Foad Shokrollahi
Option pricing in 
fractional models
aaa
ACTA  WASAENSIA 425
ACADEMIC
DISSERTATION
To
be
presented,
with
the
permission
of
the
Board
of
the
School
of
Technology
and

Innovation
of
the
University
of
Vaasa,
for
public
examination
in
Auditorium
Nissi

(,)
on
the
28UI
of
June,
2019,
at
noon.
Reviewers Docent Ehsan Azmoodeh
Ruhr Universitat Bochum
Fakulta¨t fur Mathematik
Universitatsstr. 150
44780 Bochum
GERMANY
Docent Dario Gasbarra
University of Helsinki
Department of Mathematics and Statistics
P.O. Box 68
FIN-00014 Helsingin yliopisto
FINLAND
III 
Julkaisija  Julkaisupäivämäärä 
Vaasan yliopisto Kesäkuu 2019 
Tekijä(t)  Julkaisun tyyppi  
Foad Shokrollahi Artikkeliväitöskirja 
ORCID tunniste Julkaisusarjan nimi, osan numero 
https://orcid.org/0000-0003-1434-0949 Acta Wasaensia, 425 
ISBN Yhteystiedot  
Vaasan Yliopisto 
Tekniikan ja innovaatiojohtamisen 
yksikkö  
MatemaattisHWWLHWHHW 
PL 700 
65101 Vaasa 
978-952-476-869-6 (painettu)
978-952-476-870-2 (verkkoaineisto)
URN:ISBN:978-952-476-870-2 
ISSN  
0355-2667 (Acta Wasaensia 425, painettu) 
2323-9123 (Acta Wasaensia 425, 
verkkoaineisto) 
Sivumäärä Kieli 
88 Englanti 
Julkaisun nimike  
Optioiden hinnoittelu fraktionallisissa Malleissa 
Tiivistelmä 
Väitöskirja tarkastelee fraktionaalisen Black–Scholes -mallin ja sekoitetun 
fraktionallisen Black–Scholes -mallin käyttöä erityyppisten optioiden 
arvottamisessa.  Tätä tutkitaan neljässä artikkelissa.   Ensimmäisessä artikkelissa 
tarkastellaan geometrisia aasialaisia optioita ja potenssioptioita, kun osakehinta 
noudattaa aikamuunnettua sekoitettua fraktionaalista mallia.  Tässä mallissa 
sekoitun fraktionaalisen Black–Scholes -mallin käänteinen subordinaattoriprosessi 
korvaa fysikaalisen ajan.  Kolmannen artikkelin tarkoitus on hinnoitella 
eurooppalainen valuuttaoptio fraktionaalisen Brownin liikkeen mallissa 
aikamuunnetulla strategialla.  Lisäksi aika-askeleen ja pitkän aikavälin riippuvuuden 
vaikutusta tutkitaan transaktiokulujen alaisuudessa. 
Ehdollinen keskiarvosuojaaminen fraktionaalisessa Black–Sholes -mallissa on toisen 
artikkelin aihe.  Ehdollinen keskiarvosuojaus eurooppalaiselle vaniljaoptiolle, jolla 
on konveksi tai konkaavi positiivinen tuottofunktio transaktiokulujen vallitessa, on 
artikkelin päätulos.  Neljännessä artikkelissa tutkitaan eurooppalaisia optioita 
diskreetissä ajassa mallissa, joka on hypyllinen sekoitettu fraktionaalinen Brownin 
liike.  Käyttäen keskiarvoista deltasuojausstrategiaa artikkelissa johdetaan 
hinnoittelumalli eurooppalaisille optioille transaktiokulujen vallitessa. 
Asiasanat 
Optioiden hinnoittelu, Stokastinen mallinnus, Matemaattinen rahoitusteoria, 
Fraktionaaliset mallit, Fraktionaalinen Brownin liike 

V 
Publisher  Date of publication 
Vaasan yliopisto June 2019 
Author(s)  Type of publication  
Foad Shokrollahi Doctoral thesis by publication 
ORCID identifier Name and number of series 
https://orcid.org/0000-0003-1434-0949 Acta wasaensia, 425 
ISBN Contact information 
University of Vaasa 
School of Technology and Innovation 
Mathematics and Statistics 
P.O. Box 700 
FI-65101 Vaasa 
Finland 
978-952-476-869-6 (print)
978-952-476-870-2 (online)
URN :ISBN :978-952-476-870-2 
ISSN  
0355-2667 (Acta wasaensia 425, print) 
2323-9123 (Acta wasaensia 425, online) 
Number of pages Language 
88 English 
Title of publication  
Option pricing in fractional models 
Abstract 
This thesis deals with application of the fractional Black-Scholes and mixed fractional 
Black-Scholes models to evaluate different type of options. These assessments are 
considered in four individual papers.  In the first articles, the problem of geometric 
Asian and power options pricing is investigated when the stock price follows a time 
changed mixed fractional model. In this model, an inverse subordinator process in the 
mixed fractional Black-Scholes model replaces the physical time. The aim of the third 
paper is to evaluate the European currency option in a fractional Brownian motion 
environment by the time-changed strategy. Also, the impact of time step and long 
range dependence are obtained under transaction costs. 
Conditional mean hedging under fractional Black-Scholes model is the propose of the 
second article. The conditional mean hedge of the European vanilla type option with 
convex or concave positive payoff under transaction costs is obtained. In the fourth 
article, the mixed fractional Brownian motion with jump process are incorporated to 
analyze European options in discrete time case. By a mean delta hedging strategy, the 
pricing model is proposed for European option under transaction costs. 
Keywords 
Option pricing , Stochastic modeling, Mathematical finance, Fractional model, 
Fractional Brownian motion 

VII
ACKNOWLEDGEMENTS
I would like to first thank my supervisor Prof Tommi Sottinen, for his constant
encouragement, generosity, and friendship during the period in which this work
was completed. I owe him almost completely for the opportunity to devote myself
to mathematics and work that was meaningful to me, and will be forever in his debt.
I also would like to extend my gratitude to all the members of the Department of
Mathematics and Statistics of the University of Vaasa for the friendly working
environment. Furthermore, I am thankful to the University of Vaasa for their
financial support which made it possible to attend a number of conferences and
workshops.
I wish to thank my pre-examiners Docent Dario Gasbarra and Docent Ehsan
Azmoodeh for reviewing my thesis and for their feedbacks.
Finally, I must thank my family for their constant support. In particular, I thank
my wife, Arezoo, and my lovely daughter, Lara for their patience, friendship, and
understanding while I completed this work.
Vaasa, May 2019
Foad Shokrollahi

IX
CONTENTS
1 INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 STOCHASTIC PROCESSES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 General facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Le´vy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Long-range dependence and self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Gaussian processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4.2 Fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4.3 Mixed fractional Brownian motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 ESSENTIALS OF STOCHASTIC ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 Itoˆ’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Girsanov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 FUNDAMENTAL ELEMENTS OF STOCHASTIC FINANCE . . . . . . . . . . . . 15
4.1 Useful financial terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Classical Black-Scholes market model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 PDE approach in option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.4 (Mixed) Fractional Black-Scholes market model . . . . . . . . . . . . . . . . . . . . . . 19
4.4.1 Fractional Black-Scholes market model . . . . . . . . . . . . . . . . . . . . . . . 19
4.4.2 Mixed fractional Black-Scholes market model . . . . . . . . . . . . . . . . 19
5 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.1 Summaries of the articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
XReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
XI
LIST OF PUBLICATIONS
The dissertation is based on the following three refereed articles:
(I) Shokrollahi, F. (2018). The evaluation of geometric Asian power options
under time changed mixed fractional Brownian motion. Journal of Computa-
tional and Applied Mathematics 344, 716–724.
(II) Shokrollahi, F., and Sottinen, T. (2017). Hedging in fractional Black-Scholes
model with transaction costs. Statistics & Probability Letters. 130, 85–91.
(III) Shokrollahi, F. (2018). Subdiffusive fractional Black-Scholes model for pri-
cing currency options under transaction costs. Cogent Mathematics & Statis-
tics 5, 1470145.
(IV) Shokrollahi, F. (2018). Mixed fractional Merton model to evaluate European
options with transaction costs. Journal of Mathematical Finance 8, 623–639.
All the articles are reprinted with the permission of the copyright owners. (permis-
sion is needed from the publisher!)

XIII
AUTHOR’S CONTRIBUTION
Publication I: “The evaluation of geometric Asian power options under time
changed mixed fractional Brownian motion”
This is an independent work of the author.
Publication II: “Hedging in fractional Black—Scholes model with transaction
costs”
This article is the outcome of a joint discussion and all the results are a joint work
with Tommi Sottinen.
Publication III: “Subdiffusive fractional Black-–Scholes model for pricing cur-
rency options under transaction costs ”
This is an independent work of the author.
Publication IV: “Mixed fractional Merton model to evaluate European options
with transaction costs”
This is an independent work of the author.

1 INTRODUCTION
In recent years, options have received increasing attention and their role has grown
rapidly in most trading exchanges. Most common examples of variables underlying
options are the price of stocks, bonds or commodities traded on the exchange. Ho-
wever, options can depend on almost any variable, from the price of pork bellies to
the amount of rainfall in a certain geographic area.
Standardized options on stock prices have been traded at the exchange since 1973.
There are two basic types of options. A call option gives the holder the right (and
not the obligation) to buy the underlying asset at a certain time in the future for a
certain price. A put option gives the holder the right to sell the underlying asset
at a certain time in the future for a certain price. The price for which the asset is
being exchanged is referred to as the strike price or the exercise price. The future
time when the exchange takes place is referred to as the maturity of the option or
the expiration date. Based on the exercise conditions, options are categorized into
European options and American options. European options can only be exercised at
maturity. American options can be exercised anytime during the life of the option.
Throughout this section options are assumed European unless otherwise specified.
Depending on the strategy, options trading can provide a variety of benefits, inclu-
ding the security of limited risk and the advantage of leverage. Another benefit
is that options can protect or enhance your portfolio in rising, falling and neutral
markets.
Since it appeared in the 1970s, the Black-Scholes (BS) model (Black & Scholes
(1990)) has become the most popular method to option pricing and its generalized
version has provided mathematically beautiful and powerful results on option pri-
cing. However, they are still theoretical adoptions and not necessarily consistent
with empirical features of financial returns, such as nonindependence, nonlinearity,
ect. For example, Hull and White (Hull & White (1987)) introduced a bivariate
diffusion model for pricing options on assets with stochastic volatilities. Heston
(Heston (1993)) proposed affine stochastic volatility. Furthermore, since discon-
tinuity or jumps is one of the significant component in financial asset pricing (see
Andersen, Benzoni & Lund (2002), Chernov, Gallant, Ghysels & Tauchen (2003),
Pan (2002), Eraker (2004)) and also some scholars have been represented pricing
models based on the jump processes (see Merton (1976), Kou (2002), Cont & Tan-
kov (2004), Ahn, Cho & Park (2007), Ma (2006)).
Many realistic models have been described long memory behavior in financial time
series (Lo, A. W. (1991), Willinger, W., Taqqu, M. S., & Teverovsky, V. (1999)
(1999), Cont, R. (2005), Dai & Singleton (2000), Berg & Lyhagen (1998), Hsieh
(1991), Huang & Yang (1995)). Since, fractional Brownian motion ( f Bm) is a self-
similar and long-range dependence process, then it can be a appropriate candidate to
capture these phenomena (Wang, Zhu, Tang & Yan (2010), Wang (2010), Sottinen
2 Acta Wasaensia
(2003), Sottinen & Valkeila (2003), Wang, Wu, Zhou & Jing (2012), Xiao, Zhang,
Zhang & Wang (2010), Zhang, Xiao & He (2009), Cartea & del Castillo-Negrete
(2007)). Kolmogorov introduced the f Bm in 1940. A representation theorem for
Kolmogorov’s process was introduced by Mandelbrot and Van Ness (Mandelbrot &
Van Ness (1968)). Nowadays, the f Bm process play a significant role in stochastic
finance and different extensions of the fractional Black-Scholes formulas for pricing
options based on the geometric fractional Brownian motion are proposed to capture
the behavior of underlying asset (Bayraktar, Poor & Sircar (2004), Meng & Wang
(2010)).
On the mathematical side, f Bm is neither a semi martingale nor a Markov process
(except in the Brownian motion case). Hence, the classical stochastic integration
theory developed for semimartingale is not handy to analyze financial markets based
on fractional Brownian motion (Hu, Y., & Øksendal, B. (2003)). Further, some aut-
hors discussed arbitrage under fractional Black-Scholes model and proposed some
restrictions to exclude arbitrage in fractional markets (see Bender, Sottinen & Val-
keila (2007), Bender, C., Sottinen, T., & Valkeila, E. (2008) (2008), Bender &
Elliott (2004), Bjo¨rk & Hult (2005)).
To better describe long memory property and fluctuations in the financial assets, the
mixed fractional Brownian motion (mfBm) was presented (see El-Nouty (2003),
Mishura (2008), Cheridito, P. (2001), Zili (2006)). A mfBm is a family of Gussian
process which is a linear combination of Brownian motion and independent f Bm
with Hurst parameter H ∈ (12 ,1). The pioneering work to apply the mfBm in finance
was presented by Cheridito, P. (2001). He proved that for H ∈ (34 ,1), the mfBm
is equivalent to one with Brownian motion, and then its free of arbitrage. For H ∈
(12 ,1), Mishura and Valkeila (Mishura & Valkeila (2002)) proved that the mixed
model is arbitrage-free.
Acta Wasaensia 3
2 STOCHASTIC PROCESSES
In this section, we provide some definitions and auxiliary facts that are needed in
this thesis (for further details, see Shiryaev, A. N., do Rosa´rio Grossinho, M.,
Oliveira, P. E., & Esquı´vel, M. L. (2006) (2006), Fo¨llmer, H., & Schied, A. (2011),
Shiryaev, A. N. (1999), Kallenberg, O. (2006), Clark & Ghosh (2004), Melnikov,
Pliska (1997), Mikosch (1998)). Throughout this section all random objects are
defined in the probability space (Ω,F ,P).
2.1 General facts
Definition 2.1. Let T ⊆ [0,+∞) be an interval. A stochastic process X indexed by
interval T is a collection of random variables (Xt)t≥0.
Also, for every ω ∈Ω, the real valued function t ∈ T → Xt(ω) is called a trajectory
or a sample path of the process X .
Definition 2.2. A filtration is a family (Ft)t≥0 of σ -algebrasFt ⊆F such that
∀0≤ s≤ t ⇒Fs ⊆Ft .
Definition 2.3. Let (Ft)t≥0 be a filtration. A stochastic process X = (Xt)t≥0 is said
to be adapted to the filtration (Ft)t≥0 if for every t ≥ 0, the random variable Xt is
Ft-measurable.
Definition 2.4. An stochastic process (Xt)t≥0 is called a martingale with respect to
the filtration (Ft)t≥0 ( and probability measure P) if the following conditions are
satisfied:
(i) Xt isFt-measurable for all t ≥ 0,
(ii) E[|Xt |]< ∞ for all t ≥ 0, and
(iii) E[Xt |Fs] = Xs for all 0≤ s≤ t.
Definition 2.5. AnFt - adapted stochastic process is called a local martingale with
respect to the given filtration (Ft)t≥0 if there exists an increasing sequence ofFt -
stopping times τk such that
τk → ∞ as k → ∞,
4 Acta Wasaensia
and
(Xt∧τk)t≥0, (2.1)
is anFt- martingale for all k, where t ∧ τk = min(t,τk).
Definition 2.6. A process X = (Xt)t≥0 is called an (Ft)t≥0- semi-martingale, if it
admits the representation
Xt = X0+Mt +At , (2.2)
where M is an (Ft)t≥0- local martingale with M0 = 0, A is a process of locally
bounded variation and adapted to filteration (Ft)t≥0, X0 isF0-measurable.
Definition 2.7. (Ho¨lder continuous)
Let α ∈ (0,1]. A function f : R → R is said to be locally α-Ho¨lder continuous at
x ∈ R, if there exists ε > 0 and c = cx such that
| f (x)− f (y)| ≤ c|x− y|α , for all y ∈ R with |y− x|< ε.
Definition 2.8. A stochastic process X = (Xt)t≥0 is said to have stationary incre-
ments if for all s≥ 0, and every h > 0,
(Xt −Xs)t≥0 f.d.= (Xt+h−Xs+h)t≥0. (2.3)
Here
f.d.
= denotes equality in finite dimensional distribution.
Definition 2.9. A stochastic process X = (Xt)t≥0 is said to have independent incre-
ments if for every t ≥ 0 and any choice ti ∈ T with t0 < t1 < ... < tn and n≥ 1,
Xt2 −Xt1 , ...,Xtn −Xtn−1 (2.4)
are independent random variables.
2.2 Le´vy processes
Le´vy processes are stochastic processes with independent and stationary incre-
ments.
Definition 2.10. Le´vy process (Xt)t>0 is a process with the following properties
Acta Wasaensia 5
(1) Independent increments,
(2) Stationary increments, and
(3) Continuous paths in probability: That is limh→0P(|Xt+h −Xt | ≥ ε) = 0 for
any ε > 0.
Definition 2.11. (Subordinator process)
A subordinator is a real-valued Le´vy process with nondecreasing sample paths.
Definition 2.12. (Stable process)
A stable process is a real-valued Le´vy process (Xt)t≥0 with initial value X0 = 0 that
satisfies the self-similarity property
(Xat)t≥0
f.d.
= (a1/αXt)t≥0 ∀t > 0.
The parameter α is called the exponent of the process.
Definition 2.13. (Poisson process)
A Poisson process (Xt)t≥0 satisfies the following conditions:
(1) X0 = 0,
(2) Xt −Xs are integer valued for 0≤ s < t < ∞ and
P(Xt −Xs = k) = λ
k(t− s)k
k!
e−λ (t−s) for k = 0,1,2, ... (2.5)
(3) The increments Xt2 −Xt1 , Xt4 −Xt3 ... and Xtn −Xtn−1 are independent for every
0≤ t1 < t2 < t3 < t4 < ... < tn.
Example 2.14. The fundamental Le´vy processes are the Brownian motion (defined
later) and the Poisson process. The Poisson process is a subordinator, but is not
stable; the Brownian motion is stable, with exponent α = 2.
2.3 Long-range dependence and self-similarity
Definition 2.15. Let (Xt)t≥0 be a process with stationary trajectories and (rn)n∈N
the autocovariance sequence defined by
∀n ∈ N, rn = E[Xn+1X1]. (2.6)
6 Acta Wasaensia
Then, the process X = (Xt)t≥0 is called long-range dependence if
∑
n∈N
rn = ∞.
Remark 2.16. Since (Xt)t≥0 is a process with stationary trajectories
∀s≥ 0,∀n ∈ N, rn = E[Xn+sXs].
Definition 2.17. Let H ∈ (0,1]. A stochastic process X = (Xt)t≥0 is said to be
self-similar with exponent H, if for any a > 0,
(Xat)t≥0
f.d.
= (aHXt)t≥0.
Acta Wasaensia 7
2.4 Gaussian processes
Definition 2.18. A stochastic process (Xt)t∈T is Gaussian if all finite dimensional
projections (Xt)t∈T0 , T0 ⊂ T finite, are multivariate Gaussian.
2.4.1 Brownian motion
Definition 2.19. Brownian motion is a process (Bt)t≥0 with the following proper-
ties:
(1) B0 = 0,
(2) Bt has independent increments,
(3) Bt −Bs ∼ N(0, t− s) for s < t, here N is the normal distribution function.
Definition 2.20. (Markov Process)
The process (Xt)t∈T is a Markov process if
E[ f (Xt)|Fs] = E[ f (Xt)|Xs], ∀t > s, t,s ∈ T,
whereFs = σ{Xu;u≤ s}, and f is a bounded Borel function.
2.4.2 Fractional Brownian motion
f Bm has recently become a useful choice for modeling in mathematical finance
and other sciences. On purely empirical data, some believe that f Bm is an ideal
candidate since it enjoys two important statistical features of long memory and
self-similarity. Even with its popularity, our understanding of the properties and
behaviour of f Bm is limited.
Kolmogorov (Kolmogorov (1941)) was the first to introduce the Gaussian process
which is now known as f Bm in the theory of probability. This class of processes
was studied by Kolmogorov in detail and it played an essential role in the series
of problem s of the statistical theory of turbulence. Yaglom (Yaglom (1955)) dis-
cussed the spectral density and correlation function of f Bm. A quadratic variation
formula for f Bm follows from a general result of Baxter (Baxter (1956)). Gladys-
hev (Gladyshev (1961)) extended Baxter’s result and provided a theoretical result
to determine the value of the Hurst effect denoted by H. However, most of the en-
comium to f Bm has been given to Mandelbrot and Van Ness (Mandelbrot & Van
8 Acta Wasaensia
Ness (1968)) who used f Bm to model natural phenomena such as the speculative
market fluctuations. For get more information about f Bm, you can see, Hu, Y.,
& Øksendal, B. (2003), Nualart (2006), Biagini, Hu, Øksendal & Zhang (2008),
Mishura (2008).
Definition 2.21. The fractional brownian motion with Hurst index H ∈ (0,1) deno-
ted by (BHt )t∈R, is the centered Gaussian process with covariance function
RH(s, t) =
1
2
(|t|2H + |s|2H −|t− s|2H) , s, t ∈ R.
Figure (1) shows the sample path of the f Bm for different parameter.
Figure 1. f Bm with different Hurst parameter H .
The f Bm can be represented in terms of the one-sided or two-sided standard Brow-
nin motion see (Nualart (2006)). First, we review some special function involved in
the representation results.
The Gamma function is defined by
Γ(α) =
∫ ∞
0
exp(−v)vα−1dv, α > 0.
The Beta function is defined by
β (α,β ) =
∫ 1
0
(1− v)α−1vβ−1dv, α,β > 0.
Acta Wasaensia 9
The Gauss hypergeometric function of parameters a,b,c and variable z ∈ R is defi-
ned by the formal power series
F(a,b,c,z) =
∞
∑
k=0
(a)k(b)kzk
(c)kk!
.
where (a)k = a(a+1)...(a+ k−1).
(1) The one-sided f Bm can be constructed from a one-sided Brownian motion:
Theorem 2.22. Molchan-Golosov representation (Molchan & Golosov
(1969))
For H ∈ (0,1), it holds that
BHt
=C(H)
∫ t
0
(t− s)H− 12F
(
1
2
,H− 1
2
,H+
1
2
,
s− t
s
)
dBs,
where C(H) = 2H
Γ(H+ 12 )
.
(2) The two-sided f Bm can be written in terms of one-sided Brownian motion:
Mandelbrot-Van Ness representation (Mandelbrot & Van Ness (1968))
Theorem 2.23. For H ∈ (0,1), it holds that
BHt
=C(H)
∫

(
(t− s)H− 121(−∞,t)(s)− (−s)H−
1
21(−∞,0)(s)
)
dB˜s,
here B˜ represents two-sided Brownian motion.
Theorem 2.24. (Mishura (2008)) For H 
= 12 , f Bm is neither a Markov process nor
a semimartingale.
Remark 2.25. Since f Bm is not a semimartngale,the classical integration theory de-
veloped for semimartingale is not available, Mishura (2008). Then, many scholars
introduced two different approaches for stochastic integral with respect to f Bm (1)
Pathwise approach (2) Malliavin calculus (Skorokhod integration) approach (Nual-
art (2006), Sottinen, T., & Viitasaari, L. (2016)).
Remark 2.26. Using stationarity increment of f Bm, it can be shown that the auto-
covariance function γn = of the sequence (Xn)n≥1 := (BHn+1−Bn)n≥1 is given by
γn =
n
∑
k=0
E[BH1 (B
H
k+1−BHk )] =
1
2
[
(n+1)2H −2n2H +(n−1)2H
]
, (2.7)
10 Acta Wasaensia
therefore
γn ≈ H(2H−1)n2H−2, as n→ ∞,H 
= 12 . (2.8)
Notice that when
(1) H = 12 ,γn = 0,∀n, therefore f Bm has independent increments.
(2) If H > 12 ,γn > 0, the increments of the f Bm process are positively correlated
and by p-series ∑∞n=1 |γn|= ∞, therefore has long-range dependence.
(3) If H < 12 ,γn < 0, the increments of the f Bm process are negatively correlated
and by p-series ∑∞n=1 |γn|= c < ∞, therefore has short-range dependence.
2.4.3 Mixed fractional Brownian motion
Let a and b be two real constants such that (a,b) 
= (0,0).
Definition 2.27. A mfBm with parameters a,b, and H is a process MH =
(MHt (a,b))t≥0, defined by
MHt = M
H
t (a,b) = aBt +bB
H
t , ∀t ≥ 0 (2.9)
where B is a Brownian motion and BH is an independent f Bm with Hurst parame-
ter H (Cheridito, P. (2001), van Zanten, H. (2007), Mishura (2008), Zili (2006),
Marinucci & Robinson (1999)).
Proposition 2.28. The m fBm has the following properties
(i) MH is a centered Gaussian process,
(ii) for all t ∈ R+,E((MHt (a,b))2) = a2t+b2t2H,
(iii) one has that
Cov
(
MHt (a,b),M
H
s (a,b)
)
= a2(t ∧ s)+ 1
2
b2
[
t2H + s2H −|t− s|2H
]
,∀s, t ∈ R+, (2.10)
where t ∧ s = 1/2(t+ s+ |t− s|),
(iv) the increments of the m fBm are stationary.
Acta Wasaensia 11
Lemma 2.29. (Zili (2006)) For any h > 0,(MHht (a,b))t≥0
f.d.
= (MHt (ah
1
2 ,bhH)))t≥0.
This property will be called the mixed-self-similarity.
Theorem 2.30. (Zili (2006)) For all H ∈ (0,1) − {12},a ∈ R and b ∈ R −{0},(MHt (a,b))t≥0 is not a Markov process.
Theorem 2.31. (Cheridito, P. (2001)) For H ∈ (34 ,1], the m fBm is equivalent to
Brownian motion.
Theorem 2.32. (Zili (2006)) For all a ∈ R and b ∈ R− {0}, the increments of
(MHt (a,b))t∈R+ are positively correlated if
1
2 < H < 1, uncorrelated if H =
1
2 , and
negatively correlated if 0 < H < 12 .
Lemma 2.33. (Zili (2006)) For all a ∈ R and b ∈ R− {0}, the increments of
(MHt (a,b))t∈R+ are long-range dependence if and only if H >
1
2 .
Theorem 2.34. (Ho¨lder continuity)
(Zili (2006)) For all T > 0 and γ < 12 ∧H, the m fBm has a modification which
sample paths have a Ho¨lder-continuity, with order γ , on the interval [0,T ].
12 Acta Wasaensia
3 ESSENTIALS OF STOCHASTIC ANALYSIS
3.1 Itoˆ’s lemma
Let (Xt)t≥0 be a stochastic process and suppose that there exists a real number X0
and two (Ft)t≥0-adapted processes μ = (μt)t≥0 and σ = (σt)t≥0 such that the fol-
lowing relation holds for all t ≥ 0,
Xt = X0+
∫ t
0
μsds+
∫ t
0
σsdBs. (3.1)
where the stochastic integral in (3.1) is an Itoˆ integral, such processes are called Itoˆ
diffusions. We can write the equation as follows
dXt = μtdt+σtdBt , (3.2)
Then, we can say X satisfies the SDE given by (3.2) with the initial condition X0
given. Note that the formal notation dXt = μtdt+σtdBt is only formal. It is simply
a shorthand version of the expression (3.2) above.
In option pricing, we often take as given a stochastic differential equation repre-
sentation for some basic quantity such as stock price. Many other quantities of
interest will be functions of that basic process. To determine the dynamics of these
other processes, we shall apply Itoˆ’s Lemma, which is basically the chain rule for
stochastic processes (Mikosch (1998), Tong (2012), Øksendal (2003)).
Theorem 3.1. (Itoˆ’s Lemma)
Assume the stochastic process Xt satisfies in the following equation
dXt = μtdt+σtdBt , (3.3)
where μ = (μt)t≥0 and σ = (σt)t≥0 are adapted processes to a filtration (Ft)t≥0.
Let Y be a new process defined byYt = f (t,Xt) where f (t,x) is a function twice diffe-
rentiable in its first argument and once in its second. Then Y satisfies the stochastic
differential equation:
dYt =
(∂ f
∂ t
(t,Xt)+μt
∂ f
∂x
(t,Xt)+
1
2
σ2t
∂ 2 f
∂x2
)
(t,Xt)dt+σt
∂ f
∂x
(t,Xt)dBt .
Theorem 3.2. For fBm, we have Skorokhod and Fo¨llmer types Itoˆ’s lemma ( Dun-
can, T. E., Hu, Y., & Pasik-Duncan, B. (2000) (2000), Sottinen & Valkeila (2003))
Acta Wasaensia 13
If f : R→ R is a twice continuously differentiable function with bounded derivatives
to order two, then the Skorokhod integral is
f (BHT )− f (BH0 ) =
∫ T
0
f ′(BHs )δB
H
s +H
∫ T
0
s2H−1 f ′′(BHs )ds, (3.4)
where δ is divergence operator connected to Brownian motion. For H > 12 , the
Fo¨llmer or pathwise integral is
f (BHT )− f (BH0 ) =
∫ T
0
f ′(BHs )dB
H
s , (3.5)
and for H < 12 ,
∫ T
0 f
′(BHs )dBHs does not exist as pathwise integral in general.
3.2 Girsanov’s Theorem
Definition 3.3. Two measures P and Q on a measurable space (Ω,F ) are equivalent
if
P(A) = 0⇔ Q(A) = 0, ∀A ∈F . (3.6)
The Radon-Nikodym derivative can be defined by using two equivalent measures
as follows:
Mt =
dQ
dP
|F t , (3.7)
which enables us to change a measure to another. It follows that for any random
variable X that isFt-measurable
EP[XM] =
∫
Ω
X(w)Mt(ω)dP(ω) =
∫
Ω
X(ω)dQ(ω) = EQ[X ]. (3.8)
To change the measures for Brownian motion we can use the Girsanov’s theorem.
Theorem 3.4. (Girsanov’s Theorem)
14 Acta Wasaensia
Assume we have (Ft)t≥0 be a filtration on interval [0,T ] where T < ∞. Define a
random process M:
Mt = exp
[
−
∫ t
0
λudBPu −
1
2
∫ t
0
λ 2u du
]
, t ∈ [0,T ]. (3.9)
where BP is a Brownian motion under probability measure P and λ is an (Ft)t≥0-
adapted process that satisfies the Novikov condition
E
{
exp
[1
2
∫ t
0
λ 2u du
]}
< ∞, t ∈ [0,T ]. (3.10)
If we define BQ as
BQt = B
P
t +
∫ t
0
λudu, t ∈ [0,T ], (3.11)
then the following outcomes holds:
(i) M is the Radon-Nikodym martingale
Mt =
dQ
dP
|Ft .
(ii) BQ is a Brownian motion under the probability measure Q.
Acta Wasaensia 15
4 FUNDAMENTAL ELEMENTS OF STOCHASTIC FI-
NANCE
4.1 Useful financial terminologies
Definition 4.1. (Financial derivatives)
A financial derivative is a contract whose value depends on one or more securities
or assets, called underlying assets. Typically the underlying asset is a stock, a bond,
a currency exchange rate or the quotation of commodities such as gold, oil or wheat.
Definition 4.2. The spot price (stock price) is the current price in the marketplace
at which a given asset—such as a security, commodity, or currency—can be bought
or sold for immediate delivery. The strike price will be denoted by S.
Definition 4.3. The strike (exercise) price is the price at which a derivative can be
exercised, and refers to the price of the derivative’s underlying asset. The strike
price will be denoted by K.
Definition 4.4. Expiration date (maturity time) is date on which the option can be
exercised. This will be denoted by T .
Definition 4.5. A European call (Put) option grants the right to purchase (sell) a
stock at a specific time called maturity T for a specific amount K called the exercise
price.
The value of a European call option is denoted by (ST − K)+ where (x)+ =
max(x,0). Similarly, the value of a European put option is (K−ST )+. This amount
is called the option payoff. Here, ST is the spot price at time T .
Definition 4.6. A risk free interest rate, denoted by r, is the rate of return on an
asset that possesses no risk.
Remark 4.7. A dividend payout during the life of an option will have the affect
of decreasing the value of a call and increasing the value of a put, the stock price
typically falls by the amount of the dividend when it is paid. This will be denoted
by D.
Definition 4.8. A currency option is a contract, which gives the owner the right but
not the obligation to purchase or sell the indicated amount of foreign currency at
a specified price within a specified period of time (American option) or on a fixed
date (European option).
16 Acta Wasaensia
Definition 4.9. Asian options are known as path dependent options whose payoff
depends on the average stock price and a fixed or floating strike price during a
specific period of time before maturity. There exist two types of Asian options
such as fixed strike and floating strike options. The payoff for a fixed strike price
option is (AT −K)+ and (K −AT )+ for a call and put option respectively where
K denotes the strike price, T is the strike time and AT is the average price of the
underlying asset over the predetermined interval. For a floating strike price option,
the payoffs are (ST −AT )+ and (AT − ST )+, for a call and put option respectively
where ST stands for the price of stock at time T . Further, Asian options can be also
categorized based on different averaging, namely that (i) geometric average that is
AT = exp{ 1T
∫ T
0
logStdt}.
(ii) arithmetic average
AT =
1
T
∫ T
0
Stdt.
Consider a financial market consisting of n assets with prices S1t , ...,S
n
t , which under
probability measure P are governed by the following stochastic differential equati-
ons:
dSit = μ
i
t dt+σ
i
t dB
i
t , i = 1,2, ...,n, (4.1)
where B = (B1, ...,Bn) is an n dimentional Brownian motion and, μ i and σ i are
adapted to the natural filtration of the Brownian motion B.
Next, we denote an n-dimensional stochastic process θt = (δ 1t , ...,δ nt ) as a trading
strategy, where δ it denotes the holding in asset i at time t. The value Vt(δ ) at time t
of a trading strategy δ is given by
Vt(δ ) =
n
∑
i=1
δ it S
i
t . (4.2)
Definition 4.10. A self-financing trading strategy is a strategy δ with the property:
Vt(δ ) =V0(δ )+
n
∑
i=1
∫ t
0
δ it dS
i
t , t ∈ [0,T ]. (4.3)
Definition 4.11. An arbitrage opportunity is a self-financing trading strategy δ with
Acta Wasaensia 17
(i) V0(δ )≤ 0 a.s.
(ii) VT (δ )≥ 0 a.s. and P[VT (δ )> 0]> 0.
In words, arbitrage is a situation where it is possible to make a profit without the
possibility of incurring a loss.
Definition 4.12. Time value of an option is basically the risk premium that the
seller requires to provide the option buyer with the right to buy/sell the stock up to
the expiration date.
Definition 4.13. Portfolio hedging describes a variety of techniques used by inves-
tment managers, individual investors and corporations to reduce risk exposure in an
investment portfolio. Hedging uses one investment to minimize the negative impact
of adverse price swings in another.
Hedging of options is one of the central problems in mathematical finance and it
has been studied extensively in various setups. The idea of hedging is to replicate
a claim f (ST ) by trading only the underlying asset. In mathematical terms, we are
interested in finding a predictableH such that
f (ST ) =C+
∫ T
0
HsdSs (4.4)
where the deterministic constant C is called the hedging cost. In arbitrage free
models the hedging cost can be interpreted as the fair price of the option.
Definition 4.14. A risk-neutral measure is a probability measure such that each
share price is exactly equal to the discounted expectation of the share price un-
der this measure. In other words, a risk neutral measure is any probability mea-
sure, equivalent to the market measure P , which makes all discounted asset prices
martingales. This is heavily used in the pricing of financial derivatives due to the
fundamental theorem of asset pricing, which implies that in a complete market a
derivative’s price is the discounted expected value of the future payoff under the
unique risk-neutral measure.
Definition 4.15. Greeks are the quantities representing the sensitivity of the price
of derivatives such as options to a change in underlying parameters on which the
value of an instrument or portfolio of financial instruments is dependent. The name
is used because the most common of these sensitivities are denoted by Greek letters.
Traders use different Greek values, such as delta, theta, and others, to assess options
risk and manage option portfolios.
Definition 4.16. Transaction costs are the costs incurred during trading – the pro-
cess of selling and purchasing – on top of the price of the product that is changing
hands. Transaction costs may also refer to a fee that a bank, broker, underwriter
or other financial intermediary charges. The difference between what a dealer and
buyer paid for a security is one of the transaction costs.
18 Acta Wasaensia
4.2 Classical Black-Scholes market model
In the classical Black-Scholes market model, two assets are traded continously over
the time interval [0,T ]. Denote by A the riskless asset, or bond, and by S the risky
asset, or stock. The dynamic of asset price is governed by a geometric Brownian
motion:
dAt = rtAtdt,
dSt = μtStdt+σStdBt ,
where r is a deterministic interest rate, σ > 0 is constant, B is a Brownian motion.
The function μ is the deterministic drift of the stock.
4.3 PDE approach in option pricing
Given the drift rate μ and the volatility σ , the geometric Brownian motion for the
stock price process St is given by:
dSt = μtStdt+σStdBt ,
where B is a Brownian motion.
Then, applying Itoˆ’s lemma and self financing strategy the value of an option
V (t,St) with a bounded payoff f (ST ) satisfies the following SDE:
dV =
(
∂V
∂ t
+μSt
∂V
∂St
+
1
2
σ2S2t
∂ 2V
∂S2t
)
dt+σSt
∂V
∂St
dBt
Setting up a hedging portfolio with one unit in V and Δ units of the stock St with
Δ= ∂V∂St gives us Black-Scholes PDE with the drift μ replaced by the risk-free rate
r, i.e.:
∂V
∂ t
+ rSt
∂V
∂St
+
1
2
σ2S2t
∂ 2V
∂S2t
= rV.
After delta hedging has removed the risk of the portfolio of one unit in V and Δ
Acta Wasaensia 19
units in stock St , the right stock price process to consider is the one in the risk
neutral measure, as:
dSt = rStdt+σStdBt ,
Feynman-Kac representation of SDE tell us that PDE of the Black- Scholes kind
have an equivalent probabilistic representation (see, Pascucci, A. (2011)). That
is, Feynman-Kac assures that one can solve for the price of the derivative V (t,St)
by either discretizing the Black-Scholes PDE using finite difference methods, or
by exploiting the probabilistic interpretation and using Monte Carlo methods. The
Black-Scholes PDE implies that the price of the derivative V (t,St) at time t is equi-
valent to the discounted value of the expected payoff at expiration (time T ). This is
the famous Feynman-Kac representation:
V (t,St) = EQ[e−r(T−t)V (T,ST )|Ft ].
4.4 (Mixed) Fractional Black-Scholes market model
4.4.1 Fractional Black-Scholes market model
The first try to involve the fractional Brownian motion in modeling of the market
was simply to replace the B with BH in the Black-Scholes model. To interpret the
integrals in the pathwise way (which is possible if H > 12 ), which is natural from the
point of view. In this case the dynamic of asset price in the fractional Black-Scholes
market model is given by
dSt = μtStdt+σStdBHt .
4.4.2 Mixed fractional Black-Scholes market model
If one wants to introduce an economically meaningful market model with long
range dependent returns, mixed models are a good option. In such models, one
can have both long range dependence and no-arbitrage in the sense of (Bender et al.
20 Acta Wasaensia
(2008)). In the mixed fractional market model the price of an asset is modeled as
dSt = μtStdt+σStdBt +σStdBHt .
where B is a Brownian motion and BH is a f Bm.
Acta Wasaensia 21
5 CONCLUSIONS
5.1 Summaries of the articles
I. The evaluation of geometric Asian power options under time changed mixed
fractional Brownian motion
This paper deals with pricing a geometric Asian option under time changed mixed
fractional Brownian motion. In this model, to get better behaving financial market
model, we replace the physical time t with the inverse α- subordinator process Tα(t)
in the mixed fractional Black-Scholes model. Then, we apply this result to propose
a new model for pricing asian power options. Finally, a lower bound for asian option
price is introduced.
II. Hedging in fractional Black—Scholes model with transaction costs
In this paper, we consider the discounted fractional Black–Scholes pricing model
where the riskless investment, or the bond, is taken as the numeraire and risky asset
S = (St)t∈[0,T ] is given by the dynamics
dSt = Stμdt+StσdBt ,
where B is the fractional Brownian motion with Hurst index H ∈ (12 ,1). Then,
we obtain the conditional mean hedge of the European vanilla type option with
transaction costs.
III. Subdiffusive fractional Black—Scholes model for pricing currency options
under transaction costs
A generalization of the fractional Black-Scholed model is proposed for pricing an
European currency option in a discrete time market model with transaction cost. We
assume that the stock price follows the subdiffusive fractional Black-Scholes model
St = S0 exp
{
(rd − r f )Tα(t)+σBHTα (t)
}
, S0 > 0,
22 Acta Wasaensia
where Tα(t) is the inverse of an α-stable subordinator, with α ∈ (12 ,1),H ∈
[12 ,1),α +αH > 1. In this paper, an analytical pricing model for European cur-
rency option under transaction costs is introduced in discrete time case.
IV. Mixed fractional Merton model to evaluate European options with tran-
saction costs
The problem of European option pricing in discrete time are discussed under a
mixed fractional version of the Merton model with transaction costs. In this case, it
has been assumed that the price dynamics St satisfies in the following
St = S0 exp
{
μt+σBt +σHBHt +Nt lnJ
}
, S0 > 0,
where S0,μ,σ ,σH are constant, Bt is a Brownian motion; BHt is a fractional Brow-
nian motion with Hurst parameter H ∈ (34 ,1), Nt is a Poisson process with intensity
λ > 0 and J is a positive random variable. We assume that Bt ,BHt ,Nt and J are
independent. Finally, we evaluate impact of parameters on the option price.
Acta Wasaensia 23
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Journal of Computational and Applied Mathematics 344 (2018) 716–724
Contents lists available at ScienceDirect
Journal of Computational and Applied
Mathematics
journal homepage: www.elsevier.com/locate/cam
The evaluation of geometric Asian power options under time
changed mixed fractional Brownian motion
Foad Shokrollahi
Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland
a r t i c l e i n f o
Article history:
Received 14 December 2017
Received in revised form 17 April 2018
MSC:
91G20
91G80
60G22
Keywords:
Mixed fractional Brownian motion
Geometric Asian option
Power option
Time changed process
a b s t r a c t
In this paper, the geometric Asian option pricing problem is investigated under the assump-
tion that the underlying stock price is assumed following a mixed fractional subdiffusive
Black–Scholes model, and the geometric average Asian option pricing formula is derived
under this assumption. We then apply the results to value Asian power options on the
stocks that pay constant dividends when the payoff is a power function. Finally, lower
bound of Asian options and some special cases are provided.
© 2018 Elsevier B.V. All rights reserved.
1. Introduction
A standard option is a financial contract which gives the owner of the contract the right, but not the obligation, to buy or
sell a specified asset at a prespecified time (maturity) for a prespecified price (strike price). The specified asset (underlying
asset) can be for example stocks, indexes, currencies, bonds or commodities. The option can be either a call option, which
gives the owner the right to buy the underlying asset, or it can be a put option, which gives the owner the right to sell
the underlying asset. There are several types of options that are traded in a market. American option allows the owner to
exercise his option at any time up to and including the strike date. European options can be exercised only on the strike
date. European options are also called vanilla options. Their payoffs at maturity depend on the spot value of the stock at the
time of exercise. There are other options whose values depend on the stock prices over a predetermined time interval. For
an Asian option, the payoff is determined by the average value over some predetermined time interval. The average price
of the underlying asset can either determine the underlying settlement price (average price Asian options) or the option
strike price (average strike Asian options). Furthermore, the average prices can be calculated using either the arithmetic
mean or the geometric mean. The type of Asian option that will be examined throughout this research is geometric Asian
option.
Over the past three decades, academic researchers and market practitioners have developed and adopted different
models and techniques for option valuation. Themost popular model on option pricing was introduced by Black and Scholes
(BS) [1] in 1973. In the BS model it has been assumed that the asset price dynamics are governed by a geometric Brownian
motion. However, a large number of empirical studies have shown that the distributions of the logarithmic returns of
financial asset usually exhibit properties of self-similarity, heavy tails, long-range dependence in both auto-correlations
and cross-correlations, and volatility clustering [2–4]. Actually, the most popular stochastic process that exhibits long-range
E-mail address: foad.shokrollahi@uva.fi.
https://doi.org/10.1016/j.cam.2018.05.042
0377-0427/© 2018 Elsevier B.V. All rights reserved.
28 Acta Wasaensia
F. Shokrollahi / Journal of Computational and Applied Mathematics 344 (2018) 716–724 717
dependence is of course the fractional Brownian motion. Moreover, the fractional Brownian motion produces a burstiness
in the sample path behavior, which is the important behavior of financial time series. Since fractional Brownian motion is
neither aMarkov process nor a semi-martingale, we cannot use the usual stochastic calculus to analyze it. Further, fractional
Brownianmotion admits arbitrage in a complete and frictionlessmarket. To get around this problem and to take into account
the long memory property, it is reasonable to use the mixed fractional Brownian motion (mfBm) to capture the fluctuations
of the financial asset [5–7].
The mfBm is a linear combination of the Brownian motion and fractional Brownian motion with Hurst index H ∈ ( 12 , 1),
defined on the filtered probability (Ω,F ,P) for any t ∈ R+ by:
MHt (a, b) = aB(t) + bBH (t), (1.1)
where B(t) is a Brownian motion, and BH (t) is an independent fractional Brownian motion with Hurst index H . Cheridito [7]
proved that, for H ∈ ( 34 , 1), the mixed model is equivalent to the Brownian motion and hence it is also arbitrage free. For
H ∈ ( 12 , 1), Mishura and Valkeila [8] demonstrated that the mixedmodel is arbitrage free. Rao [9] discussed geometric Asian
power option undermfBm. To see more about the mixed model, one can refer to Refs. [6,7,10,11].
Analysis of various real-life data shows thatmanyprocesses observed in economics display characteristic periods inwhich
they stay motionless [12]. This feature is most common for emerging markets in which the number of participants, and thus
the number of transactions, is rather low. Notably, similar behavior is observed in physical systems exhibiting subdiffusion.
The constant periods of financial processes correspond to the trapping events in which the subdiffusive test particle gets
immobilized [13]. Subdiffusion is a well known and established phenomenon in statistical physics. Its usual mathematical
description is in terms of the celebrated Fractional Fokker–Planck equation (FFPE). This equation was first derived from the
continuous-time random walk scheme with heavy-tailed waiting times [14,15,10], and since then became fundamental in
modeling and analysis of complex systems exhibiting slow dynamics. Following this line, and to model the observed long
range dependence and fluctuations in the financial price time series, we introduce a time-changedmixed fractional BSmodel
to value Asian power option when the underlying stock is
St = X(Tα(t))
= S0e(r−q)Tα (t)+M
H
α (t)(σ ,σ )− 12 σ2 t
α
Γ (α+1)− 12 σ2
(
tα
Γ (α+1)
)2H
, S0 = X(0) > 0, (1.2)
whereMHα (t)(σ , σ ) = σB(Tα(t)) + σBH (Tα(t)), α ∈ ( 12 , 1), 2α − αH > 1 and Tα(t) is the inverse α-stable subordinator.
We then apply the result to price geometric Asian power options that pay constant dividends when the payoff is a power
function. We also provide some special cases and lower bound for the Asian option price. The rest of the paper is organized
as follows. In Section 2, some useful concepts and theorems of time changed mixed fractional process are introduced. In
Section 3, a brief introduction of Asian options is given. Analytical valuation formula for geometric Asian options is derived
in Section 4 and then applied to geometric Asian power options in Section 5. The lower bound on the price of the Asian option
is proposed in Section 6.
2. Auxiliary facts
In this section, we recall some definitions and results about mixed fractional time changed process. More information
about mixed fractional process can be found in [16,10].
The time-changed process Tα(t) is the inverse α-stable subordinator defined as below
Tα(t) = inf{τ > 0,Uα(t) ≥ t}
here Uα(τ )τ≥0 is a strictly increasing α-stable Lévy process [17] with Laplace transform: E(e−uUα (τ )) = e−τuα , α ∈ (0, 1).
Uα(t) is 1α self-similar and Tα(t) is α self-similar, that is, for every h > 0, Uα(ht)  h
1
α Uα(t) Tα(ht)  hαTα(t), here 
indicates that the random variables on both sides have the same distribution. Specially, when α ↑ 1, Tα(t) reduces to the
physical time t . You can find more details about subordinator and its inverse processes in [18,19].
Consider the subdiffusion process
MHα (t)(a, b) = aWα(t) + bWHα (t) = aB(Tα(t)) + bBH (Tα(t)),
where B(τ ) is a Brownian motion, BH (τ ) is a fractional Brownian motion with Hurst index H and Tα(t) is inverse α-
subordinator which are supposed to be independent. When a = 0, b = 1, then it is the process considered in [20] and
if b = 0, a = 1, then it is the process considered in [21]. In this research, we assume that H ∈ ( 34 , 1) and (a, b) = (1, 1).
Remark 2.1. When α ↑ 1, the processes Wα(t) and WHα (t) degenerate to B(t) and BH (t), respectively. Then, MHα (t)(a, b)
reduces to themfBm in Eq. (1.1).
Acta Wasaensia 29
718 F. Shokrollahi / Journal of Computational and Applied Mathematics 344 (2018) 716–724
Remark 2.2. From [20,21], we know that E(Tα(t)) = tαΓ (α+1) . Then, by applying α-self-similar and non-decreasing sample
path of Tα(t), we have
E[(B(Tα(t)))2] = t
α
Γ (α + 1) (2.1)
E[(BH (Tα(t)))2] =
(
tα
Γ (α + 1)
)2H
. (2.2)
3. Asian options
The payoff of an Asian option is based on the difference between an asset’s average price over a given time period, and a
fixed price called the strike price. Asian options are popular because they tend to have lower volatility than options whose
payoffs are based purely on a single price point. It is also harder for big traders to manipulate an average price over an
extended period than a single price, so Asian option offers further protection against risk. The Asian call and put options
have a payoff that is calculated with an average value of the underlying asset over a specific period. The payoff for an Asian
call and put option with strike price K and expiration time T is (S¯(T ) − K )+ and (K − S¯(T ))+ respectively, where S¯(T ) is
the average price of the underlying asset over the prespecified interval. Since Asian options are less expensive than their
European counterparts, they are attractive to many different investors. Apart from the regular Asian option there also exists
Asian strike option. An Asian strike call option guarantees the holder that the average price of an underlying asset is not
higher than the final price. The option will not be exercised if the average price of the underlying asset is greater than the
final price. The holder of an Asian strike put optionmakes sure that the average price received for the underlying asset is not
less thanwhat the final pricewill provide. The payoff for anAsian strike call and put option is (S¯(T )−S(T ))+ and (S(T )−S¯(T ))+
respectively, where S(T ) is the value of underlying stock at maturity date T .
Asian options are divided into two different types, when calculating the average, the geometric Asian option
G(T ) = exp
{
1
T
∫ T
0
ln S(t)dt
}
,
and the arithmetic Asian option.
A(T ) = 1
T
∫ T
0
S(t)dt.
We assume that the prespecified interval [0, T ] is fixed, thenwill price the geometric Asian option in the continuous average
case under time changed mixed fractional Brownian motion environment.
4. Pricing model of geometric Asian option
In order to derive an Asian option pricing formula in a time changed mixed fractional market, we make the following
assumptions:
(i) The price of underlying stock at time t is given by Eq. (1.2).
(ii) There are no transaction costs in buying or selling the stocks or option.
(iii) The risk free interest rate r and dividend rate q are known and constant through time.
(iv) The option can be exercised only at the maturity time.
From Eq. (1.2), we know that ln St  N(u, v), where
u = ln S(0) + (r − q)Tα(t) − 12σ
2 t
α
Γ (α + 1) −
1
2
σ 2
(
tα
Γ (α + 1)
)2H
(4.1)
v = σ 2 t
α
Γ (α + 1) + σ
2
(
tα
Γ (α + 1)
)2H
. (4.2)
Let C(S(0), T ) be the price of a European call option at time 0 with strike price K and that matures at time T . Then,
from [16], we can get
C(S(0), T ) = S(0)e−qTφ(d1) − Ke−rTφ(d2),
where
d1 = ln
S0
K + (r − q + σˆ
2
2 )T
σˆ
√
T
, d2 = d1 − σˆ
√
T ,
30 Acta Wasaensia
F. Shokrollahi / Journal of Computational and Applied Mathematics 344 (2018) 716–724 719
σˆ 2 = σ 2 T
α−1
Γ (α)
+ σ 2
(
Tα−1
Γ (α)
)2H
,
and φ(.) denotes cumulative normal density function.
Under the above assumptions (i)–(iv), we obtain the value of the geometric Asian call option by the following theorem
Theorem4.1. Suppose the stock price St satisfied Eq. (1.2). Then, under the risk-neutral probabilitymeasure, the value of geometric
Asian call option C(S(0), T ) with strike price K and maturity time T is given by
C(S(0), T ) = S(0) exp
{
−rT + (r − q) T
α
Γ (α + 2) +
σ 2(−T )α
2Γ (α + 3)
− σ
2T 2αH
4(2αH + 1)(αH + 1)(Γ (α + 1))2H
}
φ(d1) − Ke−qTφ(d2), (4.3)
where
d2 = μG − ln K
σG
, d1 = d2 + σG,
μG = ln S(0) + (r − q − σ
2
2
)
Tα
Γ (α + 2) −
σ 2T 2αH
2(2αH + 1)(Γ (α + 1))2H ,
σ 2G =
σ 2Tα
Γ (α + 2) +
σ 2(−T )α
Γ (α + 3) +
σ 2T 2αH
(2αH + 2)(Γ (α + 1))2H ,
the interest rate r and the dividend rate q are constant over time and φ(.) denotes cumulative normal density function.
Proof. Suppose
L(T ) = 1
T
∫ T
0
ln S(t)dt.
Then
G(T ) = eL(T ). (4.4)
We know that ln St  N(u, v), then it is clear that the random variable L(T ) has Gaussian distribution under the risk-neutral
probability measure. We will now compute its mean and variance under the risk-neutral probability measure. Let E denote
the expectation and,μG and σ 2G denote themean and the variance of the randomvariableE under the risk-neutral probability
measure. Note that
μG = E[L(T )] = 1T
∫ T
0
E[ln S(t)]dt
= ln S(0) + 1
T
∫ T
0
(r − q) t
α
Γ (α + 1)dt −
σ 2
2T
∫ T
0
[
tα
Γ (α + 1) +
t2αH
(Γ (α + 1))2H
]
dt
= ln S(0) + (r − q) T
α
Γ (α + 2) −
σ 2Tα
2Γ (α + 2) −
σ 2T 2αH
(4αH + 2)(Γ (α + 1))2H ,
and
σ 2G = Var[L(T )] = E[(L(T ) − μG)2]
= σ
2
T 2
∫ T
0
∫ T
0
(
E[Wα(t)Wα(τ )] + E[WHα (t)WHα (τ )]
)
dtdτ ,
by independence of the processes B(t), BH (t) and Tα(t), we obtain
= σ
2
T 2
∫ T
0
∫ T
0
(
| t
α
Γ (α + 1) | + |
τα
Γ (α + 1) | − |
(t − τ )α
Γ (α + 1) |
)
dtdτ
+ σ
2
T 2
∫ T
0
∫ T
0
(
| t
α
Γ (α + 1) |
2H
+ | τ
α
Γ (α + 1) |
2H
− | (t − τ )
α
Γ (α + 1) |
2H)
dtdτ
= σ
2Tα
Γ (α + 2) +
σ 2(−T )α
Γ (α + 3) +
σ 2T 2αH
(2αH + 2)(Γ (α + 1))2H .
From (4.4), we know that the random variable G(T ) is log-normally distributed, then lnG(T )  N(μG, σ 2G ). Let I = {x :
ex > K } and φ(.) be the probability density function of a standard normal distribution, then the price of geometric Asian call
Acta Wasaensia 31
720 F. Shokrollahi / Journal of Computational and Applied Mathematics 344 (2018) 716–724
option is given by the following computations
C(S(0), T ) = e−rTE[(G(T ) − K )+]
= e−rT
∫
I
(ex − K ) 1√
2πσG
exp
{
− (x − μG)
2
2σ 2G
}
dx
= e−rT
∫
I
(eμG+zσG − K ) 1√
2πσG
exp
{
− (x − μG)
2
2σ 2G
}
ϕ(z)dz
= e−rT+μG+ 12 σ2G
∫ ∞
−d2
e−
1
2 (z−σG)2dz − Ke−rT
∫ −∞
−d2
ϕ(z)dz
= e−rT+μG+ 12 σ2G
∫ ∞
−d2−σG
ϕ(z)dz − Ke−rT
∫ d2
−∞
ϕ(z)dz
= e−rT+μG+ 12 σ2G
∫ d2+σG
−∞
ϕ(z)dz − Ke−rT
∫ d2
−∞
ϕ(z)dz
= e−rT+μG+ 12 σ2Gφ(d1) − Ke−rTφ(d2),
= S(0) exp
{
−rT + (r − q) T
α
Γ (α + 2) + σ
2 (−T )α
2Γ (α + 3)
− σ 2 T
2αH
4(2αH + 1)(αH + 1)(Γ (α + 1))2H
}
φ(d1) − Ke−qTφ(d2),
here
I = {x : ex > K } = {z : eμG+zσG > K }
= {z : μG + zσG > ln K } = {z : z > −d2},
thus we obtain the pricing formula. 
Moreover, using the put–call parity, the valuation model for a geometric Asian put option under time changed mixed
fractional BS model can be written
P(S(0), T ) = Ke−qTφ(−d2) − S(0) exp
{
−rT + (r − q) T
α
Γ (α + 2) +
σ 2(−T )α
2Γ (α + 3)
− σ
2T 2αH
4(2αH + 1)(αH + 1)(Γ (α + 1))2H
}
φ(−d1), (4.5)
where d1 and d2 are defined previously.
Letting α ↑ 1, then the stock price follows themfBm shown below
St = S0 exp
{
(r − q)T + σB(t) + σBH (t)
− 1
2
σ 2t − 1
2
σ 2t2H
}
, 0 < t < T , (4.6)
and the result is presented below.
Corollary 4.1. The value of geometric Asian call option with maturity T and strike K , whose stock price follows Eq. (4.6), is given
by
C(S(0), T ) =
S(0) exp
{
−1
2
(r + q)T − σ
2T
12
− σ
2T 2H
4(2H + 1)(H + 1)
}
φ(d1) − Ke−qTφ(d2), (4.7)
where
d2 = μG − ln K
σG
, d1 = d2 + σG,
μG = ln S(0) + 12 (r − q −
σ 2
2
)T − σ
2T 2H
2(2H + 1) ,
32 Acta Wasaensia
F. Shokrollahi / Journal of Computational and Applied Mathematics 344 (2018) 716–724 721
σ 2G =
σ 2T
3
+ σ
2T 2H
(2H + 2) ,
which is consistent with result in [9].
5. Pricing model of Asian power option
In this section, we consider the pricing model of Asian power call option with strike price K and maturity time T under
time changed mixed fractional BS model where the payoff function is (Gn(T ) − K )+ for some constant integer n ≥ 1.
Theorem5.1. Suppose the stock price St satisfied Eq. (1.2). Then, under the risk-neutral probabilitymeasure the value of geometric
Asian power call option C(S(0), T ) with strike price K , maturity time T and payoff function (Gn(T ) − K )+ is given by
C(S(0), T ) = S(0) exp
{
−rT + (r − q) nT
α
Γ (α + 2) −
(n − n2)σ 2Tα
2Γ (α + 2) +
n2σ 2(−T )α
2Γ (α + 3)
− nσ
2T 2αH
(4αH + 2)(Γ (α + 1))2H −
n2σ 2T 2αH
(4αH + 4)(Γ (α + 1))2H
}
φ(f1)
− Ke−qTφ(f2), (5.1)
where
f2 = μG −
1
n ln K
σG
, f1 = f2 + nσG,
μG = ln S(0) + (r − q − σ
2
2
)
Tα
Γ (α + 2) −
σ 2T 2αH
2(2αH + 1)(Γ (α + 1))2H ,
σ 2G =
σ 2Tα
Γ (α + 2) +
σ 2(−T )α
Γ (α + 3) +
σ 2T 2αH
(2αH + 2)(Γ (α + 1))2H ,
the interest rate r and the dividend rate q are constant over time and ϕ(.) denotes cumulative normal density function.
Proof. The payoff function for Asian power option is (Gn(T ) − K )+ = (enL(T ) − K )+, then applying similar computation in
Theorem 4.1, we obtain
C(S(0), T ) = e−rTE[(Gn(T ) − K )+]
= e−rT
∫
I
(enx − K ) 1√
2πσG
exp
{
− (x − μG)
2
2σ 2G
}
dx
= e−rT
∫
I
(en(μG+zσG) − K ) 1√
2πσG
exp
{
− (x − μG)
2
2σ 2G
}
ϕ(z)dz
= e−rT+nμG+ 12 n2σ2G
∫ ∞
−f2
e−
1
2 (z−nσG)2dz − Ke−rT
∫ −∞
−f2
ϕ(z)dz
= e−rT+nμG+ 12 n2σ2G
∫ ∞
−f2−nσG
ϕ(z)dz − Ke−rT
∫ f2
−∞
ϕ(z)dz
= e−rT+nμG+ 12 n2σ2G
∫ f2+nσG
−∞
ϕ(z)dz − Ke−rT
∫ f2
−∞
ϕ(z)dz
= e−rT+nμG+ 12 n2σ2Gφ(f1) − Ke−rTφ(f2),
= S(0) exp
{
−rT + (r − q) nT
α
Γ (α + 2) −
(n − n2)σ 2Tα
2Γ (α + 2) +
n2σ 2(−T )α
2Γ (α + 3)
− nσ
2T 2αH
(4αH + 2)(Γ (α + 1))2H −
n2σ 2T 2αH
(4αH + 4)(Γ (α + 1))2H
}
φ(f1)
− Ke−qTφ(f2),
Acta Wasaensia 33
722 F. Shokrollahi / Journal of Computational and Applied Mathematics 344 (2018) 716–724
here
I = {x : enx > K } = {z : en(μG+zσG) > K }
= {z : μG + zσG > 1n ln K } = {z : z > −f2},
thus the proof is completed. 
The time changed mixed fractional BS model includes the jump behavior of price process because the subordinator
process is a pure jump Levy process so it can capture the random variations in volatility. Also, it can be used for data with
long range dependence and visible constant time periods characteristic for processes delayed by inverse subordinators.
6. Lower bound of the Asian option price
The aim of this section is to obtain the lower bound on the price of the Asian option. The next theorem shows that the
normal distribution is stable when the random variables are jointly normal.
Theorem 6.1 ([22]). The conditional distribution of ln Sti given lnG(T ) is a normal distribution
(ln Sti | lnG(T ) = z)  N(μi + (z − μG)
λi
σ 2G
, σ 2i −
λ2i
σ 2G
), i = 1, . . . , n,
where
μi = ln S(0) + (r − q)Tα(ti) − 12σ
2 t
α
i
Γ (α + 1) −
1
2
σ 2
(
tαi
Γ (α + 1)
)2H
σ 2i = σ 2
tαi
Γ (α + 1) + σ
2
(
tαi
Γ (α + 1)
)2H
,
λi = Cov(ln Sti , lnG(T )), 0 ≤ t1 < t2 < · · · < tn ≤ T , Tα(t) is inverse α-stable subordinator and, μG and σ 2G are defined in
Theorem 4.1.
Moreover, (Sti | lnG(T )) has a lognormal distribution and
E
[
Sti | lnG(T ) = z
]
= exp
{
μi + (z − μG) λi
σ 2G
+ 1
2
(σ 2i −
λ2i
σ 2G
)
}
i = 1, . . . , n. (6.1)
Now, we condition on the geometric average G(T ) in the pricing expression of the Asian option
C(S(0), T ) = e−rTE[(A(T ) − K )+] = e−rTE[E[(A(T ) − K )+|G(T )]]
= e−rT
∫ ∞
0
E[(A(T ) − K )+|G(T ) = z]g(z)dz,
where g is the lognormal density function of G. Let
C1 =
∫ K
0
E[(A(T ) − K )+|G(T ) = z]g(z)dz,
C2 =
∫ ∞
K
E[(A(T ) − K )+|G(T ) = z]g(z)dz,
then C(S(0), T ) = e−rT (C1 + C2). Since the geometric average is less than arithmetic average A(T ) ≥ G(T ),
C2 =
∫ ∞
K
E[A(T ) − K |G(T ) = z]g(z)dz, (6.2)
from Theorem 6.1, we can calculate C2. Applying Jensen’s inequality we obtain a lower bound on C1
C1 =
∫ K
0
E[(A(T ) − K )+|G(T ) = z]g(z)dz
≥
∫ K
0
(E[A(T ) − K |G(T ) = z])+g(z)dz
=
∫ K
K˜
E[A(T ) − K |G(T ) = z]g(z)dz = C˜1 (6.3)
where K˜ = {z|E[A(T )|G(T ) = z] = K }.
34 Acta Wasaensia
F. Shokrollahi / Journal of Computational and Applied Mathematics 344 (2018) 716–724 723
Eq. (6.1) enables us to obtain K˜ , then we calculate the following expectation
E[A(T )|G(T ) = z] = E
[
1
n
n∑
i=1
Sti |G(T ) = z
]
= 1
n
n∑
i=1
E
[
Sti |G(T ) = z
]
= 1
n
n∑
i=1
exp
(
μi + (log z − μG) λi
σ 2G
+ 1
2
(σ 2i −
λ2i
σ 2G
)
)
.
Theorem 6.2. A lower bound on the price of the Asian option with strike price K and maturity time T is given by
C˜(S(0), T ) = e−rT (C˜1 + C2)
= e−rT
{1
n
n∑
i=1
exp(μi + 12σ
2
i )φ
(
μG − ln K˜ + γi
σG
)
− Kφ
(
μG − ln K˜
σG
)}
,
where all parameters are defined previously.
Proof. Collecting Eqs. (6.2) and (6.3) gives
C˜1 + C2 =
∫ ∞
K˜
E[A(T ) − K |G(T ) = z]g(z)dz
=
∫ ∞
K˜
E[A(T )|G(T ) = z]g(z)dz − K
∫ ∞
K˜
g(z)dz
=
∫ ∞
K˜
E
[
1
n
n∑
i=1
Sti |G(T ) = z
]
g(z)dz − K
∫ ∞
K˜
g(z)dz
=
∫ ∞
K˜
1
n
n∑
i=1
E
[
Sti |G(T ) = z
]
g(z)dz − K
∫ ∞
K˜
g(z)dz
= 1
n
n∑
i=1
∫ ∞
K˜
E
[
Sti | lnG(T ) = ln z
]
g(z)dz − K
∫ ∞
K˜
g(z)dz.
From the proof of Theorem 4.1, we obtain
K
∫ ∞
K˜
g(z)dz = Kφ
(
μG − ln K˜
σG
)
,
and from Eq. (6.1)∫ ∞
K˜
E
[
Sti | lnG(T ) = ln z
]
g(z)dz
=
∫ ∞
K˜
exp
(
μi + (ln z − μG) λi
σ 2G
+ 1
2
(σ 2i −
λ2i
σ 2G
)
)
g(z)dz
= exp
(
μi + 12σ
2
i
)∫ ∞
K˜
exp
(
(ln z − μG) λi
σ 2G
− 1
2
λ2i
σ 2G
)
g(z)dz.
Using the density of the lognormal distribution, we have∫ ∞
K˜
1√
2πσGz
exp
(
(ln z − μG) λi
σ 2G
− 1
2
λ2i
σ 2G
− 1
2
(
μG − ln z
σG
)2
)
dz.
Making the change of variables y = μG−ln z+λi
σG
and dydz = − 1σGz , then we have∫ −∞
μG−ln z+λi
σG
− 1√
2π
exp
(
(
λi
σG
− y) λi
σG
− 1
2
λ2i
σ 2G
− 1
2
(y − λi
σG
)2
)
dy
=
∫ μG−ln z+λi
σG
−∞
1√
2π
exp
(
−y λi
σG
+ 1
2
λ2i
σ 2G
− 1
2
y2 − 1
2
λ2i
σ 2G
+ y λi
σG
)
dy
=
∫ μG−ln z+λi
σG
−∞
1√
2π
exp
(
−1
2
y2
)
dy = φ
(
μG − ln K˜ + γi
σG
)
,
by collecting C˜1 and C2 the proof is completed. 
Acta Wasaensia 35
724 F. Shokrollahi / Journal of Computational and Applied Mathematics 344 (2018) 716–724
Acknowledgments
This paper is supported by the University of Vaasa, Finland. The author is deeply grateful to the anonymous referees for
reading the present paper carefully and giving very helpful comments which contributed to improving the quality of the
paper.
References
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36 Acta Wasaensia
Statistics and Probability Letters 130 (2017) 85–91
Contents lists available at ScienceDirect
Statistics and Probability Letters
journal homepage: www.elsevier.com/locate/stapro
Hedging in fractional Black–Scholes model with transaction
costs
Foad Shokrollahi *, Tommi Sottinen
Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland
a r t i c l e i n f o
Article history:
Received 11 June 2017
Received in revised form 23 July 2017
Accepted 25 July 2017
Available online 4 August 2017
MSC:
91G20
91G80
91G22
Keywords:
Delta-hedging
Fractional Black–Scholes model
Transaction costs
Option pricing
a b s t r a c t
We consider conditional-mean hedging in a fractional Black–Scholes pricing model in
the presence of proportional transaction costs. We develop an explicit formula for the
conditional-mean hedging portfolio in terms of the recently discovered explicit conditional
law of the fractional Brownian motion.
© 2017 Elsevier B.V. All rights reserved.
1. Introduction
We consider discrete hedging in fractional Black–Scholes models where the asset price is driven by a long-range
dependent fractional Brownian motion. For a convex or concave European vanilla type option, we construct the so-
called conditional-mean hedge. This means that at each trading time the value of the conditional mean of the discrete
hedging strategy coincides with the frictionless price. By frictionless wemean the continuous trading hedging price without
transaction costs. The key ingredient in constructing the conditional mean hedging strategy is the recent representation for
the regular conditional law of the fractional Brownian motion given in Sottinen and Viitasaari (2017). Let us note that there
are arbitrage strategies with continuous trading without transaction costs, but not with discrete trading strategies, even
in the absence of trading costs. For details of the use of fractional Brownian motion in finance and discussion on arbitrage
see Bender et al. (2011).
For the classical Black–Scholes model driven by the Brownian motion, the study of hedging under transaction costs goes
back to Leland (1985). See also Denis and Kabanov (2010) and Kabanov and Safarian (2009) for a mathematically rigorous
treatment. For the fractional Black–Scholes model driven by the long-range dependent fractional Brownian motion, the
study of hedging under transaction costs was studied in Azmoodeh (2013). In the series of articles (Shokrollahi et al., 2016;
Wang, 2010a, b; Wang et al., 2010a, b) the discrete hedging in the fractional Black–Scholes model was studied by using the
economically dubious Wick–Itô–Skorohod interpretation of the self-financing condition. Actually, with the economically
solid forward-type pathwise interpretation of the self-financing condition, these hedging strategies are valid, not for the
geometric fractional Brownianmotion, but for a geometric Gaussian processwhere the driving noise is a Gaussianmartingale
* Corresponding author.
E-mail addresses: foad.shokrollahi@uva.fi (F. Shokrollahi), tommi.sottinen@iki.fi (T. Sottinen).
http://dx.doi.org/10.1016/j.spl.2017.07.014
0167-7152/© 2017 Elsevier B.V. All rights reserved.
Acta Wasaensia 37
86 F. Shokrollahi, T. Sottinen / Statistics and Probability Letters 130 (2017) 85–91
with the same variance function as the corresponding fractional Brownian motion would have, see Gapeev et al. (2011),
Shokrollahi and Kılıçman (2014), Shokrollahi and Kılıçman (2015) and Shokrollahi et al. (2015).
2. Preliminaries
We are interested in pricing of European vanilla options f (ST ) of a single underlying asset S = (St )t∈[0,T ], where T > 0 is
a fixed time of maturity of the option.
We consider the discounted fractional Black–Scholes pricingmodel where the ‘‘riskless’’ investment, or the bond, is taken
as the numéraire and the risky asset S = (St )t∈[0,T ] is given by the dynamics
dSt
St
= μ dt + σ dBt , (2.1)
where B is the fractional Brownian motion with Hurst index H ∈ ( 12 , 1). Recall that, qualitatively, the fractional Brownian
motion is the (up to amultiplicative constant) unique Gaussian process with stationary increments and self-similarity index
H . Quantitatively, the fractional Brownian motion is defined by its covariance function
r(t, s) = 1
2
[
t2H + s2H − |t − s|2H] .
Since the fractional Brownian motion with index H ∈ ( 12 , 1) has zero quadratic variation, the classical change-of-variables
rule applies. Consequently, the pathwise solution to the stochastic differential equation (2.1) is
St = S0eμt+σBt . (2.2)
Also, it follows from the classical change-of-variables rule that
f (ST ) = f (S0) +
∫ T
0
f ′(St ) dSt , (2.3)
where f is a convex or concave function and f ′ is its left-derivative. We refer to Azmoodeh et al. (2009) for details. The
economic interpretation of (2.3) is that under continuous trading and no transaction costs, the replication price of a European
vanilla option f (ST ) is f (S0) and the replicating strategy is given by πt = f ′(St ), where πt denotes the number of the shares
of the risky asset S held by the investor at time t . Furthermore, we note that the value Vπ of the hedging strategy π = f ′(S·)
at time t is
Vπt = Vπ0 +
∫ t
0
πu dSu
= f (S0) +
∫ t
0
f ′(Su) dSu
= f (St ).
Indeed, the first equality is simply the self-financing condition and the rest follows immediately from (2.3). Note that this
is very different from the value in the classical Black–Scholes model, where the value is determined by the Black–Scholes
partial differential equation, which in turn comes to the Itô’s change-of-variables rule.
We assume that the trading only takes place at fixed preset time points 0 = t0 < t1 < · · · < tN = T . We denote by πN
the discrete trading strategy
πNt = πN0 1{0}(t) +
N∑
i=1
πNti−11(ti−1,ti](t).
The value of the strategy πN is given by
Vπ
N ,k
t = Vπ
N ,k
0 +
∫ t
0
πNu dSu −
∫ t
0
kSu|dπNu |, (2.4)
where k ∈ [0, 1) is the proportional transaction cost.
Under transaction costs perfect hedging is not possible. In this case, it is natural to try to hedge on average in the sense
of the following definition:
Definition 2.1 (Conditional-Mean Hedge). Let f (ST ) be a European vanilla type optionwith convex or concave payoff function
f . Let π be its Markovian replicating strategy: πt = f ′(St ). We call the discrete-time strategy πN a conditional-mean hedge, if
for all trading times ti,
E
[
Vπ
N ,k
ti+1 |Fti
]
= E
[
Vπti+1 |Fti
]
. (2.5)
Here Fti is the information generated by the asset price process S up to time ti.
38 Acta Wasaensia
F. Shokrollahi, T. Sottinen / Statistics and Probability Letters 130 (2017) 85–91 87
Remark 2.1 (Conditional-Mean Hedge as Tracking Condition). Criterion (2.5) is actually a tracking requirement. We do not
only require that the conditional means agree on the last trading time before the maturity, but also on all trading times.
In this sense the criterion has an ‘‘American’’ flavor in it. From a purely ‘‘European’’ hedging point of view, one can simply
remove all but the first and the last trading times.
Remark 2.2 (Arbitrage and Uniqueness of Conditional-Mean Hedge). The conditional-mean hedging strategy πN depends
on the continuous-time hedging strategy π . Since there is strong arbitrage in the fractional Black–Scholes model (zero
can be perfectly replicated with negative initial wealth), the replicating strategy π is not unique. However, the strong
arbitrage strategies are very complicated. Indeed, it follows directly from the change-of-variables formula that in the class
of Markovian strategies πt = g(t, St ), the choice πt = f ′(St ) is the unique replicating strategy for the claim f (ST ).
We stress that the expectation in (2.5) is with respect to the true probability measure; not under any equivalent
martingale measure. Indeed, equivalent martingale measures do not exist in the fractional Black–Scholes model.
To find the solution to (2.5) one must be able to calculate the conditional expectations involved. This can be done
by using Sottinen and Viitasaari (2017, Theorem 3.1), a version of which we state below as Lemma 2.1 for the readers’
convenience.
Lemma 2.1 (Conditional Fractional Brownian Motion). The fractional Brownian motion B with index H ∈ ( 12 , 1) conditioned on
its own past FBu is the Gaussian process B(u) = B|FBu with FBu -measurable mean
Bˆt (u) = Bu −
∫ u
0
Ψ (t, s|u) dBs,
where
Ψ (t, s|u) = − sin(π (H −
1
2 ))
π
s
1
2−H (u − s) 12−H
∫ t
u
zH−
1
2 (z − u)H− 12
z − s dz,
and deterministic covariance function
rˆ(t, s|u) = r(t, s) −
∫ u
0
k(t, v)k(s, v)dv,
where
k(t, s) =
(
H − 1
2
)√
2HΓ
( 3
2 − H
)
Γ
(
H − 12
)
Γ (2 − 2H) s
1
2−H
∫ t
s
zH−
1
2 (z − s)H− 23 dz;
Γ is the Euler’s gamma function.
Remark 2.3 (Conditional Asset Process). By (2.2) we have the equality of filtrations: FBt = FSt = Ft , for t ∈ [0, T ].
Consequently, the conditional process S(u) = S|Fu is, informally, given by
St (u) = S0eμt+σBt (u)
= Sueμ(t−u)+σ(Bt (u)−Bu).
More formally, this means, in particular, that for t > u,
E
[
f (St )
∣∣FSu ] = E [f (S0eμt+σBt ) ∣∣FBu ]
=
∫ ∞
−∞
f
(
S0eμt+σ Bˆt (u)+σ
√
rˆ(t|u) z
)
φ(z)dz
=
∫ ∞
−∞
f
(
Sue
μ(t−u)+σ
(
Bˆt (u)−Bu
)
+σ
√
rˆ(t|u) z
)
φ(z)dz,
where we have denoted
rˆ(t|u) = rˆ(t, t|u),
and φ is the standard normal density function.
3. Conditional-Mean hedging strategies
Denote
ΔBˆti+1 (ti) = Bˆti+1 (ti) − Bti .
Acta Wasaensia 39
88 F. Shokrollahi, T. Sottinen / Statistics and Probability Letters 130 (2017) 85–91
In Theorem 3.1 we will calculate the conditional-mean hedging strategy in terms of the following conditional gains:
ΔSˆti+1 (ti) = Sˆti+1 (ti) − Sti
= E [Sti+1 |Fti]− Sti ,
ΔVˆπti+1 (ti) = Vˆπti+1 (ti) − Vπti
= E
[
Vπti+1 |Fti
]
− Vπti ,
ΔVˆπ
N ,k
ti+1 (ti) = Vˆπ
N ,k
ti+1 (ti) − Vπti
= E
[
Vπ
N ,k
ti+1 |Fti
]
− VπN ,kti .
Lemma 3.1 states that all these conditional gains can be calculated explicitly by using the prediction law of the fractional
Brownian motion.
Lemma 3.1 (Conditional Gains).
ΔSˆti+1 (ti) = Sti
(∫ ∞
−∞
eμΔti+1+σΔBˆti+1 (ti)+σ
√
rˆ(ti+1|ti)zφ(z) dz − 1
)
,
ΔVˆπti+1 (ti) =
∫ ∞
−∞
f
(
Stie
μΔti+1+σΔBˆti+1 (ti)+σ
√
rˆ(ti+1|ti)z
)
φ(z) dz − f (Sti ),
ΔVˆπ
N ,k
ti+1 (ti) = πNti ΔSˆti+1 (ti) − kSti |ΔπNti |.
Proof. Let g : R → R be such that E [∣∣g(Bti+1 )∣∣] < ∞. Then, by Lemma 2.1,
E
[
g(Bti+1 ) |Fti
] = ∫ ∞
−∞
g
(
Bˆti+1 (ti) +
√
rˆ(ti+1|ti)z
)
φ(z) dz.
Consider ΔSˆti+1 (ti). By choosing
g(x) = S0eμt+σx,
we obtain
Sˆti+1 (ti) = E
[
Sti+1
∣∣Fti]
= E [g (Bti+1) ∣∣Fti]
=
∫ ∞
−∞
S0e
μti+1+σ
(
Bˆti+1 (ti)+
√
rˆ(ti+1|ti)z
)
φ(z) dz.
The formula for ΔSˆti+1 (ti) follows from this.
Consider then ΔVπti+1 (ti). By choosing
g(x) = f (S0eμt+σx)
we obtain
Vˆπti+1 (ti) = E
[
Vπti+1 |Fti
]
= E [f (Sti+1 ) |Fti]
= E [g(Bti+1 ) |Fti]
=
∫ ∞
−∞
g
(
Bˆti+1 (ti) +
√
rˆ(ti+1|ti)z
)
φ(z) dz
=
∫ ∞
−∞
f
(
S0e
μti+1+σ
(
Bˆti+1 (ti)+
√
rˆ(ti+1|ti)z
))
φ(z) dz.
The formula for ΔVˆπti+1 (ti) follows from this.
40 Acta Wasaensia
F. Shokrollahi, T. Sottinen / Statistics and Probability Letters 130 (2017) 85–91 89
Finally, we calculate
Vˆπ
N ,k
ti+1 (ti) = E
[
Vπ
N ,k
ti+1
∣∣Fti]
= VπN ,kti + E
[∫ ti+1
ti
πNu dSu −
∫ ti+1
ti
kSu|dπNu |
∣∣∣Fti
]
= VπN ,kti + πNti
(
E
[
Sti+1
∣∣Fti]− Sti)− kSti |ΔπNti |
= VπN ,kti + πNti ΔSˆti+1 (ti) − kSti |ΔπNti |.
The formula for ΔVˆπ
N ,k
ti+1 (ti) follows from this. 
Now we are ready to state and prove our main result. We note that, in principle, our result is general: it is true in any
pricingmodelwhere the option f (ST ) can be replicated. In practice, our result is specific to the fractional Black–Scholesmodel
via Lemma 3.1.
Theorem 3.1 (Conditional-Mean Hedging Strategy). The conditional mean hedge of the European vanilla type option with convex
or concave positive payoff function f with proportional transaction costs k is given by the recursive equation
πNti =
ΔVˆπti+1 (ti) + (Vπti − Vπ
N ,k
ti ) + kSti |ΔπNti |
ΔSˆti+1 (ti)
, (3.1)
where Vπ
N ,k
ti is determined by (2.4).
Proof. Let us first consider the left hand side of (2.5). We have
E
[
Vπ
N ,k
ti+1
∣∣Fti] = E
[
Vπ
N ,k
ti +
∫ ti+1
ti
πNu dSu − k
∫ ti+1
ti
Su|dπNu |
∣∣∣Fti
]
= VπN ,kti + πNti E
[
Sti+1 (ti) − Sti
∣∣Fti]− kSti |ΔπNti|
= VπN ,kti + πNti ΔSˆti+1 (ti) − kSti |ΔπNti |.
For the right-hand-side of (2.5), we simply write
E
[
Vπti+1
∣∣Fti] = ΔVˆπti+1 (ti) + Vπti .
Equating the sides we obtain (3.1) after a little bit of simple algebra. 
Remark 3.1. Taking the expected gains ΔSˆti+1 (ti) to be the numéraire, one recognizes three parts in the hedging formula
(3.1). First, one invests on the expected gains in the time-value of the option. This ‘‘conditional-mean Delta-hedging’’ is
intuitively themost obvious part. Indeed, a naïve approach to conditional-mean hedgingwould only give this part, forgetting
to correct for the tracking-errors already made, which is the second part in (3.1). The third part in (3.1) is obviously due to
the transaction costs.
Remark 3.2. Eq. (3.1) for the strategy of the conditional-mean hedging is recursive: in addition to the filtration Fti , the
position πNti−1 is needed to determine the position π
N
ti . Consequently, to determine the conditional-mean hedging strategy
by using (3.1), the initial position πN0 must be fixed. The initial position is, however, not uniquely defined. Indeed, let β
N
0 be
the position in the riskless asset. Then the conditional-mean criterion (2.5) only requires that
βN0 + πN0 E[St1 ] − kS0|πN0 | = E[f (St1 )].
There are of course infinite number of pairs (βN0 , π
N
0 ) solving this equation. A natural way to fix the initial position (β
N
0 , π
N
0 )
for the investor interested in conditional-mean hedging would be the one with minimal cost. If short-selling is allowed, the
investor is then faced with the minimization problem
min
πN0 ∈R
v(πN0 ),
where the initial wealth v is the piecewise linear function
v(πN0 ) = βN0 + πN0 S0
=
⎧⎨
⎩
E[f (St1 )] −
(
ΔSˆt1 (0) − kS0
)
πN0 , if π
N
0 ≥ 0,
E[f (St1 )] −
(
ΔSˆt1 (0) + kS0
)
πN0 , if π
N
0 < 0.
Acta Wasaensia 41
90 F. Shokrollahi, T. Sottinen / Statistics and Probability Letters 130 (2017) 85–91
Clearly, the minimal solution πN0 is independent of E[f (St1 )], and, consequently, of the option to be replicated. Also, the
minimization problem is bounded if and only if
k ≥
∣∣∣∣∣ΔSˆt1 (0)S0
∣∣∣∣∣ ,
i.e. the proportional transaction costs are bigger than the expected return on [0, t1] of the stock. In this case, the minimal
cost conditional mean-hedging strategy starts by putting all the wealth in the riskless asset.
We end this note by applying Theorem 3.1 to European call options.
Corollary 3.1 (European Call Option). Denote
dˆ+ti+1 (ti) =
ln
Sti
K − μΔti+1 − σΔBˆti+1 (ti)
σ
√
rˆ(ti+1|ti)
− σ
√
rˆ(ti+1|ti),
dˆ−ti+1 (ti) =
ln
Sti
K − μΔti+1 − σΔBˆti+1 (ti)
σ
√
rˆ(ti+1|ti)
,
Xˆti+1 (ti) = μΔti+1 + σΔBˆti+1 (ti) +
1
2
σ 2 rˆ(ti+1|ti),
and let Φ be the cumulative distribution function of the standard normal law. Then the conditional-mean hedging strategy for the
European call option with strike-price K is given by
πNti =
Stie
Xˆti+1 (ti)Φ(dˆ+ti+1 (ti)) − KΦ(dˆ−ti+1 (ti)) − Vπ
N ,k
ti + kSti |ΔπNti |
ΔSˆti+1 (ti)
. (3.2)
Proof. First we note that
Vˆ callti+1 (ti) =
∫ ∞
−∞
(
Stie
μΔti+1+σΔBˆti+1 (ti)+σ
√
rˆ(ti+1|ti)z − K
)+
φ(z)dz
= StieXˆti+1 (ti)Φ
(
dˆ+ti+1 (ti)
)
− KΦ
(
dˆ−ti+1 (ti)
)
.
Next we note that
V callti = (Sti − K )+.
So,
ΔVˆ callti+1 (ti) = StieXˆti+1 (ti)Φ(dˆ+ti+1 (ti)) − KΦ(dˆ−ti+1 (ti)) − (Sti − K )+,
and (3.2) follows from this. 
Acknowledgments
F. Shokrollahi was funded by the graduate school of the University of Vaasa.We thank the referee for valuable comments.
References
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Acta Wasaensia 43
APPLIED & INTERDISCIPLINARY MATHEMATICS | RESEARCH ARTICLE
Subdiffusive fractional Black–Scholes model for
pricing currency options under transaction costs
Foad Shokrollahi1*
Abstract: A new framework for pricing European currency option is developed in
the case where the spot exchange rate follows a subdiffusive fractional Black–
Scholes. An analytic formula for pricing European currency call option is proposed by
a mean self-financing delta-hedging argument in a discrete time setting. The
minimal price of a currency option under transaction costs is obtained as time-step
Δt ¼ tα1ΓðαÞ
 1
2
π
  1
2H k
σ
 1
H, which can be used as the actual price of an option. In addition,
we also show that time-step and long-range dependence have a significant impact
on option pricing.
Subjects: Science; Mathematics & Statistics; Applied Mathematics; Statistics & Probability
Keywords: subdiffusion process; currency option; transaction costs; inverse subordinator
process
MR Subject classifications: 91G20; 91G80; 60G22
1. Introduction
The standard European currency option valuation model has been presented by Garman and
Kohlhagen ðG KÞ (Garman & Kohlhagen, 1983). However, some papers have provided evidence
of the mispricing for currency options by the G K model. The most important reason why this
ABOUT THE AUTHOR
Foad Shokrollahi is a researcher at Department
of Mathematics and Statistics, University of
Vaasa, Finland. His research interests are
Stochastic Processes, Stochastic Analysis,
Fractional Brownian motion, and Mathematical
Finance.
PUBLIC INTEREST STATEMENT
Subdiffusion refers to a well-known and estab-
lished phenomenon in statistical physics. One
description of subdiffusion is related to subordi-
nation, where the standard diffusion process is
time-changed by the so-called inverse subordi-
nator. According to the features of the subdiffu-
sion process and the fractional Brownian motion,
we propose the new model for pricing European
currency options by using the fractional Brownian
motion, subdiffusive strategy, and scaling time in
discrete time setting, to get behavior from finan-
cial markets. Motivated by this objective, we
illustrate how to price a currency options in dis-
crete time setting for both cases: with and with-
out transaction costs by applying subdiffusive
fractional Brownian motion model. By considering
the empirical data, we will demonstrate that the
proposed model is further flexible in comparison
with the previous models and it obtains suitable
benchmark for pricing currency options.
Additionally, impact of the parameters on our
pricing formula is investigated.
Shokrollahi, Cogent Mathematics & Statistics (2018), 5: 1470145
https://doi.org/10.1080/25742558.2018.1470145
© 2018 The Author(s). This open access article is distributed under a Creative Commons
Attribution (CC-BY) 4.0 license.
Received: 16 November 2017
Accepted: 24 April 2018
First Published: 29 May 2018
*Corresponding author: Foad
Shokrollahi, Department of
Mathematics and Statistics,
University of Vaasa, P.O. Box 700,
FIN-65101 Vaasa, Finland
E-mail: foad.shokrollahi@uva.fi
Reviewing editor:
Carlo Cattani, University of Tuscia,
Italy
Additional information is available at
the end of the article
Page 1 of 14
44 Acta Wasaensia
model may not be entirely satisfactory could be that currencies are different from stocks in
important respects and the geometric Brownian motion cannot capture the behavior of currency
return (Ekvall, Jennergren, & Näslund, 1997). Since then, many methodologies for currency option
pricing have been proposed by using modifications of the G K model (Garman & Kohlhagen,
1983; Ho, Stapleton, & Subrahmanyam, 1995).
All this research above assumes that the logarithmic returns of the exchange rate are indepen-
dent identically distributed normal random variables. However, in general, the assumptions of the
Gaussianity and mutual independence of underlying asset log returns would not hold. Moreover,
the empirical research has also shown that the distributions of the logarithmic returns in the
financial market usually exhibit excess kurtosis with more probability mass near the origin and in
the tails and less in the flanks than would occur for normally distributed data (Dai & Singleton,
2000). That is to say the features of financial return series are non-normality, non-independence,
and nonlinearity. To capture these non-normal behaviors, many researchers have considered other
distributions with fat tails such as the Pareto-stable distribution and the Generalized Hyperbolic
Distribution. Moreover, self-similarity and long-range dependence have become important con-
cepts in analyzing the financial time series.
There is strong evidence that the stock return has little or no autocorrelation. As fractional
Brownian motion ðFBMÞ has two important properties called self-similarity and long-range depen-
dence, it has the ability to capture the typical tail behavior of stock prices or indexes (Borovkov,
Mishura, Novikov, & Zhitlukhin, 2018; Shokrollahi & Sottinen, 2017).
The fractional Black–Scholes ðFBSÞ model is an extension of the Black–Scholes ðBSÞ model, which
displays the long-range dependence observed in empirical data. This model is based on replacing the
classic Brownian motion by the fractional Brownian motion ðFBMÞ in the Black–Scholes model. That is
V^ðtÞ ¼ V^0 exp μtþ σ B^HðtÞ
n o
; V^0 > 0; (1:1)
where μ; and σ are fixed, and B^HðtÞ is a FBM with Hurst parameter H 2 ½12 ;1Þ. It has been shown
that the FBS model admits arbitrage in a complete and frictionless market (Cheridito, 2003;
Shokrollahi & Kılıçman, 2014; Sottinen & Valkeila, 2003; Wang, Zhu, Tang, & Yan, 2010; Xiao,
Zhang, Zhang, & Wang, 2010). Wang (2010) resolved this contradiction by giving up the arbitrage
argument and examining option replication in the presence of proportional transaction costs in
discrete time setting (Mastinšek, 2006).
Magdziarz (2009a) applied the subdiffusive mechanism of trapping events to describe properly
financial data exhibiting periods of constant values and introduced the subdiffusive geometric
Brownian motion
VαðtÞ ¼ VðTαðtÞÞ; (1:2)
as the model of asset prices exhibiting subdiffusive dynamics, where VαðtÞ is a subordinated
process (for the notion of subordinated processes please refer to Refs. Janicki and Weron (1993,
1995), Kumar, Wyłomańska, Połoczański, and Sundar (2017), Piryatinska, Saichev, and Woyczynski
(2005), in which the parent process VðτÞ is a geometric Brownian motion and TαðtÞ is the inverse
α-stable subordinator defined as follows:
TαðtÞ ¼ inffτ>0 : QαðτÞ> tg; 0< α<1: (1:3)
Here, QαðtÞ is a strictly increasing α-stable subordinator with Laplace transform: E eηQαðτÞ
  ¼ eτηα ,
0< α<1, where E denotes the mathematical expectation.
Magdziarz (2009a) demonstrated that the considered model is free-arbitrage but is incomplete
and proposed the corresponding subdiffusive BS formula for the fair prices of European options.
Shokrollahi, Cogent Mathematics & Statistics (2018), 5: 1470145
https://doi.org/10.1080/25742558.2018.1470145
Page 2 of 14
Acta Wasaensia 45
Subdiffusion is a well-known and established phenomenon in statistical physics. The usual
model of subdiffusion in physics is developed in terms of FFPE (fractional Fokker-Planck equations).
This equation was first derived from the continuous-time random walk scheme with heavy-tailed
waiting times (Metzler & Klafter, 2000). It provides a useful way for the description of transport
dynamics in complex systems (Magdziarz, Weron, & Weron, 2007). Another description of sub-
diffusion is in terms of subordination, where the standard diffusion process is time-changed by the
so-called inverse subordinator (Gu, Liang, & Zhang, 2012; Guo, 2017; Janczura, Orzeł, &
Wyłomańska, 2011; Magdziarz, 2009b, Magdziarz et al., 2007; Scalas, Gorenflo, & Mainardi, 2000,
Shokrollahi & Kılıçman, 2014; Yang, 2017).
The objective of this paper is to study the European call currency option by a mean self financing
delta hedging argument. The main contribution of this paper is to derive an analytical formula for
European call currency option without using the arbitrage argument in discrete time setting when
the exchange rate follows a subdiffusive FBS
St ¼ V^ðTαðtÞÞ ¼ S0exp μTαðtÞ þ σB^HðTαðtÞÞ
n o
; (1:4)
S0 ¼ V^ð0Þ>0:
We then apply the result to value European put currency option. We also provide representative
numerical results.
Making the change of variable, BHðtÞ ¼ μþrfrdσ tþ B^HðtÞ, under the risk-neutral measure, we have
that
St ¼ V^ðRβðtÞÞ ¼ S0exp ðrd  rf ÞðTαðtÞÞ þ σBHððTαðtÞÞ
 
; (1:5)
S0 ¼ V^ð0Þ>0:
This formula is similar to the Black–Scholes option pricing formula, but with the volatility being
different.
We denote the subordinated process Wα;HðtÞ ¼ BHðTαðtÞÞ, here the parent process BHðτÞ is a FBM
and TαðtÞ is assumed to be independent of BHðτÞ. The process Wα;HðtÞ is called a subdiffusion
process. Particularly, when H ¼ 12 , it is a subdiffusion process presented in Karipova and Magdziarz
(2017), Kumar et al. (2017), and Magdziarz (2010).
Figure 1 shows typically the differences and relationships between the sample paths of the spot
exchange rate in the FBS model and the subdiffusive FBS model.
The rest of the paper proceeds as follows: In Section 2, we provide an analytic pricing formula for
the European currency option in the subdiffusive FBS environment and some Greeks of our pricing
model are also obtained. Section 3 is devoted to analyze the impact of scaling and long-range
dependence on currency option pricing. Moreover, the comparison of our subdiffusive FBS model
and traditional models is undertaken in this section. Finally, Section 4 draws the concluding
remarks. The proof of Theorems are provided in Appendix.
2. Pricing model for the European call currency option
In this section, we derive a pricing formula for the European call currency option of the subdiffusive
FBS model under the following assumptions:
(i) We consider two possible investments: (1) a stock whose price satisfies the equation:
St ¼ S0exp ðrd  rf ÞTαðtÞ þ σWα;HðtÞ
 
; S0 >0; (2:1)
Shokrollahi, Cogent Mathematics & Statistics (2018), 5: 1470145
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Page 3 of 14
46 Acta Wasaensia
where α 2 12 ;1
 
, H 2 ½12 ;1Þ, αþ αH>1, and rd; and rf are the domestic and the foreign interest
rates, respectively. (2) A money market account:
dFt ¼ rdFtdt; (2:2)
where rd shows the domestic interest rate.
(ii) The stock pays no dividends or other distributions, and all securities are perfectly divisible.
There are no penalties to short selling. It is possible to borrow any fraction of the price of a
security to buy it or to hold it, at the short-term interest rate. These are the same valuation
policy as in the BS model.
(iii) There are transaction costs that are proportional to the value of the transaction in the
underlying stock. Let k denote the round trip transaction cost per unit dollar of transaction.
Suppose U shares of the underlying stock are bought ðU>0Þ or sold ðU<0Þ at the price St,
then the transaction cost is given by k2 Uj jSt in either buying or selling. Moreover, trading takes
place only at discrete intervals.
(iv) The option value is replicated by a replicating portfolio  with UðtÞ units of stock and riskless
bonds with value FðtÞ. The value of the option must equal the value of the replicating portfolio
to reduce (but not to avoid) arbitrage opportunities and be consistent with economic
equilibrium.
(v) The expected return for a hedged portfolio is equal to that from an option. The portfolio is
revised every Δt and hedging takes place at equidistant time points with rebalancing intervals
of (equal) length Δt, where Δt is a finite and fixed, small time-step.
Remark 2.1. From Guo and Yuan (2014), Magdziarz (2009c), we have EðTmα ðtÞÞ ¼ t
mαm!
Γðmαþ1Þ . Then, by
using α-self-similar and non-decreasing sample paths of TαðtÞ, we can obtain that α-self-similar
and non-decreasing sample paths of TαðtÞ,
E ΔTαðtÞð Þ ¼ E Tαðtþ ΔtÞ  TαðtÞð Þ
¼ 1Γð1þαÞ ðtþ ΔtÞα  tα½  ¼ t
α1
ΓðαÞΔt:
(2:3)
and
E ðΔBHðTαðtÞÞ2
 
¼ t
α1
ΓðαÞ
 	2H
Δt2H: (2:4)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
t
V
t
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
t
S
t
Figure 1. Comparison of the
spot exchange rate’ sample
paths in the FBS model (left) and
the subdiffusive FBS model
(right) for rd ¼ 0:03; rf ¼ 0:02;
α¼ 0:9; H¼ 0:8; σ¼ 0:1; S0 ¼ 1.
Shokrollahi, Cogent Mathematics & Statistics (2018), 5: 1470145
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Acta Wasaensia 47
Let C ¼ Cðt; StÞ be the price of a European currency option at time t with a strike price K that
matures at time T. Then, the pricing formula for currency call option is given by the following
theorem.
Theorem 2.1. C ¼ Cðt; StÞ is the value of the European currency call option on the stock St satisfied
(1.5) and the trading takes place discretely with rebalancing intervals of length Δt. Then, C satisfies
the partial differential equation
@C
@t
þ ðrd  rf ÞSt
@C
@St
þ 1
2
σ^2S2t
@2C
@S2t
 rdC ¼ 0; (2:5)
with boundary condition CðT; STÞ ¼ max ST  K;0f g. The value of the currency call option is
Cðt; StÞ ¼ Sterf ðTtÞΦðd1Þ  KerdðTtÞΦðd2Þ; (2:6)
and the value of the put currency option is
Pðt; StÞ ¼ KerdðTtÞΦðd2Þ  Sterf ðTtÞΦðd1Þ; (2:7)
where
d1 ¼
ln StK
 þ rd  rf ðT  tÞ þ σ^22 ðT tÞ
σ^
ffiffiffiffiffiffiffiffiffiffi
T tp ;
d2 ¼ d1  σ^ðtÞ
ffiffiffiffiffiffiffiffiffiffi
T t
p
; (2:8)
σ^2 ¼ σ2 t
α1
ΓðαÞ
 2H
Δt2H1 þ
ffiffiffi
2
π
r
k
σ
tα1
ΓðαÞ
 H
ΔtH1
" #
; (2:9)
where Φð:Þ is the cumulative normal distribution function.
In what follows, the properties of the subdiffusive FBS model are discussed, such as
Greeks, which summarize how option prices change with respect to underlying variables and
are critically important to asset pricing and risk management. The model can be used to
rebalance a portfolio to achieve the desired exposure to certain risk. More importantly, by
knowing the Greeks, particular exposure can be hedged from adverse changes in the market by
using appropriate amounts of other related financial instruments. In contrast to option prices
that can be observed in the market, Greeks cannot be observed and must be calculated given a
model assumption. The Greeks are typically computed using a partial differentiation of the
price formula.
Theorem 2.2. The Greeks can be written as follows:
Δ ¼ @C
@St
¼ erf ðTtÞΦðd1Þ; (2:10)
 ¼ @C
@K
¼ erdðTtÞΦðd2Þ; (2:11)
ρrd ¼
@C
@rd
¼ KðT  tÞerdðTtÞΦðd2Þ; (2:12)
ρrf ¼
@C
@rf
¼ StðT  tÞerf ðTtÞΦðd1Þ; (2:13)
Shokrollahi, Cogent Mathematics & Statistics (2018), 5: 1470145
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48 Acta Wasaensia
Θ ¼ @C
@t
¼ Strf erf ðTtÞΦðd1Þ  KrderdðTtÞΦðd2Þ
þ Sterf ðTtÞσ2ðα 1Þ t
α2
ΓðαÞ
H t
α1
ΓðαÞ
 2H1
Δt2H1
bσ ffiffiffiffiffiffiffiffiffiffiT tp ðT tÞΦ0ðd1Þ
þ Sterf ðTtÞ
ffiffiffi
2
π
r
kσðβ 1Þ t
α2
ΓðαÞ
H t
α1
ΓðαÞ
 H1
ΔtH1
2bσ ffiffiffiffiffiffiffiffiffiffiT tp ðT  tÞΦ0ðd1Þ
 Sterf ðTtÞ bσ
2
ffiffiffiffiffiffiffiffiffiffi
T tp Φ
0ðd1Þ;
(2:14)
Γ ¼ @
2C
@S2t
¼ erf ðTtÞ Φ
0ðd1Þ
Stσ^
ffiffiffiffiffiffiffiffiffiffi
T tp ; (2:15)
#σ^ ¼ @C
@σ^
¼ Sterf ðTtÞ
ffiffiffiffiffiffiffiffiffiffi
T  t
p
Φ0ðd1Þ: (2:16)
Remark 2.2. The modified volatility without transaction costs ðk ¼ 0Þ is given by
σ^2 ¼ σ2 t
α1
ΓðαÞ
 2H
Δt2H1
" #
; (2:17)
specially if α " 1,
σ^2 ¼ σ2Δt2H1; (2:18)
which is consistent with the result in Necula (2002).
Furthermore, from Equation (2.18), if H # 12 , then σ^2 ¼ σ2, which is according to the results with
the G K model (Garman & Kohlhagen, 1983).
Letting α " 1, from Equation (2.9), we obtain
Remark 2.3. The modified volatility under transaction costs is given by
σ^2 ¼ σ2 Δt2H1 þ
ffiffiffi
2
π
r
k
σ
ΔtH1
" #
; (2:19)
that is in line with the findings in Wang (2010).
3. Empirical studies
The objective of this section is to obtain the minimal price of an option with transaction costs and
to show the impact of time scaling Δt, transaction costs k, and subordinator parameter α on the
subdiffusive FBSmodel. Moreover, in the last part, we compute the currency option prices using our
model and make comparisons with the results of the G K and FBS models.
As kσ<
ffiffi
π
2
p
often holds (for example: σ ¼ 0:1; k ¼ 0:01), from Equation (2.9), we have
σ^2
σ2
¼ tα1ΓðαÞ
 2H
Δt2H1 þ
ffiffi
2
π
q
k
σ
tα1
ΓðαÞ
 H
ΔtH1
 2 tα1ΓðαÞ
 3
2H
Δt
3
2H1 2
π
 1
4 k
σ
 1
2;
(3:1)
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Acta Wasaensia 49
where H> 12 . Then, the minimal volatility σ^min is
ffiffiffi
2
p
σ t
α1
ΓðαÞ
 1
2 2
π
 1
2 14H k
σ
 1 12H as Δt ¼ tα1ΓðαÞ 1 2π  12H kσ 1H.
Thus, the minimal price of an option under transaction costs is represented as Cminðt; StÞ with σ^min
in Equation (2.8).
Moreover, the option rehedging time interval for traders to take is Δt ¼ tα1ΓðαÞ
 1
2
π
  1
2H k
σ
 1
H. The
minimal price Cminðt; StÞ can be used as the actual price of an option.
In particular, as Δt<1;α 2 12 ;1
 
and @C@σ^ ¼ Sterf ðTtÞ
ffiffiffiffiffiffi
Tt
p ffiffiffiffi
2π
p e
d2
2 >0;
@bσ
@H
¼ σ 2 t
α1
ΓðαÞ
 2H
Δt2H1 þ
ffiffiffi
2
π
r
k
σ
tα1
ΓðαÞ
 H
ΔtH1
" #
ln
tα1
ΓðαÞ
 
þ lnΔt
 	
 2 t
α1
ΓðαÞ
 2H
Δt2H1 þ
ffiffiffi
2
π
r
k
σ
tα1
ΓðαÞ
 H
ΔtH1
" #12
¼ 2 t
α1
ΓðαÞ
 2H
Δt2H1 þ
ffiffiffi
2
π
r
k
σ
tα1
ΓðαÞ
 H
ΔtH1
" #

σ2 ln t
α1
ΓðαÞ
 
þ lnΔt
h i
2bσ <0;
(3:2)
and @C@H ¼ @C@σ^ @σ^@H , then we have
@C
@H
<0 asH 2 ½1
2
;1Þ; (3:3)
which displays that an increasing Hurst exponent comes along with a decrease of the option
value (see Figure 2).
On the other hand, if H # 12 , then
σ^min ¼
ffiffiffi
2
p
σ
tα1
ΓðαÞ
 1
2 2
π
 1
2 14H k
σ
 1 12H
! σ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
tα1
ΓðαÞ
 s
(3:4)
and if α " 1, then σ^min !
ffiffiffi
2
p
σ as H # 12 :
In addition, if H # 12
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.05
0.1
0.15
0.2
0.25
k
O
pt
io
n 
V
al
ue
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
3
4
5
6
7
8
9
10
x 10
−3
α
O
pt
io
n 
V
al
ue
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.005
0.01
0.015
0.02
0.025
0.03
Δ t
O
pt
io
n 
V
al
ue
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0
0.005
0.01
0.015
0.02
0.025
H
O
pt
io
n 
V
al
ue
Figure 2. Call currency option
values.
Shokrollahi, Cogent Mathematics & Statistics (2018), 5: 1470145
https://doi.org/10.1080/25742558.2018.1470145
Page 7 of 14
50 Acta Wasaensia
Δt ¼ t
α1
ΓðαÞ
 1 2
π
  1
2H k
σ
 1
H
! t
α1
ΓðαÞ
 1 2
π
 
k
σ
 2
; (3:5)
and if α " 1, then Δt! 2π
 
k
σ
 2
as H # 12 :
Lux and Marchesi (1999) have shown that Hurst exponent H ¼ 0:51 0:004 in some cases, so
Equations (3.4) and (3.5) have a practical application in option pricing. For example: if H # 12 ; α " 1; k ¼
2% and σ ¼ 20%, then σ^min !
ffiffi
2
p
20 , and Δt! 0:02π ; and if H " 12 ; α " 1; k ¼ 0:2% and σ ¼ 20%, then
σ^min !
ffiffi
2
p
20 , and Δt ! 2π  104:
In the following, we investigate the impact of scaling and long-range dependence on option
pricing. It is well known that Mantegna and Stanley (1995) introduced the method of scaling
invariance from the complex science into the economic systems for the first time. Since then, a lot
of research for scaling laws in finance has begun. If H ¼ 12 and k ¼ 0, from Equation (2.9), we know
that σ^2 ¼ σ2 tα1ΓðαÞ
 
shows that fractal scaling Δt has not any impact on option pricing if a mean self-
financing delta-hedging strategy is applied in a discrete time setting, while subordinator para-
meter β has remarkable impact on option pricing in this case. In particular, from Equations (3.4)
and (3.5), we know that σ^min ! σ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 tα1ΓðαÞ
 r
as H  12 and Δt ! t
α1
ΓðαÞ
 1
2
π
 
k
σ
 2
; as H  12 . Therefore,
Cminðt; StÞ is approximately scaling free with respect to the parameter k, if H  12 , but is scaling
dependent with respect to subordinator parameter α. However, Δt ! tα1ΓðαÞ
 1
2
π
 
k
σ
 2
; is scaling
dependent with respect to parameters k and α, if H  12 . On the other hand, if H> 12 and k ¼ 0, from
Equation (2.17), we know that σ^2 ¼ σ2 tα1ΓðαÞ
 2H
Δt2H1
 	
, which displays that the fractal scaling Δt
and sabordinator parameter α have a significant impact on option pricing. Furthermore, for k0,
from Equation (2.8), we know that option pricing is scaling dependent in general.
Now, we present the values of currency call option using subdiffusive FBS model for different
parameters. For the sake of simplicity, we will just consider the out-of-the-money case. Indeed,
using the same method, one can also discuss the remaining cases: in-the-money and
at-the-money. First, the prices of our subdiffusive FBS model are investigated for some Δt and
prices for different exponent parameters. The prices of the call currency option versus its
parameters H;Δt;α and k are revealed in Figure 2. The selected parameters are St ¼ 1:4; K ¼
1:5; σ ¼ 0:1; rd ¼ 0:03; rf ¼ 0:02; T ¼ 1; t ¼ 0:1; Δt ¼ 0:01; k ¼ 0:01; H ¼ 0:8; α ¼ 0:9: Figure 2
indicates that the option price is an increasing function of k and Δt, while it is a decreasing
function of H and α.
For a detailed analysis of our model, the prices calculated by the G K, FBS and subdiffusive FBS
models are compared for both out-of-the-money and in-the-money cases. The following para-
meters are chosen: St ¼ 1:2; σ ¼ 0:5; rd ¼ 0:05; rf ¼ 0:01; t ¼ 0:1;Δt ¼ 0:01; k ¼ 0:001, and H ¼ 0:8,
along with time maturity T 2 ½0:1;2, strike price K 2 ½0:8;1:19 for the in-the-money case and K 2
½1:21;1:4 for the out-of-the-money case. Figures 3 and 4 show the theoretical values difference by
the G K, FBS, and our subdiffusive FBS models for the in-the-money and out-of-the-money,
respectively. As indicated in these figures, the values computed by our subdiffusive FBS model
are better fitted to the G K values than the FBS model for both in-the-money and out-of-the
money cases. Hence, when compared to these figures, our subdiffusive FBS model seems
reasonable.
4. Conclusion
Without using the arbitrage argument, in this paper, we derive a European currency option pricing
model with transaction costs to capture the behavior of the spot exchange rate price, where the
Shokrollahi, Cogent Mathematics & Statistics (2018), 5: 1470145
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Page 8 of 14
Acta Wasaensia 51
spot exchange rate follows a subdiffusive FBS with transaction costs. In discrete time case, we
show that the time scaling Δt and the Hurst exponent H play an important role in option pricing
with or without transaction costs and option pricing is scaling dependent. In particular, the
minimal price of an option under transaction costs is obtained.
Acknowledgments
I would like to thank the referees and the editor for their
careful reading and their valuable comments.
Funding
The author received no direct funding for this research.
This paper is supported by the university of Vaasa, Finland.
Author details
Foad Shokrollahi1
E-mail: foad.shokrollahi@uva.fi
1 Department of Mathematics and Statistics, University of
Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland.
Citation information
Cite this article as: Subdiffusive fractional Black–Scholes
model for pricing currency options under transaction
costs, Foad Shokrollahi, Cogent Mathematics & Statistics
(2018), 5: 1470145.
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Acta Wasaensia 53
Appendix
Proof of Theorem 2.1. The movement of St on time interval ½t; tþ ΔtÞ of length Δt is
ΔSt ¼ StþΔt  St ¼ Stðeðrdrf ÞΔTαðtÞþ σΔWα;HðtÞ  1Þ
¼ Stððrd  rf ÞΔTαðtÞ þ σΔWα;HðtÞ
þ 1
2
ððrd  rf ÞΔTαðtÞ þ σΔWα;HðtÞÞ2Þ
þ 1
6
Ste½θððrdrf ÞΔTαðtÞþ σΔWα;HðtÞÞ
 ððrd  rf ÞΔTαðtÞ þ σΔWα;HðtÞÞ3;
(4:1)
here θ ¼ θðt;ΔtÞ 2 ð0;1Þ is a random variable corresponding to process St.
Based on Lemmas 2.1 and 2.2 of Gu et al. (2012), we can get
ððrd  rf ÞΔTαðtÞ þ σΔWα;HðtÞÞ2 ¼ ðoðΔtαεÞ þ oðΔtαHεÞÞ2
¼ oðΔtαþ αH2εÞ þ oðΔt2αH2εÞ;
(4:2)
e½θððrdrf ÞΔTαðtÞþ σΔWα;HðtÞÞððrd  rf ÞΔTαðtÞ þ σΔWα;HðtÞÞ3
¼ oðΔt3α3εÞ þ oðΔt2αþ αH3εÞ þ oðΔt2αHþ α3εÞ þ oðΔt3α3εÞ
¼ oðΔt3αH3εÞ;
(4:3)
oðΔtαþ αH2εÞ þ oðΔt3αH3εÞ ¼ oðΔtαþ αH2εÞ: (4:4)
From the above equations, Equation (4.1) can be rewritten as follows
ΔSt ¼ ðrd  rf ÞStΔTαðtÞ þ σStΔWα;HðtÞ
þ 1
2
σ2StðΔWα;HðtÞÞ2 þ oðΔtαþ αH2εÞ:
(4:5)
By the assumption αHþ α>1, we obtain
ΔSt ¼ ðrd  rf ÞStΔTαðtÞ þ σStΔWα;HðtÞ (4:6)
Applying the Taylor expansion to Cðt; StÞ, we have
ΔCðt; StÞ ¼ @C
@t
Δt þ @C
@St
ΔSt þ 12
@2C
@S2t
ΔS2t
þ 1
2
@2C
@t2
Δt2 þ @
2C
@St@t
ΔtΔSt þ oðΔt3αHεÞ
¼ @C
@t
Δt þ @C
@St
ΔSt þ 12
@2C
@S2t
ΔS2t þ oðΔtÞ
¼ @C
@t
Δt þ ðrd  rf ÞSt
@C
@St
ΔTαðtÞ þ σSt @C
@St
ΔWα;HðtÞ
þ 1
2
σ2St
@C
@St
ðΔWα;HðtÞÞ2
þ 1
2
σ2S2t
@2C
@S2t
ðΔWα;HðtÞÞ2 þ oðΔtÞ:
(4:7)
From Equations (4.1)–(4.5), we obtain that @
2C
@S2t
, @
3C
@S3t
, @
2C
@C@t is oðΔt
1
2ð1HαÞε Þ and
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54 Acta Wasaensia
Δ
@C
@St
 
¼ @
2C
@St@t
Δtþ @
2C
@S2t
ΔSt þ 12
@3C
@S3t
ΔS2t þ oðΔtÞ; (4:8)
and
Δ
@C
@St
 



 



:StþΔt ¼ σS2t @2C@S2t




 



 ΔWα;HðtÞ

 

 þ oðΔtÞ: (4:9)
Moreover, from assumptions (iii) and (iv), it is found that the change in the value of portfolio t is
Δt ¼ UtðΔSt þ rf StΔtÞ þ ΔFt  k2 ΔUtj jStþΔt¼ UtðΔSt þ rf StΔtÞ þ rdFtΔt
 k2 ΔUtj jStþΔt þ oðΔtÞ;
(4:10)
where the number of bonds Ut is constant during time-step Δt. From assumption (v), Cðt; StÞ is
replicated by portfolio ðtÞ. Thus, at time points Δt, 2Δt, 3Δt; :::; we have Cðt; StÞ ¼ UtSt þ Ft and
Ut ¼ @C@St . Therefore, according to Equations (4.5)–(4.10), we have
Δ ¼ @C
@St
ðrd  rf ÞStΔTαðtÞ þ σStΔWα;HðtÞ þ
1
2
σ2StðΔWα;HðtÞÞ2 þ rf StΔt
 	
þ rdFtΔt
k
2
Δ
@C
@St
 



 



:StþΔt þ oðΔtÞ
¼ @C
@St
ðrd  rf ÞStΔTαðtÞ þ σStΔWα;HðtÞ þ
1
2
σ2StðΔWα;HðtÞÞ2 þ rf StΔt
 	
þ C t; Stð Þ  St @C
@St
 
rdΔt
k
2
σS2t
@2C
@S2t




 



ΔWα;HðtÞ þ oðΔtÞ:
(4:11)
Consequently,
Δ ΔC ¼ rdC rd  rf
 
St
@C
@St
 @C
@t
 
Δt 1
2
σ2S2t
@2C
@S2t
ΔWα;HðtÞ
 2
 k
2
σS2t
@2C
@S2t




 



ΔWα;HðtÞ þ oðΔtÞ: (4:12)
The time subscript, t, has been suppressed. As expected, using Equation (4.12), (iv), Remark 2.1,
and (4.13), we infer
EðΔ ΔCÞ ¼ rdC rd  rf
 
St
@C
@St
 @C
@t
 
Δt
 1
2
tα1
ΓðαÞ
 	2H
Δt2Hσ2S2t
@2C
@S2t
 1
2
ffiffiffi
2
π
r
kσS2t
tα1
ΓðαÞ
 	H
ΔtH
@2C
@S2t




 



¼ rdC ðrd  rf ÞSt
@C
@St
 @C
@t
 1
2
tα1
ΓðαÞ
 	2H
Δt2H1σ2S2t
@2C
@S2t
 
1
2
ffiffiffi
2
π
r
kσS2t
tα1
ΓðαÞ
 	H
ΔtH1
@2C
@S2t




 



!
Δt ¼ 0:
(4:13)
Thus, from Equation (4.13), we can derive
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Acta Wasaensia 55
rdC ¼ ðrd  rf ÞSt
@C
@St
þ @C
@t
þ 1
2
tα1
ΓðαÞ
 	2H
Δt2H1σ2S2t
@2C
@S2t
þ 1
2
ffiffiffi
2
π
r
kσS2t
tα1
ΓðαÞ
 	H
ΔtH1
@2C
@S2t




 



:
(4:14)
We define σ^2ðtÞ as follows:
σ^2 ¼ σ2 t
α1
ΓðαÞ
 	2H
Δt2H1 þ
ffiffiffi
2
π
r
kσ1
tα1
ΓðαÞ
 	H
ΔtH1
 !
: (4:15)
where @
2C
@S2t
is ever positive for the ordinary European currency call option without transaction costs,
if the same conduct of @
2C
@S2t
is postulated here and σ^ðtÞ remains fixed during the time-step ½t;ΔtÞ.
Then, from Equations (4.14) and (4.15), we obtain
@C
@t
þ ðrd  rf ÞSt
@C
@St
þ 1
2
σ^2S2t
@2C
@S2t
 rdC ¼ 0: (4:16)
Followed by
C ¼ Cðt; StÞ ¼ Sterf ðTtÞΦðd1Þ  KerdðTtÞΦðd2Þ; (4:17)
and
d1 ¼ ln
St
K
 
þðrdrf ÞðTtÞþσ^22 ðTtÞ
σ^
ffiffiffiffiffiffi
Tt
p ;
d2 ¼ d1  σ^
ffiffiffiffiffiffiffiffiffiffi
T  tp :
(4:18)
Proof of Theorem 2.2. First, we derive a general formula. Let y be one of the influence factors. Thus
@C
@y ¼ @Ste
 rfð ÞðTtÞ
@y Φðd1Þ þ Sterf ðTtÞ @Φðd1Þ@y
 @KerdðTtÞ@y Φðd2Þ  KerdðTtÞ @Φðd2Þ@y
(4:19)
But
@Φðd2Þ
@y
¼ Φ0ðd2Þ @d2
@y
¼ 1ffiffiffiffiffi
2π
p e
d2
2
2
@d2
@y
¼ 1ffiffiffiffiffi
2π
p exp ðd1  bσ ffiffiffiffiffiffiffiffiffiffiT  tp Þ2
2
 !
@d2
@y
¼ 1ffiffiffiffiffi
2π
p e
d2
1
2 expðd1bσ ffiffiffiffiffiffiffiffiffiffiffiffiT  tÞp Þ exp bσ2ðT  tÞ2
 !
@d2
@y
¼ 1ffiffiffiffiffi
2π
p e
d2
1
2 exp ln
St
K
þ ðrd  rf ÞðT  tÞ
 
@d2
@y
¼ 1ffiffiffiffiffi
2π
p e
d2
1
2
S
K
expððrd  rf ÞðT  tÞÞ
@d2
@y
:
(4:20)
Then
Shokrollahi, Cogent Mathematics & Statistics (2018), 5: 1470145
https://doi.org/10.1080/25742558.2018.1470145
Page 13 of 14
56 Acta Wasaensia
@C
@y ¼ @Ste
ðrf ÞðTtÞ
@y Φðd1Þ  @Ke
rdðTtÞ
@y Φðd2Þ
þ Sterf ðTtÞΦ0ðd1Þ @σ^
ffiffiffiffiffiffiffi
TtÞ
p
@y :
(4:21)
Substituting in (4.21), we get the desired Greeks.
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Shokrollahi, Cogent Mathematics & Statistics (2018), 5: 1470145
https://doi.org/10.1080/25742558.2018.1470145
Page 14 of 14
Acta Wasaensia 57
Journal of Mathematical Finance, 2018, 8, 623-639 
http://www.scirp.org/journal/jmf 
ISSN Online: 2162-2442 
ISSN Print: 2162-2434 
 
DOI: 10.4236/jmf.2018.84040  Nov. 7, 2018 623 Journal of Mathematical Finance 
 
 
 
 
Mixed Fractional Merton Model to Evaluate 
European Options with Transaction Costs 
Foad Shokrollahi 
Department of Mathematics and Statistics, University of Vaasa, Vaasa, Finland 
 
 
 
Abstract 
This paper deals with the problem of discrete-time option pricing by the 
mixed fractional version of Merton model with transaction costs. By a 
mean-self-financing delta hedging argument in a discrete-time setting, a Eu-
ropean call option pricing formula is obtained. We also investigate the effect 
of the time-step tG  and the Hurst parameter H on our pricing option mod-
el, which reveals that these parameters have high impact on option pricing. 
The properties of this model are also explained. 
 
Keywords 
Transaction Costs, Mixed Fractional Brownian Motion, European Option, 
Merton Model 
 
1. Introduction 
Over the last few years, the financial markets have been regarded as complex and 
nonlinear dynamic systems. A series of studies has found that many financial 
market time series display scaling laws and long-range dependence. Therefore, it 
has been proposed that the Brownian motion in the classical Black-Scholes (BS) 
model [1] should be replaced by a process with long-range dependence. 
Nowadays, the BS model is the one most commonly used for analyzing 
financial data, and some scholars have presented modified forms of the BS 
model which have influential and significant outcomes on option pricing. 
However, they are still theoretical adaptations and not necessarily consistent 
with the empirical features of financial return series, such as nonnormality, 
long-range dependence, etc. For example, some scholars [2] [3] [4] [5] [6] have 
showed that returns are of long-range (or short-range) dependence, which 
suggests strong time-correlations between different events at different time 
How to cite this paper: Shokrollahi, F. 
(2018) Mixed Fractional Merton Model to 
Evaluate European Options with Transac-
tion Costs. Journal of Mathematical Finance, 
8, 623-639. 
https://doi.org/10.4236/jmf.2018.84040 
 
Received: May 7, 2018 
Accepted: November 4, 2018 
Published: November 7, 2018 
 
Copyright © 2018 by author and  
Scientific Research Publishing Inc. 
This work is licensed under the Creative 
Commons Attribution-NonCommercial 
International License (CC BY-NC 4.0). 
http://creativecommons.org/licenses/by-nc/4.0/ 
  Open Access
58 Acta Wasaensia
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 624 Journal of Mathematical Finance 
 
scales [7] [8] [9]. In the search for better models for describing long-range 
dependence in financial return series, a mixed fractional Brownian (MFBM) 
model has been proposed as an improvement of the classical BS model [10]-[18]. 
The advantage of using the MFBM is that the markets are free of arbitrage.  
Moreover, Cheridito [10] has proved that, for 
3 ,1
4
H § ·¨ ¸© ¹ , the MFBM is  
equivalent to one with Brownian motion, and hence time-step and long-range 
dependence in return series have no impact on option pricing in a complete 
financial market without transaction costs. In addition, a number of empirical 
studies show that the paths of asset prices are discontinuous and that there are 
jumps in asset prices, both in the stock market and foreign exchange [9] [19] 
[20] [21] [22]. 
The above researches have an important implication for option pricing. 
Merton [23] created a revolution in option pricing when the underlying asset 
was governed by a diffusion process. Based on this theory, Kou [24], Cont and 
Tankov [25] also considered the problems of pricing options under a jump 
diffusion environment in a larger setting. In this paper, to capture jumps or 
discontinuities, fluctuations and to take into account the long memory property 
of financial markets, a mixed fractional version of the Merton model is 
introduced, which is based on a combination of Poisson jumps and MFBM. The 
mixed fractional Merton (MFM) model is based on the assumption that the 
underlying asset price is generated by a two-part stochastic process: 1) small, 
continuous price movements are generated by a MFBM, and 2) large, infrequent 
price jumps are generated by a Poisson process. This two-part process is 
intuitively appealing, as it is consistent with an impressive market in which 
major information arrives infrequently and randomly. This process may provide 
a description for empirically observed distributions of exchange rate changes 
that are skewed, leptokurtic, have long memory and fatter tails than comparable 
normal distributions and apparent nonstationary variance. Further, we will show 
the impact of the time-step and long-range dependence in return series exactly 
on option pricing, regardless of whether proportional transaction costs are 
considered or not in a discrete time setting. 
Leland [26] is a pioneer scholar, who investigated option replication where 
transaction costs exist in a discrete time setting. In this view, the arbitrage-free 
arguments presented by Black and Scholes [1] are not applicable in a model 
where transaction costs occur at all moments of trading of the stock or bond. 
The problem is that perfect replication incurs an infinite number of transaction 
costs because of the infinite variation which exists in the geometric Brownian 
motion. In this regard, a delta hedge strategy is constructed in accordance with 
revision conducted a discrete number of times. Transaction costs lead to the 
failure of the no arbitrage principle and the continuous time trade in general: 
instead of no arbitrage, the principle of hedge pricing, according to which the 
price of an option is defined as the minimum level of initial wealth needed to 
hedge the option, comes into force. 
Acta Wasaensia 59
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 625 Journal of Mathematical Finance 
 
According to the empirical findings obtained before and the views of 
behavioral finance and econophysics, we are motivated to examine the problem 
that exists in option pricing, while the dynamics of price tS  follows a mixed 
fractional jump-diffusion process under the transaction costs. We assume that 
tS  satisfies 
    ln
0e .H H t
t B t B t N J
tS S
P V V                       (1.1) 
where 0 , ,S P V  and HV  are fixed;  B t  is a Brownian motion;  HB t  is a  
fractional Brownian motion with Hurst parameter 
3 ,1
4
H § ·¨ ¸© ¹ ; tN  is a Poisson  
process with intensity 0O ! ; and J is a positive random variable. We assume 
that    , ,H tB t B t N  and J are independent. 
This paper is organized into several sections. In Section 2, we will study the 
problem of option pricing with transaction costs by applying delta hedging 
strategy. In addition, a new framework for pricing European option is obtained 
when the stock price tS  is satisfied in Equation (1.1). Section 3 is devoted to 
empirical studies and simulations to show the performance of the MFM model. 
A conclusion is presented in Section 4. 
2. Pricing Option by Mixed Fractional Version of Merton  
Model with Transaction Costs 
Suppose  ^ ` 0tB t t  be a standard Brownian motion and  ^ ` 0H tB t t  be a 
fractional Brownian motion with the Hurst parameter 
3 ,1
4
H § ·¨ ¸© ¹ , both defined  
on complete probability space  , , ,t P: F F , the absolute price jump size J is a 
nonnegative random variable drawn from lognormal distribution, i.e. 
   ln ,J JJ N P V , which implies  
 2 2 222~ e ,e e 1JJ J J JJ Lognormal VP P V V § ·¨ ¸¨ ¸© ¹  
and a Poisson process   0t tN N t  with rate O . Additionally, the processes 
, ,HB B N  and J are independent, P is the real world probability measure and   > @0,t t TF  denotes the P-augmentation of filtration generated by      , ,HB B tW W W d . 
The objective of this section is to derive a stock pricing formula under 
transaction costs in a discrete time setting. Consider  ,D S -market with a 
bond tD  and a stock tS , where  
0e .
rt
tD D                          (2.1) 
and 
    ln
0 0 0e , , , , , , .H H t
t B t B t N J
t HS S R D S t R
P V V P V V            (2.2) 
The groundwork of modeling the effects of transaction costs was done by 
Leland [26]. He adopted the hedging strategy of rehedging at every time-step 
60 Acta Wasaensia
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 626 Journal of Mathematical Finance 
 
tG . That is, with every tG  the portfolio is rebalanced, whether or not this is 
optimal in any sense. In the following proportional transaction cost option 
pricing model, we follow the other usual assumptions in the Black-Scholes 
model, but with the following exceptions: 
1) The price tS  of the underlying stock at time t satisfies Equation (2.2). 
2) The portfolio is revised every tG  where tG  is a finite and fixed, small 
time-step. 
3) Transaction costs are proportional to the value of the transaction in the 
underlying. Let k denote the round trip transaction cost per unit dollar of 
transaction. Suppose 0U !  shares are bought  0U !  or sold  0U   at the 
price tS , then the transaction cost is given by 2 t
k U S  in either buying or 
selling, where k is a constant. The value of k will depend on the individual 
investor. In the MFM model, where transaction costs are incurred at every time 
the stock or the bond is traded, the no arbitrage argument used by Black and 
Scholes no longer applies. The problem is that due to the infinite variation of the 
MFBM, perfect replication incurs an infinite amount of transaction costs. 
4) The hedge portfolio has an expected return equal to that from an option. 
This is exactly the same valuation policy as earlier on discrete hedging with no 
transaction costs. 
5) Traditional economics assumes that traders are rational and maximize their 
utility. However, if their behaviour is assumed to be boundedly rational, the 
traders' decisions can be explained both by their reaction to the past stock price, 
according to a standard speculative behaviour, and by imitation of other traders’ 
past decisions, according to common evidence in social psychology. It is well 
known that the delta-hedging strategy plays a central role in the theory of option 
pricing and that it is popularly used on the trading floor. Therefore, based on the 
availability heuristic, suggested by Tversky and Kahneman [27], traders are 
assumed to follow, anchor, and imitate the Black-Scholes delta-hedging strategy 
to price an option. In this case, delta-hedging argument is a partial and 
imperfect hedging strategy, which does not eliminate all of the risk. However, as 
mentioned in the paper [28], in most models of stock fluctuations, except for 
very special cases, risk in option trading cannot be eliminated and strict 
arbitrage opportunities do not exist, whatever be the price of the option. The risk 
cannot be eliminated is furthermore the fundamental reason for the very 
existence of option markets. 
Delta hedging is an options strategy that aims to reduce, or hedge, the risk 
associated with price movements in the underlying asset, by offsetting long and 
short positions. For example, a long call position may be delta hedged by 
shorting the underlying stock. This strategy is based on the change in premium, 
or price of option, caused by a change in the price of the underlying security. In 
this section we use the delta hedging strategy to obtain a pricing formula for 
European call option. 
Let the price of European call option be denoted with expiration T and strike 
Acta Wasaensia 61
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 627 Journal of Mathematical Finance 
 
price K by  , tC t S  with boundary conditions:  
       , , ,0 0, , as .T T t t tC T S S K C t C t S S S   o of       (2.3) 
Then,  , tC t S  is derived by the following theorem. 
Theorem 2.1. The price at every > @0,t T  of a European call option with 
strike price K that matures at time T is given by  
 
          1 2
0
e
, e .
!
nT t
r T t
t t
n
T t
C t S S d K d
n
O O I I
c f  
 
c  ª º ¬ ¼¦       (2.4) 
Moreover,  , tC t S  satisfies the following equation  
   
> @
2 2 2
2
ˆ
, ,
2
1 0,
t
t t t
t t
t
t
SC C CrS rC E C t JS C t S
t S S
CE J S
S
V O
O
w w w ª º    ¬ ¼w w w
w   w
       (2.5) 
where  
   2
1 2 1
ln
2 , ,
t n
n
n
n
S r T t T t
Kd d d T t
T t
V
VV
§ ·    ¨ ¸© ¹            (2.6) 
 
2 2
2 22 ˆe , ,
J
J J
n
nE J
T t
VP VO O O V Vc                    (2.7) 
   
2
2
2
2ln
1 e 1 ,
J
J
J
J
n
n
n E J
r r E J r
T t T t
VP
VP
O O 
§ ·¨ ¸§ · © ¹¨ ¸       ¨ ¸ © ¹
     (2.8) 
     22 1 2 22 2 2 22ˆ ,ʌ
H H
H Ht k t signt
VV V V G V GG
 § ·    *¨ ¸© ¹
        (2.9) 
 sign *  is the signum function of 2 2
t
C
S
w
w ; n is the number of prices jumps;  
tG  is a small and fixed time-step; k is the transaction costs and  .I  is the 
cumulative normal distribution.  
Moreover, using the put call parity, we can easily obtain the valuation model 
for a put currency option, which is provided by the following corollary.  
Corollary 2.1. The value of European put option with transaction costs is 
given by 
 
          2 1
0
e
, e .
!
nT t
r T t
t t
n
T t
P t S K d S d
n
O O I I
c f  
 
c  ª º   ¬ ¼¦  
3. Properties of Pricing Formula 
In this section, we present the properties of MFM’s log-return density. The 
effects of Hurst parameter and time-step on our modified volatility  2nV  are 
also discussed in the discrete time and continuous time cases. Then we show that 
these parameters play a significant role in a discrete time setting, both with and 
62 Acta Wasaensia
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 628 Journal of Mathematical Finance 
 
without transaction costs.  
3.1. Log-Return Density 
In the case of MFM the log return jump size is assumed to be  
     2ln ~ ,i i J JY J N P V  and the probability density of log return  lnt tx S S  
is achieved as a quickly converging series of the following form: 
     
0
|t t n t
n
P x A P N n P x A N n
f
 
     ¦  
     2 2 2 2
0
e
; , ,
!
nt
H
t t J H J
n
t
P x N x t n t t n
n
O O P P V V V
f
 
   ¦       (3.1) 
where  
 
 
  
 
2 2 2 2
2
2 2 2 22 2 2 2
; ,
1 exp
22ʌ
H
t J H J
t J
HH
H JH J
N x t n t t n
x t n
t t nt t n
P P V V V
P P
V V VV V V
  
ª º « » « »   ¬ ¼
      (3.2) 
The term    e
!
nt
t
t
P N n
n
O O   is the probability that the asset price jumps  
n times during the time interval of length t. And  
   2 2 2 2| ; , Hn t t J H JP x A N n N x t n t t nP P V V V       is the mixed fractional 
normal density of log-return. It supposes that the asset price jumps i times in the 
time interval of t. As a result, in the MFM model, the log-return density can be 
described as the weighted average of the mixed fractional normal density by the 
probability that the asset price jumps n times. 
The outstanding properties of log-return density  tP x  are observed in the 
MFM. Firstly, the JP  sign refers to the expected log-return jump size, 
   ln JE Y E J P  , which indicates the skewness sign. If 0JP  , the 
log-return density  tP x  shows negatively skewed, and if 0JP  , it is 
symmetric as displayed in Figure 1 (Table 1). 
Secondly, larger value of intensity O  (which means that jumps are expected 
to occur more frequently) makes the density fatter-tailed as illustrated in Figure 
2. Note that the excess kurtosis in the case 20O   is much smaller than in the 
case 1O   or 10O  . This is because excess kurtosis is a standardized measure 
(by standard deviation) (Table 2). 
3.2. The Impact of Parameters 
Mantegna and Stanley [29] as pioneer scholars proposed the scaling invariance 
method from the complex science of economic systems which led to numerous 
investigations into scaling laws in finance. The major question in economics is 
whether the price impact of scaling law and long-range dependence is significant 
in option pricing. The answer to this question is assured. For instance, one of the 
significant issues in finance concerning the modeling of high-frequency data is 
related to analyzing the volatility in different time scales. 
Acta Wasaensia 63
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 629 Journal of Mathematical Finance 
 
 
Figure 1. MFM’s log-return density. Fixed parameters are 0.25V  , 0.25HV  , 
0.76H  , 0.1JV  , 0JP  , 0.009P  , and 0.5t  . 
 
 
Figure 2. MFM’s log-return density. Fixed parameters are 0.25V  , 0.25HV  , 
0.76H  , 0.1JV  , 3O  , 0.009P  , and 0.5t  . 
 
Table 1. Annualized moments of Merton’s log-return density in Figure 1. 
Model Mean Standard Deviation Skewness Excess Kurtosis 
0.4JP    −1.1910 0.6161 −0.5082 0.2806 
0JP   0.0090 0.1361 0 0.706 
0.4JP   1.2090 0.6161 0.5082 0.2806 
64 Acta Wasaensia
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 630 Journal of Mathematical Finance 
 
Table 2. Annualized moments of Merton’s log-return density in Figure 2. 
Model Mean Standard Deviation Skewness Excess Kurtosis 
1O   0.0040 0.1161 0 0.0223 
10O   −0.0411 0.2061 0 0.706 
20O   −0.0913 0.3061 0 0.0640 
 
Remark 3.1. In a continuous time setting  0, 0tG O z  without transaction 
costs the implied volatility is 
2
2 2ˆ Jn
n
T t
VV V   , thus the option value is similar to  
the Merton jump diffusion model [19]. Moreover, if 0tG   in the absence of 
transaction costs and jump case, the MFM model reduces to the BS model  
2 2 2
2 0,2
t
t
t t
SC C CrS rC
t S S
Vw w w    w w w                  (3.3) 
which shows that the Hurst parameter H and time-step tG  have no effect on 
option pricing model in a continuous time setting  0tG  .  
Remark 3.2. In a discrete time setting without transaction costs  
 0, 0k tG z , if jump occurs, the modified volatility is  
  22 12 2 2ˆ H Jn H nt T t
VV V V G      and when jump does not occur  0O  , from 
Equation (2.5), we have 
   2 22 12 2 2 0,2H tt Ht tSC C CrS t rCt S SV V G w w w     w w w           (3.4) 
which demonstrates that the delta hedging strategy in a discrete time case is 
fundamentally different in comparison with a continuous time case. It also 
indicates that the scaling exponent 2 1H   and time-step tG  play a significant 
role in option pricing theory. Figure 3 illustrates the impacts of the time-step, 
Hurst parameter, mean jump, and jump intensity on our option pricing model. 
Remark 3.3. From [30] we infer there exists 
10,t
M
G § ·¨ ¸© ¹  such that  
2
10,
ˆmin ,
t
M
G
V§ ·¨ ¸© ¹
                           (3.5) 
Holds, 
where 1M ! , k is small enough 
   22 1 2 22 2 2 22ˆ .ʌ 2
H H
H Ht k tt
VV V V G V GG
 § ·   ¨ ¸© ¹
          (3.6) 
Indeed,  
   
   
2
2 1 2 22 2
1
1 2 41 2 2222
2
ʌ
22 .ʌ
H H
H H
H H
H H
t k t
t
t k t
t
VV G V GG
VV G V GG
 
 
§ · ¨ ¸© ¹
§ ·§ ·t ¨ ¸¨ ¸¨ ¸© ¹© ¹
             (3.7) 
Acta Wasaensia 65
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 631 Journal of Mathematical Finance 
 
 
Figure 3. Modified volatility. Fixed parameters are 0.1V  , 0.1HV  , 0.76H  , 
0.03JV  , 0.2T  , 0.2T  , and 0.1t  . 
Set 
   
22 2
2 12
2 2
2 .ʌ
H H
H H
kt
t t
VVV G G G


§ ·¨ ¸ ¨ ¸© ¹
                (3.8) 
Thus  
 
22 2 2
2
22
2 2 8
ʌ ʌ ʌ
.
2
H
H
k k k t
t
V G
V G
§ · ¨ ¸© ¹               (3.9) 
Suppose  
 
22 2 2
2
2 2
2 2 8
ʌ ʌ ʌ
.
2
H
H
k k k x
f x x
V
V
§ · ¨ ¸© ¹              (3.10) 
Since  0 0f   and 
22 2 2
2
2
2
2 2 8 1
ʌ ʌ ʌ1 1 0,
2
H
H
k k k
M
f
M M
V
V
§ · ¨ ¸§ · § · © ¹  !¨ ¸ ¨ ¸© ¹ © ¹        (3.11) 
as k is small enough. 
Hence, there exists a 
10,t
M
G § ·¨ ¸© ¹  such that 
2
10,
ˆmin
t
M
G
V§ ·¨ ¸© ¹
 holds. 
Suppose  
 2 2
10,
ˆ ˆmin min ,
t
M
G
V V§ ·¨ ¸© ¹
                      (3.12) 
so 
  22 2 2
1 10, 0,
ˆmin min min .n n
t t
M M
n
T tG G
GV V V§ · § · ¨ ¸ ¨ ¸© ¹ © ¹
                (3.13) 
Then the minimal price of an option with respect to transaction costs is 
displayed as  min , tC t S  with  2 minnV  in Equation (2.4).  min , tC t S  can be 
66 Acta Wasaensia
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 632 Journal of Mathematical Finance 
 
applied to the real price of an option.  
4. Conclusion 
To capture the long memory and discontinuous property, this article focuses on 
the problem of pricing European option in a mixed fractional Merton 
environment without using the arbitrage argument. We obtain a mixed 
fractional version of Merton model for pricing European option with transaction 
costs. Some properties of mixed fractional Merton’s log-return density are 
discussed. Moreover, we derive that the Hurst parameter H and time-step tG  
play a significant role in pricing option in a discrete time setting for cases both 
with and without transaction costs. But these parameters have no impact on 
option pricing in a continuous time setting. 
Conflicts of Interest 
The authors declare no conflicts of interest regarding the publication of this pa-
per. 
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DOI: 10.4236/jmf.2018.84040 635 Journal of Mathematical Finance 
 
Appendix 
Proof of Theorem 2.1. We consider a replicating portfolio with  t\  units of 
financial underlying asset and one unit of the option. Then, the value of the 
portfolio at time t is  
   , .t t tP t S C t S\                       (4.1) 
Now, the movement in tS  and tP  is considered under discrete time interval 
tG . In view of this, we suppose that trading takes place at the specific time 
points of t and t tG . It can be said that the number of shares through the use 
of delta-hedging strategy and the present stock price tS  are constantly held 
during the rebalancing interval > ,t t tG . Then, the movement in the value of 
the portfolio after time interval tG  is defined as follows: 
     , .
2t t t t
kP t S C t S t SG \ G G G\                (4.2) 
where tSG  is the movement of the underlying stock price,  tG\  is the 
movement of the number of units of stock held in the portfolio, and tPG  is the 
change in the value of the portfolio. 
Since the time-step tG  and the asset change are both small, according to 
Taylor’s formulae we have if 0tNG   with probability 1 tOG , so 
        
        
2
3
2
e ,
6
H H
t
t t t t H H H H
t B t B tt
H H
SS S t S B t S B t t B t B t
S t B t B tT PG VG V G
G PG GV GV PG VG V G
PG VG V G ª º¬ ¼
     
  
 (4.3) 
where  , ,t w wT T : , and 0 1T  . 
Since  B t  and  HB t  are continuous, then from [28] we have 
     1 1log ,HHt B t O t tG G G G
§ · ¨ ¸¨ ¸© ¹
                (4.4) 
     32 1log ,t B t O t
t
G G G G
§ · ¨ ¸¨ ¸© ¹
                  (4.5) 
 
  0 as 0,
HB t t
B t
G GG o o                     (4.6) 
and 
       
           
         
3
53 1 2 1
2
3 33 31 12 22 2
e
log log
log log .
H Ht B t B t
H Ht B t B t
O t O t t O t t
O t t O t t
T PG VG V G PG VG V G
G G G G G
G G G G
 ª º¬ ¼
 
 
 ª º¬ ¼
§ ·  ¨ ¸© ¹
§ · § ·  ¨ ¸ ¨ ¸¨ ¸ ¨ ¸© ¹ © ¹
 
Thus, we can get  
   
       32 12 log ,
2
t t t H H
t
H H
S S t S B t B t
S B t B t O t t
G P G VG V G
VG V G G G 
  ª º¬ ¼
§ ·  ª º ¨ ¸¬ ¼ © ¹
      (4.7) 
70 Acta Wasaensia
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 636 Journal of Mathematical Finance 
 
         322 12 2 log ,t t H HS S B t B t O t tG VG V G G G § ·  ª º ¨ ¸¬ ¼ © ¹      (4.8) 
         2 32 1221, log ,2t t tt t
C C CC t S t S S O t t
t S S
G G G G G G w w w § ·    ¨ ¸w w w © ¹   (4.9) 
and 
       2 32 1221 log .2t tt tt t S S O t tt S S
\ \ \G\ G G G G G w w w § ·    ¨ ¸w w w © ¹    (4.10) 
If 1tNG   with probability tOG  and the jump of tN  in > @,t t t '  is 
assumed to occur at current time t, then 
    ln
0e ,H H
t B t B t J
t
S S P V V
                       (4.11) 
      ln
0e ,H H
t t B t t B t t J
t tS S
P G V G V G
G
     
                  (4.12) 
   e 1 ,H Ht B t B tt tt t tS S S S
P V V
GG      ª º   ¬ ¼             (4.13) 
     e 1H H
t t t t t t tt t
t B t B t
tt t
S S S S S S S
S S S
G G
P V V
G  
 
 
 
     
ª º   ¬ ¼
             (4.14) 
         
     
     2 32 122
, , , , ,
, ,
1 log ,
2
t t t tt t
tt t
t
t
t
C t S C t t S C t S C t S C t S
C CC t S C t S t S
t S
C S O t t
S
GG G
G G
G G G
 
 





    
w w   w w
w § ·  ¨ ¸w © ¹
      (4.15) 
           
       2 32 122
, ,
, , ,
,1 log ,
2
t t
t tt t
t
t
t
t
t S t S
t S t S t S t S
t S
t S
S O t t
S
\ \G\ \ \ G G
\ G G G
 
 





w w   w w
w § ·  ¨ ¸w © ¹
   (4.16) 
where t tt tS S SGG    . 
Based on the above assumptions iv and v, we have t t
t t
P DE
P D
G G§ ·  ¨ ¸© ¹
, i.e. 
  2t tE P rP O tG G  . Then  
     
              
       
         
 
2 32 1
2
2
2
2
1 ,
2
e 1 , ,
1 log
2
, ,
, ,
2
,1
2
H H
t
t t
t B t B t
t tt t t
t t
t t
t tt
tt t
t
t
t
kSt E S C t S t
tE S t S S t C t S C t S
C C Ct S S O t t
t S S
t S t SkS t S t S t S
t S
t S
S
S
P V V
OG \G G G\
OG \ \
G G G G G
\ \\ \ G G
\ G
  
 
 
 
 



 

ª º  « »¬ ¼
ª     «¬
·w w w § ·    ¸¨ ¸w w w © ¹¹
w w   w w
w w      
32 1
2 log ,tt O t t rP tG G G 
º§ ·  »¨ ¸© ¹¼
 
Acta Wasaensia 71
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 637 Journal of Mathematical Finance 
 
i.e.  
          
             
           
   
2
32 1
2
2
1 , e 1
, ,
, ,
,1 log 1
2 2
,
, ,
2
H Ht B t B t
t t t
t t
t tt t t
t
t t
t
t
t
tt
t E S C t S tE S t
C t S C t S
S S t C t S C t S t S
t S
C t S kSS O t t t E t
S
tkStE t S t S
P V VOG \G G OG \
\ G G
G G G OG G\
\OG \ \

 
  





 

ªª º   «¬ ¼ ¬
§ w w¨     ¨ w w©
º·w § · ª º»¸   ¨ ¸ « »»¸w ¬ ¼© ¹¹¼
wª  «¬
     
          2 32 1 222
,
,1 log ,
2
t t
t
t
t
tt
t
S t S
t S
t S
t S
S O t t rP t O t
S
\G G
\ G G G G G
 






ww w
w º§ ·   »¨ ¸w © ¹ ¼
 
where the current stock price tS  is given. Since  
         
       
       
         
     
2
32 1
2
2
2
2
2
3 1
2
1
,
2 2 2
, ,
,
,1 log
2
, , ,1 1
2 2
log ,
2
t tt
t
t t
t t
t
t
t
t
t t tt
t t
t t
t
t
t kS t kSkSE t E t t S
t S t S
t S t S
t S
t S
S O t t
S
t S t S t St kS
E t S S
t S S
t kS
O t t t S
OG OGG\ G\ \
\ \\ G G
\ G G G
\ \ \OG G G G
OGG G \

 







ª º  « »¬ ¼
w w  w w
w § ·  ¨ ¸w © ¹
w w w  w w w
§ · ¨ ¸© ¹      
,
, tt
t S
t S t
t
\\ G
w  w
 
           
         
        
2
32 1
2
2
2
2
22 2
, ,1 log
2
, ,
2 2
2 , , ,
2 ʌ 2
t t
t t
t t
t t
H H tt
t
Ht t
H tt
t
t S t S
S S O t t
S S
kS k S tE B t B t t S t S O t
S
kS k S tt t t S t S O t
S
\ \G G G G
O G\ VG V G \ \ G
O G\ V G V G \ \ G
 
 
 



w w § ·   ¨ ¸w w © ¹
w|    w
w    w
(4.17) 
and 
t
C
S
\ w w , from Equations (4.1) - (4.17), we can get 
  
    > @
   
2 2
2 12 2
2
2 2 2
2 22
2
2
, , 1
2 0.
2 ʌ
Ht
t H
t t
t t t t
t
Ht
H
t
SC C CrS t
t S S
CrC E C t J S C t S E J S
S
kS Ct t O t
t S
V V G
O O
V V G G GG


ªw w w  « w w w¬
wª º    ¬ ¼ w
º§ · w »    ¨ ¸ w »© ¹ ¼
       (4.18) 
72 Acta Wasaensia
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 638 Journal of Mathematical Finance 
 
Hence, we assume that  
  
    > @
 
2 2
2 12 2
2
2 2 2
2 22
2
2
, , 1
2 0.
2 ʌ
Ht
t H
t t
t t t t
t
Ht
H
t
SC C CrS t
t S S
CrC E C t J S C t S E J S
S
kS Ct
t S
V V G
O O
V V GG


w w w  w w w
wª º    ¬ ¼ w
§ · w   ¨ ¸ w© ¹
         (4.19) 
Note that the term  2 2 2 222
2 ʌ
Ht
H
kS t
t
V V GG
§ ·¨ ¸© ¹
 is nonlinear, except when 
2
2
t
C
S
w*  w  does not change sign for all tS . Since it represents the degree of  
mishedging of the portfolio, it is not surprising to observe that *  is involved in 
the transaction cost term. We may rewrite Equation (4.19) in the form which 
resembles the Merton equation:  
   
> @
2 2 2
2
ˆ
, ,
2
1 0.
t
t t t t
t t
t
t
SC C CrS rC E C t J S C t S
t S S
CE J S
S
V O
O
w w w ª º    ¬ ¼w w w
w   w
      (4.20) 
where > @
2
21 e 1
J
JE J
VP     and the implied volatility is given by  
     22 1 2 22 2 2 22ˆ .ʌ
H H
H Ht k t signt
VV V V G V GG
 § ·    *¨ ¸© ¹
      (4.21) 
If 2Vˆ , Equation (4.20) becomes mathematically ill-posed. This occurs when 
0*   and 0tG o . However, it is known that *  is always positive for the 
simple European call and put options in the absence of transaction costs. If we 
postulate the same sign behaviour for *  in the presence of transaction costs, 
Equation (4.20) becomes linear under such an assumption so that the Merton 
formula becomes applicable except that the modified volatility Vˆ  should be 
used as the volatility parameter. Moreover, if 0* !  from Equation (4.20) we 
obtain 
 
          1 2
0
e
, e ,
!
nT t
r T t
t t
n
T t
C t S S d K d
n
O O I I
c f  
 
c  ª º ¬ ¼¦  
where  
   2
1 2 1
ln
2 , ,
t n
n
n
n
S r T t T t
Kd d d T t
T t
V
VV
§ ·    ¨ ¸© ¹     
 
2 2
2 22 ˆe , ,
J
J J
n
nE J
T t
VP VO O O V Vc       
   
2
2
2
2ln
1 e 1 ,
J
J
J
J
n
n
n E J
r r E J r
T t T t
VP
VP
O O 
§ ·¨ ¸§ · © ¹¨ ¸       ¨ ¸ © ¹
 
Acta Wasaensia 73
F. Shokrollahi 
 
 
DOI: 10.4236/jmf.2018.84040 639 Journal of Mathematical Finance 
 
     22 1 2 22 2 2 22ˆ ,ʌ
H H
H Ht k t signt
VV V V G V GG
 § ·    *¨ ¸© ¹
 
 sign *  is the signum function of 2 2
t
C
S
w
w , tG  is a small and fixed time-step, 
k is the transaction costs and  .I  is the cumulative normal distribution. 
 
 
 
 
 
 
74 Acta Wasaensia