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Efficient Numerical Schemes for Computations of European Options with Transaction Costs

Md. Shorif Hossan1 , Md. Shafiqul Islam1 , Md. Kamrujjaman2,∗
1Department of Applied Mathematics, University of Dhaka, Dhaka, Bangladesh

shorif@du.ac.bd, mdshafiqul@du.ac.bd
2Department of Mathematics, University of Dhaka, Dhaka, Bangladesh

kamrujjaman@du.ac.bd
∗Correspondence: kamrujjaman@du.ac.bd

Abstract. This paper aims to find numerical solutions of the non-linear Black-Scholes partial dif-ferential equation (PDE), which often appears in financial markets, for European option pricing inthe appearance of the transaction costs. Here we exploit the transformations for the computationalpurpose of a non-linear Black-Scholes PDE to modify as a non-linear parabolic type PDE with reli-able initial and boundary conditions for call and put options. Several schemes are derived rigorouslyusing the finite volume method (FVM) and finite difference method (FDM), which is the novelty of thispaper. Stability and consistency analysis assure the convergence of these schemes. We apply theseschemes to various volatility models, such as the Leland, Boyle and Vorst, Barles and Soner, andRisk-adjusted pricing methodology (RAPM). All the schemes are tested numerically. The convergenceof the obtained results is observed, and we find that they are also reliable. Finally, we display allthe approximate results together with the exact values through graphical and tabular representations.

1. Introduction
Understanding and accurately evaluating transaction costs in a financial market is vital forsecurity trading, asset pricing, stock market regulation, and many other issues. During the last fewdecades, pricing options more accurately after including realistic assumptions-such as transactioncost, getting more importance from both the traders and the investors.The literature’s [1–6], contains descriptive discussions of options. Fischer Black and MyronScholes [7] worked jointly, and first disclosed the concept of the Black-Scholes model for options

pricing and corporate liabilities, and was published in 1973, while Robert Merton [8] advanced thismodel in the article "Theory of rational option pricing" in the same year. Their derived equation isbased on the assumption that there are no fees for buying and selling options and stocks, as wellas no trade barriers (i.e., no commissions and transaction costs). In other words, this model makes
Received: 20 Dec 2021.
Key words and phrases. nonlinear Black-Scholes PDE; option pricing; volatility model; finite volume method; finitedifference method. 1

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 2
a friction-less assumption (which is indispensable, as actual costs correlated with practical marketapplications) to implement a hedging plan for any contingent claim of the European type.Various studies have been conducted about the linear Black-Scholes model [9–15] though itadopts the unrealistic assumption of no transaction costs. Several studies have been attempted toevaluate the price of European options [16–23], American options [24–28], Asian options [29, 30],and Barrier options [31] in a completely friction-less market. Recently, the fractional Black-Scholesmodel [32–34] received some attention.Contrastingly, the non-linear Black-Scholes PDE, where the non-linear term denotes the pres-ence of transaction costs, is of great importance to our contemporary world over some time both interms of approach and applicability. Several models [35] consider transaction costs: Leland model,Paras, and Avellaneda model, Boyle and Vorst model, Hodges and Neuberger model, Barles andSoner model, and RAPM (Risk-adjusted pricing methodology) model. If the transaction cost pa-rameters are equal to zero, all of these non-linear transaction cost models are unvarying with thelinear model.Soner et al. [36] showed that there is no nontrivial hedging portfolio for option pricing withtransaction costs. They also suggested that the best hedging strategy is buying an asset andtaking on it for a certain period as a call or put option. Leland [37] inaugurates the idea of usingtransaction costs at discrete times. He also indicated that the hedging error could be minimized ifthe length of re-balancing frequency approaches zero. Later, Boyle and Vorst [38] demonstrated fur-ther in a discrete-time framework with a binomial tree model for the option prices with proportionaltransaction costs, and it is pretty accurate for possible parameter values. Besides, Dewynne etal. [39] considered path-dependent and exotic options with transaction costs. Recently, asymptoticanalysis [40] and Markov chain approximation [41] were also studied for pricing European optionswith transaction costs in some previous literature.On the other hand, few researchers [42–46, 49–51] paid their attention to solve the non-linearBlack-Scholes equation numerically. For example, the exponential time differencing (ETD) method[44] was applied to solve the non-linear Black–Scholes model for pricing American options with ahighly stable and efficient transaction cost. Lesmana and Wang [45] developed the numerical methodbased on an upwind finite difference scheme for a non-linear parabolic PDE, and they attemptedto pricing European options under transaction costs. Ankudinova and Ehrhardt [46] focused on thenon-linear Black-Scholes equation for European call options using several transaction cost modelsas well as Crank–Nicolson and Rigal compact schemes. R. L. Valkov [47] has solved the non-linearBlack–Scholes-Bellman model numerically as well as discuss the monotonicity and consistency ofhis suggested scheme in considerable detail. A monotone finite volume spatial discretization and asecond-order predictor-corrector scheme in time are considered by Radoslas Valkov [48] to handlethe Black–Scholes equation with uncertain volatility and dividend. The applicability of implicit

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 3
numerical schemes for the valuation of contingent claims in non-linear Black–Scholes models hasbeen discussed by Pascal Heider [49]. He also studied the practical implications of the derivedstability criteria on relevant numerical examples. He claimed that if certain stability requirementsare satisfied, it is possible to construct convergent implicit algorithms for non-linear Black–Scholesequations. Ekaterina Dremkova and Matthias Ehrhardt [50] have solved non-linear Black–Scholesequations for American options with a non-linear volatility function using various compact finitedifference techniques to improve the order of the accuracy. The existence and uniqueness ofsolutions to the well-known non-linear Black-Scholes equation have been demonstrated by NaoyukiIshimura [51] for both in the classical and weak senses.However, in this paper, we work on approximating non-linear Black-Scholes PDE for valuingEuropean options when there are transaction costs. For this, we organize the present research workas follows: we modify the original model into parabolic type PDE exploiting the transformations [46]which are written in section 2. A brief description of different volatility models is given in section3 subsequently. Section 4 is devoted to discretize the transformed parabolic type equation byusing some numerical schemes. Stability and consistency analysis are included in sections 5 and6, respectively. In section 7, numerical examples are given to show the efficacy of the proposedschemes. Subsequently, a general conclusion is drawn in section 8. Finally, all relevant referencesare included.

2. The Model Equation
This section considers a non-linear Black-Scholes PDE and modifies it to a non-linear parabolictype equation with appropriate and available transformations, which would be easy to computenumerically. Let us consider the non-linear Black-Scholes PDE [46],

∂F

∂t
+ rS

∂F

∂S
+

1

2
σ̃2S2

∂2F

∂S2
− rF = 0, 0 < S < ∞,t ∈ (0,T ) (1)

subject to the terminal and boundary conditions for European call and put options:
F (S,T ) = max(S − K, 0),F (S,t) = 0 when S = 0,F (S,t) = S − Ke−r(T−t), when S → ∞and F (S,T ) = max(K − S, 0),F (S,t) = Ke−r(T−t), when S = 0,F (S,t) = 0, when S → ∞respectively. Throughout this paper, we use the notations: F = F (S,t) = the option price, S =stock price, K = strike price, T = maturity time, r = interest rate, t = time in years, and
σ̃ = σ̃

(
t,S, ∂F

∂S
, ∂

2F
∂S2

) depends on the volatility model.
Now consider the transformations [46] as given below,

y = ln
(
K−1S

)
,τ =

1

2
σ2(T − t) and u(y,t) = K−1e−yF (S,t)

and substituting these into Equation (1) to obtain the following non-linear parabolic PDE

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 4
∂u

∂τ
=

2r

σ2
∂u

∂y
+

(
σ̃

σ

)2 (
∂2u

∂y2
+
∂u

∂y

)
,ymin < y < ymax,τ ∈

(
0,
σ2

2
T

) (2)
with the modified initial and boundary conditions for European call and put options:
u(y, 0) = max

(
1 −e−y, 0

) as y ∈ (−∞,∞),u(y,τ) = 0 as y →−∞,u(y,τ) = 1−e−(y+2rτ/σ2)as y →∞, and u(y, 0) = max (e−y − 1, 0) as y ∈ (−∞,∞),u(y,τ) = e−(y+2rτ/σ2) as y →−∞,
u(y,τ) = 0 as y →∞, respectively.

3. Volatility Models
This section concerns four stochastic volatility models to discretize the non-linear Black-ScholesPDE, whose solution provides the option price for transaction fees. We give a short description,but details are available in some previous literature [46].

Leland Volatility Model (LVM). Leland [37] developed a technique for replicating options in thepresence of transactions costs for a small time interval. He proposed that the option price isthe solution of the non-linear Black-Scholes Equation (1) but with the adjusted volatility [46] asfollows:
σ̃ = σ

√
1 +

√
2

π

µ

σ
√

∆t
sign (FSS) (3)

where, σ is the original volatility, µ is the round-trip transaction cost per unit dollar of the trans-action, and ∆t is the transaction frequency. In this formula, both µ and ∆t are assumed to be smallwhile keeping the ratio µ√
∆t

of order one.
Boyle and Vorst Volatility Model (BVVM). Boyle and Vorst [38] derived a method for calculatingoption prices in a discrete-time where option price meets to Black-Scholes price with the modifiedvolatility [46] given by

σ̃ = σ

√
1 +

µ

σ
√

∆t
sign (FSS) (4)

where, σ,µ, and ∆t represents the same meaning as Leland.
Barles and Soner Volatility Model (BSVM). Barles and Soner [43] evolved a model using theutility function approach of Hodges and Neuberger [52] along with asymptotic analysis of partialdifferential equations. For this case, the formula for the modified volatility [46] is given by

σ̃ = σ
√

1 + er(T−t)a2S2FSS (5)
where, µ = a√∈ is the round-trip transaction cost per unit dollar of the transaction for someconstant a > 0 and ∈→ 0.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 5
RAPM Volatility Model (RAPMVM). Kratka [53] took the first step for this model and laterimproved by Jandačka and Ševčovič [54]. Here the modified volatility is of the form [46]

σ̃ = σ

√
1 + 3 × 3

√
C2M

2π
SFSS (6)

where, M ≥ 0 is the transaction cost measure and C ≥ 0 is the risk premium measure.
4. Derivations of Computational Schemes

In this section, we derive five computational schemes, in detail, for Equation (2) using twowell-known numerical methods.
4.1. Dufort-Frankel Finite Difference Scheme. The Dufort-Frankel FD scheme [55] can be appliedto solve various kinds of problems which occur in finance. This scheme is a multi-step method, andrequires another scheme for simulating the first temporal vector. In this formulation ∂u

∂τ
, ∂u
∂y

, and ∂2u
∂y2are discretized by central difference and uj

i
is replaced by (uj+1

i
+ u

j−1
i

)
/2. Thus, discretizingEquation (2) by Dufort-Frankel FDM, we obtain

1

2∆τ

(
u
j+1
i
−uj−1

i

)
=

(
σ̃

σ

)2 [
1

(∆y)2

(
u
j
i−1 −

(
u
j+1
i

+ u
j−1
i

)
+ u

j
i+1

)]
+

(
σ̃

σ

)2 [
1

2∆y

(
u
j
i+1
−uj

i−1

)]
+

r

σ2∆y

(
u
j
i+1
−uj

i−1

)
or, equivalently

u
j+1
i

=u
j−1
i

+
2r(∆τ)(∆y)

σ2

(
u
j
i+1
−uj

i−1

)
+

∆τ

∆y

(
σ̃

σ

)2 (
u
j
i+1
−uj

i−1

)
+

2(∆τ)

(∆y)2

(
σ̃

σ

)2 (
u
j
i−1 −u

j+1
i
−uj−1

i
+ u

j
i+1

)
which can be written as

u
j+1
i

= aiu
j
i−1 + biu

j
i+1

+ ciu
j−1
i

; i = 0, 1, 2, . . . ,n− 1; j = 0, 1, 2, . . . ,m− 1 (7)
where

ai =

[
(∆y)2 + 2(∆τ)

(
σ̃

σ

)2]−1
×

[
(∆τ)

(
σ̃

σ

)2
(2 − ∆y) − (∆y)(∆τ)

2r

σ2

]
,

bi =

[
(∆y)2 + 2(∆τ)

(
σ̃

σ

)2]−1
×

[
(∆τ)

(
σ̃

σ

)2
(2 + ∆y) + (∆y)(∆τ)

2r

σ2

]
,

and
ci =

[
(∆y)2 + 2(∆τ)

(
σ̃

σ

)2]−1
×

[
(∆y)2 − 2(∆τ)

(
σ̃

σ

)2]
which is our proposed Dufort-Frankel Finite Difference Scheme (DFFDS).

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 6
4.2. Laasonen Finite Difference Scheme. The Laasonen finite difference scheme [55] can be ap-plied to solve linear and non-linear partial differential equations. This method metamorphosedpartial differential equations into a system of linear algebraic equations. In this formulation ∂u

∂τis approximated by a central differencing at a step ∆τ
2

, and ∂u
∂y
, ∂

2u
∂y2

are approximated by centraldifferences at time levels j + 1. Now the discretized form of Equation (2) is as follows
1

∆τ

(
u
j+1
i
−uj

i

)
=

1

2(∆y)2

(
σ̃

σ

)2 [
2
(
u
j+1
i−1 − 2u

j+1
i

+ u
j+1
i+1

)
+ ∆y

(
u
j+1
i+1
−uj+1

i−1

)]
+

r

σ2∆y

(
u
j+1
i+1
−uj+1

i−1

)
After simplification, we get[

r∆τ

∆yσ2
+

∆τ

2(∆y)2

(
σ̃

σ

)2
(∆y − 2)

]
u
j+1
i−1 +

[
1 +

2∆τ

(∆y)2

(
σ̃

σ

)2]
u
j+1
i

−

[
r∆τ

∆yσ2
+

∆τ

2(∆y)2

(
σ̃

σ

)2
(∆y + 2)

]
u
j+1
i+1

= u
j
i

The above equation reduces to
diu

j+1
i−1 + (1 + ei ) u

j+1
i

+ fiu
j+1
i+1

= u
j
i
; i = 0, 1, 2, . . . ,n− 1; j = 0, 1, 2, . . . ,m− 1 (8)

where
di =

r∆τ

∆yσ2
+

∆τ

2(∆y)2

(
σ̃

σ

)2
(∆y − 2),ei =

2∆τ

(∆y)2

(
σ̃

σ

)2
and

fi = −
r∆τ

∆yσ2
−

∆τ

2(∆y)2

(
σ̃

σ

)2
(∆y + 2)

4.3. Finite Volume Schemes. The finite volume scheme is a scheme of solving different kinds oftime-dependent or independent partial differential equations in algebraic equations. In this scheme,we divide the physical space into a finite number of control volumes. In this section, we describeit in a few lines, but details are available in the previous study [56] conducted by Malalasekera etal.
Applying the finite volume integration in Equation (2) over a control volume (CV) with a finite timestep ∆τ, we obtain

∫ τ+∆τ
τ

∫
CV

∂u

∂τ
dV dτ =

(
2r

σ2
+

(
σ̃

σ

)2)∫ τ+∆τ
τ

∫
CV

∂u

∂y
dV dτ

+

(
σ̃

σ

)2 ∫ τ+∆τ
τ

∫
CV

∂2u

∂y2
dV dτ

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 7
After rearranging, we get∫

CV

[∫ τ+∆τ
τ

∂u

∂τ
dτ

]
dV =

(
2r

σ2
+

(
σ̃

σ

)2)∫ τ+∆τ
τ

[∫
CV

∂u

∂y
dV

]
dτ

+

(
σ̃

σ

)2 ∫ τ+∆
τ

[∫
CV

∂2u

∂y2
dV

]
dτ

Applying Gauss’s divergence theorem, the above equation leads
(
uP −u0P

)
∆V =

1

2

(
2r

σ2
+

(
σ̃

σ

)2)
A

∫ τ+∆τ
τ

(uE −uW ) dτ

+

(
σ̃

σ

)2 ∫ τ+∆τ
τ

[(
A
uE −uP
δyPE

)
−
[(
A
uP −uW
δyWP

)]]
dτ

(9)
For 0 ≤ θ ≤ 1, we assume ∫ τ+∆τ

τ

uPdτ =
[
θuP + (1 −θ)u0P

]
∆τ (10)

Applying Equation (10) into Equation (9) and dividing by we get
(
uP −u0P

) ∆y
∆τ

=
1

2

(
2r

σ2
+

(
σ̃

σ

)2)[
θ (uE −uW ) + (1 −θ)

(
u0E −u

0
W

)]
+

(
σ̃

σ

)2
θ

(
uE −uP
δyPE

−
uP −uW
δyWP

)
+

(
σ̃

σ

)2
(1 −θ)

(
u0E −u

0
P

δyPE
−
u0P −u

0
W

δyWP

)
(11)

For convenience, we put δyWP = δyPE = ∆y on the following three schemes.
Explicit Scheme. Substitution of θ = 0 into Equation (11) gives the following explicit discretizedequation,

(
uP −u0P

) ∆y
∆τ

=
1

2

(
2r

σ2
+

(
σ̃

σ

)2)(
u0E −u

0
W

)
+

1

∆y

(
σ̃

σ

)2 (
u0E − 2u

0
P + u

0
W

)
This equation may be re-writtens as

uP = αiu
0
W + (1 + βi ) u

0
P + γiu

0
E (12)

where
αi =

∆τ

(∆y)2

(
σ̃

σ

)2
−

∆τ

2∆y

(
2r

σ2
+

(
σ̃

σ

)2)
, βi = −

2∆τ

(∆y)2

(
σ̃

σ

)2
and

γi =
∆τ

(∆y)2

(
σ̃

σ

)2
+

∆τ

2∆y

(
2r

σ2
+

(
σ̃

σ

)2)
which is the desired Finite Volume Explicit Scheme (FVES).

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 8
Crank-Nicolson Scheme. Putting θ = 1

2
into Equation (11), we get the following Crank-Nicolsondiscretized equation,

(
uP −u0P

) ∆y
∆τ

=
1

4

(
2r

σ2
+

(
σ̃

σ

)2)(
uE −uW + u0E −u

0
W

)
+

1

2∆y

(
σ̃

σ

)2 (
uE − 2uP + uW + u0E − 2u

0
P + u

0
W

)
After simplification, we get the following equation

λiuw + (1 + ξi ) uP + ηiuE = −λiu0π + (1 −ξi ) u
0
P −ηiu

0
E (13)

where
λi =

∆τ

4∆y

(
2r

σ2
+

(
σ̃

σ

)2)
−

∆τ

2(∆y)2

(
σ̃

σ

)2
,ξi =

∆τ

(∆y)2

(
σ̃

σ

)2
and

ηi = −

[
∆τ

4∆y

(
2r

σ2
+

(
σ̃

σ

)2)
+

∆τ

2(∆y)2

(
σ̃

σ

)2]
which is the proposed Finite Volume Crank-Nicolson Scheme (FVCNS).
Fully Implicit Scheme. Substitution of θ = 1 into Equation (11) leads to the following form:

(
uP −u0P

) ∆y
∆τ

=
1

2

(
2r

σ2
+

(
σ̃

σ

)2)
(uE −uW ) +

1

∆y

(
σ̃

σ

)2
(uE − 2uP + uW )

and the reduced formula is then
qiuW + (1 + ri ) uP + siuE = u

0
P (14)

where
qi =

∆τ

2∆y

(
2r

σ2
+

(
σ̃

σ

)2)
−

∆τ

(∆y)2

(
σ̃

σ

)2
, ri =

2∆τ

(∆y)2

(
σ̃

σ

)2
and

si = −
∆τ

2∆y

(
2r

σ2
+

(
σ̃

σ

)2)
−

∆τ

(∆y)2

(
σ̃

σ

)2
which is our proposed Finite Volume Fully Implicit Scheme (FVFIS).

5. Stability of the numerical schemes
To test the stability of the derived schemes in section 4, with the help of the Von-Neumannstability method [55], let us consider a Fourier component for uj

i
and u0P as

u
j
i

= UjeIθi and u0P = UjeIθi (15)
where I = √−1, i.e., imaginary unit, Uj is the amplitude at a time level j,θ(= R∆y) is the phaseangle, R is the wave number in the x-direction, and i represents the index of the node.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 9
Similarly,

u
j∓1
i±1 = U

j∓1eIθ(i±1) u0W = U
jeIθ(i−1) u0E = U

jeIθ(i+1) uP = U
j+1eIθi

uW = U
j+1eIθ(i−1) uE = U

j+1eIθ(i+1)
(16)

For convenience, let us suppose that G = Uj+1
Uj

. Thus, the stability requirement is |G|2 ≤ 1.Applying Equation (15) and Equation (16) into Equation (7), and dividing by eIθi , we get
|G|2 =

1

4

{4∆τ (σ̃
σ

)2
cos θ±

√
A

}2
+ 4(∆τ)2(∆y)2

(
2r

σ2
+

(
σ̃

σ

)2)2 (
1 − cos2 θ

)
×

(
(∆y)2 + 2∆τ

(
σ̃

σ

)2)−2 (17)
where

A =16(∆τ)2
(
σ̃

σ

)4
cos2 θ− 4(∆τ)2(∆y)2

(
2r

σ2
+

(
σ̃

σ

)2)2 (
1 − cos2 θ

)
+ 4(∆y)4

− 16(∆τ)2
(
σ̃

σ

)4
+ I16(∆τ)2(∆y)

(
σ̃

σ

)2 (
2r

σ2
+

(
σ̃

σ

)2)
cos θ

√
1 − cos2 θ

For extremum value of |G|2, solving d|G|2
d(cos θ)

= 0 for cos θ, and substituting it into d2|G|2
d(cos θ)2

< 0. Thenfrom Equation (17), we cannot confirm that the maximum value of |G|2 would occur. However, theextreme values of cos θ must yet be investigated. For cos θ = 1, Equation (17) gives |G|2 = 1, andthe stability requirement is satisfied. For cos θ = −1, Equation (17) also yields |G|2 = 1 and, andthe stability requirement is satisfied. Thus, the DFFDS proposed in Equation (7) is unconditionallystable.
Similarly, we can show that LFDS and FVFIS wrote in Equation (8) and Equation (14), respectively,both are unconditionally stable.
Again, applying Equation (15) and Equation (16) into Equation (12) and dividing by eIθi , we get

G = 1 +
2∆τ

(∆y)2

(
σ̃

σ

)2
(cos θ− 1) + I

∆τ

∆y

(
2r

σ2
+

(
σ̃

σ

)2)
sin θ

Then we may obtain easily,
|G|2 =

{
1 +

2∆τ

(∆y)2

(
σ̃

σ

)2
(cos θ− 1)

}2
+

(
∆τ

∆y

)2 (
2r

σ2
+

(
σ̃

σ

)2)2 (
1 − cos2 θ

) (18)
For extremum value of |G|2 such that d|G|2

d(cos θ)
= 0, we can find

cos θ =
1

∆τ
×

[
2

(
σ̃

σ

)2
− 4

∆τ

(∆y)2

(
σ̃

σ

)4]
×

(2r
σ2

+

(
σ̃

σ

)2)2
− 4

∆τ

(∆y)2

(
σ̃

σ

)4−1 (19)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 10
Considering d2|G|2

d(cos θ)2
< 0, and substituting the value of cos θ from Equation (19) into Equation (18),which does not provide us the maximum value of |G|2. But, the extreme values of cos θ must beinvestigated. For cos θ = 1, Equation (18) gives |G|2 = 1, and the stability requirement is satisfied.For cos θ = −1, Equation (18) yields |G|2 = {1 − 4∆τ

(∆y)2

(
σ̃
σ

)2}2 and, imposing the requirement of
|G|2 ≤ 1, yields, FVES in Equation (12) is conditionally stable and the condition is(

σ̃

σ

)2
≤

(∆y)2

2∆τ
(20)

Similarly, we can state that FVCNS, Equation (13) is also conditionally stable and the conditionis (
σ̃

σ

)2
≤

(∆y)2

∆τ
(21)

6. Consistency of the numerical schemes
For consistency, the finite difference equation (FDE) approximation of a PDE must reduce tothe original PDE as the step sizes approach zero [55].

Now expanding each u(y,τ) in a Taylor series expansion about uj
i
, we get

u
j+1
i

= u
j
i

+ ∆τ
∂u

∂τ
+

(∆τ)2

2!

∂2u

∂τ2
+

(∆τ)3

3!

∂3u

∂τ3
+ O(∆τ)4 (22)

u
j+1
i+1

=u
j
i

+ ∆τ
∂u

∂τ
+ ∆y

∂u

∂y
+

1

2!

(
∆τ

∂

∂τ
+ ∆y

∂

∂y

)2
u

+
1

3!

(
∆τ

∂

∂τ
+ ∆y

∂

∂y

)3
u + O

[
(∆τ)4, (∆y)4

] (23)
u
j+1
i−1 =u

j
i

+ ∆τ
∂u

∂τ
− ∆y

∂u

∂y
+

1

2!

(
∆τ

∂

∂τ
−∆y

∂

∂y

)2
u

+
1

3!

(
∆τ

∂

∂τ
− ∆y

∂

∂y

)3
u + O

[
(∆τ)4, (∆y)4

] (24)
Applying Equations (22), (23), and (24) into Equation (8) yields

(di + ei + fi ) u
j
i

+ (1 + di + ei + fi ) ∆τ
∂u

∂τ
+ (1 + di + ei + fi )

(∆τ)2

2

∂2u

∂τ2

+ (−di + fi ) ∆y
∂u

∂y
+ (−di + fi ) ∆τ∆y

∂2u

∂τ∂y
+ (di + fi )

(∆y)2

2

∂2u

∂y2

+ O
[
(∆τ)3, (∆y)3

]
= 0

from which we get
∂u

∂τ
+

∆τ

2

∂2u

∂τ2
−

(
2r

σ2
+

(
σ̃

σ

)2)
∂u

∂y
− ∆τ

(
2r

σ2
+

(
σ̃

σ

)2)
∂2u

∂y∂τ

−
(
σ̃

σ

)2
∂2u

∂y2
+ O

[
(∆τ)2, (∆y)2

]
= 0

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 11
It is obvious that if ∆τ, ∆y → 0, then the original PDE (2) is recovered. Therefore, the Laasonenfinite difference scheme, Equation (8), is consistent. Now according to Lax’s equivalence theorem,[55], LFDS is convergent for all values of the parameters. Similar arguments hold for DFFDS andFVFIS. On the other hand, FVES and FVCNS are also convergent if the conditions (20) and (21)respectively, are satisfied.

Table 1. Call option prices using the Leland volatility model.
S0 Exact Finite Difference Schemes Finite Volume Schemes(Linear) DFFDS LFDS FVES FVFIS FVCNS37.00 0.00001 0.04734 0.04893 0.00000 0.00054 0.0005047.00 0.00182 0.30914 0.31368 0.00006 0.01340 0.0129757.00 0.05078 1.09349 1.09949 0.00036 0.10771 0.1058867.00 0.45226 2.93576 2.94472 0.00335 0.65191 0.6479577.00 1.97686 6.06630 6.07104 0.01709 2.26632 2.2623087.00 5.46222 10.56460 10.56947 1.25649 5.63768 5.6368897.00 11.17037 16.34912 16.34757 6.49278 11.07370 11.07714107.00 18.71972 23.33442 23.33541 16.15278 18.66210 18.66616117.00 27.48006 31.19401 31.19406 26.33380 27.42028 27.42359127.00 36.91158 39.65579 39.65486 36.72393 36.83405 36.83640137.00 46.67034 48.63231 48.63285 46.68664 46.59745 46.59938147.00 56.57397 57.89847 57.89955 56.61703 56.45705 56.45879157.00 66.53723 67.45340 67.45432 66.59903 66.46595 66.46767167.00 76.52370 77.08904 77.08984 76.50332 76.39766 76.39940177.00 86.51886 86.88858 86.88931 86.48557 86.40846 86.41023187.00 96.51716 96.75424 96.75478 96.48843 96.43330 96.43508197.00 106.51657 106.59706 106.59728 106.40288 106.36023 106.36202207.00 116.51636 116.67091 116.67078 116.55261 116.52319 116.52499217.00 126.51629 126.48888 126.48906 126.39973 126.37558 126.37738227.00 136.51627 136.46008 136.46065 136.39791 136.37870 136.38049237.00 146.51626 146.57401 146.57489 146.53847 146.52419 146.52597247.00 156.51626 156.41475 156.41484 156.38607 156.37369 156.37545257.00 166.51626 166.40053 166.39979 166.37904 166.36868 166.37039267.00 176.51626 176.52571 176.52411 176.51173 176.50347 176.50515

7. Results and Discussions
In this section, we choose the same parameters: r = 0.1, σ = 0.2, K = 100, T = 1, µ = 0.05,

∆t = 0.01, a = 0.02, M = 0.01, and C = 30, as illustrated in the literature [46]. Then wecalculate the call option values using the proposed schemes, described in previous section 4, fordifferent volatility models. We compare the approximate results with the exact value of the linearBlack-Scholes model and among themselves also.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 12

Figure 1. Approximate results of Equation (1) by using (a) Boyle and Vorst volatilitymodel, and (b) Barles and Soner volatility model.
From Table 1 and Figure 8.7 (see Appendix), we observe that fully implicit FVS and Crank-Nicolson FVS provide comparatively better results than the other methods. Note that all of themethods provide poor results when the initial stock price is less than the strike price (here strikeprice, in comparison with the exact value of the linear Black-Scholes model.From Table 8.3 in Appendix 8 and Figure 1 (a), we can make similar comments, but here theFVES gives a very poor approximation than the other methods when the initial stock price is lessthan the strike price (K = 100). Table 8.4 in Appendix 8 and Figure 1(b) show that all of themethods provide a closer approximation to the exact value of the linear Black-Scholes model forall of the initial stock price, whether it is greater than the strike price, K = 100.

Table 2. Call option prices using RAPM volatility model.
S0 Exact Finite Difference Schemes Finite Volume Schemes(Linear) DFFDS LFDS FVES FVFIS FVCNS37.00 0.00001 0.09975 0.10198 0.00075 0.00054 0.0005047.00 0.00182 0.64320 0.64859 0.02029 0.01340 0.0129757.00 0.05078 2.01164 2.01581 0.16518 0.10771 0.1058867.00 0.45226 4.58788 4.59820 0.97606 0.65191 0.6479577.00 1.97686 8.35995 8.36458 3.28335 2.26632 2.2623087.00 5.46222 13.24558 13.25523 7.65193 5.63768 5.6368897.00 11.17037 19.14277 19.14296 13.90023 11.07370 11.07714107.00 18.71972 25.95302 25.96004 21.59995 18.66210 18.66616117.00 27.48006 33.49710 33.50138 30.05441 27.42028 27.42359127.00 36.91158 41.58132 41.58229 39.02457 36.83405 36.83640137.00 46.67034 50.16133 50.16454 48.32523 46.59745 46.59938147.00 56.57397 59.07941 59.08280 57.80665 56.45705 56.45879157.00 66.53723 68.31796 68.31997 67.49623 66.46595 66.46767167.00 76.52370 77.71942 77.72089 77.20105 76.39767 76.39941177.00 86.51886 87.32671 87.32812 87.03280 86.40846 86.41023187.00 96.51716 97.04281 97.04399 96.91429 96.43330 96.43509197.00 106.51657 106.79810 106.79881 106.75046 106.36023 106.36202207.00 116.51636 116.77932 116.77954 116.81690 116.52319 116.52498217.00 126.51629 126.56491 126.56478 126.62193 126.37555 126.37734227.00 136.51627 136.50685 136.50637 136.57895 136.37864 136.38041237.00 146.51626 146.59198 146.59115 146.67807 146.52410 146.52585247.00 156.51626 156.42589 156.42489 156.50633 156.37350 156.37521257.00 166.51626 166.40454 166.40336 166.47896 166.36838 166.37004267.00 176.51626 176.52231 176.52094 176.59035 176.50307 176.50468

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 13
Finally, from Table 2 and the corresponding Figure 8.8 in Appendix 8, we may observe thatfor the RAPM volatility model, the FVCNS and FVFIS give better approximation than the othernumerical schemes when the initial stock price is closer to and/or greater than the strike price.On the other hand, from Figures 2, 3, 4, it is clear that FVFIS and FVCNS produce comparativelybetter results than the other schemes for all of the volatility models. Figures 5, 6 depict the optionprices at various time periods (from initial time t = 0 to maturity time, t = T) with different initialstock values. The similar results of solution surface for option price by using Barles and Sonervolatility model and RAPM volatility model are presented in Appendix 8, see Figures 8.9,8.10.

Figure 2. Approximate results of Equation (1) using (a) Dufort-Frankel Finite Dif-ference Scheme, and (b) Laasonen Finite Difference Scheme.

Figure 3. Approximate results of Equation (1) using (a) Finite Volume ExplicitScheme, and (b) Finite Volume Fully Implicit Scheme.

Figure 4. Approximate results of Equation (1) using Finite Volume Crank-NicolsonScheme.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 14

(a) LVM (DFFDS) (b) LVM (LDS) (c) LVM (FVES)

(d) LVM (FVCNS) (e) LVM (FVFIS)
Figure 5. Solution surface for option price by using Leland volatility model.

(a) BVVM (DFFDS) (b) BVVM (LDS) (c) BVVM (FVES)

(d) BVVM (FVCNS) (e) BVVM (FVFIS)
Figure 6. Solution surface for option price by using Boyle and Vorst volatilitymodel.

8. Conclusion
In this research work, we have derived some numerical schemes using the FVM and FDM tosolve the non-linear Black-Scholes PDE for European option pricing with the transaction costs byexploiting the transformations available in the existing literature [46]. Thus we have modified themodel equation accordingly to a non-linear parabolic PDE. For the convergence of these schemes,stability and consistency have been shown rigorously. Then these schemes have been applied tovarious volatility models. According to the visible results, as presented in the earlier sections, it

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 15
is noted that all of the proposed schemes provide the best approximation to the exact value of thelinear Black-Scholes model for all initial stock prices, regardless of whether they are closer to orgreater than the strike price; particularly in the case of Barles and Soner Volatility Model. Wemay claim that the FVFIS and FVCNS approximate better than the other methods for all four-volatility models. Thus, it is observed that the FVFIS and FVCNS are very effective and proficientin locating approximate solutions to non-linear Black-Scholes models. Notice that the limitationof these schemes is that they may offer poor results sometimes when the initial stock price is lessthan the strike price compared to the exact value of the linear Black-Scholes model. Finally, wemay conclude that the proposed schemes may be applied to other non-linear partial differentialequations to compute the numerical solutions with the desired accuracy.
Conflicts of Interest. The authors declare no competing interests exist.
Funding Statement. This research received no external funding.

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https://www.jstor.org/stable/2245301
https://doi.org/10.1111/j.1540-6261.1985.tb02383.x
https://doi.org/10.1111/j.1540-6261.1985.tb02383.x
https://doi.org/10.1111/j.1540-6261.1992.tb03986.x
https://www.jstor.org/stable/54362
 https://doi.org/10.1111/1467-9965.00034
https://doi.org/10.1016/S0165-1889(03)00059-9
https://doi.org/10.1016/j.camwa.2008.02.010
https://doi.org/10.1016/j.camwa.2008.02.010
https://doi.org/10.1007/s007800050046
https://doi.org/10.1080/00207160.2012.688115
https://doi.org/10.1080/00207160.2012.688115
https://doi.org/10.1016/j.amc.2012.12.077
https://doi.org/10.1016/j.amc.2012.12.077
https://doi.org/10.1016/j.camwa.2008.02.005
https://doi.org/10.1080/00207160.2015.1069818
https://doi.org/10.1515/cmam-2015-0029
https://doi.org/10.1080/13504860903075670
https://doi.org/10.1080/00207160.2011.558574
https://doi.org/10.1080/00207160.2011.558574
https://doi.org/10.1007/s10690-010-9115-3
https://doi.org/10.1155/JAM.2005.235


Eur. J. Math. Anal. 10.28924/ada/ma.2.9 18
Appendix

This section contains the supporting figures and tables to observe the accuracy of the solutionmethodologies.

Figure 8.7. Approximate results of Equation (1) by using Leland volatility model.

Figure 8.8. Approximate results of Equation (1) by using RAPM volatility model.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 19
Table 8.3. Call option prices using Boyle and Vorst volatility model.

S0 Exact Finite Difference Schemes Finite Volume Schemes(Linear) DFFDS LFDS FVES FVFIS FVCNS37.00 0.00001 0.08130 0.08362 0.00000 0.00054 0.0005047.00 0.00182 0.44919 0.45475 0.00016 0.01340 0.0129757.00 0.05078 1.43187 1.43827 0.00137 0.10771 0.1058867.00 0.45226 3.52436 3.53389 0.00181 0.65191 0.6479577.00 1.97686 6.88032 6.88512 -0.01095 2.26632 2.2623087.00 5.46222 11.52515 11.53065 -0.03314 5.63768 5.6368897.00 11.17037 17.36380 17.36252 1.18226 11.07370 11.07714107.00 18.71972 24.30605 24.30805 14.49730 18.66210 18.66616117.00 27.48006 32.07120 32.07189 26.66264 27.42028 27.42359127.00 36.91158 40.41514 40.41449 37.98090 36.83405 36.83640137.00 46.67034 49.26375 49.26507 47.25031 46.59745 46.59938147.00 56.57397 58.41457 58.41649 56.80881 56.45705 56.45879157.00 66.53723 67.86188 67.86337 66.66025 66.46595 66.46767167.00 76.52370 77.41353 77.41510 76.50821 76.39767 76.39940177.00 86.51886 87.14056 87.14251 86.47640 86.40846 86.41023187.00 96.51716 96.94695 96.94884 96.47214 96.43330 96.43509197.00 106.51657 106.75019 106.75144 106.38908 106.36023 106.36202207.00 116.51636 116.78198 116.78255 116.54146 116.52319 116.52500217.00 126.51629 126.57938 126.58055 126.39087 126.37558 126.37738227.00 136.51627 136.53059 136.53250 136.39143 136.37869 136.38048237.00 146.51626 146.62463 146.62715 146.53436 146.52418 146.52596247.00 156.51626 156.45839 156.45983 156.38285 156.37367 156.37541257.00 166.51626 166.43684 166.43715 166.37675 166.36863 166.37034267.00 176.51626 176.55434 176.55347 176.51042 176.50341 176.50508

Table 8.4. Call option prices using Barles and Soner volatility model.
S0 Exact Finite Difference Schemes Finite Volume Schemes(Linear) DFFDS LFDS FVES FVFIS FVCNS37.00 0.00001 0.00056 0.00060 0.00047 0.00054 0.0005047.00 0.00182 0.01448 0.01496 0.01255 0.01340 0.0129757.00 0.05078 0.11766 0.11964 0.10402 0.10771 0.1058867.00 0.45226 0.70853 0.71289 0.64387 0.65191 0.6479577.00 1.97686 2.42414 2.42846 2.25831 2.26632 2.2623087.00 5.46222 5.90671 5.90840 5.63756 5.63768 5.6368897.00 11.17037 11.38951 11.38681 11.08596 11.07370 11.07714107.00 18.71972 18.90018 18.89694 18.68219 18.66210 18.66616117.00 27.48006 27.57177 27.56903 27.44285 27.42028 27.42359127.00 36.91158 36.90926 36.90718 36.85630 36.83405 36.83640137.00 46.67034 46.63600 46.63420 46.61611 46.59745 46.59938147.00 56.57397 56.47483 56.47316 56.47209 56.45705 56.45879157.00 66.53723 66.47486 66.47318 66.47729 66.46595 66.46767167.00 76.52370 76.40243 76.40071 76.40662 76.39766 76.39940177.00 86.51886 86.41173 86.40997 86.41573 86.40846 86.41022187.00 96.51716 96.43561 96.43381 96.43938 96.43329 96.43508197.00 106.51657 106.36234 106.36053 106.36582 106.36022 106.36202207.00 116.51636 116.52509 116.52327 116.52827 116.52319 116.52499217.00 126.51629 126.37745 126.37563 126.38047 126.37557 126.37737227.00 136.51627 136.38055 136.37873 136.38340 136.37869 136.38048237.00 146.51626 146.52603 146.52421 146.52870 146.52418 146.52596247.00 156.51626 156.37561 156.37380 156.37798 156.37368 156.37543257.00 166.51626 166.37066 166.36886 166.37271 166.36866 166.37038267.00 176.51626 176.50552 176.50374 176.50725 176.50346 176.50514

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Eur. J. Math. Anal. 10.28924/ada/ma.2.9 20

(a) BSVM (DFFDS) (b) BSVM (LDS) (c) BSVM (FVES)

(d) BSVM (FVCNS) (e) BSVM (FVFIS)
Figure 8.9. Solution surface for option price by using Barles and Soner volatilitymodel.

(a) RAPM (DFFDS) (b) RAPM (LDS) (c) RAPM (FVES)

(d) RAPM (FVCNS) (e) RAPM (FVFIS)
Figure 8.10. Solution surface for option price by using RAPM volatility model.

https://doi.org/10.28924/ada/ma.2.9

	1. Introduction
	2. The Model Equation
	3. Volatility Models
	Leland Volatility Model (LVM)
	Boyle and Vorst Volatility Model (BVVM)
	Barles and Soner Volatility Model (BSVM)
	RAPM Volatility Model (RAPMVM)

	4. Derivations of Computational Schemes 
	4.1. Dufort-Frankel Finite Difference Scheme
	4.2. Laasonen Finite Difference Scheme
	4.3. Finite Volume Schemes

	5. Stability of the numerical schemes
	6. Consistency of the numerical schemes
	7. Results and Discussions
	8. Conclusion
	Conflicts of Interest
	Funding Statement

	References
	Appendix

