The aim of this paper is to develop a framework for evaluating derivatives if the underlying of the derivative contract is supposed to be driven by a fractional Brownian motion with Hurst parameter greater than 0.5. For this purpose we first prove some results regarding the quasi-conditional expectation, especially the behavior to a Girsanov transform. We obtain the risk-neutral valuation formula and the fundamental evaluation equation in the case of the fractional Black-Scholes market.fractional Brownian motion, fractional Black-Scholes market, quasiconditional expectation, mathematical finance, contingent claim

Ciprian Necula

English

Research Papers in Economics

 
1
 
 
 
 
A Framework for Derivative Pricing in the Fractional Black-Scholes Market 
 
 
 
 
 
Ciprian Necula 
DOFIN 
Academy of Economic Studies 
Bucharest, Romania 
email: ciprian.necula@fin.ase.ro 
 
 
First draft: September 12 2007 
 
 
 
Abstract: The aim of this paper is to develop a framework for evaluating 
derivatives if the underlying of the derivative contract is supposed to be driven by a 
fractional Brownian motion with Hurst parameter greater than 0.5. For this purpose 
we first prove some results regarding the quasi-conditional expectation, especially 
the behavior to a Girsanov transform. We obtain the risk-neutral valuation formula 
and the fundamental evaluation equation in the case of the fractional Black-Scholes 
market.  
 
Keywords: fractional Brownian motion, fractional Black-Scholes market, quasi-
conditional expectation, mathematical finance, contingent claim 
 
JEL Classification: C02, C60, G12, G13 
AMS Classification: 60H05, 60H10, 91B26, 91B26, 91B70 
 
 
  
2
 
 
1. INTRODUCTION 
 
The fractional Brownian motion (fBm) with Hurst parameter H 
( 1 0 < < H ) is the continuous Gaussian process  ( ) { }  ∈ t t BH , ,  ( ) 0 = t BH  with 
mean  ( ) [] 0 = t B E H  and whose covariance is given by: 
              ( ) () () [] { }
H H H
H H H s t s t t B t B E s t C
2 2 2
  
2
1
, − − + = =  
If 
2
1
= H  then  ( ) t BH  coincides with the standard Brownian motion  () t B . 
The fractional Brownian motion is a self-similar process meaning that for 
any  0 > α   () t BH α  has the same law as  ( ) t BH
H α . 
The parameter H  determines the sign of the covariance of the future and 
past increments. This covariance is positive when 
2
1
> H , zero when 
2
1
= H  (i.e. 
the classical Brownian motion ) and negative when 
2
1
< H . 
Another property of the fractional Brownian motion is that for 
2
1
> H  it 
has long-range dependence in the sense that if we put 
 
                           () ( ) ( ) ( ) ( ) n B n B B Cov n r H H H − + = 1 , 1  
then 
                                               () ∞ = ∑
∞
=1 n
n r  
The self-similarity and long-range dependence properties make the 
fractional Brownian motion a suitable tool in different applications like 
mathematical finance. Since for 
2
1
≠ H  the fractional Brownian motion is neither 
a Markov process, nor a semimartingale (see for exemple Rogers, 1997), we cannot 
use the usual stochastic calculus to analyse it. Worse still, after a pathwise 
integration theory for fractional Brownian motion was developed by Lin (1995) 
and Decreusefond and Ustunel (1999), it was proven by Rogers (1997) that the 
market mathematical models driven by  ( ) t BH  could have arbitrage. The fractional 
Brownian motion was no longer considered fit for mathematical modeling in 
finance. However after the development by Duncan, Hu and Pasik-Duncan (2000) 
and Hu and Oksendal (2003) of a new kind of integral based on the Wick product  
3
called the fractional Ito integral, it was proved in Hu and Oksendal (2003) that the 
corresponding Ito type fractional Black-Schools market has no arbitrage. 
Equivalent definitions of the fractional Ito integral were introduced by Alos, Mazet 
and Nualart (2000), Perez-Abreau and Tudor (2002) and Bender (2002). Hu and 
Oksendal (2003) introduced the concept of quasi-conditional expectation and 
quasi-martingales. In the same paper a formula for the price of a European option 
at  0 = t  is derived.  
The aim of this paper is to build a framework for evaluating derivatives if 
the underlying of the derivative contract is supposed to be driven by a fractional 
Brownian motion with Hurst parameter greater than 0.5. 
This paper is organized as follows: in the first section we prove some 
results regarding the quasi-conditional expectation, especially the behaviour to a 
Girsanov transform. In the third section we apply these results to obtain the risk-
neutral valuation formula and the fundamental evaluation equation in the case of 
the fractional Black-Scholes market. The final section concludes. 
 
 
2. SOME RESULTS REGARDING THE QUASI-CONDITIONAL 
EXPECTATION 
 
Let  () P , ,K Ω  a probability field such that  ( ) ω , t BH  is a fractional 
Brownian motion with respect to P ,  ( ) ( ) t s s BH
H
t ≤ = , : B F  and  [ ] t E
~
 the quasi-
conditional expectation with respect to 
H
t F  (definition 4.9 in Hu and Oksendal, 
2003). 
 
THEOREM 2.1. For every  T t < < 0  and   ∈ λ  we have that 
                                           
() []
() ⎟
⎠
⎞ ⎜
⎝
⎛ − +
=
H t H T t H B T H B
t e e E
2 2
2
2
~
λ
λ λ
 
 
Proof: 
 
Consider the fractional differential equation:  
 
                              ( ) ( ) ( ) ( ) 1 0      , = = X t dB t X t dX H λ  
 
The solution of this equation is (Hu and Oksendal, 2003): 
 
                             () () ⎟
⎠
⎞
⎜
⎝
⎛ − =
H
H t t B t X
2 2
2
1
exp λ λ                                         
Since  
4
                                  () () () ∫ =
t
H s dB s X t X
0
λ  
we know that  () t X  is a quasi-martingale (Hu and Oksendal, 2003). So it 
follows that: 
 
                                            ( ) [ ] ( ) t X T X Et =
~
 
or 
                                  
() []
() ⎟
⎠
⎞ ⎜
⎝
⎛ − +
=
H t H T t H B T H B
t e e E
2 2
2
2
~
λ
λ λ
                                           
 
q.e.d. 
 
THEOREM 2.2. Let    → : f  be a function such that 
() () [] ∞ < T B f E H . Then for every  T t ≤   
() () []
()
( ) ( )
() () dx x f
t T
t B x
t T
T B f E H H
H
H H H t ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
−
−
−
=∫ 2 2
2
2 2 2
exp  
2
1 ~
 π
 . 
 
Proof: 
 
Let  f ˆ  be the Fourier transform of  f : 
 
                                           ( ) ( ) ∫
− =

dx x f e f
ixξ ξ ˆ  
Then  f  is the inverse Fourier transform of  f ˆ : 
 
                                           () () ∫ =

ξ ξ
π
ξ d f e x f
ix ˆ
2
1
 
We have that: 
                              () ()
() () ∫ =

ξ ξ
π
ξ d f e T B f
T H iB
H ˆ
2
1
  
 
It follows that: 
                  () () []
() () ⎥
⎦
⎤
⎢
⎣
⎡
= ∫

ξ ξ
π
ξ d f e E T B f E
T H B i
t H t ˆ
2
1 ~ ~  
  
5
                                        
() [ ] () ∫ =

ξ ξ
π
ξ d f e E
T H B i
t ˆ ~
2
1  
 
                                       
() () ∫
⎟
⎠
⎞ ⎜
⎝
⎛ − −
=

ξ ξ
π
ξ
ξ
d f e
H t H T t H B i ˆ
2
1
2 2
2
2
 
 
                                        ( ) ( ) t B h H =                                                             
 
where  h  is the inverse Fourier transform of the product between 
⎟
⎠
⎞ ⎜
⎝
⎛ − − H t H T
e
2 2
2
2 ξ
 and  () ξ f ˆ . 
 
But the first function is the Fourier transform of  
 
                             ()
() () ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
−
−
= H H H H T t t T
x
t T
x n 2 2
2
2 2 , 2
exp
2
1
π
           
 
Using the fact that the Fourier transform of a convolution is the product of 
the Fourier transform of the two functions it follows that 
 
                                     ( ) ( ) ( ) ( ) ( ) ∫ − =

dy y f y t B n t B h H T t H ,  
q.e.d. 
                                                                                                                      
COROLLARY 2.3.  Let  ( )  B ∈ A . Then 
             () () []
()
( ) ( )
()
dx
t T
t B x
t T
T B E H H
H
A
H H H A t ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
−
−
−
=∫ 2 2
2
2 2 2
exp  
2
1 ~
π
            
 
 
 
Let   ∈ θ . Consider the process 
                          () () () T t d H t B t t B t B
t
H
H
H
H H ≤ ≤ + = + = ∫
− 0     ,   2  
0
1 2 2 * τ τ θ θ   
Theorem 3.18 in Hu and Oksendal (2003) assures us that there is a measure 
* P  such that  () t BH
*  is a fractional Brownian motion under 
* P . 
We will denote  [] ⋅
* ~
t E  the quasi-conditional expectation with respect to 
* P . 
 
Consider  
6
                                          () () ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− − =
H
H t t B t Z
2
2
2
exp :
θ
θ                    
 
THEOREM 2.4. Let  f  be a function such that  ( ) ( ) [ ] ∞ < T B f E H . Then 
for every  T t ≤    
                               () () [] () () () ( ) [] T Z T B f E
t Z
T B f E H t H t
~ 1 ~* =  
Proof: 
 
Again we will denote by  f ˆ  the Fourier transform of  f . 
We have 
            () () ( ) []
() ( ) ()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
= ∫
− −
R
H T T H B i
t H t d f e E T Z T B f E ξ ξ
π
θ
θ ξ ˆ
2
1 ~ ~
2
2
2
 
 
                              
() ( ) [] () ∫
− −
=
R
T H B i
t
H T
d f e E e ξ ξ
π
θ ξ
θ
ˆ ~
2
1  
2
2
2
 
                              
() ( )
() ∫
⎟
⎠
⎞ ⎜
⎝
⎛ − ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+ − − + −
−
=
R
H t H T i t H B i H T
d f e e ξ ξ
π
θ
ξθ
ξ
θ ξ θ
ˆ
2
1
2 2
2
2
2
2
  2
2
2
 
                               ()
()
() ∫
⎟
⎠
⎞ ⎜
⎝
⎛ − ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− − +
=
R
H t H T i t H B i
d f e t Z ξ ξ
π
ξθ
ξ
ξ
ˆ
2
1
2 2
2
2
 
               
 
On the other hand 
 
     () () []
() () ⎥
⎦
⎤
⎢
⎣
⎡
= ∫
⎟
⎠
⎞ ⎜
⎝
⎛ −
R
H T T H B i
t H t d f e E T B f E ξ ξ
π
θ ξ ˆ
2
1 ~ ~
2   *   * *  
                                        
() () ∫
−
⎥ ⎦
⎤
⎢ ⎣
⎡ =
R
H T i T H B i
t d f e e E ξ ξ
π
θ ξ ξ ˆ ~
2
1 2    
*   *  
                                        
() () ∫
− ⎟
⎠
⎞ ⎜
⎝
⎛ − −
=
R
H T i
H t H T t H B i
d f e e ξ ξ
π
θ ξ
ξ
ξ ˆ
2
1 2    
2 2
2
2
*  
 
                                       
()
() ∫
− ⎟
⎠
⎞ ⎜
⎝
⎛ − − ⎟
⎠
⎞ ⎜
⎝
⎛ +
=
R
H T i
H t H T H t t H B i
d f e e ξ ξ
π
θ ξ
ξ
θ ξ ˆ
2
1 2    
2 2
2
2
2    
  
7
                                       
()
() ∫
⎟
⎠
⎞ ⎜
⎝
⎛ − ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− − +
=
R
H t H T i t H B i
d f e ξ ξ
π
ξθ
ξ
ξ
ˆ
2
1
2 2
2
2
 
                      
 
The result follows immediately. 
q.e.d. 
 
 
3. RISK NEUTRAL VALUATION IN THE FRACTIONAL BLACK-
SCHOLES MARKET 
 
Consider the fractional Black-Scholes market consisting of two investment 
possibilities: 
1.  a money market account: 
                              ( ) ( ) ( ) T t M dt t rM t dM ≤ ≤ = = 0     , 1 0      ,                       
where r  represent the constant riskless interest rate. 
 
2.  a stock whose price satisfies the equation: 
               () ( ) ( ) ( ) ( ) T t S S t B d t S dt t S t dS H ≤ ≤ > = + = 0    , 0 0      ,     σ μ              
where  0 , ≠ σ μ  are constants and  ( ) t BH  is a fractional Brownian motion 
with respect to the market measure. 
 
In Hu and Oksendal (2003) it was shown that this market does not have 
arbitrage and is complete, the same properties as the classical Black-Scholes model 
based on the Brownian motion.  
Under the risk-neutral measure (denoted P ) we have that: 
                 () ( ) ( ) ( ) ( ) T t S S t dB t S dt t rS t dS H ≤ ≤ > = + = 0    , 0 0      , σ  
where   () t BH  is a fractional Brownian motion with respect to P . 
The solution of this stochastic differential equation is (Hu and Oksendal, 
2003): 
                              () () ⎟
⎠
⎞
⎜
⎝
⎛ − + =
H
H t t r t B S t S
2 2
2
1
    exp σ σ  
We will denote by  [] ⋅ t E
~
 the quasi-conditional expectation with respect to 
the risk-neutral measure. 
 
THEOREM 3.1. (fractional risk-neutral evaluation) 
The price at every  [ ] T t , 0 ∈  of a bounded 
H
T F - measurable contingent 
claim  () P L F
2 ∈  is given by  
8
                                                  ( ) ( ) [ ] F E e t F t
t T r ~ − − =                                      
 
Proof: 
 
Since the market is complete there is a replicating portfolio of the claim 
() () () t s t m ,  whose value is: 
                                         ( ) ( ) ( ) ( ) ( ) t S t s t M t m t F + =  
and  
                                                      ( ) F T F =  
We have that 
                         () ( ) ( ) ( ) ( ) t dS t s t dM t m t dF + =  
                                   ( ) ( ) ( ) ( ) t dB t S t s dt t rF H   σ + =  
 
By multiplying with 
rt e
−  and integrating it follows that 
  () () ()() () T t dB S s e F t F e
t
H
r rt ≤ ≤ + = ∫
− − 0    ,   0
0
τ τ τ σ
τ  (3.1) 
By the fractional Clark-Ocone theorem (theorem 4.15 in Hu and Oksendal, 
2003) we have that 
                                 [] []() ∫
− − − + =
T
H
rT rT rT dB F D E e F e E F e
0
~
τ τ τ                        
where  F Dτ  is the Malliavin derivative of F (definition 4.3 in Hu and 
Oksendal, 2003). 
From the completeness of the market we get 
 
                               [ ] ( ) ( ) ( ) T S s e F D E
T r ≤ ≤ =
− τ τ τ σ
τ
τ τ 0    ,  
~
                         
So we have that 
                           [] ()() () ∫
− − − + =
T
H
r rT rT dB S s e F e E F e
0
  τ τ τ σ
τ  
It follows that 
                       [] [] ()() () ⎥
⎦
⎤
⎢
⎣
⎡
+ = ∫
− − −
T
H
r
t
rT rT
t dB S s e E F e E F e E
0
 
~ ~
τ τ τ σ
τ                   
Using the fact that  ()() () ∫
−
t
H
r dB S s e
0
  τ τ τ σ
τ  is a quasi-martingale we have:  
9
  [] [] ()() () ∫
− − − + =
t
H
r rT rT
t dB S s e F e E F e E
0
 
~
τ τ τ σ
τ  (3.2) 
From 3.1 and 3.2 we have that 
                                                ( ) ( ) [ ] F E e t F t
t T r ~ − − =  
q.e.d. 
 
THEOREM 3.2 (fractional fundamental evaluation equation) 
The price of a derivative on the stock price with a bounded payoff 
() () T S f  is given by  ( ) () t S t F , , where  ( ) S t F ,  is the solution of the PDE: 
                       0 2
2
2 1 2 2 = −
∂
∂
+
∂
∂
+
∂
∂ − rF
S
F
rS
S
F
S t H
t
F H σ  
                      ( ) S f S T F = ) , (                                                                      
 
Proof: 
 
From Theorem 3.1 and Theorem 2.2 it follows that the price at a moment t 
of the derivative with payoff  ( ) ( ) T S f  is a function of t and  ( ) t S . 
As in the classical Black-Scholes model (Black and Scholes, 1973) we 
consider a portfolio that contains a derivative and  Δ −  stock. 
The value of this portfolio is 
( ) ( ) ( ) ( ) t S t S t F t Δ Π − = ,                                    
Under the market measure P  using the fractional Ito formula (theorem 4.5 
in Duncan, Hu and Pasik-Duncan, 2000) and the fact that 
                      () () () ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− + =
H
H u B D S S D
2
2
2
τ
σ
μτ τ σ τ τ τ  
                                  ( ) ( ) ( ) τ σ τ H u B D S =  
                                  ( ) [] ( ) u S τ τ σ , 0   =  
and 
                            () () ( ) () ∫
− = =
τ
φ
τ τ τ σ τ φ τ σ τ
0
1 2 ,
H HS du u S S D  
we get that  
 
           () () () ( ) t dS t S t dF t d Δ Π − = ,                   
10
                    
() t B d S
S
F
S dt S
S
F
S
S
F
S t H
t
F
H
H ⎟
⎠
⎞
⎜
⎝
⎛ −
∂
∂
+ ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
∂
∂
+
∂
∂
+
∂
∂
=
− σ Δ σ μ Δ μ σ     2
2
2 1 2 2  
 
 
We want this portfolio to be riskless. So 
 
                                    
S
F
∂
∂
= Δ          and         ( ) ( )dt t r t d Π = Π                        
 
It follows that the evaluation equation is given by: 
 
                                       0 2
2
2 1 2 2 = −
∂
∂
+
∂
∂
+
∂
∂ − rF
S
F
S t H
S
F
rS
t
F H σ  
q.e.d. 
 
   
      
4. CONCLUSION 
 
In this paper it was developed a framework for evaluating derivatives if the 
underlying of the derivative contract is supposed to be driven by a fractional 
Brownian motion with Hurst parameter greater than 0.5.  
We proved that in the fractional Black-Scholes market one can use the 
risk-neutral evaluation methodology but by using the quasi-conditional 
expectation. Thus, in this context, the price of a contingent claim is the quasi-
conditional expectation of the present value of the future cash-flows generated by 
this financial product. 
We also obtained for the fractional Black-Scholes market the fundamental 
evaluation equation of a contingent claim. As in the classical Black-Scholes model 
the fundamental equation does not depend on the expected return (μ ), but on the 
riskless interest rate (r ).  
The risk-neutral evaluation methodology or the fundamental evaluation 
equation can be used to price a large class of derivatives in the context of fractional 
Black-Scholes market. For example, the price of an European call option with 
strike price K  cab be obtained by solving the PDE in Theorem 3.3 with the 
boundary condition  () ( ) 0 , max , K S S T F − = . 
                      
 
 
  
11
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A Framework for Derivative Pricing in the Fractional Black-Scholes Market

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