Introduction to Fractional Brownian Motion in Finance By Zhongyin John Daye

May 2003 Advanced FE Prof. S. G. Kou

Table of Contents 1. Motivation

3

2. Introduction to Fractional Brownian Motion

3

3. Elements of Wick-Ito Stochastic Calculus

4

3.1 Construction On Hilbert Space Lφ2 (ℜ)

5

3.2 Extension on Schwartz Space S (ℜ)

5

3.3 Wick Operations

8

3.4 Fractional Wick-Ito Calculus

9

3.5 Quasi-conditional Expectation

12

4. Applications of Wick-Ito Stochastic Calculus in Finance

14

4.1 No Arbitrage and Completeness of Market

14

4.2 Fractional Black-Scholes Pricing

16

5. Further Developments

18

5.1 Wick-Ito Stochastic Calculus for arbitrary Hurst parameter

18

5.2 Multidimensional Wick-Ito Stochastic Calculus

19

5.3 Other Applications

19

References

20

2

1. Motivation Brownian motion (BM) has been well established in finance. Indeed, introduction of the BM-based Black-Scholes formulation of vanilla options by Black, Scholes, and Merton marked the advent of mathematical finance. Nevertheless, classical mathematical models of financial assets are far from perfect. Two apparent problems exist in the Black-Scholes formulation, namely financial processes are not wholly Gaussian and Markovian in distribution. After the 1987 market crash, industry and researchers began to take note of the heavy-tail distribution of financial assets. A series of models has been developed using more general or heavy-tailed Levy processes. [Boyarchenko&Levendorskii] The second problem leads to long-range dependence. For a couple decades, the general consensus is to assume that all information is contained within current asset price and hence it is reasonable to assume a Markovian process. However, technical traders have consistently beaten the market using long-term memory strategies. This motivated a series of academic studies further purporting the existence of a non-Markovian market. [Lo&MacKinlay] To compensate, stochastic volatility models have been developed that can produce quasi long-range dependence. However, these models are awfully intractable as they lead to high-dimensional PDE’s with variable coefficients. Hence is the circumstance. Fractional Brownian motion (fBM) deals with the second problem while still assuming a Gaussian process. Nevertheless, it offers the promise of giving simple, tractable solutions to pricing financial options and presents a natural way of modeling long-range dependence. 2. Introduction to Fractional Brownian Motion (The following summarizes results from [Embrechts&Maejima].) Definition 1: self-similar (ss) A stochastic process {X(t)}t≥0 is self-similar if

{ X (ct )} ≈ {kX (t )} where c and k are positive, real-valued constants. The Definition 54 basically states that if a process is self-similar, then with a stochastic scaled in time one can always find another one scaled in sample paths that has the same distribution. Definition 2: H-self-similar (H-ss) A self-similar process with scaling factors as defined above is H-self-similar if k=cH, and H is known as the Hurst parameter. Definition 3: fractional Brownian motion (fBM) For H ∈ (0, 1), a Gaussian process {BH(t)}t≥0 is a fractional Brownian Motion if it has

3

Mean: E[BH(t)] = 0 Covariance: E[BH(t)BH(s)] = (1/2) {|t|2H + |s|2H - | t – s|2H} For all t, s ∈ ℜ Note that one may recover the standard Brownian motion by replacing H with ½. Theorem 4: properties of fBM A fBM process {BH(t)}t≥0 has the following properties (i) Unique (ii) H-ss, e.g. BH(t) ~ tHBH(1) (iii) Has stationary increments, e.g. E[(BH(t+h) - BH(h))(BH(s+h) - BH(h))] (iv) If H=1/2, it has independent increments (v) If H>1/2, then it has long-range dependence, e.g. ∑n=1∞ Cov(BH(1), BH(n+1) - BH(n)) = ∞ (vi) If H≠1/2, then it is non-Markovian (vii) If H≠1/2, then it is not a semimartingale (viii) The covariance between future and past increments is positive if H>1/2 and negative if H<1/2 The numerous properties of fBM easily illuminate both the benefits and shortcomings of using fBM in modeling financial instruments. H-ss, where H can be between 0 and 1, makes fBM more flexible as a modeling tool than standard Brownian motion that only allows ½-ss. The existence of long-range dependence and positive correlation of future and past increments makes fBM especially an attractive pricing tool. Statistical analysis has indicated that most markets are monofractal have Hurst parameters between 0.5 and 1. [Razdan] Thus, for financial purposes, Hurst parameters are usually assumed to be between ½ and 1, exclusively. However, if H is not equal to ½, we cannot use semimartingale and non-Markovian properties to construct its integration. This makes the task infinitely harder since few results from classical stochastic calculus can be directly used. The following section discusses various approaches and their important results. 3. Elements of Wick-Ito Stochastic Calculus A myriad of methods have been developed for integrating fBM. Two systems bear the most importance in finance. The fractional pathwise integral has the form defined by b

∫ a

n −1

f (t , w)δB H (t ) = lim ∑ f (t k , w)( B H (t k +1 ) − B H (t k )) |Π | → 0

k =0

where Π is a partition of the interval [a,b] and |∆| = max0≤k≤n-1(tk+1-tk). The fractional Wick-Ito integral has the form, b

∫ a

n −1

f (t , w)dB H (t ) = lim ∑ f (t k , w)◊( B H (t k +1 ) − B H (t k )) |Π | → 0

k =0

4

where ◊ represents a Wick product and ½ < H < 1. In relations to BM calculus, fractional pathwise integral mirrors a Stratonovich integral defined in [Karatzas&Shreve, p148], whereas fractional Wick-Ito integral emulates an Ito BM calculus. Fractional pathwise integral was developed by [Lin] during 1995 a nd found to produce an arbitrage market by [Rogers] in 1997 due to misbehavior of Gaussian kernel near 0. Apparently, pricing in an arbitrageable system is unreasonable. And the use of fBM has been deterred till only recently in 2000 when [Duncan&etal] and [Hu&Oksendal] developed the Wick-Ito integral. This new type of integration surprisingly produced a no-arbitrage market. [Wick] introduced the Wick product in 1950. [ Hida&Ikeda] introduced Wick product in analyzing stochastic processes, whereas [Holden&etal] provides a systemic introduction to the subject. (The following summarizes results from [Duncan&etal], [Holden&etal], and [Hu&Oksendal].) 3.1 Construction On Hilbert Space Lφ2 (ℜ)

1 2

Definition 5: φ ( s, t ) = H (2 H − 1) | s − t | 2 H − 2 , s, t ∈ ℜ, H ∈ ( ,1)

Definition 6: f ∈ Lφ2 (ℜ) , where f : ℜ → ℜ measurable, if | f |φ2 =

∫∫ f (s) f (t )φ (s, t )dsdt < ∞

ℜ×ℜ

Definition 7: ( f , g ) φ an inner product defined as

∫∫ f (s) g (t )φ (s, t )dsdt , f , g ∈ Lφ (ℜ) 2

ℜ×ℜ

Lemma 8: Lφ2 (ℜ) is a separable Hilbert space, e.g. has countable yet dense subsets. Theorem 9: Γφ , is an isometry from Lφ2 (ℜ) to L2 (ℜ) defined by, ∞

Γφ f (u ) = c H ∫ (t − u ) H −3 / 2 f (t )dt , where u

c H = {H (2 H − 1)Γ(1.5 − H )} {Γ( H − .5)Γ(2 − 2 H )} and Γ a gamma function. Proof: see [Hu&Oksendal, Lemma 2.1] Definition 10: ε ( f ) = e

1

∫ℜ fdBH − 2 | f |φ

2

, f ∈ Lφ2 (ℜ)

3.2 Extension On Schwartz Space S (ℜ)

5

Now, let S (ℜ) be a Schwartz space of rapidly decreasing real-valued smooth functions and

Ω = S ' (ℜ) the space of tempered distributions w on ℜ . Lemma 11: There exists probability measure µ φ ∈ Ω such that

∫

Ω

e i < w, f > dµ φ ( w) = e

1 − | f |φ2 2

,∀ f ∈ S (ℜ) and E µφ [< ⋅, f >] = 0 , E µφ [< ⋅, f > 2 ] =| f |φ2

Proof: The existence of µ φ is guaranteed through the Bochner-Minlos theorem, whereas the expectations are derived from the characteristic function of . For a statement of BochnerMinlos theorem consult [Holden&etal, Theorem 2.1.1] and its proof [Holden&etal, Appendix A]. Apparently, we need to construct a new Schwartz space S (ℜ) rather than using the Hilbert space

Lφ2 (ℜ) due to lack of an existence theorem for probability measure µ φ in the latter case. The herein obtained probability measure µ φ allows the acclamation of the following important theorems. Theorem 12: To define the integration of a function f ∈ Lφ2 (ℜ) , let the sequence of approximating m →∞ functions { f m (t )} → f (t ) be f m (t ) =

∑c

(m) i

χ [t ,t ] (t ) , then there exists i

i +1

i

ℜ

f m (t )dB H (t ) = ∑i ai( m ) ( BH (t i +1 ) − BH (t i )) and

ℜ

f (t )dB H (t ) = lim ∫ f m (t )dBH (t ) ∈ L2 ( µ φ ) , where µ φ represents the probability measure of

∫ ∫

m →∞ ℜ

BH Proof: The existence of the latter limit is guaranteed through Theorem 9. Theorem 13: The linear span of {ε ( f ); f ∈ Lφ2 (ℜ)} is dense in L2 ( µ φ ) Proof: see [Duncan&etal, Theorem 3.1] ~

~

Definition 14: B H (t ) = B H (t , w) =< w, χ [ 0,t ] (⋅) >∈ L2 ( µ φ ), ∀t ∈ ℜ , where

0≤s≤t  1,  χ [ 0,t ] ( s ) = − 1, t < s ≤ 0  0 otherwise  ~

Theorem 15: There exists a t-continuous version B H (t ) of B H (t ) that is also a fractional Brownian motion Proof: Kolmogorov-Centsov continuity theorem [Karatzas&Shreve, Theorem 1.2.8] guarantees the existence of B H (t ) , whereas Lemma 11 yields B H (t ) as a Gaussian process with

6

1 Eµφ [ BH (t )] = 0 and E µφ [ BH ( s ) BH (t )] = {| t | 2 H + | s | 2 H − | t − s | 2 H } . A comparison with 2 Definition 3 gives the result. Theorem 16: < w, f >=

∫

ℜ

f (t )dB H (t , w)

Proof: A direct consequence of Theorem 12, Definition 14, and theorem 15. Definition 17: Hermite polynomials, hn ( x) = (−1) e n

x2 / 2

d n − x2 / 2 (e ), n ∈ Ν 0 dx n ~

Lemma 18: There exists an orthonormal basis {ei }i∞=1 of Lφ2 (ℜ) , where en (u ) = Γφ−1 ( h n )(u ) for ~

h n ( x) = π

−

1 4

−

1 2

((n − 1)!) hn −1 ( x 2 )e

| ∫ en ( s )φ ( s, t )ds |< C t n

−

x2 2

, n ∈ Ν , and C t < ∞, ∀t ∈ ℜ such that

1 6

ℜ

Proof: see [Hu&Oksendal, Lemma 3.1] Lemma 19: The orthonormal basis {ei }i∞=1 of Lφ2 (ℜ) is smooth in Lφ2 (ℜ) and

t → ∫ ei ( s )φ ( s, t )ds continuous ℜ

Proof: This is a directly consequence of the boundedness of the integral in Lemma 18. Definition 20: H α ( w) = hα1 (< w, e1 >)  hα m (< w, em >) , where α = (α 1 ,..., a m ) a sequence of indices of nonnegative integers This representation can be used to reach the integration form of Theorem 16 with appropriate choices of indices α . Theorem 21: fractional Wiener-Ito chaos expansion theorem There exists constants cα ∈ ℜ and α ∈ I , where I represents the space of all indices of nonnegative integers, such that F ( w) =

∑ cα H α (w) and || F || α ∈I

2 L2 ( µφ )

= ∑ (α 1!α 2 !⋅ ⋅ ⋅α m !)cα2 for α ∈I

the function F ∈ L2 ( µ φ ) Proof: see [Hu&etal, Theorem 2.6] A comparison of Theorem 21 with Theorem 12 will illuminate the importance of the fractional Wiener-Ito chaos expansion theorem in defining a Wick-Ito integral.

7

Lemma 22:

∞

f ( s )dBH ( s ) = ∑ ( f , ei ) φ H ε ( i ) ( w) for f ∈ Lφ2 (ℜ)

∫

ℜ

i =1

Proof: Set cα =

1 E µ [ FH α ] and F ( w) =< w, f >= ∫ f ( s )dBH ( s ) in Theorem 20, ℜ α 1!α 2 !⋅ ⋅ ⋅α m ! φ

and apply Theorem 9 to obtain the result. For detail see [Hu&Oksendal, p. 8]. Definition 23: fractional Hida test function spaces ( S) H is the set of all ψ ( w) =

∑ aα H α (w) ∈ L (µφ ) such that || ψ || α 2

∈I

finite for all k ∈ Ν , where (2 Ν )

= ∏ (2 j )

(γ 1 ,γ 2 ,...,γ m )

2 H ,k

= ∑ α 1!α 2 !⋅ ⋅ ⋅α m !aα2 (2 Ν ) kα

γj

j

Definition 24: fractional Hida distribution spaces ( S ) *H is the set of all formal expansions G ( w) =

bβ H β ( w) , such that ∑ β ∈I

|| G ||

2 H ,−q

= ∑ β !bβ (2 Ν ) 2

− qβ

< ∞ , for some q ∈ N

β ∈I

Definitions 23 and 24 are important in providing spaces where integrability can be easily guaranteed. Definition 25: a function Z : ℜ → ( S ) *H is integrable in ( S ) *H if

〈〈 Z (t ),ψ 〉〉 ∈ L1 (ℜ, dt ),∀ψ ∈ ( S ) H and

〈〈 ∫ Z (t )dt ,ψ 〉〉 = ∫ 〈〈 Z (t ),ψ 〉〉 dt ,L1 (ℜ, dt ),∀ψ ∈ ( S ) H and ℜ

ℜ

∫

ℜ

Z (t )dt the unique element of

( S ) *H Lemma 26: The fractional white noise WH (t ) is integrable in ( S ) *H Proof: Define WH (t ) =

∞

∑ [∫

ℜ

i =1

ei (v)φ (t , v)dv]H ε ( i ) ( w) and assume q > 4 / 3 , then we have

|| WH (t ) || 2H , − q < ∞ by Lemma 18 and integrable in ( S ) *H by continuity guaranteed through Lemma 19. 3.3 Wick Operations Definition 27: Wick Product F◊G Let F ( w) =

∑ aα H α (w) ∈ (S ) α ∈I

( F◊G )( w) =

∑ aα bβ H α α β , ∈I

+β

* H

and G ( w) =

bβ H β ( w) ∈ ( S ) ∑ β ∈I

* H

, then

( w) ∈ ( S ) *H

8

Note that the Wick Product is analogous to a vector multiplication except that all elements of a sequence of indices must be enumerated in a sum of products. Definition 28: Wick Product of the Integrations of two functions f , g ∈ Lφ2 (ℜ)

( ∫ fdB H )◊( ∫ gdB H ) = ( ∫ fdB H )( ∫ gdB H ) − ( f , g ) φ ∈ ( S ) *H ℜ

ℜ

ℜ

ℜ

Proof: Obtained through algebraic manipulations after redefinition using Lemma 22 and Definition 27. Definition 29: Wick Power of X ∈ ( S ) *H

X ◊n =  X◊ X◊ ⋅ ⋅ ⋅ ◊X ∈ ( S ) *H n

Definition 30: Wick Operation of a function g : C → C for X = ( X 1 , X 2 ,..., X n ) ∈ (( S ) *H ) n If the function g has a power expansion such that g ( z1 , z 2 ,..., z n ) =

cα z α , then ∑ α

g ◊ ( X 1 , X 2 ,..., X n ) = ∑ cα X ◊α defines the function as a Wick Operation if the power series α

converges in ( S )

* H

In particular, Definition 31: Wick Exponential of X ∈ ( S ) *H

X ◊n If the series converges in ( S ) , then exp ( X ) = ∑ n = 0 n! * H

◊

∞

Note the Wick exponential is defined in tantamount to the power series expansion of an ordinary exponential, except Wick power is used instead. Definition 32: Wick Exponential of < w, f > , f ∈ Lφ2 (ℜ)

1 exp ◊ (< w, f >) = exp(< w, f > − | f |φ2 ) = ε ( f ) = exp ◊ ( ∫ f (t )dB H (t , w)) ℜ 2 Compare with Definition 10 and Theorem 17. The former offers clue to the usage of Theorem 13, whereas the latter facilitates calculation involving Geometric fBM. 3.4 Fractional Wick-Ito Calculus

9

Definition 33: Fractional Wick-Ito Integral of a Function Y : ℜ → ( S ) *H If Y (t ) ◊WH (t ) is integrable in ( S ) *H , then the fractional Wick-Ito Integral is defined as

∫ Y (t )dB ℜ

H

(t ) = ∫ Y (t )◊WH (t )dt ℜ

t

Lemma 34:

∫B

H

( s )dBH ( s ) =

0

1 2 1 BH (t ) − t 2 H 2 2

Proof: see [Holden&etal] for general methodology Definition 35: Given a function F : S ' (ℜ) → ℜ and

Dγ F ( w) = lim

F ( w + εγ ) − F ( w)

ε

ε →0

γ ∈ S ' (ℜ) . If

exists in ( S ) *H , then F has the directional derivative in the

direction γ , Dγ F . Definition 36: If, for F : S ' (ℜ) → ℜ , there exists a mapping ψ : ℜ → ( S ) *H such that

ψ (t )γ (t ) = ψ (t , w)γ (t ) is integrable in ( S ) *H and Dγ F ( w) = ∫ ψ (t , w)γ (t )dt , ∀γ ∈ L2 (ℜ) , ℜ

then Dt F ( w) =

dF (t , w) = ψ (t , w) is the stochastic gradient (or Hida / Malliavin derivative) of dw

F at t. The foundation of the derivative of fractional Wick-Ito Calculus is based upon Malliavin calculus for BM. It is used in the BM setting when derivatives cannot be well defined. [Oksendal] provides a good introduction for the subject, whereas [Ocone&Karatzas] gives an example in Finance. Definition 37: L1φ, 2 (ℜ) is the completion of the set of all ℑt( H ) − adapted processes

f (t ) = f (t , w) such that || f || 2L1, 2 (ℜ ) = E µφ [ ∫∫ f ( s ) f (t )φ ( s, t )dsdt ] + E µφ [( ∫ Dsφ f ( s )ds ) 2 ] < ∞ , where φ

ℜ×ℜ

ℜ

Dsφ F = ∫ φ ( s, t ) Dt Fdt ℜ

Compare with Definition 6. Note, L1φ, 2 (ℜ) is both stronger and weaker than Lφ2 (ℜ) ; the former because of the extra derivative, and the latter due to boundedness in expectation. Lemma 38: Fractional Wick-Ito Isometry If f (t ) = f (t , w) is an ℑt( H ) − adapted process, then E[(

∫

ℜ

f (t , w)dBH (t )) 2 ] =|| f || 2L1, 2 ( ℜ ) . φ

Proof: see [Dasgupta&Kallianpur, Theorem 3.7]

10

Note the difference in regards to the standard Ito isometry. The standard Ito’s isometry simply pushes in the power into the integral, whereas the fBM version requires a double integral to account for longmemory and the Malliavin derivative to offset directional change in the limit. Lemma 39: Expectation of an Integral of a function f ∈ L1φ, 2 (ℜ)

E[ ∫ f (t , w)dBH (t )] = 0 ℜ

Note that this is analogous to the BM case since both are defined with a linear summation of zeromean Gaussian processes. Lemma 40: Geometric Fractional Brownian Motion The fBM SDE, dX (t ) = µX (t )dt + σX (t )dB H (t ), X (0) = x > 0 , can be written as

dX (t ) = µX (t )dt + σX (t )◊WH (t ) in ( S ) *H and possesses the solution dt 1 X (t ) = x exp(σB H (t ) + µt − σ 2 t 2 H ) 2 Proof: Wick Calculus gives X (t ) = x exp ◊ ( µt + σB H (t )) . Now, set < w, f >= µt + σB H (t ) to obtain the result using Definition 32 and Lemma 34. Theorem 41: Fractional Ito Rule If f ∈ C 2 (ℜ × ℜ) and dX (t ) = µ (t , w)dt + σ (t , w)dB H (t ) µ , σ ∈ L1φ, 2 , then

f (t , X (t )) − f (0, X (0)) t ∂f t ∂f ∂f ∂2 f ( s, X ( s))ds + ∫ ( ( s, X ( s)) µ ( s) + 2 ( s, X ( s ))σ ( s ) Dsφ X ( s ))ds + ∫ ( s, X ( s ))σ ( s )dBH ( s ) 0 ∂s 0 ∂x 0 ∂x ∂x

=∫

t

Compare with [Karatzas&Shreve, Theorem 3.3.3]. The difference is apparent. A coefficient of replaced by the Malliavin derivative Dsφ X (s ) for

1 is 2

∂2 f . ∂x 2

Theorem 42: Fractional Girsanov Formula If γ is a continuous function such that Supp γ ⊂ [0, T ] , T > 0 and K a function with

Supp K ⊂ [0, T ] so that γ (t ) = ∫ K ( s )φ ( s, t )ds, 0 ≤ t ≤ T , we can define a probability measure ℜ

µ φ ,γ on σ-algebra ℑT( H ) generated by {BH ( s );0 ≤ s ≤ T } , such that

dµ φ ,γ dµ φ

= exp ◊ {− < w, K >} ,

t

∫

and Bˆ H (t ) = BH (t ) + γ ( s ) ds, 0 ≤ t ≤ T is a fBM under µ φ ,γ . 0

11

Proof: see [Hu&Oksendal, Theorem 3.18] Lemma 43: Wick Products of Different White Noise Spaces For F , G ∈ ( S ) *H , let ◊ P be the Wick product with respect to µ φ , ◊ Q the Wick product of µ φ ,γ , then F◊ P G = F◊ Q G . 3.5 Quasi-conditional Expectation The conditional expectation of a fBM process, E ( B H (t ) | ℑ (sH ) ) , is extremely difficult to compute due to correlation with past. [Gripenberg&Norros] gives such computations. Hence, a system of quasi-conditional expectations is developed for fBM. A new space is introduced to provide regularity conditions. Lemma 44: (i)

If ψ ( w) =

∞

∑∫ n =0

ℜn

f n dB H⊗n (t ) ∈ L2 ( µ φ ), f n ∈ Lˆφ2 (ℜ n ) and

∞

|| ψ ||℘2 K = ∑ n!|| f n || 2L2 (ℜ n ) e 2 kn < ∞ , then ψ ∈℘k = ℘k ( µ φ ) and φ

n =0

∞

℘ = ℘( µ φ ) = k =1℘k ( µ φ ) (ii)

If the formal expansion G =

∞

∑∫

ℜn

n =0

g n dBH⊗n (t ), g n ∈ Lˆφ2 (ℜ n ) is such that

∞

|| G ||℘2 − q = ∑ n!|| g n || 2L2 (ℜ n ) e − 2 qn < ∞ , then G ∈℘− q = ℘− q ( µ φ ) and φ

n =0

∞

℘* = ℘* ( µ φ ) = q =1℘− q ( µ φ ) (iii)

∞

∑ n!( g

The action of G ∈℘* on ψ ∈℘ is as following 〈〈 G ,ψ 〉〉 =

n =0

Definition 45: The fractional quasi-conditional expectation of G =

∞

∑∫ n =0

~

with respect to ℑt( H ) is defined as E µφ [G | ℑt

(H )

ℜn

n

, f n ) L2 ( ℜ n ) φ

g n ( s )dBH⊗n ( s ) ∈℘*

∞

] = ∑ ∫ n g n ( s ) χ {0≤ s ≤t } ( s )dB H⊗n ( s ) n =0

ℜ

Definition 46: An ℑ (sH ) − adapted stochastic process X (t , w) is defined as a quasi-martingale if

~ X (t ) ∈℘* and E µφ [ X (t ) | ℑ(sH ) ] = X ( s ) .

12

Analogous to the definition of a regular martingale [Karatzas&Shreve, Definition 1.3.1] , the quasimartingale has stricter boundedness condition given by Lemma 44 and a new type of conditional expectation given by Definition 45. Lemma 47:

~

a) If F ∈℘* , then E µφ [ F | ℑt

] ∈℘*

(H )

~

b) If F , G ∈℘* , then E µφ [ F◊G | ℑt

(H )

~

c) If F ∈ L2 ( µ φ ) , E µφ [ F | ℑt

(H )

~ ~ ] = E µφ [ F | ℑt( H ) ]◊E µφ [G | ℑt( H ) ]

] = F if and only if F is ℑt( H ) − measurable

Lemma 48: a)

B H (t ) is a quasi-martingale

b)

ε (t ) = exp( ∫ f (t )dBH −

c)

X (t ) = ∫ f ( s, w)dBH ( s ), f ∈ L1φ, 2 is a quasi-martingale

t

0

1 | fχ {[ 0,t ]} |φ2 ), f ∈ Lφ2 (ℜ) is a quasi-martingale 2

t

0

Theorem 49: Fractional Clark-Ocone Theorem

~

a) If G ( w) ∈℘* is ℑT( H ) − measurable , then Dt G ∈℘* , E µφ [ Dt G | ℑT ] ∈℘* , (H )

~ E µφ [ Dt G | ℑT( H ) ]◊WH (t ) integrable in ( S ) *H , T ~ G ( w) = E µφ [G ] + ∫ E µφ [ Dt G | ℑT( H ) ]◊WH (t )dt 0

b) If G ( w) ∈ Lφ ( µ φ ) is ℑT( H ) − measurable , then 1, 2

~

ψ (t , w) = E µφ [ Dt G | ℑT( H ) ] ∈ f ∈ L1φ, 2 (ℜ) , T ~ G ( w) = E µφ [G ] + ∫ E µφ [ Dt G | ℑT( H ) ]dBH (t ) 0

Proof: See [Aase&etal, Theorem 3.15] and [Aase&etal, Theorem 3.11]

~

Lemma 50: E µφ [e

λBH (T )

| ℑt( H ) ] = exp(λBH (t ) +

λ2

(T 2 H − t 2 H ))

2 Proof: Starting with dX (t ) = λX (t )dB H (t ), X (0) = 1 , Lemma 40 gives the solution of X (t ) ,

whereas Lemma 48 (c) indicates X (t ) as a quasi-martingale. The solution is apparent. Lemma 51: If E ( f ( B H (T ))) < ∞ , then ∀t ≤ T ,

~ E µφ [ f ( B H (T )) | ℑt( H ) ] = ∫

ℜ

1 2π (T 2 H − t 2 H )

exp(−

( x − B H (t )) 2 ) f ( x)dx 2(T 2 H − t 2 H )

Proof: Let,

13

f ( B H (T )) =

1 2π

∫

ℜ

e iBH (T )ξ fˆ (ξ ) dξ .

Using Lemma 50, we obtain,

~ E µφ [ f ( B H (T )) | ℑt( H ) ] 1 ~ = E µφ [ ∫ e iBH (T )ξ fˆ (ξ )dξ | ℑt( H ) ] 2π ℜ 1 ~ = E µφ [e iBH (T )ξ | ℑt( H ) ] fˆ (ξ )dξ ∫ 2π ℜ ξ 2 2H 2H ˆ 1 = exp(iξB H (t ) − (T − t )) f (ξ )dξ 2π ∫ℜ 2 This is an inverse Fourier transform of e as the Fourier transform of z ( x) =

−

ξ2 2

(T 2 H − t 2 H )

and fˆ (ξ ) . A transformation gives the former

1 2π (T 2 H − t 2 H )

exp(−

x2 ) . Then, 2(T 2 H − t 2 H )

~ E µφ [ f ( B H (T )) | ℑt( H ) ] = ∫ z ( B H (t ) − y ) f ( y )dy and the result follows apparently. ℜ

Lemma 52: If E ( f ( B H (T ))) < ∞ and

~ E µφ ,γ [ f ( BH (t )) | ℑt( H ) ] =

1

ε (−θχ [ 0,t ] )

ε (−θχ [ 0,t ] ) = exp(−θBH (t ) −

θ2 2

t 2 H ) , then

~ E µφ [ f ( BH (T ))ε (−θχ [ 0,T ] ) | ℑt( H ) ], ∀t ≤ T , where µ φ ,γ is

a new probability measure defined as in Theorem 42. Proof: Similar to Lemma 51. For detail see [Necula, Theorem 3.4]. 4. Applications of Wick-Ito Stochastic Calculus in Finance (The following summarizes results from [Hu&Oksendal] and [Necula].) Assume the market compose of two instruments. (1) Money with dM (t ) = rM (t )dt , 0 ≤ t ≤ T , M (0) = 1 , where r is the riskless interest rate (2) Stock with dS (t ) = µS (t )dt + σS (t ) dB H (t ), S (0) = s > 0 , where µ and σ ≠ 0 constants 4.1 No Arbitrage and Completeness of Market Lemma 53: S (t ) = s exp(σB H (t ) + µt −

Definition 54: Portfolio

1 2 2H σ t ), t ≥ 0 2

θ (t )

14

θ (t ) = (u (t ), v(t )) is an ℑt( H ) − adapted process with value process V θ (t , w) = u (t ) M (t ) + v(t )◊S (t ) Definition 55: Self-financing A portfolio is self-financing if

dV θ (t , w) = u (t )dM (t ) + µv(t )◊S (t )dt + σv(t )◊S (t )dB H (t ),0 ≤ t ≤ T Compare with [Karatzas&Shreve2, p. 6]. Lemma 56: A self-financing portfolio has value process dV θ (t ) = rV θ (t )dt + σv(t ) ◊S (t )dBˆ H (t ) ,

µ −r where Bˆ H (t ) = dt + dB H (t ) is a fBM σ

Proof: dV θ (t ) and Bˆ H (t ) are obtained by assuming self-financing portfolio and substituting

V θ (t ) of Definition 54 into Definition 55. Bˆ H (t ) is proven to be a fBM using Theorem 36 by defining it’s probability measure µˆ φ with dµˆ φ ( w) = exp(− and

∫

T

0

K (T , s )φ (t , s )ds =

∫

T

0

K ( s )dBH ( s ) −

1 | K |φ2 )dµ φ ( w) 2

µ −r ,0 ≤ t ≤ T . For an explicit form for K (T , s ) see [Hu&Oksendal, σ

Appendix] This is equivalent to the BM version of pricing under change of measure, except a Wick product is introduced in dV θ (t ) .

θ (t ) 1, 2 1, 2 is self-financing and v◊S ∈ Lˆφ (ℜ) , where Lˆφ (ℜ) is defined as Definition 37 with probability

Definition 57: An Admissible Portfolio

measure µˆ φ . Apparently, this imposes conditions on the wealth process of fBM to make it well behaved. Compare with [Karatzas&Shreve2, Definition 3.3.2]. Definition 58: A portfolio

θ (t ) is an arbitrage for a market ( M (t ), S (t )) if there exists for a value

process V θ (t ), t ∈ [0, T ] , such that V θ (0) ≤ 0 or V θ (T ,⋅) ≥ 0 , and µ φ ({w;V θ (T , w) > 0}) > 0 . This is equivalent to the BM case. Compare with [Karatzas&Shreve2, Definition 1.4.1]. Lemma 59: No Arbitrage exists in the fBM market ( M (t ), S (t ))

15

Proof: Consult Lemma 56 to obtain a stochastic integral for e − ρT V θ (t ) . Alleviate the Wick product using Lemma 43. Lemma 39 then yields E µˆφ (e Definition 60: If there exists v ∈ ℜ and

− ρT

V θ (T )) = V θ (0) = 0 .

θ (t ) such that F ( w) = V θ ,v (T , w), µ φ − a.s. for every

ℑT( H ) − measurable bounded random variable F (w) , then the market ( M (t ), S (t )) is complete. Lemma 61: The fBM market ( M (t ), S (t )) is complete. Proof: Definition 60 implies that we must have e − ρT F ( w) = v +

∫

T

0

e − ρt σv(t )◊S (t )dBˆ H (t ) .

Applying Theorem 49, we obtain that T ~ e − ρT F ( w) = E µˆφ (e − ρT F ) + ∫ E µˆφ (e − ρT Dˆ t F | ℑt( H ) )dBˆ H (t ) . A comparison of the herein 0

θ

obtained equations yields that unique v ∈ ℜ and θ (t ) exists with v = V (0) = E µˆφ ( F ) . 4.2 Fractional Black-Scholes Pricing Lemma 62: The fBM market ( M (t ), S (t )) has the replicating portfolio instrument C, where v(t ) = e

− ρ (T −t )

θ (t ) = (u (t ), v(t )) for an

~

σ −1 X ◊ ( −1) (t )◊E µφ [ Dt G | ℑT( H ) ] and

V θ (t ) − v(t )◊S (t ) . u (t ) = M (t ) Theorem 63: fractional Black-Scholes formula The European call price at t ∈ [0, T ] with strike price K and maturity T is given by

σ 2 2H 2H S (t ) ln( ) + r (T − t ) + (T − t ) 2 K C (t , S (t )) = S (t ) N (d1 ) − Ke − r (T −t ) N (d 2 ) , where d1 = σ T 2H − t 2H and d 2 =

ln(

σ 2 2H 2H S (t ) ) + r (T − t ) − (T − t ) 2 K . σ T 2H − t 2H

Proof: It is apparent that

~ ~ ~ C (t , S (t )) = E µφ [e − r (T −t ) ( S (T ) − K ) + | ℑt( H ) ] = e − r (T −t ) E µφ [ S (T ) χ {S (t ) > K } | ℑt( H ) ] − Ke − r (T −t ) E µφ [ χ {S (t ) > K } | ℑt( H ) ] ~ ~ (H ) Now, E µφ [ χ {S ( t ) > K } | ℑt( H ) ] = E µφ [ χ ] = N (d 2 ) by applying 1 2 2 H ( B H (T )) | ℑ t K {x>

ln(

S (T )

) − rT + σ T 2

σ

}

Lemma 51.

16

Theorem 42 (Fractional Girsanov Formula) guarantees that there exists a probability measure µ φ ,γ such that Bˆ H (t ) = B H (t ) − σt 2 H , 0 ≤ t ≤ T is a fBM. Applying Lemma 52 with θ = −σ yields

~ E µφ ,γ [ χ {S (T ) > K } | ℑt( H ) ] = e − rT

1

ε (σχ [ 0,t ] )

~ ~ E µφ ,γ [ χ {S (T ) > K } | ℑt( H ) ] = E µφ ,γ [ χ {x>

ln( K

~ E µφ [ S (T ) χ {S (T ) > K } | ℑt( H ) ] . We also have

1 ) − rT − σ 2T 2 H S (T ) 2 }

( Bˆ H (T )) | ℑt( H ) ] = N (d1 ) . Now the result

σ

follows immediately by algebraic manipulations. The result reveals a surprising yet natural difference between fBS and the classical Black-Scholes formula. Compare with the classical Black-Scholes formula,

C (t , S (t )) = S (t ) N ( d1 ) − Ke − r (T −t ) N ( d 2 ) 1 ln( S (t ) / K ) + ( r + σ 2 )(T − t ) 2 d1 = σ T −t 1 log(S (t ) / K ) + (r − σ 2 )(T − t ) 2 d2 = σ T −t Clearly, the difference lies with parameters (T − t ) and (T 2 H − t 2 H ) of the classical and fractional versions, respectively. A close examination will reveal that option prices evaluated at different times t but same increment T − t will give different results with fractional BS formula, whereas the classical BS will give identical prices as long as the increment T − t is the same. Apparently, fBM’s non-Markovian property, Theorem 4, forces the derivative price to be dependent on the stock process evolution during the increment between different t ’s. Theorem 64: fractional Black-Scholes PDE A derivative with bounded payoff f ( S (T )) has price given by C (t , S (t )) that satisfies,

∂C ∂ 2C ∂C + Hσ 2 t 2 H −1 S 2 + rS − rC = 0 2 ∂t ∂S ∂S C (T , S ) = f ( S ) Proof: see [Necula, Theorem 4.3]

Compare with the classical Black-Scholes PDE

∂C 1 2 2 ∂ 2 C ∂C + σ S + rS − rC = 0 of 2 ∂t 2 ∂S ∂S

[Wilmott&etal]. One notes that the Black-Scholes PDE’s may be recovered from each other with the H−

1 2

transformation of σ ↔ 2 H σt . Apparently, the fractional Black-Scholes PDE scales the concavity of the option price with respect to the stock process by a function dependent on both t and H.

17

The Greeks with there analogies in the classical BM setting is given as the following. Theorem 65: fractional Black-Scholes Greeks classical Black-Scholes

∆=

∂C = N (d1 ) ∂S

Γ=

N ' (d1 ) ∂ 2C = 2 ∂S Sσ T − t

Θ=

SN ' (d 1 )σ ∂C =− − rK exp(−r (T − t ) N (d 2 ) ∂t 2 T −t

fractional Black-Scholes

N (d1 ) N ' (d1 ) Sσ T 2 H − t 2 H Θ=

SN ' (d1 )σ ∂C = − Ht 2 H −1 − rK exp(−r (T − t ) N (d 2 ) ∂t T 2H − t 2H

∂C S T 2 H − t 2 H N ' (d1 ) = S T − t N ' (d1 ) ∂σ ∂C K (T − t ) exp(−r (T − t )) N (d 2 ) ρ= = K (T − t ) exp(− r (T − t )) N (d 2 ) ∂r Note that d1 and d 2 are defined differently for classical and fractional versions. Refer to Theorem V =

63 and its associated remark. Proof: See [Necula, Theorem 5.4] 5. Further Developments With Section 3 and 4, the essence of fractional Stochastic Calculus in Finance is introduced. Numerous extensions and applications are possible. (This section summarizes numerous papers that I found during the end of the semester, and others that present results obtained using repetitive procedures as already included in previous sections.) 5.1 Wick-Ito Stochastic Calculus for arbitrary Hurst parameter An apparent inconvenience of the Wick-Ito Calculus developed in Section 3 is its restraint to

1 H ∈ ( ,1) . Though most markets have Hurst parameter in this range, a general tool is more 2 desirable nonetheless. Numerous papers deal with this subject. Of note is [Elliott&Hoek] and [Bender] that achieved fBM integration with arbitrary Hurst parameter in 2001 a nd 2003 respectively. [Biagini&etal] offers a recent summary and improvement in 2003. The idea starts with defining a new operator, M H , such that for f ∈ S (ℜ) ,

18

π 1 1 −1 f ( x − t ) − f ( x)  dt 1 3 (2Γ( H − 2 ) cos( 2 ( H − 2 ))) ∫ℜ −H H , 0 < < 2  |t | 2  π f (t ) 1 1 −1 1 M H f ( x) =  (2Γ( H − ) cos( ( H − ))) ∫ dt , < H < 1 3 ℜ −H 2 2 2 2  | t − x |2 1  H= f x ( )  2  1

This operator is equivalent to that of Theorem 9. It is chosen so that Mˆ H f (ξ ) =| ξ | 2

−H

fˆ (ξ ) .

This is necessary for computation of the quasi-conditional expectation. Refer to Lemma 51. This general type of Wick-Ito Stochastic Integral is constructed identically henceforth as in Section 3. [Elliott&Hoek] also yield equivalent results of Section 4. 5.2 Multidimensional Wick-Ito Stochastic Calculus Analogous to the BM case, multidimensional Wick-Ito Stochastic Calculus can be easily constructed using similar methods as of Section 3. [Biagini&etal, Section 6] provides an updated summary of the multidimensional case. The idea is exactly the same, using Wick product to define an integral and Malliavin Calculus for the derivative. [Hu&etal2] discusses solving Stochastic Differential Equations in the multidimensional setting. [Biagini&Oksendal] is an interesting paper that uses multidimensional fBM optimal control theory to solve a minimal variance-hedging problem. 5.3 Other Applications [Hu&etal] and [Hu&etal4] are important papers that provide the foundations of optimal theory and consumption for a fBM market. [Hu&etal3] studies the stop-loss-start-gain portfolio and obtains a Carr-Jarrow decomposition of the vanilla option prices into intrinsic and time values. [Brody&etal] derives a Weather derivative pricing formula based upon an application of the general Wick-Ito Stochastic Calculus of Section 5.1 on the Ornstein-Uhlenbeck process.

19

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