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Pricing of basket options in subdiffusive fractional Black–Scholes model
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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 xxx 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. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction During the last few decades the scientists have had a keen interest to the problem of financial markets’ modelling and derivation of prices of different financial derivatives. The breakthrough happened when in 1973 the papers of Black and Scholes [1] and Merton [14] were published. These papers had a great impact to the financial market and arouse huge academic interest, setting up the key principles of arbitrage option pricing. However, the original model has its disadvantages as it sets a number of restrictions on the real state of life [19]. Many of assumptions were relaxed in the ongoing researches. In this work the subdiffusive phenomena is of specific interest. The fact is that in some financial markets the number of participants and, therefore, performed transactions, is so low that the price of the asset may remain constant during the period of time. This phenomenon is the most inherent for the emerging markets and it breaks the assumption on liquidity set in the original Black– Scholes (B-S) model. The idea to model the price of such assets using subdiffusive Geometric Brownian Motion (GBM) comes from physics: the so-called stagnation periods in the market are associated with the trapping events of the subdiffusive test particle [4], which is manifested by the fractional derivatives in Fokker–Planck equations [15,16].
∗
Corresponding author. E-mail addresses: [email protected] [email protected] (M. Magdziarz).
(G.
Karipova),
The generalized to the subdiffusive regime B-S model for one dimensional case was already introduced in details in the literature [9]. However, such model cannot define the price of such important financial derivatives as, for instance, European basket options. For these reasons the multidimensional B-S model was applied in the standard approach. Here we generalize the multidimensional B-S model by adding the periods of stagnation, which are characteristic for subdiffusion and can be modelled by equations involving fractional derivatives. We underline the extension from one- to multidimensional setting of the subdiffusive B-S model is at some points not straightforward. Property of lack of arbitrage depends on the parameters of the volatility matrix. Also, the martingale measure is not unique, its choice has crucial influence on the price. We use the relative entropy criterion to solve this problem. Additionally, there is no closed-form formula for the fair price of the basket option. It has to be approximated using Monte Carlo methods or applying deterministic approach. Let us remind that the celebrated B-S formula was derived by solving certain heat equation, so the motivation came clearly from physics. We expect that the similar process will be observed in financial engineering. Fractional operators, which are successfully used in statistical physics in the description of anomalous fractional dynamics will find important applications also in finance. The first chapter of this work includes the basic knowledge about the options, classical one-dimensional and multidimensional B-S models and such characteristics of the financial market as lack of arbitrage and market completeness. The second chapter introduces the concept of multidimensional B-S model, generalized to the subdiffusive case. At the beginning
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Please cite this article as: G. Karipova, M. Magdziarz, Pricing of basket options in subdiffusive fractional Black–Scholes model, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.05.013
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the inverse stable subordinator is defined and its basic characteristics are given. The subdiffusive multidimensional GBM is introduced, i.e. the subordinated process used to model the prices of the underlying assets with specific periods of stagnation. The fractional multidimensional Fokker–Planck equation is derived, which gives the information about the dynamics of the probability density function (PDF) of the studied process [13,15,16]. In the following it is proved that the considered market is arbitrage-free and incomplete. It should be added that the lack of arbitrage for subdiffusive B-S model was also analyzed in [7]. The B-S formula of option pricing is derived, the price of basket call option depending on the stability parameter α , maturity time and the strike price is approximated by the methods of numerical integration and MonteCarlo simulations. The obtained price functions are compared with the classical case. We end the paper with concluding chapter. For the sake of clarity we moved the most technical proofs to the Appendix.
2.2. Classical B-S model
2. B-S: classical approach
Here, Wt is the standard Brownian motion with respect to P, σ > 0 is the diffusion (volatility) parameter and μ ∈ R is the drift. The celebrated B-S price of European call option equals [1]:
2.1. Options Let us recall the definition of an option. Option is a financial instrument such that its buyer (owner) has the right to buy (call option) or to sell (put option) an asset Z(t) at some prespecified maturity time T for a prespecified strike price K. The underlying asset is usually a stock but some other assets are also possible. A European option, in contrary to the American option, can be exercised at the maturity time only, whereas it is possible to exercise American option at any time up to expiration [1,17]. The payoffs of the European call option and the European put option are equal to (Z (T ) − K )+ and (K − Z (T ))+ , respectively. Here (x )+ = max(x, 0 ). There is a relationship between price of the put option P and the price of a call option C. This relationship is called put-call parity [17]:
Z (0 ) − Ke−rT = C − P.
(1)
Here r ≥ is the interest rate. Further on, we will assume for simplicity that r = 0. Let us now introduce the concept of a basket option, which is defined as financial instrument, where the underlying asset is a portfolio of various assets Z(i) (t), i = 1, . . . , m, for instance, single stocks. Basket option among others belongs to the class of exotic options. The basket option implies higher cost-effectiveness than a simple collection of single options, as it is a diversification instrument which also takes into account the interdependences between diverse risk factors. As an example, there might be a negative correlation between stocks in the “basket”, strong enough to significantly reduce the total risk or even make it disappear [17]. Basket option, exercised in European call style, has the following payoff
m i=1
(i )
ωi Z ( T ) − K +,
(2)
m where {ωi }m are constant weights, such that i=1 ωi = 1. Here i=1 the constant K > 0 is the strike price. Trading of option contracts has had a long historical life. However, the prevalence of the option market increased rapidly in 1973 when options regulations became standardized and transactions on the stock options were performed on the Chicago Board Options Exchange. Simultaneously in 1973 Black and Scholes [1] and Merton [14] published their celebrated papers. These papers had a great impact to the financial market and arouse huge academic interest, setting up the key principles of arbitrage option pricing. While the fair prices of European options can be found using the classical B-S formula, to find the price of a basket option one needs to use the multidimensional models.
Over the last forty years the B-S [2,14,17,19] model is being a powerful tool for pricing derivatives. One may define it as a mathematical model that allows us to simulate the prices of financial instruments, such as stocks, as well as derive the fair prices of certain financial derivatives such as European call options. Let us consider such a market, that its development up to time T is defined on the probability space (, F, P), where is called the sample space, F is a set of all events and possible statements about the prices on the market and P is the usual probability measure. The price of an asset Zt in classical B-S model is assumed to follow GBM given by
dZt =
1 μ + σ 2 Zt dt + Zt σ dWt , Z0 = z0 2
(3)
or equivalently
Zt = Z0 exp{μt + σ Wt }.
(4)
CB−S (Z0 , K, T , σ ) = Z0 (d+ ) − K (d− ),
(5)
with
d± =
± 12 σ 2 T , √ σ T
Z0 K
log
Here is the distribution function of Gaussian distribution with mean zero and variance equal to one. This fair price was originally derived by Black and Scholes by solving the well-known in physics heat equation. By the mentioned put-call parity we have for the price of put option
PB−S (Z0 , K, T , σ ) = CB−S (Z0 , K, T , σ ) + K − Z0 .
(6)
The B-S model was certainly a break-through in the option pricing apparatus. However, it has number of limiting assumptions that do not reflect the real market conditions and behavior of assets’ prices. This gives a fruitful area for academic purposes: already existing and ongoing researches are aimed to possibly relax these assumptions. 2.3. Multidimensional B-S model The classical B-S model can be generalized to the multidimensional case, i.e. the number of assets m > 1 and the asset prices Z (t ) = (Zt(1 ) , . . . , Zt(m ) ) follow multidimensional GBM as
(i )
dZt =
n n 1 2 (i ) μi + σi j Zt dt + Zt(i) σi j dWt( j ) , Z0(i) = z0i 2 j=1
or equivalently
(i )
(i )
Zt = Z0 exp
(7)
j=1
μi t +
n
( j)
σi jWt
,
i = 1, . . . m,
j=1
where W (t ) = (Wt(1 ) , . . . , Wt(n ) ) is n-dimensional Brownian motion with respect to P, {σ ij }m×n , σ ij ≥ 0, is non-singular volatility matrix and (μ1 , . . . , μm ) is a drift vector. Multidimensional B-S model allows to find the fair price of basket options defined in (2). Unfortunately, there is no explicit B-S formula for such purpose. The price of a basket option in the classical multidimensional B-S model can be found using Gentle’s approximation by geometric average [6,17]. It is given by
CB−S =
m
(i )
ωi Z0
(c(l1 (T )) − (K¯ + c − 1 )(l2 (T ))),
(8)
i=1
Please cite this article as: G. Karipova, M. Magdziarz, Pricing of basket options in subdiffusive fractional Black–Scholes model, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.05.013
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where
c = exp
3. Subdiffusive multidimensional B-S model
m 1 2 1 v − ωˆ j σ j2 T , 2 2 j=1
v2 =
m
ρi, j ωˆ i ωˆ j σi σ j ,
i, j=1
ρi, j =
n
k=1 σik σ jk , σi = σi σ j
n
σik2 ,
k=1
ω Z (i ) K ωˆ i = m i 0 ( j ) , K¯ = m , ( j) j=1 ω j Z0 j=1 ω j Z0 where {ρ ij }m×n are the instantaneous correlation coefficients and
ln c − ln(K¯ + c − 1 ) ± √ v t
l1,2 (t ) =
1 2 t 2
v
.
2.4. Arbitrage free market and market completeness The lack of arbitrage for a market model is crucial requirement for pricing regulations, i.e. it should not be possible to get a profit without any risk. The Fundamental Theorem of Asset Pricing [3] states that the market model is arbitrage-free if the asset price Z(t) is a martingale with respect to some measure Q equivalent to P. In other words, Z(t) is a fair-game process w.r.t. Q. The measure Q is such a measure under which the expected rate of return on all assets existing in arbitrage-free market is equal for all financial instruments despite the variability of assets’ price, whereas under the real-life measure P the higher is the risk, the larger is expected rate of return [9]. Another vital characteristic of the market model is market completeness which assures the uniqueness of the fair price for each financial instrument. The Second Fundamental Theorem of Asset Pricing [3] states that if there is only one martingale measure Q then the market is complete. One can show that the classical one-dimensional B-S model responds to both arbitrage-free and completeness criteria, therefore, its B-S formula for option pricing defines a unique fair price of single European option. The martingale measure Q for the B-S model is given by
Q (A ) =
μ + σ 2 / 2 2 T μ + σ 2 / 2 − WT dP, A ∈ F . σ 2 σ
A
exp −
(9) In the multidimensional B-S model the case looks a bit different. Let us introduce constants {γ j }nj=1 as the solution of the following system
r − μi −
n n 1 2 σi j = σi j γ j , 2 j=1
i = 1, . . . m,
(10)
j=1
If the solution of such system exists, then there exists Q defined as
Q (A ) =
A
exp −
n i=1
γiWT(i) −
3
n 1 2 γi T dP, 2
A ∈ F,
(11)
i=1
such that Z(t) is a martingale w.r.t. Q. This implies the lack of arbitrage. One can show that the considered market model is complete, therefore, all financial derivatives have unique fair prices.
The generalized to the subdiffusive regime B-S model for one dimensional case was already introduced in details in the literature [9]. However, such model cannot define the price of such important financial derivatives as, for instance, European basket options. For these reasons the multidimensional B-S model was applied in the classical approach. The following chapter introduces the concept of multidimensional B-S model, generalized to the subdiffusive case. It consists of three sections. The first defines the inverse α -stable subordinator and gives its basic characteristics. The second introduces subdiffusive multidimensional GBM, i.e. subordinated process used to model the prices of the underlying assets with specific periods of stagnation. There is also a fractional multidimensional Fokker– Plank equation derived, which gives the information about the behavior of the PDF of the studied process [13,15,16]. The third section shows that the introduced subdiffusive market has no arbitrage and is not complete. The subdiffusive B-S formula of option pricing is derived here, the price of basket options depending on the α , maturity time and the strike price is approximated by the methods of numerical integration and Monte-Carlo simulations. The obtained price functions are compared with the classical case. 3.1. Inverse alpha-stable subordinator The inverse α -stable subordinator is defined as
Sα (t ) = inf{τ > 0 : Uα (τ ) > t },
(12)
0 < α < 1. Here {Uα (τ )}τ > 0 is the strictly increasing α -stable Lévy process (subordinator). It has the following Laplace transform α E (e−uUα (t ) ) = e−tu [8,22,23]. Here E denotes the expectation. Using the fact that Uα (t) is 1/α -self-similar process, we get that Sα (t) has the same distribution as [t/Uα (1)]α . The moments of the considered process can be found in [12]
E[Sαn (t )] =
t nα n !
(nα + 1 )
,
(13)
where (·) is the gamma function. Moreover, the Laplace transform of Sα (t) equals
E (e−uSα (t ) ) = Eα (−ut α ),
(14) ∞
zn
where the function Eα (z ) = n=0 (nα +1 ) is the Mittag-Leffler function [11,21]. Simulated trajectories of the processes Uα (t) and corresponding Sα (t) are presented in Fig. 1. As it can be seen, consecutive jumps of Uα (t) are reflected by the flat periods (the periods of stagnation) of Sα (t), which is characteristic for subdiffusion. 3.2. Subdiffusive geometric Brownian motion Let us consider such a market model in which the price of m different assets is given by the following subdiffusive process
Zα (t ) = Z (Sα (t )),
(15)
where Sα (t) is inverse α -stable subordinator and Z (t ) = (Zt(1) , . . . , Zt(m) ) is the GBM given in (15). Such defined pro-
cess Zα (t ) = (Zα(1 ) (t ), . . . , Zα(m ) (t )) will describe the prices of m different assets. We will call it subdiffusive GBM. Contrary to the classical one, it will capture the property of subdiffusion (periods of stagnation), characteristic for emerging markets or interest rates. The next theorem determines the multidimensional fractional Fokker–Plank equation (fFPE), which gives the information about the behavior of the PDF of Zα (t).
Please cite this article as: G. Karipova, M. Magdziarz, Pricing of basket options in subdiffusive fractional Black–Scholes model, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.05.013
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Fig. 1. Trajectories of the α -stable and inverse α -stable subordinators, α = 0.9.
Theorem 1. Let Zα (t) be given by (15). Then its PDF is the solution of the fFPE
n ∂ω (x, t ) ∂ 1 2 1 −α = 0 Dt − σi j xi ω (x, t ) μi + ∂t ∂ xi 2 i=1 j=1 m m ∂2 + (K (x )ω (x, t )) , ∂ xi ∂ x j i j i=1 j=1 m
ω (x, 0 ) = δZ0 (x ), Ki j (x ) = 1 −α g 0 Dt
(t ) =
1
d
(α ) dt
t
0
1 2
n
k=1
∞
(16)
0
∞ 0
f (x, τ )gˆ(τ , k )dτ .
(17)
m n ∂ f (x, t ) ∂ 1 2 =− σi j xi f (x, t ) μi + ∂t ∂ xi 2 i=1 j=1 +
i=1 j=1
i=1
+
n ∂ 1 2 σi j xi fˆ(x, t ) μi + ∂ xi 2 j=1
m m i=1 j=1
∂2 (K (x ) fˆ(x, t )). ∂ xi ∂ x j i j
t 0
u(y, τ )dy =
t
ατ
(18)
u(t, τ ),
where by u(t, τ ) we denote the PDF of Uα (τ ). This implies that α gˆ(τ , k ) = kα −1 e−τ k
m
m n ∂ ∂ p(x, t ) 1 2 μi + = 0 Dt1−α − σi j xi p(x, t ) ∂t ∂ xi 2 i=1 j=1 m m ∂2 + (Ki j (x ) p(x, t )) , ∂ xi ∂ x j i=1 j=1
and the proof is completed.
In Fig. 2 we compare trajectory of the first coordinate of standard multidimensional GBM and trajectory of the first coordinate of the corresponding subdiffusive multidimensional GBM.
Theorem 2. If C(t) is a linear combination of independent Brownian motions, i.e. C (t ) = ni=1 γiW (i ) (t ), γi = const, ∀i, then the subdiffusion process A(t ) = C (Sα (t )) is a martingale. Moreover, the stochastic exponential of A(t) defined as
Y (t ) = exp
Using selfsimilarity of Sα (t) and Uα (t) we obtain
∂ g( τ , t ) = − ∂τ
n ∂ 1 2 σi j xi pˆ (x, t ) μi + ∂ xi 2 i=1 j=1 m m ∂2 + (Ki j (x ) pˆ (x, t )) . ∂ xi ∂ x j i=1 j=1
k pˆ (x, k ) − p(x, 0 ) = kα −1 −
The following theorems verify the existence of martingale measures for Zα (t).
Putting the Laplace transform on both sides of the above we arrive at m
α f (x, τ )kα −1 e−τ k dτ = kα −1 fˆ(x, kα ).
3.3. Subdiffusive B-S formula
∂2 (K (x ) f (x, t )), ∂ xi ∂ x j i j
k fˆ(x, k ) − f (x, 0 ) = −
0
Lastly, if we invert the Laplace transform, we arrive at the equation
The above functions f(x, τ ) and g(τ , t) are the respective PDFs of the processes Z(τ ) and Sα (t). We use the convention that hˆ is the Laplace transform of a function h w.r.t. the time variable. Recall that Z(τ ) is defined in (15), therefore its PDF solves the Fokker– Plank equation
m m
∞
σik σ jk xi x j . Here, the operator
(t − s )α−1 g(s )ds,
e−kt p(x, t )dt =
Proof. We will apply the technique from the one-dimensional case. Let p(x, t) be the PDF of Zα (t). The total probability formula implies that
pˆ (x, k ) =
If we change the variables k → kα , we arrive at.
0 < α < 1, is the Riemann–Liouville fractional derivative [21].
pˆ (x, k ) =
and consequently
λ2 λA(t ) − A(t ), A(t ) , λ = 0 2
is also a martingale. Here < A(t), A(t) > is the quadratic variation of A ( t). For the proof see the Appendix. Let us introduce constants {γ j }nj=1 as the solution of the following system
r − μi −
n n 1 2 σi j = σi j γ j , 2 j=1
i = 1, . . . , m.
(19)
j=1
Please cite this article as: G. Karipova, M. Magdziarz, Pricing of basket options in subdiffusive fractional Black–Scholes model, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.05.013
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5
Fig. 2. Trajectories of Zt(1) and Zα(1) (t ). Here, the parameters are: α = 0.9, T = 1, m = n = 10, Z0(i ) = 1, μi ∈ [−2, 3], σ ij ∈ [0, 0.6], ∀i, j = 1, . . . , 10.
Theorem 3. Let ≥ 0. Let Q be the probability measure defined as
The relative entropy of Q is equal to
Q ( A ) = C
exp A
n
n 1 2 γiW (i) (Sα (T )) − + γi Sα (T ) dP, 2
i=1
D=−
2
i=1
i=1
On the other hand, the relative entropy of Q is equal to
i=1
(20) where 1 n
dQ 1 dP = E (Sα (T )) γi2 . dP 2 n
log
n
A ∈ F and C = [E (exp{ i=1 γiW (i ) (Sα (T )) − ( + γi2 )Sα (T )} )]−1 is the normalizing constant. Then Zα (t), t
dQ dP dP n n 1 2 = log E exp γiW (i) (Sα (T )) − + γi Sα (T ) 2
D = −
∈ [0, T], is Q -martingale.
For the proof see the Appendix. Two Fundamental Theorems of Asset Pricing imply that
+
log
i=1
Corollary 1. The market model in which the asset prices follow the multidimensional subdiffusive GBM Zα (t), has no arbitrage and is incomplete.
= log E E
Q (A ) =
exp A
n i=1
γiW (i) (Sα (T )) −
1 2
n
γi2 Sα (T ) dP, A ∈ F.
i=1
(21) Then the relative entropy for the measure Q is less than for the measure Q , > 0. Proof. Clearly, Q is the special case of Q , defined in (20) for = 0 and C = 1. Thus, from Theorem 2 we have that Zα (t) is a Qmartingale.
n
γiW (i) (Sα (T )) −
+
n 1 2 γi Sα (T ) 2
i=1
n 1 2 + γi E (Sα (T )) 2 i=1
n 1 2 = log E (exp{− Sα (T )}E (Y (t ))) + + γi E (Sα (T )) 2
i=1
n 1 2 = log E (exp{− Sα (T )} ) + + γi E (Sα (T )) 2
1 ≥ E (Sα (T )) 2
exp
× exp{− Sα (T )}|Gt
Lemma 1. Let us define a probability measure Q as
i=1
i=1
Market incompleteness means that there is no unique fair price of financial derivatives. Unfortunately, the incontrovertible approach to the best choice of the corresponding martingale measure does not exist. However the martingale measure can be chosen according to the criterion of minimal relative entropy, meaning that the best choice of measure Q minimizes the distance from measure P [10].
i=1
n 1 2 + γi E (Sα (T )) 2
n
i=1
γi2 = D.
i=1
Thus, Q minimizes relative entropy.
Remark 1. Note that for α = 1 the measure Q defined in (21) reduces to the martingale measure in the standard B-S model. Under the above arguments, further on we will find the fair prices using the martingale measure Q. In the next theorem we determine the fair price of a basket option in the subdiffusive B-S model. Theorem 4. Let the assets prices follow Zα (t). Then the corresponding fair price CBsub (Z , K, T , σ , α ) of a basket option satisfies −S 0
Please cite this article as: G. Karipova, M. Magdziarz, Pricing of basket options in subdiffusive fractional Black–Scholes model, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.05.013
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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], Z0(i ) = 1, ∀i, j, K = 40. The results were obtained using (22) from 10 0 0 simulated independent realizations of the random variable Sα (T).
CBsub −S (Z0 , K, T , σ , α ) = E (CB−S (Z0 , K, Sα (T ), σ )) ∞ = CB−S (Z0 , K, x, σ )T −α gα (x/T α )dx,
(22)
0
where CB−S (Z0 , K, T , σ ) is price of the basket option in the standard multidimensional B-S model, gα (z) stands for the PDF of Sα (1) and can be expressed using Fox function
10 gα (z ) = H11 z|((10,−1α) ,α ) .
(Z0 , K, T , σ , α ) = E
Q
ωi Zα(i) (T ) − K
exp
n
i=1
=E E
exp
n
n 1 2 γi Sα (T ) 2
×
i=1
m i=1
m
ωi Zα(i) (T ) − K
i=1
+
γiW (i) (Sα (T ))
i=1
sub PBsub −S (Z0 , K, T , σ , α ) = CB−S (Z0 , K, T , σ , α ) + K − Zα (0 ).
γiW (i) (Sα (T ))
i=1
The put-call parity applied to the multidimensional subdiffusive B-S model, with r = 0 is as follows
Thus the subdiffusive price of the put option yields
+
n 1 2 − γi Sα (T ) 2
−
i=1
=E
m
mate CBsub (Z , K, T , σ , α ) using Monte Carlo methods. −S 0
sub CBsub −S (Z0 , K, T , σ , α ) − PB−S (Z0 , K, T , σ , α ) = Zα (0 ) − K, .
Proof. The arbitrage-free rule of pricing requires that
CBsub −S
to approximate first the classical price CB−S (Z0 , K, x, σ ) using Gentle’s approximation (already given in (8), see [6,17]). Next, one ∞ needs to approximate the integral 0 CB−S (Z0 , K, x, σ )gα (x, T )dx using some standard method. Another possibility is to use the fact that CBsub (Z , K, T , σ , α ) = E (CB−S (Z0 , K, Sα (T ), σ )) and to approxi−S 0
ωi Zα(i) (T ) − K |Sα ( T ) +
= E (CB−S (Z0 , K, Sα (T ), σ )) ∞ = CB−S (Z0 , K, x, σ )gα (x, T )dx. 0
Here by gα (x, T) we denote the PDF of Sα (T). since Sα (T) is α selfsimilar, we obtain gα (x, T ) = T −α gα (x/T α ). Thus, the statement follows. Remark 2. There is no explicit formula for CB−S (Z0 , K, T , σ ), therefore, in order to find the basket option price in the subdiffusive case, it is necessary to use the approximation methods. One needs
Remark 3. The derivatives prices with payoff functions, different than European basket option, can be calculated using analogous approach. The above result allows us to find fair price of basket option in multidimensional subdiffusive B-S model. For instance, for α = 1/2 the Fox function can be evaluated as follows [18]
z2 1 g1/2 (z ) = √ exp − , z ≥ 0. 4 π The price of a basket option in the classical Multidimensional BS model can be found using Gentle’s approximation by geometric average, mentioned in previous chapter. The approximate value of the classical B-S formula is given by (8). Thus, one can estimate the value of CBsub by numerical integration of expression in formula −S (22). Another method of finding the price of the basket option is by using well-known Monte Carlo methods. It is only needed to simulate the realization of Sα (T). From the self-similarity property α of Sα (t) it is evident that Sα (T) is equal in distribution to U T(1 ) , α where Uα (τ ) is α -stable subordinator. Uα (1) can be generated in the standard way, see [5]. Thus, one only needs to simulate Uα (1) and estimate the expectations in formula (22) using Monte-Carlo method. In Fig. 3 prices of the basket option, exercised in European call style, depending on the maturity time for different α are
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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], Z0(i ) = 1, ∀i, j, K = 40. The results were obtained using (22) from 10 0 0 simulated independent realizations of the random variable Sα (T).
presented. Recall that when α = 1 we recover a classical B-S formula for multidimensional case. In Fig. 4 prices of the European basket option, depending on the strike price for different α are presented. Obviously, the price of call option decreases when the strike price increases: if the buyer of the call option gets the right to buy the portfolio of assets for the higher strike price, then the price if the option should be lower. As Fig. 4 shows, the price of the option in subdiffusive regime is lower than the classical one for any strike price while the other parameters are fixed. It should be noted that the results for α = 1/2 in Figs. 3 and 4, obtained by method of numerical approximation of the integral and Monte-Carlo method, coincide.
jectories of heavy-tailed waiting times (periods of stagnation). The other parameters can be estimated under the same approach as in the classical multidimensional model after elimination of the subordinating effects. It should be noted that the work can be extended to the arbitrary choice of inverse subordinators, whose nature, parameters and characteristics can be estimated empirically from given specific market. Since the inspiration for the derivation of the celebrated B-S formula came from physics, we believe that the similar situation can be observed in financial engineering in the context of fractional calculus. Fractional operators, which are successfully used in statistical physics to model anomalous fractional dynamics should find important applications also in finance.
4. Conclusions In this paper we introduced the concept of multidimensional B-S model generalized to the subdiffusive case. We derived the corresponding multidimensional fFPE, which gives the information about the behavior of the PDF of the analyzed process [13,15,16]. One can take advantage of this equation in order to investigate any other specific properties of the process. We showed that the considered market is arbitrage-free. Since there is more than one martingale measure, the market is incomplete. The latter means that there is no unique fair price of the financial derivatives on the specified market. Unfortunately, the incontrovertible approach to the best choice of the corresponding martingale measure does not exist. However the martingale measure was chosen according to the criterion of minimal relative entropy. Moreover, the chosen martingale measure extends in a natural way the martingale measure from the standard B-S model. It can be recovered as α → 1. We derived the subdiffusive B-S formula for basket option. It was necessary to use the approximation methods to find its value. Here we used Monte Carlo methods and numerical integration. In order to apply the introduced model to a real financial market, one needs to estimate required parameters α , {σ ij }m×n and { μi } m . The parameter α can be estimated from the extracted trai=1
Acknowledgement The research of Marcin Magdziarz was partially supported by the Ministry of Science and Higher Education of Poland program Iuventus Plus no. IP2014 027073.
Appendix Proof of Theorem 2 Let us introduce the filtration {Hτ }τ where
Hτ = ∩s>τ {σ (W (z ) : 0 ≤ z ≤ s ) ∨ σ (Sα (z ) : z ≥ 0 )}.
≥ 0,
(23)
Firstly, note that {Hτ } is right-continuous, thus the random variable Sα (t0 ) is a Markov stopping time w.r.t. {Hτ } for each t0 ∈ (0, T]. Define
Gt = HSα (t ) .
(24)
Clearly, Wt is {Ht }-martingale. Let us show that C(t) is also {Ht }martingale.
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E (C (t )|Hs ) = E
n
are martingales w.r.t. Ht . Next, for every A ∈ Ht the following holds
(i )
γiW (t )|Hs
i=1 n
=
n
E (γiW (i ) (t )|Hs ) =
i=1
γiW (i) (s ) = C (s ).
i=1
Therefore, C(t) is {Ht }-martingale, C (t ) ∼ N (0, t the following {Hτ }-stopping times
n
i=1
E {C (Tn ∧ Sα (t ))|Gs } → E {C (Sα (t ))|Gs }
Y (t ) = exp
γ jWt( j ) −
j=1
1 2
Z (t ) = (Z
(t ), . . . , Z
=
E (1A e− Sα (T )Y (t )),
t < Sα ( T )
− S α ( T )
t ≥ Sα ( T )
E (1A e
= E (1A e
− S α ( T )
E Q sup ZS(αi )(T ) (t )
= E Q
t≥0
=E
Y (t )Z (t ) = exp
−
j=1
n
n
E (exp{λSα (T )} ) =
γi2 Sα (t )
(25)
= exp
n 1 2 γi Sα (T ) 2
exp
i=1
n
n
sup Z (i ) (t )
t≤Sα (T )
γiW (i) (Sα (T )) e|μi |Sα (T ) ( j)
σi jW (Sα (t )) .
(27)
j=1
j=1
( j)
(σi j + γ j )Wt
E
n
∞
( T α λ )n <∞
( nα + 1 ) n=0
γiW (i) (Sα (T ))
i=1
E
exp
sup exp t≤T
1 Sα ( T ) γ j2 2 n
< ∞.
n 1 − ( σi j + γ j ) 2 t . 2
n
2
σi jW ( j ) (Sα (t ))
j=1
≤ 4E
exp 2
n
σi jW ( j ) (Sα (T ))
< ∞.
j=1
This implies
ZSα (T ) (t ) = Z (t ∧ Sα (T )).
E Q sup ZS(αi )(T ) (t ) < ∞. t≥0
Therefore, ZS(i )(T ) (t ) are martingales. Additionally, they are uniα formly integrable. Thus there must exist a sequence {X (i ) }m with i=1 the following property ZS(i )(T ) (t ) = E Q (X (i ) |Ht ) and α
Then we get that the processes t≥0
exp
j=1
j=1
n!
=
Additionally, as exp{ nj=1 σi j W ( j ) (Sα (t ))} is a non-negative submartingale, we apply the Doob’s inequality and obtain the following
This implies that the processes Y(t)Z(i) (t), ∀i are {Ht }-martingales w.r.t. P. Introduce the process
e− Sα (T )Y (t ∧ Sα (T ))ZS(αi )(T ) (t )
λn E (Sαn (T ))
j=1
∞
for any λ > 0. Now, using the conditioning argument we arrive at
=E
n n n 1 2 1 2 σi j + σi j γ j + γj t 2 2
j=1
+
n=0
j=1
σi j γ j ), then
(σi j + γ j )Wt( j )
n
γiW (i) (Sα (T ))
i=1
t≤T
j=1
n
n (T )) = Using the already mentioned formula for moments E (Sα T nα n !
(nα +1 ) , n ∈ N, we obtain
j=1
+
× sup
j=1
n
t≤Sα (T )
exp
−
n n 1 2 Y (t )Z (i ) (t ) = exp γ j t + (σi j + γ j )Wt( j ) . μi −
(i )
sup Z (i ) (t )
i=1
and setting λ = 1, from Theorem 1 we know that Y(Sα (t)) is a (Gt , P)-martingale. The following holds
σi2j +
≤E
i=1
j=1
Y (t ∧ Sα (T ))).
j=1
A(t ), A(t ) = C (Sα (t )), C (Sα (t )) =
n
Y (Sα (T ))),
This implies that ZSα (T ) (t ) is a martingale w.r.t. (Ht , Q ). Moreover
γ j2t ,
Using the fact
Set μi = −( 21
γiW (i) (Sα (T ))
|Ht
(t )).
2
i=1
= E (1A e− Sα (T ) (E (Y (Sα (T ))|Ht ))
j=1
(m )
exp
2
n
i=1
n ( j) (i ) Z (t ) = exp μi t + σi jWt , (1 )
i=1
as n → ∞. Finally, we obtain E {C (Sα (t ))|Gs } = C (Sα (s ))), thus A(t ) = C (Sα (t )) is a {Gt }-martingale. Using Prop. 3.4, Chap. 4 in [20] we obtain that the process Y(t) is a local martingale. Additionally, E (supo≤u≤t Y (u )) < ∞. This implies that Y(t) is also a martingale. Proof of Theorem 3: Let us put n
i=1
n 1 2 − γi Sα (T ) 2
E {C (Tn ∧ Sα (t ))|Gs } = C (Tn ∧ Sα (s )). Now, we are in position to use the Lebesgue dominated convergence theorem, which yields
n 1 2 γiW (i) (Sα (T )) − + γi Sα (T )
= E 1A e− Sα (T ) E
γi2 ). Define
Note that Tn ∞ when n → ∞. Additionally C(Tn ∧τ ) is a martingale, Moreover, it is bounded by n. Therefore using Doob’s theorem we get for s < t
n
n
Tn = inf{τ > 0 : |C (τ )| = n}.
Q (A ) = E 1A exp
(26)
Zα(i ) (t ) = ZS(αi )(T ) (Sα (t )) = E Q (X (i ) |HSα (t ) ).
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Lastly, Zα (t) is a martingale w.r.t. (HSα (t ) , Q ) Note that for each ≥ >0 we obtain different measure Q . So, there is no unique martingale measure for Zα (t). References [1] Black F, Scholes M. The pricing of options and corporate liabilities. J Pol Econ 1973;81:637–54. [2] Company R, Jodar L. Numerical solution of Black–Scholes option pricing with variable yield discrete dividend payment. Banach Cent Publ 2008;83:37–47. [3] Cont R, Tankov P. Financial modeling with jump processes. Chapman & Hall/CRC, Boca Raton; 2004. [4] Elizar I, Klafler J. Spatial gliding, temporal trapping and anomalous transport. Physica D 2004;187:30–50. [5] Janicki A, Weron A. Simulation and chaotic behaviour of α -stable stochastic processes. New York: Marcel Dekker; 1994. [6] Krekel M, Kock J, Korn R, Man T-K. An analysis of pricing methods for baskets options. Wilmott Mag 2004;3:82–9. [7] Li GH, Zhang H, Luo MK. A multidimensional subdiffusion model: an arbitrage-free market. Chin Phys B 2012;21:128901. [8] Magdziarz M, Weron A. Anomalous diffusion and semimartingales. EPL 20 09;86:60 010. [9] Magdziarz M. Black–Scholes formula in subdiffusive regime. J Stat Phys 2009;136:553–64. [10] Magdziarz M, Gajda J. Anomalous dynamics of Black–Scholes model time changed by inverse subordinators. ACTA Physica Polonica 2012;B 43:1093–110. [11] Magdziarz M. Path properties of subdiffusion - a martingale approach. Stochastic Models 2010;26:256–71.
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[12] Magdziarz M. Stochastic representation of subdiffusion processes with time-dependent drift. Stochastic Process Appl 2009;119:3238–52. [13] Martens W, Wagner U. Advances in solving high dimensional Fokker–Planck equations. Rome: ENOC; 2011. [14] Merton RC. Theory of rational option pricing. Bell J Econ Manag Sci 1973;4:141–83. [15] Metzler R, Barkai E, Klafter J. Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker–Planck equation approach. Phys Rev Lett 1999;82:3563–7. [16] Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 20 0 0;339:1–77. [17] Musiela M, Rutkowski M. Martingale methods in financial modelling. Berlin: Springer; 1997. [18] Piryatinska A, Saichev AI, Woyczynski WA. Models of anomalous diffusion: the subdiffusive case. Physica 2005;A 349:375–420. [19] Ray S. A close look into Black–Scholes option pricing model. J Sci (JOS) 2012;2:172–8. [20] Revuz D, Yor M. Continuous martingales and Brownian motion. Berlin: Springer; 1999. [21] Samko SG, Kilbas AA, Maritchev DI. Integrals and derivatives of the fractional order and some of their applications. Amsterdam: Gordon and Breach; 1993. [22] Xu L. Ergodicity of the stochastic real Ginzburg–Landau equation driven by α -stable noises. Stochastic Process Appl 2013;123:3710–36. [23] Zhang X. Derivative formulas and gradient estimates for SDEs driven by α -stable processes. Stochastic Process Appl 2013;123:1213–28.
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Chaos, Solitons and Fractals 0 0 0 (2017) 1–9
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 xxx 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. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction During the last few decades the scientists have had a keen interest to the problem of financial markets’ modelling and derivation of prices of different financial derivatives. The breakthrough happened when in 1973 the papers of Black and Scholes [1] and Merton [14] were published. These papers had a great impact to the financial market and arouse huge academic interest, setting up the key principles of arbitrage option pricing. However, the original model has its disadvantages as it sets a number of restrictions on the real state of life [19]. Many of assumptions were relaxed in the ongoing researches. In this work the subdiffusive phenomena is of specific interest. The fact is that in some financial markets the number of participants and, therefore, performed transactions, is so low that the price of the asset may remain constant during the period of time. This phenomenon is the most inherent for the emerging markets and it breaks the assumption on liquidity set in the original Black– Scholes (B-S) model. The idea to model the price of such assets using subdiffusive Geometric Brownian Motion (GBM) comes from physics: the so-called stagnation periods in the market are associated with the trapping events of the subdiffusive test particle [4], which is manifested by the fractional derivatives in Fokker–Planck equations [15,16].
∗
Corresponding author. E-mail addresses: [email protected] [email protected] (M. Magdziarz).
(G.
Karipova),
The generalized to the subdiffusive regime B-S model for one dimensional case was already introduced in details in the literature [9]. However, such model cannot define the price of such important financial derivatives as, for instance, European basket options. For these reasons the multidimensional B-S model was applied in the standard approach. Here we generalize the multidimensional B-S model by adding the periods of stagnation, which are characteristic for subdiffusion and can be modelled by equations involving fractional derivatives. We underline the extension from one- to multidimensional setting of the subdiffusive B-S model is at some points not straightforward. Property of lack of arbitrage depends on the parameters of the volatility matrix. Also, the martingale measure is not unique, its choice has crucial influence on the price. We use the relative entropy criterion to solve this problem. Additionally, there is no closed-form formula for the fair price of the basket option. It has to be approximated using Monte Carlo methods or applying deterministic approach. Let us remind that the celebrated B-S formula was derived by solving certain heat equation, so the motivation came clearly from physics. We expect that the similar process will be observed in financial engineering. Fractional operators, which are successfully used in statistical physics in the description of anomalous fractional dynamics will find important applications also in finance. The first chapter of this work includes the basic knowledge about the options, classical one-dimensional and multidimensional B-S models and such characteristics of the financial market as lack of arbitrage and market completeness. The second chapter introduces the concept of multidimensional B-S model, generalized to the subdiffusive case. At the beginning
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the inverse stable subordinator is defined and its basic characteristics are given. The subdiffusive multidimensional GBM is introduced, i.e. the subordinated process used to model the prices of the underlying assets with specific periods of stagnation. The fractional multidimensional Fokker–Planck equation is derived, which gives the information about the dynamics of the probability density function (PDF) of the studied process [13,15,16]. In the following it is proved that the considered market is arbitrage-free and incomplete. It should be added that the lack of arbitrage for subdiffusive B-S model was also analyzed in [7]. The B-S formula of option pricing is derived, the price of basket call option depending on the stability parameter α , maturity time and the strike price is approximated by the methods of numerical integration and MonteCarlo simulations. The obtained price functions are compared with the classical case. We end the paper with concluding chapter. For the sake of clarity we moved the most technical proofs to the Appendix.
2.2. Classical B-S model
2. B-S: classical approach
Here, Wt is the standard Brownian motion with respect to P, σ > 0 is the diffusion (volatility) parameter and μ ∈ R is the drift. The celebrated B-S price of European call option equals [1]:
2.1. Options Let us recall the definition of an option. Option is a financial instrument such that its buyer (owner) has the right to buy (call option) or to sell (put option) an asset Z(t) at some prespecified maturity time T for a prespecified strike price K. The underlying asset is usually a stock but some other assets are also possible. A European option, in contrary to the American option, can be exercised at the maturity time only, whereas it is possible to exercise American option at any time up to expiration [1,17]. The payoffs of the European call option and the European put option are equal to (Z (T ) − K )+ and (K − Z (T ))+ , respectively. Here (x )+ = max(x, 0 ). There is a relationship between price of the put option P and the price of a call option C. This relationship is called put-call parity [17]:
Z (0 ) − Ke−rT = C − P.
(1)
Here r ≥ is the interest rate. Further on, we will assume for simplicity that r = 0. Let us now introduce the concept of a basket option, which is defined as financial instrument, where the underlying asset is a portfolio of various assets Z(i) (t), i = 1, . . . , m, for instance, single stocks. Basket option among others belongs to the class of exotic options. The basket option implies higher cost-effectiveness than a simple collection of single options, as it is a diversification instrument which also takes into account the interdependences between diverse risk factors. As an example, there might be a negative correlation between stocks in the “basket”, strong enough to significantly reduce the total risk or even make it disappear [17]. Basket option, exercised in European call style, has the following payoff
m i=1
(i )
ωi Z ( T ) − K +,
(2)
m where {ωi }m are constant weights, such that i=1 ωi = 1. Here i=1 the constant K > 0 is the strike price. Trading of option contracts has had a long historical life. However, the prevalence of the option market increased rapidly in 1973 when options regulations became standardized and transactions on the stock options were performed on the Chicago Board Options Exchange. Simultaneously in 1973 Black and Scholes [1] and Merton [14] published their celebrated papers. These papers had a great impact to the financial market and arouse huge academic interest, setting up the key principles of arbitrage option pricing. While the fair prices of European options can be found using the classical B-S formula, to find the price of a basket option one needs to use the multidimensional models.
Over the last forty years the B-S [2,14,17,19] model is being a powerful tool for pricing derivatives. One may define it as a mathematical model that allows us to simulate the prices of financial instruments, such as stocks, as well as derive the fair prices of certain financial derivatives such as European call options. Let us consider such a market, that its development up to time T is defined on the probability space (, F, P), where is called the sample space, F is a set of all events and possible statements about the prices on the market and P is the usual probability measure. The price of an asset Zt in classical B-S model is assumed to follow GBM given by
dZt =
1 μ + σ 2 Zt dt + Zt σ dWt , Z0 = z0 2
(3)
or equivalently
Zt = Z0 exp{μt + σ Wt }.
(4)
CB−S (Z0 , K, T , σ ) = Z0 (d+ ) − K (d− ),
(5)
with
d± =
± 12 σ 2 T , √ σ T
Z0 K
log
Here is the distribution function of Gaussian distribution with mean zero and variance equal to one. This fair price was originally derived by Black and Scholes by solving the well-known in physics heat equation. By the mentioned put-call parity we have for the price of put option
PB−S (Z0 , K, T , σ ) = CB−S (Z0 , K, T , σ ) + K − Z0 .
(6)
The B-S model was certainly a break-through in the option pricing apparatus. However, it has number of limiting assumptions that do not reflect the real market conditions and behavior of assets’ prices. This gives a fruitful area for academic purposes: already existing and ongoing researches are aimed to possibly relax these assumptions. 2.3. Multidimensional B-S model The classical B-S model can be generalized to the multidimensional case, i.e. the number of assets m > 1 and the asset prices Z (t ) = (Zt(1 ) , . . . , Zt(m ) ) follow multidimensional GBM as
(i )
dZt =
n n 1 2 (i ) μi + σi j Zt dt + Zt(i) σi j dWt( j ) , Z0(i) = z0i 2 j=1
or equivalently
(i )
(i )
Zt = Z0 exp
(7)
j=1
μi t +
n
( j)
σi jWt
,
i = 1, . . . m,
j=1
where W (t ) = (Wt(1 ) , . . . , Wt(n ) ) is n-dimensional Brownian motion with respect to P, {σ ij }m×n , σ ij ≥ 0, is non-singular volatility matrix and (μ1 , . . . , μm ) is a drift vector. Multidimensional B-S model allows to find the fair price of basket options defined in (2). Unfortunately, there is no explicit B-S formula for such purpose. The price of a basket option in the classical multidimensional B-S model can be found using Gentle’s approximation by geometric average [6,17]. It is given by
CB−S =
m
(i )
ωi Z0
(c(l1 (T )) − (K¯ + c − 1 )(l2 (T ))),
(8)
i=1
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where
c = exp
3. Subdiffusive multidimensional B-S model
m 1 2 1 v − ωˆ j σ j2 T , 2 2 j=1
v2 =
m
ρi, j ωˆ i ωˆ j σi σ j ,
i, j=1
ρi, j =
n
k=1 σik σ jk , σi = σi σ j
n
σik2 ,
k=1
ω Z (i ) K ωˆ i = m i 0 ( j ) , K¯ = m , ( j) j=1 ω j Z0 j=1 ω j Z0 where {ρ ij }m×n are the instantaneous correlation coefficients and
ln c − ln(K¯ + c − 1 ) ± √ v t
l1,2 (t ) =
1 2 t 2
v
.
2.4. Arbitrage free market and market completeness The lack of arbitrage for a market model is crucial requirement for pricing regulations, i.e. it should not be possible to get a profit without any risk. The Fundamental Theorem of Asset Pricing [3] states that the market model is arbitrage-free if the asset price Z(t) is a martingale with respect to some measure Q equivalent to P. In other words, Z(t) is a fair-game process w.r.t. Q. The measure Q is such a measure under which the expected rate of return on all assets existing in arbitrage-free market is equal for all financial instruments despite the variability of assets’ price, whereas under the real-life measure P the higher is the risk, the larger is expected rate of return [9]. Another vital characteristic of the market model is market completeness which assures the uniqueness of the fair price for each financial instrument. The Second Fundamental Theorem of Asset Pricing [3] states that if there is only one martingale measure Q then the market is complete. One can show that the classical one-dimensional B-S model responds to both arbitrage-free and completeness criteria, therefore, its B-S formula for option pricing defines a unique fair price of single European option. The martingale measure Q for the B-S model is given by
Q (A ) =
μ + σ 2 / 2 2 T μ + σ 2 / 2 − WT dP, A ∈ F . σ 2 σ
A
exp −
(9) In the multidimensional B-S model the case looks a bit different. Let us introduce constants {γ j }nj=1 as the solution of the following system
r − μi −
n n 1 2 σi j = σi j γ j , 2 j=1
i = 1, . . . m,
(10)
j=1
If the solution of such system exists, then there exists Q defined as
Q (A ) =
A
exp −
n i=1
γiWT(i) −
3
n 1 2 γi T dP, 2
A ∈ F,
(11)
i=1
such that Z(t) is a martingale w.r.t. Q. This implies the lack of arbitrage. One can show that the considered market model is complete, therefore, all financial derivatives have unique fair prices.
The generalized to the subdiffusive regime B-S model for one dimensional case was already introduced in details in the literature [9]. However, such model cannot define the price of such important financial derivatives as, for instance, European basket options. For these reasons the multidimensional B-S model was applied in the classical approach. The following chapter introduces the concept of multidimensional B-S model, generalized to the subdiffusive case. It consists of three sections. The first defines the inverse α -stable subordinator and gives its basic characteristics. The second introduces subdiffusive multidimensional GBM, i.e. subordinated process used to model the prices of the underlying assets with specific periods of stagnation. There is also a fractional multidimensional Fokker– Plank equation derived, which gives the information about the behavior of the PDF of the studied process [13,15,16]. The third section shows that the introduced subdiffusive market has no arbitrage and is not complete. The subdiffusive B-S formula of option pricing is derived here, the price of basket options depending on the α , maturity time and the strike price is approximated by the methods of numerical integration and Monte-Carlo simulations. The obtained price functions are compared with the classical case. 3.1. Inverse alpha-stable subordinator The inverse α -stable subordinator is defined as
Sα (t ) = inf{τ > 0 : Uα (τ ) > t },
(12)
0 < α < 1. Here {Uα (τ )}τ > 0 is the strictly increasing α -stable Lévy process (subordinator). It has the following Laplace transform α E (e−uUα (t ) ) = e−tu [8,22,23]. Here E denotes the expectation. Using the fact that Uα (t) is 1/α -self-similar process, we get that Sα (t) has the same distribution as [t/Uα (1)]α . The moments of the considered process can be found in [12]
E[Sαn (t )] =
t nα n !
(nα + 1 )
,
(13)
where (·) is the gamma function. Moreover, the Laplace transform of Sα (t) equals
E (e−uSα (t ) ) = Eα (−ut α ),
(14) ∞
zn
where the function Eα (z ) = n=0 (nα +1 ) is the Mittag-Leffler function [11,21]. Simulated trajectories of the processes Uα (t) and corresponding Sα (t) are presented in Fig. 1. As it can be seen, consecutive jumps of Uα (t) are reflected by the flat periods (the periods of stagnation) of Sα (t), which is characteristic for subdiffusion. 3.2. Subdiffusive geometric Brownian motion Let us consider such a market model in which the price of m different assets is given by the following subdiffusive process
Zα (t ) = Z (Sα (t )),
(15)
where Sα (t) is inverse α -stable subordinator and Z (t ) = (Zt(1) , . . . , Zt(m) ) is the GBM given in (15). Such defined pro-
cess Zα (t ) = (Zα(1 ) (t ), . . . , Zα(m ) (t )) will describe the prices of m different assets. We will call it subdiffusive GBM. Contrary to the classical one, it will capture the property of subdiffusion (periods of stagnation), characteristic for emerging markets or interest rates. The next theorem determines the multidimensional fractional Fokker–Plank equation (fFPE), which gives the information about the behavior of the PDF of Zα (t).
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Fig. 1. Trajectories of the α -stable and inverse α -stable subordinators, α = 0.9.
Theorem 1. Let Zα (t) be given by (15). Then its PDF is the solution of the fFPE
n ∂ω (x, t ) ∂ 1 2 1 −α = 0 Dt − σi j xi ω (x, t ) μi + ∂t ∂ xi 2 i=1 j=1 m m ∂2 + (K (x )ω (x, t )) , ∂ xi ∂ x j i j i=1 j=1 m
ω (x, 0 ) = δZ0 (x ), Ki j (x ) = 1 −α g 0 Dt
(t ) =
1
d
(α ) dt
t
0
1 2
n
k=1
∞
(16)
0
∞ 0
f (x, τ )gˆ(τ , k )dτ .
(17)
m n ∂ f (x, t ) ∂ 1 2 =− σi j xi f (x, t ) μi + ∂t ∂ xi 2 i=1 j=1 +
i=1 j=1
i=1
+
n ∂ 1 2 σi j xi fˆ(x, t ) μi + ∂ xi 2 j=1
m m i=1 j=1
∂2 (K (x ) fˆ(x, t )). ∂ xi ∂ x j i j
t 0
u(y, τ )dy =
t
ατ
(18)
u(t, τ ),
where by u(t, τ ) we denote the PDF of Uα (τ ). This implies that α gˆ(τ , k ) = kα −1 e−τ k
m
m n ∂ ∂ p(x, t ) 1 2 μi + = 0 Dt1−α − σi j xi p(x, t ) ∂t ∂ xi 2 i=1 j=1 m m ∂2 + (Ki j (x ) p(x, t )) , ∂ xi ∂ x j i=1 j=1
and the proof is completed.
In Fig. 2 we compare trajectory of the first coordinate of standard multidimensional GBM and trajectory of the first coordinate of the corresponding subdiffusive multidimensional GBM.
Theorem 2. If C(t) is a linear combination of independent Brownian motions, i.e. C (t ) = ni=1 γiW (i ) (t ), γi = const, ∀i, then the subdiffusion process A(t ) = C (Sα (t )) is a martingale. Moreover, the stochastic exponential of A(t) defined as
Y (t ) = exp
Using selfsimilarity of Sα (t) and Uα (t) we obtain
∂ g( τ , t ) = − ∂τ
n ∂ 1 2 σi j xi pˆ (x, t ) μi + ∂ xi 2 i=1 j=1 m m ∂2 + (Ki j (x ) pˆ (x, t )) . ∂ xi ∂ x j i=1 j=1
k pˆ (x, k ) − p(x, 0 ) = kα −1 −
The following theorems verify the existence of martingale measures for Zα (t).
Putting the Laplace transform on both sides of the above we arrive at m
α f (x, τ )kα −1 e−τ k dτ = kα −1 fˆ(x, kα ).
3.3. Subdiffusive B-S formula
∂2 (K (x ) f (x, t )), ∂ xi ∂ x j i j
k fˆ(x, k ) − f (x, 0 ) = −
0
Lastly, if we invert the Laplace transform, we arrive at the equation
The above functions f(x, τ ) and g(τ , t) are the respective PDFs of the processes Z(τ ) and Sα (t). We use the convention that hˆ is the Laplace transform of a function h w.r.t. the time variable. Recall that Z(τ ) is defined in (15), therefore its PDF solves the Fokker– Plank equation
m m
∞
σik σ jk xi x j . Here, the operator
(t − s )α−1 g(s )ds,
e−kt p(x, t )dt =
Proof. We will apply the technique from the one-dimensional case. Let p(x, t) be the PDF of Zα (t). The total probability formula implies that
pˆ (x, k ) =
If we change the variables k → kα , we arrive at.
0 < α < 1, is the Riemann–Liouville fractional derivative [21].
pˆ (x, k ) =
and consequently
λ2 λA(t ) − A(t ), A(t ) , λ = 0 2
is also a martingale. Here < A(t), A(t) > is the quadratic variation of A ( t). For the proof see the Appendix. Let us introduce constants {γ j }nj=1 as the solution of the following system
r − μi −
n n 1 2 σi j = σi j γ j , 2 j=1
i = 1, . . . , m.
(19)
j=1
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Fig. 2. Trajectories of Zt(1) and Zα(1) (t ). Here, the parameters are: α = 0.9, T = 1, m = n = 10, Z0(i ) = 1, μi ∈ [−2, 3], σ ij ∈ [0, 0.6], ∀i, j = 1, . . . , 10.
Theorem 3. Let ≥ 0. Let Q be the probability measure defined as
The relative entropy of Q is equal to
Q ( A ) = C
exp A
n
n 1 2 γiW (i) (Sα (T )) − + γi Sα (T ) dP, 2
i=1
D=−
2
i=1
i=1
On the other hand, the relative entropy of Q is equal to
i=1
(20) where 1 n
dQ 1 dP = E (Sα (T )) γi2 . dP 2 n
log
n
A ∈ F and C = [E (exp{ i=1 γiW (i ) (Sα (T )) − ( + γi2 )Sα (T )} )]−1 is the normalizing constant. Then Zα (t), t
dQ dP dP n n 1 2 = log E exp γiW (i) (Sα (T )) − + γi Sα (T ) 2
D = −
∈ [0, T], is Q -martingale.
For the proof see the Appendix. Two Fundamental Theorems of Asset Pricing imply that
+
log
i=1
Corollary 1. The market model in which the asset prices follow the multidimensional subdiffusive GBM Zα (t), has no arbitrage and is incomplete.
= log E E
Q (A ) =
exp A
n i=1
γiW (i) (Sα (T )) −
1 2
n
γi2 Sα (T ) dP, A ∈ F.
i=1
(21) Then the relative entropy for the measure Q is less than for the measure Q , > 0. Proof. Clearly, Q is the special case of Q , defined in (20) for = 0 and C = 1. Thus, from Theorem 2 we have that Zα (t) is a Qmartingale.
n
γiW (i) (Sα (T )) −
+
n 1 2 γi Sα (T ) 2
i=1
n 1 2 + γi E (Sα (T )) 2 i=1
n 1 2 = log E (exp{− Sα (T )}E (Y (t ))) + + γi E (Sα (T )) 2
i=1
n 1 2 = log E (exp{− Sα (T )} ) + + γi E (Sα (T )) 2
1 ≥ E (Sα (T )) 2
exp
× exp{− Sα (T )}|Gt
Lemma 1. Let us define a probability measure Q as
i=1
i=1
Market incompleteness means that there is no unique fair price of financial derivatives. Unfortunately, the incontrovertible approach to the best choice of the corresponding martingale measure does not exist. However the martingale measure can be chosen according to the criterion of minimal relative entropy, meaning that the best choice of measure Q minimizes the distance from measure P [10].
i=1
n 1 2 + γi E (Sα (T )) 2
n
i=1
γi2 = D.
i=1
Thus, Q minimizes relative entropy.
Remark 1. Note that for α = 1 the measure Q defined in (21) reduces to the martingale measure in the standard B-S model. Under the above arguments, further on we will find the fair prices using the martingale measure Q. In the next theorem we determine the fair price of a basket option in the subdiffusive B-S model. Theorem 4. Let the assets prices follow Zα (t). Then the corresponding fair price CBsub (Z , K, T , σ , α ) of a basket option satisfies −S 0
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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], Z0(i ) = 1, ∀i, j, K = 40. The results were obtained using (22) from 10 0 0 simulated independent realizations of the random variable Sα (T).
CBsub −S (Z0 , K, T , σ , α ) = E (CB−S (Z0 , K, Sα (T ), σ )) ∞ = CB−S (Z0 , K, x, σ )T −α gα (x/T α )dx,
(22)
0
where CB−S (Z0 , K, T , σ ) is price of the basket option in the standard multidimensional B-S model, gα (z) stands for the PDF of Sα (1) and can be expressed using Fox function
10 gα (z ) = H11 z|((10,−1α) ,α ) .
(Z0 , K, T , σ , α ) = E
Q
ωi Zα(i) (T ) − K
exp
n
i=1
=E E
exp
n
n 1 2 γi Sα (T ) 2
×
i=1
m i=1
m
ωi Zα(i) (T ) − K
i=1
+
γiW (i) (Sα (T ))
i=1
sub PBsub −S (Z0 , K, T , σ , α ) = CB−S (Z0 , K, T , σ , α ) + K − Zα (0 ).
γiW (i) (Sα (T ))
i=1
The put-call parity applied to the multidimensional subdiffusive B-S model, with r = 0 is as follows
Thus the subdiffusive price of the put option yields
+
n 1 2 − γi Sα (T ) 2
−
i=1
=E
m
mate CBsub (Z , K, T , σ , α ) using Monte Carlo methods. −S 0
sub CBsub −S (Z0 , K, T , σ , α ) − PB−S (Z0 , K, T , σ , α ) = Zα (0 ) − K, .
Proof. The arbitrage-free rule of pricing requires that
CBsub −S
to approximate first the classical price CB−S (Z0 , K, x, σ ) using Gentle’s approximation (already given in (8), see [6,17]). Next, one ∞ needs to approximate the integral 0 CB−S (Z0 , K, x, σ )gα (x, T )dx using some standard method. Another possibility is to use the fact that CBsub (Z , K, T , σ , α ) = E (CB−S (Z0 , K, Sα (T ), σ )) and to approxi−S 0
ωi Zα(i) (T ) − K |Sα ( T ) +
= E (CB−S (Z0 , K, Sα (T ), σ )) ∞ = CB−S (Z0 , K, x, σ )gα (x, T )dx. 0
Here by gα (x, T) we denote the PDF of Sα (T). since Sα (T) is α selfsimilar, we obtain gα (x, T ) = T −α gα (x/T α ). Thus, the statement follows. Remark 2. There is no explicit formula for CB−S (Z0 , K, T , σ ), therefore, in order to find the basket option price in the subdiffusive case, it is necessary to use the approximation methods. One needs
Remark 3. The derivatives prices with payoff functions, different than European basket option, can be calculated using analogous approach. The above result allows us to find fair price of basket option in multidimensional subdiffusive B-S model. For instance, for α = 1/2 the Fox function can be evaluated as follows [18]
z2 1 g1/2 (z ) = √ exp − , z ≥ 0. 4 π The price of a basket option in the classical Multidimensional BS model can be found using Gentle’s approximation by geometric average, mentioned in previous chapter. The approximate value of the classical B-S formula is given by (8). Thus, one can estimate the value of CBsub by numerical integration of expression in formula −S (22). Another method of finding the price of the basket option is by using well-known Monte Carlo methods. It is only needed to simulate the realization of Sα (T). From the self-similarity property α of Sα (t) it is evident that Sα (T) is equal in distribution to U T(1 ) , α where Uα (τ ) is α -stable subordinator. Uα (1) can be generated in the standard way, see [5]. Thus, one only needs to simulate Uα (1) and estimate the expectations in formula (22) using Monte-Carlo method. In Fig. 3 prices of the basket option, exercised in European call style, depending on the maturity time for different α are
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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], Z0(i ) = 1, ∀i, j, K = 40. The results were obtained using (22) from 10 0 0 simulated independent realizations of the random variable Sα (T).
presented. Recall that when α = 1 we recover a classical B-S formula for multidimensional case. In Fig. 4 prices of the European basket option, depending on the strike price for different α are presented. Obviously, the price of call option decreases when the strike price increases: if the buyer of the call option gets the right to buy the portfolio of assets for the higher strike price, then the price if the option should be lower. As Fig. 4 shows, the price of the option in subdiffusive regime is lower than the classical one for any strike price while the other parameters are fixed. It should be noted that the results for α = 1/2 in Figs. 3 and 4, obtained by method of numerical approximation of the integral and Monte-Carlo method, coincide.
jectories of heavy-tailed waiting times (periods of stagnation). The other parameters can be estimated under the same approach as in the classical multidimensional model after elimination of the subordinating effects. It should be noted that the work can be extended to the arbitrary choice of inverse subordinators, whose nature, parameters and characteristics can be estimated empirically from given specific market. Since the inspiration for the derivation of the celebrated B-S formula came from physics, we believe that the similar situation can be observed in financial engineering in the context of fractional calculus. Fractional operators, which are successfully used in statistical physics to model anomalous fractional dynamics should find important applications also in finance.
4. Conclusions In this paper we introduced the concept of multidimensional B-S model generalized to the subdiffusive case. We derived the corresponding multidimensional fFPE, which gives the information about the behavior of the PDF of the analyzed process [13,15,16]. One can take advantage of this equation in order to investigate any other specific properties of the process. We showed that the considered market is arbitrage-free. Since there is more than one martingale measure, the market is incomplete. The latter means that there is no unique fair price of the financial derivatives on the specified market. Unfortunately, the incontrovertible approach to the best choice of the corresponding martingale measure does not exist. However the martingale measure was chosen according to the criterion of minimal relative entropy. Moreover, the chosen martingale measure extends in a natural way the martingale measure from the standard B-S model. It can be recovered as α → 1. We derived the subdiffusive B-S formula for basket option. It was necessary to use the approximation methods to find its value. Here we used Monte Carlo methods and numerical integration. In order to apply the introduced model to a real financial market, one needs to estimate required parameters α , {σ ij }m×n and { μi } m . The parameter α can be estimated from the extracted trai=1
Acknowledgement The research of Marcin Magdziarz was partially supported by the Ministry of Science and Higher Education of Poland program Iuventus Plus no. IP2014 027073.
Appendix Proof of Theorem 2 Let us introduce the filtration {Hτ }τ where
Hτ = ∩s>τ {σ (W (z ) : 0 ≤ z ≤ s ) ∨ σ (Sα (z ) : z ≥ 0 )}.
≥ 0,
(23)
Firstly, note that {Hτ } is right-continuous, thus the random variable Sα (t0 ) is a Markov stopping time w.r.t. {Hτ } for each t0 ∈ (0, T]. Define
Gt = HSα (t ) .
(24)
Clearly, Wt is {Ht }-martingale. Let us show that C(t) is also {Ht }martingale.
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E (C (t )|Hs ) = E
n
are martingales w.r.t. Ht . Next, for every A ∈ Ht the following holds
(i )
γiW (t )|Hs
i=1 n
=
n
E (γiW (i ) (t )|Hs ) =
i=1
γiW (i) (s ) = C (s ).
i=1
Therefore, C(t) is {Ht }-martingale, C (t ) ∼ N (0, t the following {Hτ }-stopping times
n
i=1
E {C (Tn ∧ Sα (t ))|Gs } → E {C (Sα (t ))|Gs }
Y (t ) = exp
γ jWt( j ) −
j=1
1 2
Z (t ) = (Z
(t ), . . . , Z
=
E (1A e− Sα (T )Y (t )),
t < Sα ( T )
− S α ( T )
t ≥ Sα ( T )
E (1A e
= E (1A e
− S α ( T )
E Q sup ZS(αi )(T ) (t )
= E Q
t≥0
=E
Y (t )Z (t ) = exp
−
j=1
n
n
E (exp{λSα (T )} ) =
γi2 Sα (t )
(25)
= exp
n 1 2 γi Sα (T ) 2
exp
i=1
n
n
sup Z (i ) (t )
t≤Sα (T )
γiW (i) (Sα (T )) e|μi |Sα (T ) ( j)
σi jW (Sα (t )) .
(27)
j=1
j=1
( j)
(σi j + γ j )Wt
E
n
∞
( T α λ )n <∞
( nα + 1 ) n=0
γiW (i) (Sα (T ))
i=1
E
exp
sup exp t≤T
1 Sα ( T ) γ j2 2 n
< ∞.
n 1 − ( σi j + γ j ) 2 t . 2
n
2
σi jW ( j ) (Sα (t ))
j=1
≤ 4E
exp 2
n
σi jW ( j ) (Sα (T ))
< ∞.
j=1
This implies
ZSα (T ) (t ) = Z (t ∧ Sα (T )).
E Q sup ZS(αi )(T ) (t ) < ∞. t≥0
Therefore, ZS(i )(T ) (t ) are martingales. Additionally, they are uniα formly integrable. Thus there must exist a sequence {X (i ) }m with i=1 the following property ZS(i )(T ) (t ) = E Q (X (i ) |Ht ) and α
Then we get that the processes t≥0
exp
j=1
j=1
n!
=
Additionally, as exp{ nj=1 σi j W ( j ) (Sα (t ))} is a non-negative submartingale, we apply the Doob’s inequality and obtain the following
This implies that the processes Y(t)Z(i) (t), ∀i are {Ht }-martingales w.r.t. P. Introduce the process
e− Sα (T )Y (t ∧ Sα (T ))ZS(αi )(T ) (t )
λn E (Sαn (T ))
j=1
∞
for any λ > 0. Now, using the conditioning argument we arrive at
=E
n n n 1 2 1 2 σi j + σi j γ j + γj t 2 2
j=1
+
n=0
j=1
σi j γ j ), then
(σi j + γ j )Wt( j )
n
γiW (i) (Sα (T ))
i=1
t≤T
j=1
n
n (T )) = Using the already mentioned formula for moments E (Sα T nα n !
(nα +1 ) , n ∈ N, we obtain
j=1
+
× sup
j=1
n
t≤Sα (T )
exp
−
n n 1 2 Y (t )Z (i ) (t ) = exp γ j t + (σi j + γ j )Wt( j ) . μi −
(i )
sup Z (i ) (t )
i=1
and setting λ = 1, from Theorem 1 we know that Y(Sα (t)) is a (Gt , P)-martingale. The following holds
σi2j +
≤E
i=1
j=1
Y (t ∧ Sα (T ))).
j=1
A(t ), A(t ) = C (Sα (t )), C (Sα (t )) =
n
Y (Sα (T ))),
This implies that ZSα (T ) (t ) is a martingale w.r.t. (Ht , Q ). Moreover
γ j2t ,
Using the fact
Set μi = −( 21
γiW (i) (Sα (T ))
|Ht
(t )).
2
i=1
= E (1A e− Sα (T ) (E (Y (Sα (T ))|Ht ))
j=1
(m )
exp
2
n
i=1
n ( j) (i ) Z (t ) = exp μi t + σi jWt , (1 )
i=1
as n → ∞. Finally, we obtain E {C (Sα (t ))|Gs } = C (Sα (s ))), thus A(t ) = C (Sα (t )) is a {Gt }-martingale. Using Prop. 3.4, Chap. 4 in [20] we obtain that the process Y(t) is a local martingale. Additionally, E (supo≤u≤t Y (u )) < ∞. This implies that Y(t) is also a martingale. Proof of Theorem 3: Let us put n
i=1
n 1 2 − γi Sα (T ) 2
E {C (Tn ∧ Sα (t ))|Gs } = C (Tn ∧ Sα (s )). Now, we are in position to use the Lebesgue dominated convergence theorem, which yields
n 1 2 γiW (i) (Sα (T )) − + γi Sα (T )
= E 1A e− Sα (T ) E
γi2 ). Define
Note that Tn ∞ when n → ∞. Additionally C(Tn ∧τ ) is a martingale, Moreover, it is bounded by n. Therefore using Doob’s theorem we get for s < t
n
n
Tn = inf{τ > 0 : |C (τ )| = n}.
Q (A ) = E 1A exp
(26)
Zα(i ) (t ) = ZS(αi )(T ) (Sα (t )) = E Q (X (i ) |HSα (t ) ).
Please cite this article as: G. Karipova, M. Magdziarz, Pricing of basket options in subdiffusive fractional Black–Scholes model, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.05.013
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Lastly, Zα (t) is a martingale w.r.t. (HSα (t ) , Q ) Note that for each ≥ >0 we obtain different measure Q . So, there is no unique martingale measure for Zα (t). References [1] Black F, Scholes M. The pricing of options and corporate liabilities. J Pol Econ 1973;81:637–54. [2] Company R, Jodar L. Numerical solution of Black–Scholes option pricing with variable yield discrete dividend payment. Banach Cent Publ 2008;83:37–47. [3] Cont R, Tankov P. Financial modeling with jump processes. Chapman & Hall/CRC, Boca Raton; 2004. [4] Elizar I, Klafler J. Spatial gliding, temporal trapping and anomalous transport. Physica D 2004;187:30–50. [5] Janicki A, Weron A. Simulation and chaotic behaviour of α -stable stochastic processes. New York: Marcel Dekker; 1994. [6] Krekel M, Kock J, Korn R, Man T-K. An analysis of pricing methods for baskets options. Wilmott Mag 2004;3:82–9. [7] Li GH, Zhang H, Luo MK. A multidimensional subdiffusion model: an arbitrage-free market. Chin Phys B 2012;21:128901. [8] Magdziarz M, Weron A. Anomalous diffusion and semimartingales. EPL 20 09;86:60 010. [9] Magdziarz M. Black–Scholes formula in subdiffusive regime. J Stat Phys 2009;136:553–64. [10] Magdziarz M, Gajda J. Anomalous dynamics of Black–Scholes model time changed by inverse subordinators. ACTA Physica Polonica 2012;B 43:1093–110. [11] Magdziarz M. Path properties of subdiffusion - a martingale approach. Stochastic Models 2010;26:256–71.
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Please cite this article as: G. Karipova, M. Magdziarz, Pricing of basket options in subdiffusive fractional Black–Scholes model, Chaos, Solitons and Fractals (2017), http://dx.doi.org/10.1016/j.chaos.2017.05.013