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Chaos, Solitons and Fractals 102 (2017) 245–253
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals
Nonlinear Science, and Nonequilibrium and Complex Phenomena
journal homepage: www.elsevier.com/locate/chaos
Pricing of basket options in subdiffusive fractional Black–Scholes
model
Gulnur Karipova, Marcin Magdziarz
∗
Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw,
Poland
a r t i c l e i n f o
Article history:
Received 26 January 2017
Revised 4 May 2017
Accepted 5 May 2017
Available online 10 May 2017
Keywords:
Black–Scholes model
Subdiffusion
Basket options
Stable process
a b s t r a c t
In this paper we generalize the classical multidimensional Black-Scholes model to the subdiffusive case.
In the studied model the prices of the underlying assets follow subdiffusive multidimensional geometric
Brownian motion. We derive the corresponding fractional Fokker–Plank equation, which describes the
probability density function of the asset price. We show that the considered market is arbitrage-free and
incomplete. Using the criterion of minimal relative entropy we choose the optimal martingale measure
which extends the martingale measure from used in the standard Black–Scholes model. Finally, we derive
the subdiffusive Black–Scholes formula for the fair price of basket options and use the approximation
methods to compare the classical and subdiffusive prices.
heorem 3. Let ε ≥ 0 . Let Q ε be the probability measure defined as
ε (A ) = C
∫ A
exp
{n ∑
i =1
γi W
(i ) (S α(T )) −(
ε +
1
2
n ∑
i =1
γ 2 i
)S α(T )
}dP,
(20)
here A ∈ F and C = [ E( exp { ∑ n i =1 γi W
(i ) (S α(T )) − (ε +1 2
∑ n i =1 γ
2 i ) S α(T ) } )] −1 is the normalizing constant. Then Z α( t ), t
[0, T ], is Q ε-martingale.
For the proof see the Appendix .
Two Fundamental Theorems of Asset Pricing imply that
orollary 1. The market model in which the asset prices follow the
ultidimensional subdiffusive GBM Z α( t ), has no arbitrage and is in-
omplete.
Market incompleteness means that there is no unique fair
rice of financial derivatives. Unfortunately, the incontrovertible
pproach to the best choice of the corresponding martingale mea-
ure does not exist. However the martingale measure can be cho-
en according to the criterion of minimal relative entropy, mean-
ng that the best choice of measure Q minimizes the distance from
easure P [10] .
emma 1. Let us define a probability measure Q as
(A ) =
∫ A
exp
{n ∑
i =1
γi W
(i ) (S α(T )) − 1
2
n ∑
i =1
γ 2 i S α(T )
}dP, A ∈ F .
(21)
hen the relative entropy for the measure Q is less than for the mea-
ure Q ε , ε > 0 .
roof. Clearly, Q is the special case of Q ε , defined in (20) for
= 0 and C = 1 . Thus, from Theorem 2 we have that Z α( t ) is a Q -
artingale.
The relative entropy of Q is equal to
= −∫ �
log dQ
dP dP =
1
2
E(S α(T )) n ∑
i =1
γ 2 i .
n the other hand, the relative entropy of Q ε is equal to
ε = −∫ �
log dQ ε
dP dP
= log E
(exp
{n ∑
i =1
γi W
(i ) (S α(T )) −(
ε +
1
2
n ∑
i =1
γ 2 i
)S α(T )
})
+
(
ε +
1
2
n ∑
i =1
γ 2 i
)
E(S α(T ))
= log E
(E
(exp
{n ∑
i =1
γi W
(i ) (S α(T )) − 1
2
n ∑
i =1
γ 2 i S α(T )
}
× exp {−εS α(T ) }| G t
))+
(ε +
1
2
n ∑
i =1
γ 2 i
)E(S α(T ))
= log E ( exp {−εS α(T ) } E (Y (t))) +
(ε +
1
2
n ∑
i =1
γ 2 i
)E(S α(T ))
= log E( exp {−εS α(T ) } ) +
(ε +
1
2
n ∑
i =1
γ 2 i
)E(S α(T ))
≥ 1
2
E(S α(T )) n ∑
i =1
γ 2 i = D.
hus, Q minimizes relative entropy. �
emark 1. Note that for α = 1 the measure Q defined in (21) re-
uces to the martingale measure in the standard B-S model.
Under the above arguments, further on we will find the fair
rices using the martingale measure Q . In the next theorem we
etermine the fair price of a basket option in the subdiffusive B-S
odel.
heorem 4. Let the assets prices follow Z α( t ) . Then the corresponding
air price C sub B −S
(Z 0 , K, T , σ, α) of a basket option satisfies
250 G. Karipova, M. Magdziarz / Chaos, Solitons and Fractals 102 (2017) 245–253
Fig. 3. Prices of the basket option, exercised in European call style, depending on the maturity time for different α. The parameters for the figure above are as follows:
m = n = 10 , σ ij ∈ [0, 1], Z (i ) 0
= 1 , ∀ i, j, K = 40 . The results were obtained using (22) from 10 0 0 simulated independent realizations of the random variable S α ( T ).
C
C
t
t
n
u
t
m
B
C
T
P
R
t
a
m
t
g
T
S
a
t
t
(
b
s
o
w
t
a
m
c
sub B −S (Z 0 , K, T , σ, α) = E(C B −S (Z 0 , K, S α(T ) , σ ))
=
∫ ∞
0
C B −S (Z 0 , K, x, σ ) T −αg α(x/T α) dx, (22)
where C B −S (Z 0 , K, T , σ ) is price of the basket option in the standard
multidimensional B-S model, g α( z ) stands for the PDF of S α(1) and
can be expressed using Fox function
g α(z) = H
10 11
(z| (1 −α,α)
(0 , 1)
).
Proof. The arbitrage-free rule of pricing requires that
sub B −S (Z 0 , K, T , σ, α) = E Q
((m ∑
i =1
ω i Z (i ) α (T ) − K
)+
)
= E
(exp
{n ∑
i =1
γi W
(i ) (S α(T ))
− 1
2
n ∑
i =1
γ 2 i S α(T )
}(m ∑
i =1
ω i Z (i ) α (T ) − K
)+
)
= E
(E
(exp
{n ∑
i =1
γi W
(i ) (S α(T ))
− 1
2
n ∑
i =1
γ 2 i S α(T )
}
×(
m ∑
i =1
ω i Z (i ) α (T ) − K
)+
)| S α(T )
)= E(C B −S (Z 0 , K, S α(T ) , σ ))
=
∫ ∞
0
C B −S (Z 0 , K, x, σ ) g α(x, T ) dx.
Here by g α( x, T ) we denote the PDF of S α( T ). since S α( T ) is α-
selfsimilar, we obtain g α(x, T ) = T −αg α(x/T α) . Thus, the statement
follows. �
Remark 2. There is no explicit formula for C B −S (Z 0 , K, T , σ ) , there-
fore, in order to find the basket option price in the subdiffusive
case, it is necessary to use the approximation methods. One needs
o approximate first the classical price C B −S (Z 0 , K, x, σ ) using Gen-
le’s approximation (already given in (8) , see [6,17] ). Next, one
eeds to approximate the integral ∫ ∞
0 C B −S (Z 0 , K, x, σ ) g α(x, T ) dx
sing some standard method. Another possibility is to use the fact
hat C sub B −S
(Z 0 , K, T , σ, α) = E(C B −S (Z 0 , K, S α(T ) , σ )) and to approxi-
ate C sub B −S
(Z 0 , K, T , σ, α) using Monte Carlo methods.
The put-call parity applied to the multidimensional subdiffusive
-S model, with r = 0 is as follows
sub B −S (Z 0 , K, T , σ, α) − P sub
B −S (Z 0 , K, T , σ, α) = Z α(0) − K, .
hus the subdiffusive price of the put option yields
sub B −S (Z 0 , K, T , σ, α) = C sub
B −S (Z 0 , K, T , σ, α) + K − Z α(0) .
emark 3. The derivatives prices with payoff functions, different
han European basket option, can be calculated using analogous
pproach.
The above result allows us to find fair price of basket option in
ultidimensional subdiffusive B-S model. For instance, for α = 1 / 2
he Fox function can be evaluated as follows [18]
1 / 2 (z) =
1 √
πexp
{− z 2
4
}, z ≥ 0 .
he price of a basket option in the classical Multidimensional B-
model can be found using Gentle’s approximation by geometric
verage, mentioned in previous chapter. The approximate value of
he classical B-S formula is given by (8) . Thus, one can estimate
he value of C sub B −S
by numerical integration of expression in formula
22) . Another method of finding the price of the basket option is
y using well-known Monte Carlo methods. It is only needed to
imulate the realization of S α( T ). From the self-similarity property
f S α( t ) it is evident that S α( T ) is equal in distribution to (
T U α (1)
)α,
here U α( τ ) is α-stable subordinator. U α(1) can be generated in
he standard way, see [5] . Thus, one only needs to simulate U α(1)
nd estimate the expectations in formula (22) using Monte-Carlo
ethod.
In Fig. 3 prices of the basket option, exercised in European
all style, depending on the maturity time for different α are
G. Karipova, M. Magdziarz / Chaos, Solitons and Fractals 102 (2017) 245–253 251
Fig. 4. Prices of the basket option, exercised in European call style, depending on the strike price for different α. The parameters for the figure above are as follows:
m = n = 10 , σ ij ∈ [0, 1], Z (i ) 0
= 1 , ∀ i, j, K = 40 . The results were obtained using (22) from 10 0 0 simulated independent realizations of the random variable S α( T ).
p
m
s
p
t
t
i
p
t
n
c
4
B
c
a
O
o
t
p
fi
c
m
s
t
u
c
w
H
k
{
j
o
i
s
t
t
f
f
c
t
s
fi
A
t
I
A
w
H
F
S
D
G
C
m
resented. Recall that when α = 1 we recover a classical B-S for-
ula for multidimensional case.
In Fig. 4 prices of the European basket option, depending on the
trike price for different α are presented.
Obviously, the price of call option decreases when the strike
rice increases: if the buyer of the call option gets the right to buy
he portfolio of assets for the higher strike price, then the price if
he option should be lower. As Fig. 4 shows, the price of the option
n subdiffusive regime is lower than the classical one for any strike
rice while the other parameters are fixed. It should be noted that
he results for α = 1 / 2 in Figs. 3 and 4 , obtained by method of
umerical approximation of the integral and Monte-Carlo method,
oincide.
. Conclusions
In this paper we introduced the concept of multidimensional
-S model generalized to the subdiffusive case. We derived the
orresponding multidimensional fFPE, which gives the information
bout the behavior of the PDF of the analyzed process [13,15,16] .
ne can take advantage of this equation in order to investigate any
ther specific properties of the process.
We showed that the considered market is arbitrage-free. Since
here is more than one martingale measure, the market is incom-
lete. The latter means that there is no unique fair price of the
nancial derivatives on the specified market. Unfortunately, the in-
ontrovertible approach to the best choice of the corresponding
artingale measure does not exist. However the martingale mea-
ure was chosen according to the criterion of minimal relative en-
ropy. Moreover, the chosen martingale measure extends in a nat-
ral way the martingale measure from the standard B-S model. It
an be recovered as α → 1.
We derived the subdiffusive B-S formula for basket option. It
as necessary to use the approximation methods to find its value.
ere we used Monte Carlo methods and numerical integration.
In order to apply the introduced model to a real financial mar-
et, one needs to estimate required parameters α, { σ ij } m ×n and
μi } m
i =1 . The parameter α can be estimated from the extracted tra-
ectories of heavy-tailed waiting times (periods of stagnation). The
ther parameters can be estimated under the same approach as
n the classical multidimensional model after elimination of the
ubordinating effects. It should be noted that the work can be ex-
ended to the arbitrary choice of inverse subordinators, whose na-
ure, parameters and characteristics can be estimated empirically
rom given specific market.
Since the inspiration for the derivation of the celebrated B-S
ormula came from physics, we believe that the similar situation
an be observed in financial engineering in the context of frac-
ional calculus. Fractional operators, which are successfully used in
tatistical physics to model anomalous fractional dynamics should
nd important applications also in finance.
cknowledgement
The research of Marcin Magdziarz was partially supported by
he Ministry of Science and Higher Education of Poland program
uventus Plus no. IP2014 027073 .
ppendix
Proof of Theorem 2 Let us introduce the filtration { H τ } τ ≥ 0 ,
here
τ = ∩ s>τ { σ (W (z) : 0 ≤ z ≤ s ) ∨ σ (S α(z) : z ≥ 0) } . (23)
irstly, note that { H τ } is right-continuous, thus the random variable
α( t 0 ) is a Markov stopping time w.r.t. { H τ } for each t 0 ∈ (0, T ].
efine
t = H S α (t) . (24)
learly, W t is { H t }-martingale. Let us show that C ( t ) is also { H t }-
252 G. Karipova, M. Magdziarz / Chaos, Solitons and Fractals 102 (2017) 245–253
Y
Y
Y
a
h
Q
T
o
E
U
E
f
E
A
m
i
E
T
E
T
f
t
Z
E(C(t) | H s ) = E
(n ∑
i =1
γi W
(i ) (t) | H s
)
=
n ∑
i =1
E(γi W
(i ) (t) | H s ) =
n ∑
i =1
γi W
(i ) (s ) = C(s ) .
Therefore, C ( t ) is { H t }-martingale, C(t) ∼ N(0 , t ∑ n
i =1 γ2
i ) . Define
the following { H τ }-stopping times
T n = inf { τ > 0 : | C(τ ) | = n } . Note that T n ↗∞ when n → ∞ . Additionally C ( T n ∧ τ ) is a martin-
gale, Moreover, it is bounded by n . Therefore using Doob’s theorem
we get for s < t
E{ C(T n ∧ S α(t)) | G s } = C(T n ∧ S α(s )) .
Now, we are in position to use the Lebesgue dominated conver-
gence theorem, which yields
E{ C(T n ∧ S α(t)) | G s } → E{ C(S α(t)) | G s } as n → ∞ . Finally, we obtain E{ C(S α(t)) | G s } = C(S α(s ))) , thus
A (t) = C(S α(t)) is a { G t }-martingale.
Using Prop. 3.4, Chap. 4 in [20] we obtain that the process Y ( t )
is a local martingale. Additionally, E( sup o≤u ≤t Y (u )) < ∞ . This im-
plies that Y ( t ) is also a martingale.
Proof of Theorem 3 :
Let us put
(t) = exp
{n ∑
j=1
γ j W
( j) t − 1
2
n ∑
j=1
γ 2 j t
},
Z (i ) (t) = exp
{μi t +
n ∑
j=1
σi j W
( j) t
},
Z(t) = (Z (1) (t ) , . . . , Z (m ) (t )) .
Using the fact
〈 A (t) , A (t) 〉 = 〈 C(S α(t)) , C(S α(t)) 〉 =
n ∑
i =1
γ 2 i S α(t) (25)
and setting λ = 1 , from Theorem 1 we know that Y ( S α( t )) is a ( G t ,
P )-martingale. The following holds
(t) Z (i ) (t) = exp
{(μi −
1
2
n ∑
j=1
γ 2 j
)t +
n ∑
j=1
(σi j + γ j ) W
( j) t
}.
Set μi = −( 1 2
∑ n j=1 σ
2 i j
+
∑ n j=1 σi j γ j ) , then
(t) Z (i ) (t) = exp
{−
(1
2
n ∑
j=1
σ 2 i j +
n ∑
j=1
σi j γ j +
1
2
n ∑
j=1
γ 2 j
)t
+
n ∑
j=1
(σi j + γ j ) W
( j) t
}
= exp
{n ∑
j=1
(σi j + γ j ) W
( j) t − 1
2
n ∑
j=1
(σi j + γ j ) 2 t
}.
This implies that the processes Y ( t ) Z ( i ) ( t ), ∀ i are { H t }-martingales
w.r.t. P .
Introduce the process
Z S α (T ) (t) = Z(t ∧ S α(T )) .
Then we get that the processes (e −εS α (T ) Y (t ∧ S α(T )) Z (i )
S α (T ) (t)
)t≥0
(26)
re martingales w.r.t. H t . Next, for every A ∈ H t the following
olds
ε (A ) = E
(1 A exp
{n ∑
i =1
γi W
(i ) (S α(T )) −(
ε +
1
2
n ∑
i =1
γ 2 i
)S α(T )
})
= E
(1 A e
−εS α (T ) E
(exp
{n ∑
i =1
γi W
(i ) (S α(T ))
− 1
2
n ∑
i =1
γ 2 i S α(T )
}| H t
))= E ( 1 A e
−εS α (T ) (E (Y (S α(T )) | H t ))
=
{E( 1 A e
−εS α (T ) Y (t)) , t < S α(T )
E( 1 A e −εS α (T ) Y (S α(T ))) , t ≥ S α(T )
= E( 1 A e −εS α (T ) Y (t ∧ S α(T ))) .
his implies that Z S α(T ) (t) is a martingale w.r.t. ( H t , Q ε ). More-
ver
Q ε
(sup
t≥0
Z (i ) S α (T )
(t) )
= E Q ε(
sup
t≤S α (T )
Z (i ) (t) )
= E
(exp
{n ∑
i =1
γi W
(i ) (S α(T ))
−(
ε +
1
2
n ∑
i =1
γ 2 i
)S α(T )
}sup
t≤S α (T )
Z (i ) (t)
)
≤ E
(exp
{n ∑
i =1
γi W
(i ) (S α(T ))
}e | μi | S α (T )
× sup
t≤T
n ∑
j=1
σi j W
( j) (S α(t))
). (27)
sing the already mentioned formula for moments E(S n α(T )) =T nαn !
(nα+1) , n ∈ N, we obtain
( exp { λS α(T ) } ) =
∞ ∑
n =0
λn E(S n α(T ))
n ! =
∞ ∑
n =0
(T αλ) n
(nα + 1) < ∞
or any λ > 0. Now, using the conditioning argument we arrive at
(exp
{n ∑
i =1
γi W
(i ) (S α(T ))
})
= E
(exp
{1
2
S α(T ) n ∑
j=1
γ 2 j
})< ∞ .
dditionally, as exp { ∑ n j=1 σi j W
( j) (S α(t)) } is a non-negative sub-
artingale, we apply the Doob’s inequality and obtain the follow-
ng
(sup
t≤T
exp
{n ∑
j=1
σi j W
( j) (S α(t))
})2
≤ 4 E
(exp
{2
n ∑
j=1
σi j W
( j) (S α(T ))
})< ∞ .
his implies
Q ε
(sup
t≥0
Z (i ) S α (T )
(t) )
< ∞ .
herefore, Z (i ) S α(T )
(t) are martingales. Additionally, they are uni-
ormly integrable. Thus there must exist a sequence { X (i ) } m
i =1 with
he following property Z (i ) S α(T )
(t) = E Q ε (X (i ) | H t ) and
(i ) α (t) = Z (i )
S α (T ) (S α(t)) = E Q ε (X
(i ) | H S α (t) ) .
G. Karipova, M. Magdziarz / Chaos, Solitons and Fractals 102 (2017) 245–253 253
L
ε
m
R
[
[
[
astly, Z α( t ) is a martingale w.r.t. (H S α(t) , Q ε ) Note that for each
≥ > 0 we obtain different measure Q ε . So, there is no unique
artingale measure for Z α( t ). �
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