f x T i+1 = f x T i+1 T i + Δx i+1 f x T i+1 35

36 G) Io formula: Deerminisic case: If we ake he sum over all inervals, we arrive o: n f x T f x = f x T i+1 f x T i n i= = f x T i+1 f x T i+1 i= + f x T i+1 f x T i n = f x T i+1 i= + Δx i+1 f x T i+1 n T i+1 + b f x d i= T i T i+1 = b f x d T i = b f x d T 36

37 G) Io formula: Case of a jump diffusion N Le X = σw + μ + i=1 ΔX i = X c + J, where J is a compound Poisson process (he jump par of X) X c is a diffusion wih drif (he coninuous par of X) Le f C 2 and le us consider Y = f(x ) Le us denoe by T i, i = 1,, N T he jump imes of X 37

38 G) Io formula: Case of a jump diffusion Then on each open inerval ]T i, T i+1 [ we can apply he Io lemma: Y Ti+1 Y Ti = = T i+1 T i+1 σ 2 T i+1 2 f X d + f X dx T i σ 2 since dx = dx c on ]T i, T i+1 [ T i T i 2 f X d + f X dx c If here is a jump a ime, wih size ΔX, hen he resuling change in Y is: Y Y = f X f X = f X + ΔX f X = Δ f X 38

39 G) Io formula: Case of a jump diffusion Hence he oal change of Y = f X on [, ] is he sum of hese wo ypes of erms: f X f X = f X s dxc s + σ 2 2 f X s ds + f X s + ΔX s f X s ΔX s s Usual Iô formula (nearly ) Addiional erm including jumps 39

40 G) Io formula: Case of a jump diffusion We can also rewrie his differenly by using dx c = dx s ΔX s : f X f X = f X s dx s + σ 2 2 f X s ds + f X s + ΔX s f X s ΔX s f (X s ) ΔX s s Boh formulaions are equivalen here as here is a finie number of jumps (case of a compound Poisson process for he jump par) Now, his las formulaion is ineresing as i is also well defined required ha X is a semi-maringale 4

41 G) Io formula: Case of a jump diffusion This resul can be exended : Proposiion Le (X ) be a jump diffusion: X = X + b s ds where B, σ are adaped processes wih E σ 2 d T + σ s dw s < N + i=1 ΔX i Then for any C 1,2 funcion f:, T R R, Y = f(, X ) can be represened as : Y = Y + f s s, X s + f x s, X s b s σ s 2 2 f x 2 s, X s ds + f x s, X s σ s dw s + f X Ti + ΔX i f X Ti T i i 1 Or, in differenial noaions: dy = f, X f + b x, X + σ 2 2 f 2 x 2, X d + f x, X σ dw + f X + ΔX f X 41

42 G) Io formula: Case of a general Levy process Proposiion Le X be a Lévy process wih riple (σ 2, ν, γ) and f: R R be a C 2 funcion such ha is 2 firs derivaives are a priori bounded by some consan C. Then Y = f X saisfies: f X f X = f X s dx s + σ 2 2 f X s ds + f X s + ΔX s f X s ΔX s f (X s ) ΔX s s 42

43 G) Io formula: Case of a general Levy process Proposiion: If now f = f, X depends on 2 variables, and is C 1,2, Y = f, X saisfies he following (in differenial noaions): dy = f, X d + f X, X dx f 2 X 2 + f X + ΔX f X f X, X, X d X, X c ΔX 43

44 H) Applicaion: Ordinary and Doleans-Dade exponenial of a Lévy process Conex If B = μ + σw is a Brownian moion wih drif, we know ha S = S exp B saisfies : S = S + S u db1 1 u ds = S db where B 1 = μ + σ2 2 + σw = B + σ2 2 is anoher Brownian moion (apply he sandard Io lemma for Brownian inegral) We will replace B by a general Lévy process exponenial Lévy process As well as B 1 sochasic exponenial or Doleans-Dade exponenial 44

45 H) Io formula - Ordinary Exponenial: Le (X ) be a Lévy process wih riple (σ 2, ν, γ), and le us consider Y = exp X Applying Io formula yields: dy = exp (X ) dx exp X σ 2 d + exp X + ΔX exp X Y ΔX = Y dx Y σ 2 d + Y exp ΔX Y Y ΔX = Y dx + σ2 2 d + exp ΔX 1 ΔX To be compared o wha we had in he pure diffusion case: S = S exp B ds = S db 1 = S db + σ2 2 d 45

46 H) Io formula - Ordinary Exponenial: One can show moreover he following resul: e y ν(dy) <, hen Y = exp X is a semi- If moreover maringale. y 1 Moreover, Y is a maringale if and only if γ + σ2 2 + ez 1 zi z 1 ν(dz) R = 46

47 H) Io formula - Doleans-Dade Exponenial: We will now prove he converse resul We depar from Y defined by: dy = Y dx where X is a Lévy process We will solve his SDE 47

48 H) Io formula - Doleans-Dade Exponenial: Proposiion: Le X be a Lévy process wih riple (σ 2, ν, γ ). Then here exiss a unique cadlag process (Z ) soluion of: Z is given by: dz = Z dx Z = 1 Z = e X σ ΔX s e ΔX s s If moreover X has a finie variaion ( 1 1 x ν dx < ), hen Z = e X c σ ΔX s s 48

49 H) Io formula - Doleans-Dade Exponenial: 49

50 H) Io formula - Doleans-Dade Exponenial: 5

51 H) Io formula - Doleans-Dade Exponenial: 51

52 H) Io formula - Doleans-Dade Exponenial: 52

53 Exercices: 1. Le N() be a Poisson process wih inensiy parameer λ. Consider he following SDE: λ λ dz = Z (dn λ λd) Z = 1 Use he Iô lemma o show ha he soluion of his SDE is Z = e λ λ λ λ N Show ha he process Z is a maringale wih E Z = 1 53