The purpose of this paper is to obtain the price of the barrier options in a fractional Brownian motion environment in the special case of zero interest rate. As a consequence we derive a reflection principle for the fractional Brownian motion.fractional Brownian motion, fractional Black-Scholes market, quasiconditional expectation

Cipian Necula

English

Research Papers in Economics

  1
 
 
 
Barrier Options and a Reflection Principle of 
the Fractional Brownian Motion  
 
 
 
Ciprian Necula 
 
 
DOFIN 
Academy of Economic Studies 
Bucharest, Romania 
 
 
Email: ciprian.necula@fin.ase.ro 
    cipnec@yahoo.com 
 
 
 
First draft: September 26, 2003 
 
 
 
Abstract 
 
 
The purpose of this paper is to obtain the price of the barrier options in a 
fractional Brownian motion environment in the special case of zero interest rate. As a 
consequence we derive a reflection principle for the fractional Brownian motion. 
 
 
 
 
 
 
 
 
 
 
   2
1. Introduction 
 
If  1 0 < < H  the fractional Brownian motion (fBm) with Hurst parameter 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 constant 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  and negative 
when 
2
1
< H . Another property of the fractional Brownian motion is that for 
2
1
> H  
it has long-range dependence. 
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, we cannot use the usual stochastic calculus to analyze it. Worse still 
after a pathwise integration theory for fractional Brownian motion was developed 
(Lin (1995), Decreusefond and Ustunel (1999)) it was proven that the market 
mathematical models driven by  () t BH  could have arbitrage (Rogers (1997)). The 
fractional Brownian motion was no longer considered fit for mathematical modeling 
in finance. However after the development of a new kind of integral based on the 
Wick product (Duncan, Hu and Pasik-Duncan (2000), Hu and Oksendal (2000)) 
called fractional Ito integral, it was proved (Hu and Oksendal (2000)) that the 
corresponding Ito type fractional Black-Schools market has no arbitrage. There are 
some other ways of defining the fractional Ito integral. See for example Alos, Mazet 
and Nualart (2001), Alos and Nualart (2002), Perez-Abreau and Tudor (2002) or 
Bender (2002). A review of the results concerning the fractional Ito integral can be 
found in Necula (2002) or Necula (2003). 
In the paper of Hu and Oksendal (2000) a formula for the price of a European 
option at  0 = t  is derived and the formula is extended for every  [] T t , 0 ∈  in the paper 
of Necula (2002). Bender (2003) generalized it for the case of a non constant but 
deterministic volatility, using not the quasi-conditional expectation but linear 
fractional BSDE. 
The purpose of this paper was to derive the price of barrier options in a 
fractional Brownian motion environment. But so far we only managed to obtain the 
price of barrier option for the case of zero interest rate. These results are not very 
appealing for the finance, but they can be used to obtain for the fractional Brownian 
motion something that can be called the reflection principle. 
   3
 
 
2.  A Reflection Principle for the Fractional Brownian Motion 
 
 
Consider a fractional Black-Scholes market with a money market account and 
a stock. 
Under the risk-neutral measure (P ) we have that the stock price satisfies the 
equation: 
 
           () () () () () T t S S t dB t S dt t rS t dS H ≤ ≤ > = + = 0    , 0 0      , σ                (2.1) 
 
where r  represent the constant riskless interest rate. 
 
We will denote by  [] ⋅ t E ~  the quasi-conditional expectation (Hu and Oksendal 
(2000)) with respect to the risk-neutral measure. 
 
We know that the price of a 
H
T F - measurable contingent claim F  is given by 
(Necula (2002)): 
 
                          () ()[] F E e t F t
t T r ~ − − =                                                (2.2) 
 
We consider the contingent claims: 
 
  Binary call and put with strike K :  } ) ( K T S BC > = {  ,   } ) ( K T S BP < = {   
   Gap call and put with strikeK : () } ) ( K T S T S GC > = {   ,  () } ) ( K T S T S GP < = {    
 
Lemma 2.1 
 
We have that: 
                   () () ()() 2 , , d N e K t S t BC
t T r − − =  ,  () () ()() 2 , , d N e K t S t BP
t T r − =
− −              (2.3) 
 
        () ()( ) 1 ) ( , , d N t S K t S t GC = ,  () ()( ) 1 ) ( , , d N t S K t S t GP − =  
 
 
where                 
() ()
H H
H H
t T
t T t T r
K
t S
d
2 2
2 2
2
1
2
) (
ln
−
− + − + 





=
σ
σ
        and 
                                       
() ()
H H
H H
t T
t T t T r
K
t S
d
2 2
2 2
2
2
2
) (
ln
−
− − − + 





=
σ
σ
 
 
and  () ⋅ N  is the cumulative probability of the standard normal distribution.   4
Proof: 
 
See the proof of Theorem 4.2 in Necula (2002). 
 
 
We will make the following notations: 
 
() {} L t S t L = = | inf : τ  
() ( ) t S T m
T t
S
< < =
0inf :                                                 (2.4) 
() ( ) t S T M
T t
S
< <
=
0
sup :  
 
First we will analyze the case of no drift in 2.1 (i.e  0 = r ) 
 
We know that  () ()






+ − = t B t S t S H
H σ
σ 2
2
2
exp                                      (2.5) 
 
Theorem 2.1 
 
Consider that  0 = r . 
 
1. If    L K > ,  L S >  and  L t τ <  then 
 
()
() () () 2 1 } , ) (
~
a N
L
t S
a N
L T m K T S
E S t − = 





> > {
                      (2.6) 
             
where 
()
H H
H H
t T
t T
K
t S
a
2 2
2 2
2
1
2
) (
ln
−
− − 





=
σ
σ
   and  
()
H H
H H
t T
t T
t LS
H
a
2 2
2 2
2 2
2
2 ) (
ln
−
− −  


 


=
σ
σ
 
 
2. If    L S >  and  L t τ <  then        
 
                         
()
() () () 2 1 }
~
}
~
b N
L
t S
b N
L T m
E T E S t
L
t − = 





>
=  

 

> { {   τ          (2.7)   
                  
where 
()
H H
H H
t T
t T
L
t S
b
2 2
2 2
2
1
2
) (
ln
−
− − 





=
σ
σ
   and  
()
H H
H H
t T
t T
t S
L
b
2 2
2 2
2
2
2 ) (
ln
−
− −  


 


=
σ
σ
 
 
3. If    L K < ,  L S <  and  L t τ <  then 
 
                            
()
()() () 2 1 } , ) (
~
a N
L
t S
a N
L T M K T S
E S t − − − = 





< < {
           (2.8) 
      5
4. If    L S <  and  L t τ <  then        
 
                     
()
()() () 2 1 }
~
}
~
b N
L
t S
b N
L T M
E T E S t
H
t − − − = 





<
=  

 

> { {   τ        (2.9)               
 
 
Proof 
 
1.  Consider a down-and-out binary call (DOBC) with strike price K , barrier L and 
maturity  T .  The payoff of this option is 
() } , ) ( L T m K T S
S > > {
 . The price of this 
contingent claim  () t DOBC  is nonzero if  L t τ <  and zero if  L t τ > . 
 
 Consider now a portfolio that consists in a long position of one binary call 
with strike K  and maturity T  and a short position of 
L
1
 gap puts with strike price 
K
L
2
 and maturity T . It can be seen from Lemma 2.1 that if  L t τ =  (i.e.  () L t S = ) the 
price of this portfolio is zero.  So if the barrier is hit previous to the maturity T  the 
value of this portfolio is equal to that of the option. If the barrier is not hit the 
portfolio and the option will have the same payoff at maturity since the gap put is out 
of the money ( () ()
K
L
T S L K L T S
2
, > ⇒ > > ). 
Since the fractional Black-Scholes does not have arbitrage and the down-and-
out binary call and the portfolio have the same payoff, their value will be the same for 
T t L ∧ <τ . So 
                                     () () () t GP
L
t BC t DOBC
1
− =  
 
2.  Consider a contingent claim that pays one unit if the stock price does not hit the 
barrier  L before T  (down-and-out bond). The payoff of this contingent claim is 
() } L T m
S > {
 . As in the previous case we look for a portfolio that has the same value 
as the contingent claim at  T L ∧ τ , and as a consequence of the no-arbitrage property 
of the fractional Black-Scholes they will have the same value for every  T t L ∧ <τ .  
In this case we can chose a portfolio consisting in a long position of one 
binary call with strike L and maturity T  and a short position of 
L
1
 gap puts with 
strike price L and maturity T . 
 
3.  In this case we consider an up-and-out binary put and a portfolio that consists in a 
long position of one binary put with strike K  and maturity T  and a short position of 
L
1
 gap calls with strike price 
K
L
2
 and maturity T .   6
4.  In this case we consider an up-and-out bond and a portfolio that consists in a long 
position of one binary put with strike L and maturity T  and a short position of 
L
1
 
gap calls with strike price L and maturity T . 
 
Theorem 2.2 
 
Consider that  0 = r . 
 
1. If    L K > ,  L S >  and  L t τ <  then 
 
                 ()
()
() ( ) ( ) 2 1  
} , ) (
 
~
c N L c N t S
L T m K T S
T S E S t − − = 





> > {
              (2.10) 
         
where 
()
H H
H H
t T
t T
K
t S
c
2 2
2 2
2
1
2
) (
ln
−
− + 





=
σ
σ
   and  
()
H H
H H
t T
t T
L
t KS
c
2 2
2 2
2
2
2
2
) (
ln
−
− − 





=
σ
σ
 
 
2. If    L K < ,  L S <  and  L t τ <  then 
 
                  ()
()
() ( ) ( ) 2 1  
} , ) (
 
~
c N L c N t S
L T M K T S
T S E S t − − = 





< < {
             (2.11)   
 
 
Proof 
 
1.  Consider a down-and-out call with strike price K , barrier L and maturity T .  The 
payoff of this option is  () ()
() } , ) (
  T S
L T m K T S
K S > >
−
{
 . The value of this option at 
T t L ∧ <τ  is given by: 
               ()
() () 





< <
− 





< < } , ) (
 
~
 
} , ) (
 
~
L T M K T S
E K
L T M K T S
T S E S t S t { {
   
 
One can see that this contingent claim has the same payoff as a portfolio that 
consists in a long position of one call with strike K  and maturity T  and a short 
position of 
L
K
 puts with strike price 
K
L
2
 and maturity T . Using Theorem 4.2 in 
Necula (2002), the value of this portfolio at  T t L ∧ <τ  is: 
 
                                    () ( ) ( ) ( ) () () 




 − − − − 2 1 2 1   a N
L
t S
a N K c N L c N t S    7
2.  In this case we consider an up-and-out put and a portfolio that consists in a long 
position of one put with strike K  and maturity T  and a short position of 
L
K
 calls 
with strike price 
K
L
2
 and maturity T . 
 
 
We know that there is a probability measures 
* P  such that 
() () t B t t B H
H
H + − =
2 *
2
σ
 is a fBm under 
* P .  
 
The reflection principle of the Brownian motion gives the common 
distribution of the Brownian motion and its minimum (or maximum). 
 
The next corollary gives a similar result for the fractional Brownian motion. 
 
Corollary 2.1 
 
1. If    0 ≤ y ,  y x >  and  y t τ <  then 
 
         
() ()
() ()
−
 




 




−
− + −
=








> >
H H
H H
H
B
H
t
t T
t T x t B
N
y T m x T B
E
H 2 2
2 2 *
*
2
} ,
~
*
σ
{
        
                                   () () {}
() ()
 




 




−
− + − −
− −
H H
H H
H
H
t T
t T t B x y
N y t B
2 2
2 2 *
* 2
2
exp
σ
σ        (2.12) 
 
2. If    0 ≥ y ,  y x <  and  y t τ <  then 
 
         
() ()
() ()
−
 




 




−
− − −
=








< <
H H
H H
H
B
H
t
t T
t T t B x
N
y T M x T B
E
H 2 2
2 2 *
*
2
} ,
~
*
σ
{
        
                                  () () {}
() ()
 




 




−
− − + −
− −
H H
H H
H
H
t T
t T t B y x
N y t B
2 2
2 2 *
* 2
2
exp
σ
σ       (2.13)   
 
 
The shortcoming of the result is that the quasi-conditional expectation it is 
under  P , not under the probability measure 
* P  ( () t BH
*  is a fBm under 
* P , not under 
P ).  
 
   8
Remarks 
 
 
For finance an important result would be a formula for the price of barrier 
options in the case in which the interest rate is not zero. But the extension to the case 
of non-zero drift in 2.1 seems very difficult. 
 
 
 
 
References 
 
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Stochastic calculus with respect to Gaussian processes,

Barrier Options and a Reflection Principle of the Fractional Brownian Motion

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Barrier Options and a Reflection Principle of the Fractional Brownian Motion

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