ITO FORMULAS FOR FRACTIONAL BROWNIAN MOTIONS. Robert J. Elliott. Haskayne School of Business, University of Calgary

Transcription

1 ITO FORMULAS FOR FRACTIONAL BROWNIAN MOTIONS Robert J. Elliott Haskayne School of Business, University of Calgary 1

2 ABSTRACT The difficulty of studying financial models with fractional Brownian motions is partly due to the fact that the stochastic calculus for fractional Brownian motions is not as well developed as for traditional models. Ito formulas play an important role in this stochastic calculus. This paper will review the current state of knowledge of such formulas and present a new and transparent approach to the derivation of these generalized Ito formulas. We will present applications to financial models. This presentation represents joint work with John van der Hoek from the University of Adelaide. 2

3 What is FBM? For each Hurst index 0 <H<1 B H = { B H (t),t R } is a Gaussian process with mean 0 for each t and covariance E [ B H (t)b H (s) ] = 1 [ t 2H + s 2H t s 2H] 2 We now take B H (0) = 0, B 1 2 is a standard Brownian motion. E [( B H (t) B H (s) ) 2 ] = t s 2H defined on a suitable probability space (Ω, F,P), with expectation operator E[ ] 3

4 Write Ω = S (R) = space of tempered distributions F = Borel sets of Ω. The probability measure P exists by Bochner-Minlos Theorem, so that where and E [ e i f, ] = e 1 2 f 2 f S(R) f 2 = f 2 (x)dx R This is the White Noise framework of Hida et al. B(t)(ω) := I(0,t),ω for ω Ω 4

5 (M H f)(x) = d dx C H (t x) t x H 3 2 f(t) dt When f = I(0,t), the characteristic function of [0,t], we will write M H (0,t) for this M H f. For H (0, 1) define B H as the continuous version of the process defined by B H (t)(ω) = M H (0,t),ω for ω Ω. The constant C H is chosen so that E [ B H (t)b H (s) ] = 1 2 [ t 2H + s 2H t s 2H] 5

6 Hermite Functions and Chaos Expansions ξ n (x) =π 1 4( (n 1)! ) 1 2 h n 1 ( 2 x)e x2 2, n =1, 2,... h n (x) =( 1) n e x2 2 dn dx n e x2 2, n =0, 1, 2,... B(t)(ω) := I(0,t),ω = n B H (t)(ω) := M H (0,t),ω = n ξ n,ω ξ n,ω 0 ξ n(s) ds 0 M Hξ n (s) ds W (t)(ω) := n ξ n,ω ξ n (t) W H (t)(ω) := n ξ n,ω M H ξ n (t) 6

7 The space L 2( S (R), F,P ) of square integrable random variables on S (R). I denotes the set of all finite multi-indices α =(α 1,...,α n )forsomen 1 of non-negative integers. Write α = α 1 + α α n and α! =α 1!α 2! α n!. For α I write H α (ω) = n i=1 h αi ( ξ i,ω ). In particular, write ε i =(0,...0, 1) (n components). Then H εj (ω) =h 1 ( ξ j,ω ) = ξ j,ω 7

8 Every F L 2( S (R), F,P ) has a unique representation F (ω) = α I c α H α (ω) = α c α h α1 ( ξ 1,ω )...h αn ( ξ n,ω ) and E[H α H β ]=α!δ α,β F 2 = α I α!c 2 α 8

9 Consider formal sums F (ω) = α c α H α (ω) G(ω) = α d α H α (ω) and define (2N) γ := m j=1 (2j) γ j with γ =(γ 1,...,γ m ) I We define (Hida) spaces (S) L (S) as follows: F (S) if for all k, α!c 2 α (2N)kα < where (2N) kα (2j) kα j α j G (S) if for some q>0, α α!d 2 α(2n) qα < and the dual pairing is F, G = α!c α d α α 9

10 Suppose Z : R (S) is such that Z(t),F L 1 (R) for all F (S) Then R Z(t)dt is defined to be that unique element of (S) such that R Z(t)dt, F = R Z(t),F dt for all F (S) We then say that Z(t) is integrable in (S) 10

11 We introduce Wick Products Suppose F (ω) = α c α H α (ω) G(ω) = α d α H α (ω) are in (S). The their Wick product is F G(ω) := α,β a α b β H α+β (ω) = γ ( α+β=γ a α b β )H γ (ω) 11

12 Hitsudu-Skorohod Integral We have W H (t) (S) for all t. Suppose Y : R (S) is such that Y (t) W H (t) is integrable in (S). Then we define Y R (t)dbh (t) := Y (t) W H (t)dt R For example, if Y (t)(ω) = α c α (t)h α (ω), then formally R Y (t)dbh (t)(ω) = α,i H α+ɛi (ω) = α,i H α+ɛi (ω) R c α(t)m H ξ i (t)dt R M Hc α (t)ξ i (t)dt This integral generalises the Itô integral. 12

13 Generalized Hida Differentiation Suppose F : S (R) R and suppose γ S (R). If D H γ F (ω) := lim ε 0 F (ω + εm H γ) F (ω) ε ( ) exists in (S), we then call Dγ HF the directional M H-derivative of F in the direction γ. If γ = δ t, then [ ] γt H (x) (M Hδ t )(x) = C H (t x) t x H 3 2 S (R) x and we write H t F (ω) for the right hand side of (*). 13

14 Remark Suppose γ S (R) and has compact support. Let ψ be any infinitely differentiable function with compact support and which is identically 1 on the support of γ, then M H γ is a tempered distribution defined by (independent of ψ) M H γ,φ γ,ψm H φ for any φ S(R). 14

15 Examples (1) If H> 1 2, then γh t (x) =C H(H 1 2 ) t x H 3 2 L 1 loc (R) (2) If 0 <H<1, then H t BH (u) = H [ t 2H 1 (t u) 2H 1] H [ t 2H 1 +(u t) 2H 1] if t>u if t<u and H t W H (u) =H(2H 1) u t 2H 2 for u t and H 1 2. (3) If H> 1 2, F = R f(u)dbh (u), then H t F = H(2H 1) R f(u) t u 2H 2 du 15

16 Examples continued (4) If H< 1 2, then if b F = a f(u)dbh (u) =f(b)b H (b) f(a)b H (a) then for a<t<b H t F = f(b) H t BH (b) f(a) H t BH (a) (5) If H = 1 2 then H t b b a f (u)b H (u)du a f (u) H t BH (u)du B(u) =1ifu>tand=0ifu<t H t W (u) =δ(t u). If F = R f(u)db(u) then H t F = f(t) tf 16

17 A Basic Lemma Φ R Ψ(u) dbh (u) = R [ΦΨ(u)] dbh (u)+ R Ψ(u) H u Φdu 17

18 A Corollary or a Definition R Y (u)dbh (u) = R Y (u) dbh (u)+ R H u Y (u)du Proof: Let Φ = Y (u i )andψ=i(u i,u i+1 ] in the basic lemma to gives Y (u i ) [ B H (u i+1 ) B H (u i ) ] ui+1 = Y (u i ) db H ui+1 (u)+ u H Y (u i)du u i u i and so on. 18

19 First Itô s Lemma for H> 1 2 Let X(t) =X(0) + 0 u(s)ds + 0 v(s)dbh (s) and f is sufficiently smooth f(x(t)) = f(x(0)) f (X(s))u(s)ds 0 [f (X(s))v(s)] db H (s) + H(2H 1) 0 f (X(s))v(s) [ s 0 v(r)(s r)2h 2 dr ] 19 ds

20 First Itô s Lemma for H< 1 2 Let X(t) =X(0) + 0 u(s)ds + 0 v(s)dbh (s) and f is sufficiently smooth f(x(t)) = f(x(0)) f (X(s))u(s)ds 0 [f (X(s))v(s)] db H (s) + Hv(0) + H 0 f (X(s))v(s)s 2H 1 ds 0 f (X(s))v(s) [ s 0 v (r)(s r) 2H 1 dr ] ds 20

21 Remark 1 The formula for H < 1 2 also holds for H > 1 2 and agrees with the usual formula when H = 1 2. f(x(t)) = f(x(0)) + 0 f (X(s))u(s)ds + [f (X(s))v(s)] db(s) v(0) f (X(s))v(s)ds f (X(s))v(s) = f(x(0)) + + [ s 0 v (r)dr 0 f (X(s))u(s)ds 0 [f (X(s))v(s)] db(s)+ ] 0 ds 1 2 f (X(s))v(s) 2 ds 21

22 Remark 2 When X(t) =B H (t), it is now well known that f(b H (t)) = f(0)+ f (B H (s)) db H (s)+h 0 0 f (B H (s))s 2H 1 ds which again agrees with the usual formula when H =

23 General Itô Formula for H> 1 2 X(t) =X(0) + implies f(x(t)) = f(x(0)) µ(x(s))ds + σ(x(s)) 0 0 dbh (s) 0 f (X(s))µ(X(s))ds 0 [f (X(s))σ(X(s))] db H (s) 0 f (X(s))σ(X(s)) H s X(s)ds s H X(t) = 0 µ (X(r)) s H X(r)dr + + H(2H 1) 0 σ(x(r)) s r 2H 2 dr 0 [σ (X(r)) H s X(r)] dbh (s) 23

24 Outline of Proof for H> 1 2 Let t j = j t, j =0, 1, 2,..., n with t n = t f(x(t)) f(x(0)) = = = = n 1 [ 1 j=0 n 1 [ j+1 Φ j j=0 n 1 j+1 j=0 = I 1 n + I2 n + I3 n n 1 j=0 f(x(t j+1 )) f(x(t j )) 0 f (θx(t j+1 )+(1 θ)x(t j ))dθ t j u(s)ds + t j Φ j u(s)ds + j+1 t j j+1 t j ] v(s)db H (s) [X(t j+1 ) X(t j )] ] [ Φj v(s) ] db H (s)+j+1 t j v(s)[ H s Φ j]ds 24

25 Outline of Proof for H> 1 2 continued s H Φ j = = f (θx(t j+1 )+(1 θ)x(t j )) [ θ H s X(t j+1)+(1 θ) H s X(t j) ] dθ 1 0 θf (θx(t j+1 )+(1 θ)x(t j ))dθ H s [X(t j+1) X(t j )] 1 0 f (θx(t j+1 )+(1 θ)x(t j ))dθ H s X(t j ) In 3 = I4 n + I5 n In 4 n 1 = In 5 n 1 = [ 1 j=0 [ 1 j=0 0 θf (...)dθ 0 f (...)dθ ] j+1 t j ] j+1 t j v(s) H s [X(t j+1) X(t j )]ds v(s) H s X(t j)ds 25

26 Outline of Proof for H> 1 2 continued H s X(t) =H(2H 1) s H X(t j)=h(2h 1) In 5 H(2H 1) 0 v(r) s r 2H 2 dr j 0 v(r)(s r)2h 2 dr if t j <s<t j+1 0 v(s)f (X(s)) [ s 0 v(r)(s r)2h 2 dr ] ds j+1 t j v(s) H s [X(t j+1) X(t j )]ds v(t j ) 2 (t j+1 t j ) 2H and 2H >1 I 4 n 0 26

27 Remark This proof also provides a new proof for establishing the usual Itô formula for standard Brownian Motion using Hitsuda-Skorohod integrals along the way. The use of generalized Hida derivatives provides a transparent approach for finding general Itô formulas. Results are equivalent to those of Duncan, Hu & Pasic-Duncan (2000), Sottinen & Valkeila (2001), Biagini & Øksendal (2004) and so on by other approaches. 27

28 Now some applications in finance We use FBM to model long-range dependence. A second order stationary process X = {X(t) t 0} has Hurst index defined by H = inf{h : lim sup t t 2h var (X(t) X(0)) < }. B H has Hurst index H, but so does B H + B K when H>K[!] 28

29 Fractional Black and Scholes Market - 1 Stock price S 1 ds 1 (t) =µs 1 (t) dt + σs 1 (t) db H (t) (or B M ) then S 1 (0) = s>0 S 1 (t) =s exp [ µt+ σb H (t) 1 2 σ2 t 2H] f(t, B H (t)) for an appropriate f = f(t, x). 29

30 Proof: By the FBM Itô formula: S 1 (t) =s + = s + + H + 0 f t(s, B H (s))ds + 0 f xx(s, B H (s))s 2H 1 ds 0 S 1(s)(µ Hσ 2 s 2H 1 )ds 0 σs 1(s) db H (s)+h 0 f x(s, B H (s)) db H (s) 0 S 1(s)σ 2 s 2H 1 ds = s + 0 µs 1(s)ds + 0 σs 1(s) db H (s) 30

31 Fractional Black and Scholes Market - 2 Stock price S 1 (andh> 1 2 ) ds 1 (t) =µs 1 (t) dt + σs 1 (t)db H (t) then S 1 (0) = s>0 S 1 (t) =s exp [ µt+ σb H (t) ] g(t, B H (t)) for an appropriate g = g(t, x). 31

32 Proof: We first observe (for H> 1 2 )that t H S 1(t) =σs 1 (t) t H BH (t) =σs 1 (t)ht 2H 1 and by the FBM Itô formula: S 1 (t) =s + = s + = s + = s + + H 0 g t(s, B H (s))ds + 0 g xx(s, B H (s))s 2H 1 ds 0 S 1(s)µds + + H 0 µs 1(s)ds + 0 g x(s, B H (s)) db H (s) 0 σs 1(s) db H (s)+h 0 S 1(s)σ 2 s 2H 1 ds 0 σs 1(s)dB H (s) 0 S 1(s)σ 2 s 2H 1 ds 0 H s S 1(s)ds µs 1(s)ds + σs 1(s)dB H (s)

33 Fractional Black and Scholes Market - 3 Stock price S 1 (andh> 1 2 ), B is Brownian motion independent of B H ds 1 (t) =(µ a2 ) S 1 (t) dt + as 1 (t)db(t)+σs 1 (t)db H (t) then S 1 (0) = s>0 S 1 (t) =s exp [ µt+ ab(t)+σb H (t) ] g(t, B(t),B H (t)) for an appropriate g = g(t, x, y). 33

34 Self Financing Portfolios Portfolio replication in continuous time: θ = { θ(t) } θ(t) = ( α(t),β(t) ) is a pair of (suitably) adapted processes which give the number of units of S 1 and S 0 held at time t. V θ = { V θ (t) } is a wealth process and with S 0 (t) exp(rt), V θ (t) =α(t) S 1 (t)+β(t) S 0 (t) 34

35 θ satisfies certain technical conditions and must be self-financing: dv θ (t) =α(t) ds 1 (t)+β(t) ds 0 (t) Changes in the wealth process come from changes in the values of the underlying assets S 0 and S 1. V θ (T )= [ S 1 (T ) K ] + for a European call option, say. 35

36 With ds 1 (t) =rs 1 (t)dt + σs 1 (t) db H (t) we can show dv θ (t) =rv θ (t)dt + σ[α(t)s 1 (t)] db H (t)+σs 1 (t) H t α(t)dt and T T e rt V θ (T )=V θ (0)+ 0 σe ru [α(u)s 1 (u)] db H (u)+ 0 σe ru S 1 (u) u H α(u)du 36

37 As equation (1) makes sense for 0.5 H 1 and (2) for all 0 <H<1, the latter can be taken as the self-financing requirement, and could be established from discrete approximations. Now in general V θ (0) E [ e rt V θ (T ) ] But when H =1/2, then H t α(t) 0. 37

38 Girsanov Transformation Suppose that under probability ˆP ˆB H (t) =B H (t) then e rt V θ (T )=V θ (0) + + T 0 σe rt T whence we are led to seek φ so that 0 φ(u) du 0 σe rt [α(t)s(t)] d ˆB H (t) ( S(t) H t α(t)+[α(t)s(t)] φ(t) ) dt S(t) H t α(t)+[α(t)s(t)] φ(t) =0 ( ) for all t. 38

39 A Lemma If for each self-financing strategy θ, there exists a probability measure change (there exists a suitable solution of (*)), then V θ (0) = Ê [ e rt V θ (T ) ] and there are no-arbitrage opportunities in this market When H = 1 2, φ 0 solves (*) for all α. 39

40 Some Issues 1. Given a (B, S) market, with S driven by FBM or by multiple FBMs, and a class of admissible self-financing strategies, does there exist for each α in this class, a suitable choice of φ for that (*) holds. 2. Given φ as in the previous item, is there ˆP so that ˆB H (t) = B H (t) 0 φ(u)du is FBM under ˆP (or a suitable generalization for multiple FBMs). One needs to use an anticipative Girsanov Theorem. 40

41 In the (B,S) market, Cheridito(2000) claimed that there are arbitrage opportunities, so one may try to construct α for which there is no φ. However for some mixture models there are noarbitrage opportunities (e.g. Kuznetsov(1999), Cheridito(2000), Androshchuk + Mishura(2005)). For FBM markets, models for S have to be specified carefully, and the specification of admissible trading strategies. Our machinery can be adapted to multiple FBM models and used to study these questions. This is on-going research. 41

42 Fractional Black and Scholes Market - 3 again Bank account S 0 = e rt,stocks 1 (t) =s exp [ µt+ab(t)+σb H (t) ] Let that is φ t (t, x)+ 1 2 a2 x 2 φ xx (t, x)+rxφ x (t, x) rφ(t, x) =0 φ(t,x)=g(x) φ(t, x) = 1 2π R g(x exp[(r 1 2 a2 )(T t)+ a z T t ])exp( 1 2 z2 )dz when a 0. Let α(t) =φ x (t, S 1 (t)) β(t) =e rt (φ(t, S 1 (t)) S 1 (t)φ x (t, S 1 (t))) 42

43 Then V θ (t) α(t)s 1 (t)+β(t)s 0 (t) is self-financing and V θ (T )= g(s 1 (T )). With a 0 it is possible to construct a measure change so that V θ (0) = Ẽ[e rt V θ (T )] = Ẽ[e rt g(s 1 (T ))] In fact e rt V θ (T )=V θ (0) + + T T 0 e rt as 1 (t)φ x (t, S 1 (t))db(t) 0 e rt σ[s 1 (t)φ x (t, S 1 (t))] db H (t) T + 0 e rt S 1 (t)φ x (t, S 1 (t))λ(t, S 1 (t))dt and now we perform a Girsanov shift on B. A call price on S 1 could in principle then be computed either as V θ (0) = φ(0,s)or by computing an appropriate expectation. 43

44 This shows that when we restrict self-financing strategies to ones which are of Markov-type [θ(t) θ(t, S 1 (t))], then there are no arbitrage opportunities. But when a =0 φ(t, x) =e r(t t) g(x e r(t t) ) and with the choice g(x) =(x s e rt ) 2 say, we have V θ (0) = 0 and V θ (T ) 0withP [V θ (T ) > 0] > 0, an arbitrage. This example was stimulated by the study of Androshchuk and Mishura(2005). 44

45 When g(x) (x K) +, then φ(s(t),t) gives the price of the European call option. That is C t = φ(s(t),t)=s(t)n (d 1 (t)) Ke r(t t) N (d 2 (t)) with d 1 (t) = ln [ S(t) K ] +(r+ 1 2 a2 )(T t) a T t and d 2 (t) =d 1 (t) a T t. This agrees with the usual Black-Scholes formula when σ 0+. But when a 0+, the expression C t converges to e r(t t) [ S(t)e r(t t) K ] + 45