This is a self-archived – parallel published version of this article in the 

publication archive of the University of Vaasa. It might differ from the original. 

The Valuation of European Option Under 

Subdiffusive Fractional Brownian Motion of the 

Short Rate 

Author(s): Shokrollahi, Foad 

Title: The Valuation of European Option Under Subdiffusive Fractional 

Brownian Motion of the Short Rate 

Year: 2020 

Version: Accepted manuscript 

Copyright © World Scientific Publishing Company, 

https://www.worldscientific.com/worldscinet/ijtaf. Electronic version 

of an article published as International Journal of Theoretical and 

Applied Finance 23(4), 1-16. 

https://doi.org/10.1142/S0219024920500223 

Please cite the original version: 

 Shokrollahi, F. (2020). The Valuation of European Option Under 

Subdiffusive Fractional Brownian Motion of the Short Rate. 

International Journal of Theoretical and Applied Finance 23(4), 1-16. 

https://doi.org/10.1142/S0219024920500223 

 

 

https://www.worldscientific.com/worldscinet/ijtaf


THE VALUATION OF EUROPEAN OPTION UNDER

SUBDIFFUSIVE FRACTIONAL BROWNIAN MOTION

MECHANISM OF THE SHORT RATE

FOAD SHOKROLLAHI

Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700,
FIN-65101 Vaasa, FINLAND

Abstract. In this paper we propose an extension of the Merton model. We
apply the subdiffusive mechanism to analyze European option in a fractional
Black-Scholes environment, when the short rate follows the subdiffusive fracti-
onal Black-Scholes model. We derive a pricing formula for call and put options
and discuss the corresponding fractional Black-Scholes equation. We present
some features of our model pricing model for the cases of α and H .

1. Introduction

The pioneer study of the option pricing was introduced by Black-Scholes [1] in
1973. In the Black-Scholes (BS) model has been assumed that the underlying as-
sets follows a geometric Broawinian motion. While, there exist a series of evidence
which show the BS model unable to cover substantial behavior from financial
markets such as: long-range dependence, heavy-tailed and periods of constant va-
lues. Hence, they proposed various modifications of the BS model to capture these
shortcomings.

One of well developed modifications of the BS model is the fractional Black-
Scholes model which, describes long-range dependence and self-similarity from fi-
nancial data. In the fractional Black-Scholes (FBS) model, the Brownian motion
is substituted with the fractional Brownian motion (FBM) in the BS model. For
more details about fractional Black-Scholes model, you can see [16, 14, 2, 13].

Furthermore, analysis of financial data displays that various processes viewed
in finance show special terms in which they are constant [8]. The same property
is observed in physical system with subduffusion. The fixed terms of financial
processes according to the trapping event in which the subdiffusive examination
particle is constant [4]. The mathematical interpretation of subdiffusion is in terms
of Fractional Fokker Planck equation (FFPE). This equation was introduced
from the continuous time random walk strategy with fat tail waiting times [12],
later used as a substantial tool to evaluate complex system with slow dynamics. In
this paper, we use the FBS model in subdiffusive mechanism to better describe

E-mail address: foad.shokrollahi@uva.fi.
Date: April 15, 2019.
2010 Mathematics Subject Classification. 91G20; 91G80; 60G22.
Key words and phrases. Merton short rate model; Subdiffusive processes; Fractional Brownian

motion; Option pricing.

1



2 SHOKROLLAHI

behaviour from financial markets. We use the same strategy in [11, 15], which
the objective time t is replaced by the inverse α-stable subordinator Tα(t) in the
FBS model. Then, the dynamic of asset price is given by the following subdiffusive
FBS

dSα(t) =

dS(Tα(t)) = µsS(Tα(t))d(Tα(t)) + σsS(Tα(t))dBH
1 (Tα(t)),(1.1)

where µs, σs are constant, BH
1 is FBM with Hurst parameter H ∈ [1

2 , 1) .
Tα(t) is the inverse α-stable subordinator with α ∈ (0, 1) defined as follows

Tα(t) = inf{τ > 0 : Uα(τ) > t},(1.2)

Tα(t) is assumed to be independent of BH
1 . {Uα(t)}t≥0 is a α-stable Levy

process with nonnegative increments and Laplace transform: E
(
e−uUα(t)

)
= e−tu

α

[5, 17, 7]. when α ↑ 1, the Tα(t) degenerates to t .

On the other hand, all above studies have assumed that the short rate is constant
during the life of an option. However, in reality the short rate is evolving randomly
over time. Hence, in order to take into account the stochastic short rate, we assume
that the short rate r(t) = Ŝ(Tα(t)) follows:

dŜα(t) =

dŜ(Tα(t)) = µrdTα(t) + σrdB
H
2 (Tα(t)),(1.3)

here µr, σr are constant, BH
2 is FBM with Hurst parameter H ∈ [1

2 , 1) and

Tα(t) is assumed to be independent of BH
2 .

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 

V̂
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. discrepancy and relation between the sample paths of
the stock price in the FBS model (left) and the subdiffusive FBS
model (right) for r = 0.01, α = 0.9, H = 0.8, σ = 0.1, S0 = 1.

The first contribution of this paper is to propose a valuation model to price a
zero-coupon bond by applying the subdiffusive mechanism of the short rate. The



FRACTIONAL SHORT RATE 3

second contribution is to value an European option when the asset price and short
rate are follow subdiffusive FBS model. This paper is organized as follows. In the
next section, we derive a new model to value a riskless zero-coupon bond paying
$1 at maturity. In Section 3, we obtain the corresponding FBS equation by
using delta hedging argument and discuss some special cases of this equation. In
Section 4, we propose a pricing model for the European call and put options. Some
particular features and simulation studies of our sudiffusive model are discussed in
Section 5. Section 6 concludes this research.

2. Pricing model for a zero-coupon bond

We assume that the short rate r(t) satisfy Equation (1.3), α ∈ (1
2 , 1) and 2α−

αH > 1, then by using the Taylor series expansion to P (r, t, T ), we obtain

P (r + ∆r, t+ ∆t) = P (r, t, T ) +
∂P

∂r
∆r +

∂P

∂t
∆t

+
1

2

∂2P

∂r2
(∆r)2 + +

1

2

∂2P

∂r∂t
∆r(∆t) +

1

2

∂2P

∂t2
(∆t)2 +O(∆t).(2.1)

From, Equation (1.3) and [17], we have

∆r = µr(∆Tα(t)) + σrB
H
1 (Tα(t))

= µr

(
tα−1

Γ(α)

)2H

(∆t)2H + σr∆B
H
1 (Tα(t)) +O((∆t)2H).(2.2)

(∆r)2 = σ2
r

(
tα−1

Γ(α)

)2H

(∆t)2H +O((∆t)2H).(2.3)

∆r(∆t) = O((∆t)2H).(2.4)

Then from the Lemma 1 in [17], we can get

dP (r, t, T ) =

[(
tα−1

Γ(α)

)2H (
µr
∂P

∂r
+

1

2
σ2
r

∂2P

∂r2

)
2Ht2H−1 +

∂P

∂t

]
dt

+σr
∂P

∂t
dBH

1 (Tα(t)).(2.5)

Assuming

µ =
1

P

[(
tα−1

Γ(α)

)2H (
µr
∂P

∂r
+

1

2
σ2
r

∂2P

∂r2

)
2Ht2H−1 +

∂P

∂t

]
,

σ =
1

P

(
∂P

∂r

)
,(2.6)



4 SHOKROLLAHI

and letting the local expectations hypothesis holds for the term structure of interest
rates (i.e. µ = r ), we have

∂P

∂t
+ 2Ht2H−1µr

(
tα−1

Γ(α)

)2H
∂P

∂r

+Ht2H−1σ2
r

(
tα−1

Γ(α)

)2H
∂2P

∂r2
− rP = 0.(2.7)

Then, zero-coupon bond P (r, t, T ) with boundary condition P (r, t, T ) = 1 satisfy
the following partial differential equation

∂P

∂t
+ 2Ht2H−1µr

(
tα−1

Γ(α)

)2H
∂P

∂r

+Ht2H−1σ2
r

(
tα−1

Γ(α)

)2H
∂2P

∂r2
− rP = 0.(2.8)

To solve Equation (2.8) for P (r, t, T ), let τ = T − t, P (r, t, T ) = exp{f1(τ) −
rf2(τ)} , then we can get

∂P

∂t
= P

(
−∂f1(τ)

∂t
+ r

∂f2(τ)

∂t

)
,(2.9)

∂P

∂r
= −Pf2(τ),(2.10)

∂2P

∂r2
= Pf2(τ)2.(2.11)

Replacing Equations (2.10) and (2.11) into Equation (2.9) and simplifying Equation
(2.8) becomes

P

[
Ht2H−1σ2

rf2(τ)2

(
tα−1

Γ(α)

)2H

− 2Ht2H−1µrf2(τ)

(
tα−1

Γ(α)

)2H

−∂f1(τ)

∂τ
+ r

(
∂f2(τ)

∂t
− 1

)]
= 0.(2.12)

From Equation (2.12), we obtain

∂f1(τ)

∂τ
= Ht2H−1

(
tα−1

Γ(α)

)2H (
σ2
rf2(τ)2 − 2µrf2(τ)

)
,

∂f2(τ)

∂τ
= 1.(2.13)

Then,

f1(τ) =
Hσ2

r

(Γ(α))2H

∫ τ

0
(T − s)(α−1)2H+2H−1s2ds

− 2Hµr
(Γ(α))2H

∫ τ

0
(T − s)(α−1)2H+2H−1sds,(2.14)

f2(τ) = τ.(2.15)

Therefore, we derive a pricing model for a riskless zero-coupon bond.

P (r, t, T ) = e−rτ+f1(τ).(2.16)



FRACTIONAL SHORT RATE 5

Corollary 2.1. When α ↑ 1, Equations (1.3) and (1.1) reduce to the FBM , we
obtain

f1(τ) = Hσ2
r

∫ τ

0
(T − s)2H−1s2ds− 2Hµr

∫ τ

0
(T − s)2H−1sds,(2.17)

specially, if t = 0

f1(τ) = σ2
r

T 2H+2

(2H + 1)(2H + 2)
− µr

T 2H+1

2H + 1
,(2.18)

then

P (r, t, T ) = exp

{
−rT + σ2

r

T 2H+2

(2H + 1)(2H + 2)
− µr

T 2H+1

2H + 1

}
.(2.19)

Corollary 2.2. If H = 1
2 , from Equation (2.14), we obtain

f1(τ) =
1

2

σ2
r

Γ(α)

∫ τ

0
(T − s)α−1s2ds

− µr
Γ(α)

∫ τ

0
(T − s)α−1sds,(2.20)

then the result is consistent with the result in [6].

Further, if α ↑ 1 and H = 1
2 , Equations (1.3) and (1.1) reduce to the geometric

Brownian motion, then we have

f1(τ) =
1

6
σ2
rτ

3 − 1

2
µrτ

2,(2.21)

then

P (r, t, T ) = e−rτ+ 1
6
σ2
rτ

3− 1
2
µrτ2 .(2.22)

which is consistent with the result in [9, 3].

3. Fractional Black-Scholes equation

This section provides corresponding FBS equation for European options when
the short rate and stock price satisfy Equations (1.3) and (1.1), respectively, here
BH

1 and BH
2 are two dependent FBM with Hurst parameter H ∈ [1

2 , 1) and
correlation coefficient ρ .

Let C = C(S, r, t) be the price of a European call option at time t with a strike
price K that matures at time T . Then we have.

Theorem 3.1. Assume that the short rate r(t) and stock price S(t) satisfy Equa-
tions (1.3) and (1.1), respectively. Then, C(S, r, t) is the solution the following
equation:

∂C

∂t
+ σ̃2

s(t)S
2∂

2C

∂S2
+ σ̃2

r (t)
∂2C

∂r2
+ 2ρσ̃r(t)σ̃s(t)

∂2C

∂S∂r

+2Ht2H−1µr

(
tα−1

Γ(α)

)2H
∂C

∂r
+ rS

∂C

∂S
− rC = 0,(3.1)



6 SHOKROLLAHI

where

σ̃2
s(t) = Ht2H−1σ2

s

(
tα−1

Γ(α)

)2H

,(3.2)

σ̃2
r (t) = Ht2H−1σ2

r

(
tα−1

Γ(α)

)2H

.(3.3)

σs, σr, µs, µs, are constant, H ∈ [1
2 , 1) and α ∈ (1

2 , 1) and 2α− αH > 1.

Proof: We consider a portfolio with D1t units of stock and D2t units of zero-
coupon bond P (r, t, T ) and one unit of C = C(r, t, T ). Then, the value of the
portfolio at current time t is

Πt = C −D1tSt −D2tPt.(3.4)

Then, from [6] we have

dΠt = Ct −D1tdSt −D2tdPt

=

[
∂C

∂t
dt+Ht2H−1σ2

sS
2
t

(
tα−1

Γ(α)

)2H
∂2C

∂S2
+Ht2H−1σ2

r

(
tα−1

Γ(α)

)2H
∂2C

∂r2

+ 2Ht2H−1ρσrσsS

(
tα−1

Γ(α)

)2H
∂2C

∂S∂r

]
dt+

[
∂C

∂t
−D1t

]
dSt

+

[
∂C

∂r
−D2t

∂P

∂r

]
dr +D2t

[
∂P

∂t
+Ht2H−1σ2

r

(
tα−1

Γ(α)

)2H
∂2P

∂r2

]
dt.(3.5)

By setting D1t = ∂C
∂S , D2t =

∂C
∂r
∂P
∂r

, to eliminate the stochastic noise, then

dΠt =

=

[
∂C

∂t
+Ht2H−1

(
tα−1

Γ(α)

)2H (
σ2
sS

2∂
2C

∂S2
+ σ2

r

∂2C

∂r2
+ 2ρσrσsS

∂2C

∂S∂r

)]
dt

−
∂C
∂r
∂P
∂r

[
rP − 2Ht2H−1µr

(
tα−1

Γ(α)

)2H
∂P

∂r

]
dt.(3.6)

The return of an amount Πt invested in bank account is equal to r(t)Πtdt at
time dt , E(dΠt) = r(t)Πtdt = r(t) (C −D1tSt −D2tPt), hence from Equation (3.6)
we have

∂C

∂t
+Ht2H−1

(
tα−1

Γ(α)

)2H (
σ2
sS

2∂
2C

∂S2
+ σ2

r

∂2C

∂r2
+ 2ρσrσsS

∂2C

∂S∂r

)

+2Ht2H−1µr

(
tα−1

Γ(α)

)2H
∂C

∂r
+ rS

∂C

∂S
− rC = 0.(3.7)

Let

σ̃2
s(t) = Ht2H−1σ2

s

(
tα−1

Γ(α)

)2H

,(3.8)

σ̃2
r (t) = Ht2H−1σ2

r

(
tα−1

Γ(α)

)2H

.(3.9)



FRACTIONAL SHORT RATE 7

Then

∂C

∂t
+ σ̃2

s(t)S
2
t

∂2C

∂S2
t

+ σ̃2
r (t)

∂2C

∂r2
+ 2ρσ̃r(t)σ̃s(t)

∂2C

∂S∂r

+2Ht2H−1µr

(
tα−1

Γ(α)

)2H
∂C

∂r
+ rS

∂C

∂S
− rC = 0,(3.10)

proof is completed.

From Theorem (3.1), we can get the following corollaries

Corollary 3.1. If ρ = 0 and r(t) be a constant, then the European call option
C = C(S, r, T ) satisfies

∂C

∂t
+Ht2H−1σ2

sS
2
t

(
tα−1

Γ(α)

)2H
∂2C

∂S2
t

+ rS
∂C

∂S
− rC = 0,(3.11)

which is a fractional BS equation considered in [10].

Corollary 3.2. When α ↑ 1, we obtain

∂C

∂t
+Ht2H−1σ2

sS
2
t

∂2C

∂S2
t

+Ht2H−1σ2
r

∂2C

∂r2
+ 2Ht2H−1ρσrσs

∂2C

∂S∂r

+2Ht2H−1µr
∂C

∂r
+ rS

∂C

∂S
− rC = 0,(3.12)

Further, if ρ = 0, H = 1
2 , and r(t) be a constant, from Equation (3.12) we have

the celebrated BS equation

∂C

∂t
+

1

2
σ2
sS

2
t

∂2C

∂S2
t

+ rS
∂C

∂S
− rC = 0,(3.13)

4. Pricing formula under subdiffusive fractionalBlack-Scholes
model

In this section, we propose an explicit formula for European call option when
its value satisfies the partial differential equation (3.1) with boundary condition
C(S, r, T ) = (ST −K)+ . Then, we can get

Theorem 4.1. Let r(t) satisfies Equation (1.3) and S(t) satisfies Equation (1.1),
then the price of European call and put options with strike price K and maturity
T are given by

C(S, r, t) = Sφ(d1)−KP (r, t, T )φ(d2),(4.1)

P (S, r, t) = KP (r, t, T )φ(−d2)− φ(−d1).(4.2)

where

d1 =
ln S

K − lnP (r, t, T ) + H
(Γ(α))2H

∫ T
t σ̂2(s)s(α−1)2H+2H−1ds

√
2H

(Γ(α))2H

∫ T
t σ̂2(s)s(α−1)2H+2H−1ds

,(4.3)

d2 = d1 −

√
2H

(Γ(α))2H

∫ T

t
σ̂2(s)s(α−1)2H+2H−1ds,(4.4)

σ̂2(t) = σ2
s + 2ρσrσs(T − t) + σ2

r (T − t)2.(4.5)



8 SHOKROLLAHI

P (r, t, T ) is given by Equation (2.16) and φ(.) is the cumulative normal distribution
function.

Proof:

Consider the partial differential equation (3.1) of the European call option with
boundary condition C(S, r, T ) = (ST −K)+

∂C

∂t
+ σ̃2

s(t)S
2
t

∂2C

∂S2
t

+ σ̃2
r (t)

∂2C

∂r2
+ 2ρσ̃r(t)σ̃s(t)

∂2C

∂S∂r

+2Ht2H−1µr

(
tα−1

Γ(α)

)2H
∂C

∂r
+ rS

∂C

∂S
− rC = 0.(4.6)

Denote

z =
S

P (r, t, T )
, Θ(z, t) =

C(S, r, t)

P (r, t, T )
,(4.7)

therefore by computing, we get

∂C

∂t
= Θ

∂P

∂t
+ P

∂Θ

∂t
− z ∂Θ

∂z

∂P

∂t
,

∂C

∂r
= Θ

∂P

∂r
− z ∂Θ

∂z

∂P

∂r
,

∂C

∂S
=

∂Θ

∂z
,(4.8)

∂2C

∂r2
= Θ

∂2P

∂r2
− z ∂Θ

∂z

∂2P

∂r2
+
z2

P

∂2Θ

∂z2

(
∂P

∂r

)2

,

∂2C

∂r∂S
= − z

P

∂2Θ

∂z2

∂P

∂r
,

∂2C

∂S2
=

1

P

∂2Θ

∂z2
.

Inserting Equation (4.8) into Equation (4.6)

∂Θ

∂t
+

∂2Θ

∂z2

[
σ̃2
s(t)

S2

P 2
+ 2ρz2σ̃r(t)σ̃s(t)

1

P

∂P

∂r
+ σ̃2

r (t)z
2

(
1

P

∂P

∂r

)2
]

− z

P

[
∂P

∂t
+ σ̃2

r (t)
∂2P

∂r2
+ 2Ht2H−1µr

(
tα−1

Γ(α)

)2H
∂P

∂r
− rS

z

]

+
Θ

P

[
∂P

∂t
+ σ̃2

r (t)
∂2P

∂r2
+ 2Ht2H−1µr

(
tα−1

Γ(α)

)2H
∂P

∂r
− rP

]
= 0.(4.9)

From Equation (2.8), we can obtain

∂Θ

∂t
+ σ2(t)z2∂

2Θ

∂z2
= 0,(4.10)

with boundary condition Θ(z, T ) = (z −K)+ ,



FRACTIONAL SHORT RATE 9

where

σ2(t) = σ̃2
s(t) + 2ρσ̃r(t)σ̃s(t)(T − t) + σ̃r(t)

2(T − t)2.(4.11)

The solution of partial differential Equation (4.10) with boundary condition
Θ(z, T ) = (z −K)+ , is given by

Θ(z, t) = zφ(d̂1)−Kφ(d̂2),(4.12)

here

d̂1 =
ln z

K +
∫ T
t σ2(s)ds√

2
∫ T
t σ̂2(s)ds

,(4.13)

d̂2 = d̂1 −

√
2

∫ T

t
σ2(s)ds.(4.14)

Thus, from Equations (4.7) and (4.12)-(4.14) we obtain

C(S, r, t) = Sφ(d1)−KP (r, t, T )φ(d2),(4.15)

where

d1 =
ln S

K − lnP (r, t, T ) + H
(Γ(α))2H

∫ T
t σ̂2(s)s(α−1)2H+2H−1ds

√
2H

(Γ(α))2H

∫ T
t σ̂2(s)s(α−1)2H+2H−1ds

,(4.16)

d2 = d1 −

√
2H

(Γ(α))2H

∫ T

t
σ̂2(s)s(α−1)2H+2H−1ds.(4.17)

Letting α ↑ 1, from Theorem 4.1, we obtain

Corollary 4.1. The price of European call and put options with strike price K
and maturity T are given by

C(S, r, T ) = Sφ(d1)−KP (r, t, T )φ(d2),(4.18)

P (S, r, T ) = KP (r, t, T )φ(−d2)− Sφ(−d1).(4.19)

where

d1 =
ln S

K − lnP (r, t, T ) +H
∫ T
t σ̂2(s)s2H−1ds√

2H
∫ T
t σ̂2(s)s2H−1ds

,(4.20)

d2 = d1 −

√
2H

∫ T

t
σ̂2(s)s2H−1ds,(4.21)

σ̂2(t) = σ2
s + 2ρσrσs(T − t) + σ2

r (T − t)2,(4.22)

P (r, t, T ) = exp

{
− rτ +Hσ2

r

∫ τ

0
(T − s)2H−1s2ds

−2Hµr

∫ τ

0
(T − s)2H−1sds

}
, τ = T − t.(4.23)



10 SHOKROLLAHI

More specifically, if H = 1
2 , we have

d1 =
ln S

K − lnP (r, t, T ) + 1
2ϕ(t, T )√

ϕ(t, T )
,(4.24)

d2 = d1 −
√
ϕ(t, T ),(4.25)

ϕ(t, T ) = σ2
s(T − t) + ρσrσs(T − t)2 +

1

3
σ2
r (T − t)3,(4.26)

P (r, t, T ) = exp

{
−r(T − t)− 1

2
µr(T − t)2 +

1

6
σ2
r (T − t)3

}
.(4.27)

which is consistent with result in [3].

Letting H = 1
2 , from Theorem 4.1, we can get

Corollary 4.2. The price of European call and put options with strike price K
and maturity T are given by

C(S, r, T ) = Sφ(d1)−KP (r, t, T )φ(d2),(4.28)

P (S, r, T ) = KP (r, t, T )φ(−d2)− φ(−d1).(4.29)

where

d1 =
ln S

K − lnP (r, t, T ) + 1
2Γ(α)

∫ T
t σ̂2(s)sα−1ds

√
1

Γ(α)

∫ T
t σ̂2(s)sα−1ds

,(4.30)

d2 = d1 −

√
1

Γ(α)

∫ T

t
σ̂2(s)sα−1ds,(4.31)

σ̂2(t) = σ2
s + 2ρσrσs(T − t) + σ2

r (T − t)2,(4.32)

P (r, t, T ) = exp

{
− rτ +

σ2
r

2Γ(α)

∫ τ

0
(T − s)α−1s2ds

− µr
(Γ(α)

∫ τ

0
(T − s)α−1sds

}
.(4.33)

Specially, If ρ = 0, from Equations (4.28)-(4.33), we have

d1 =
ln S

K − lnP (r, t, T ) + 1
2Γ(α)

∫ T
t σ̂2(s)sα−1ds

√
1

Γ(α)

∫ T
t σ̂2(s)sα−1ds

,(4.34)

d2 = d1 −

√
1

Γ(α)

∫ T

t
σ̂2(s)sα−1ds,(4.35)

σ̂2(t) = σ2
s + σ2

r (T − t)2,(4.36)

P (r, t, T ) = exp

{
− rτ +

1

2

σ2
r

Γ(α)

∫ τ

0
(T − s)α−1s2ds

− µr
Γ(α)

∫ τ

0
(T − s)α−1sds

}
.(4.37)

which is similar with results mentioned in [6].



FRACTIONAL SHORT RATE 11

5. Simulation studies

Let us first discuss about the implied volatility of the subdiffusive FBS model,
then we will show some simulation findings.

Corollary 5.1. If t = 0, the value of European call option C(K,T ) and put option
P (K,T ) can be written as

C(K,T ) = S0φ(d1)−KP0φ(d2),(5.1)

P (K,T ) = KP0φ(−d2)− S0φ(−d1).(5.2)

where

P0 = exp

{
− r0T +

2HT (α−1)2H+2H+1

(Γ(α))2H((α− 1)2H + 2H)((α− 1)2H + 2H + 1)

×
(

σ2
rT

(α− 1)2H + 2H + 2
− µr

)}
(5.3)

d1 =
ln S0

K + rT + 1
2σ

2T

σ
√
T

,(5.4)

d2 = d1 − σ
√
T ,(5.5)

r = r0 +
2HT (α−1)2H+2H

(Γ(α))2H((α− 1)2H + 2H)((α− 1)2H + 2H + 1)
(5.6)

×
(
µr −

σ2
rT

(α− 1)2H + 2H + 2

)
,

σ2 =
2HT (α−1)2H+2H−1

(Γ(α))2H((α− 1)2H + 2H)

(
σ2
s +

ρσrσsT

(α− 1)2H + 2H + 1

+
σ2
rT

2

((α− 1)2H + 2H + 1)((α− 1)2H + 2H + 2)

)
.(5.7)

and φ(.) is the cumulative normal distribution function.

Table 1 indicates the theoretical prices from our FBS and subdiffusive FBS
models and Merton and subdiffusive BS models, where S0 shows the stock price,
PM presents the prices evaluated by the Merton model, PSBS denotes the price
simulated by the subdiffusive BS model, PFBS and PSFBS show the price obtained
by the FBS and subdiffusive FBS models, respectively.



12 SHOKROLLAHI

Table 1. Results by different pricing models. Here, α = 0.9, H =
0.6,K = 3, σr = 0.3, σs = 0.4, ρ = 0.4, µr = 0.5, r0 = 0.3, T =
0.3, t = 0.

T = 0.2 T = 1
S PM PSBS PFBS PSFBS PM PSBS PFBS PSFBS
2 0.0174 0.0334 0.0012 0.0036 1.8826 1.9129 1.7986 1.8347

2.25 0.0638 0.0979 0.0122 0.0236 2.1326 2.1629 2.0486 2.0847
2.5 0.1598 0.2126 0.0587 0.0859 2.3826 2.4129 2.2986 2.3347
2.75 0.3094 0.3754 0.1687 0.2094 2.6326 2.6629 2.5486 2.5847

3 0.5023 0.5752 0.3440 0.3900 2.8826 2.1929 2.7986 2.8347
3.25 0.7235 0.7988 0.5630 0.6086 3.1326 3.1629 3.0486 3.0847
3.5 0.9604 1.0360 0.8026 0.8466 3.3826 3.4129 3.2986 3.3347
3.75 1.2094 1.2801 1.0498 1.0926 3.6326 3.6629 3.5486 3.5847

4 1.4527 1.5275 1.2991 1.3414 3.8826 3.9129 3.7986 3.8347

By comparing columns PM , PSBS , PFBS and PSFBS in Table 1, we conclude
the call option prices obtained by four pricing models are close to each other in
the both in-the-money and out-of-the-money cases with low and high maturities.
Meanwhile, we can see that the prices given by the our FBS and subdiffusive FBS
models are smaller than the prices given by the Merton and subdiffusive Merton
models [3, 6].

0
0.1

0.2
0.3

0.4
0.5

0.6
0.7

0.8
0.9

1

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

T
K

D
if
fe

re
n

c
e

Figure 2. The European call option under subdiffusive FBS .
Where r0 = 0.1, α = 0.9, H = 0.8, σr = 0.3, σs = 0.4, S0 = 3, µr =
0.2, ρ = 0.2.



FRACTIONAL SHORT RATE 13

0

5

0.1

0.2

4.5

0.3

14

0.4

0.9

0.5

D
if
fe

re
n

c
e

3.5
0.8

0.6

K

3

0.7

0.7

0.8

0.6

T

2.5

0.9

0.5

2 0.4

0.3
1.5

0.2

1 0.1

Subdiffusive FBS versus subdiffusive BS

Subdiffusive FBS versus Merton

Figure 3. The difference between the price of the European call
option under subdiffusive FBS , subdiffusive Merton and Merton
models. Where r0 = 0.1, α = 0.9, H = 0.8, σr = 0.3, σs = 0.4, S0 =
3, µr = 0.2, t = 0, ρ = 0.3.

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

α

O
p

ti
o

n
 V

a
lu

e

 

 

H=0.6

H=0.7

H=0.8

Figure 4. The European call option under subdiffusive FBS .
Where r0 = 0.3, σr = 0.1, σs = 0.3, S0 = 4, µr = 0.2, ρ = 0.2, t =
0, T = 0.2.

From Equations (5.1)-(5.7), it is easy to see that σ and r is the implied volatility
and implied short rate connected to the FBS model, respectively (See Fig 2, 3
and 4 ).

6. Conclusion

Most of prior pricing models have assumed the constant short rate during the life
of an option. However, in real life the short rate is evolving randomly through time.
For this purpose, we apply the subdiffusive mechanism to get better characteristic



14 SHOKROLLAHI

property of stock markets. We propose a pricing model for a zero-coupon bond
when the short rate is governed by the subdiffusive fractional Black-Scholes model.
Then, we exert these results to develop analytical valuation formulas for European
option and corresponding fractional Black-Scholes equation.

We allow to referees to evaluate our manuscript.

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