DEPARTMENT OF MATHEMATICSAND

APPLIED MATHEMATICS

TECHNICAL REPORTUPWT 2013/24

APPLYING THE BARYCENTRIC JACOBI SPECTRALMETHOD TO PRICE OPTIONS WITH TRANSACTION

COSTS IN A FRACTIONAL BLACK-SCHOLESFRAMEWORK

BF NTEUMAGNE, E PINDZA AND EBEN MARE

August 2013

Applying the barycentric Jacobi spectral method to price options

with transaction costs in a fractional Black-Scholes framework

B F Nteumagne, E Pindza and E Mare

Department of Mathematics and Applied Mathematics, University of Pretoria,

Pretoria 002, Republic of South Africa

Abstract

The aim of this paper is to show how options with transaction costs under fractional,mixed Brownian-fractional, and subdiusive fractional Black-Scholes models can be ef-ciently computed using the barycentric Jacobi spectral method. The reliability of thebarycentric Jacobi spectral method for space (asset) direction discretization is demon-strated by solving partial dierential equations (PDEs) arising from pricing Europeanoptions with transaction costs under these models. The discretization of these PDEsin time relies on the implicit Runge-Kutta Radau IIA method. We conducted variousnumerical experiments and compare our numerical results with existing analytical so-lutions. It was found that the proposed method is ecient, highly accurate, reliableand is an alternative to some existing numerical methods for pricing nancial options.

2010 Mathematics Subject Classication: 39A05, 65M06, 65M12, 91G60.

Keywords : Jacobi spectral method, European option, Fractional Black-Scholes model, MixedBrownian-fractional Brownian Black-Scholes model, transaction cost, Subdiusive fractionalBlack-Scholes model, Scaling.

1 Introduction

Modeling of nancial derivatives has been of great interest in the past three decades. Numer-ous mathematical models have been developed in the classical Black-Scholes [1] frameworkto help investors in their decision making process. These models are based on an arbitrageargument, i.e., by continuously adjusting a portfolio consisting of a stock and a risk-freebond, an investor can exactly replicate the return to any option on the stock.

In the presence of transaction costs in capital markets the presence of arbitrage argument[4] is no longer valid [5], since perfect hedging is impossible. Due to the innite variation ofthe geometric Brownian motion, the continuous replication policy incurs an innite amountof transaction costs over any trading interval no matter how small it might be. Therefore, inrecent years one observes many generalizations of the Black-Scholes model to deal with the

Corresponding author. E-mail: eben.mare@up.ac.za. Fax: +27-12-4203893.

1

problem of option pricing and hedging with transaction costs. This leads to the Black-Scholesmodel but with an adjusted volatility. Leland [5] was the rst to examine option replicationin a discrete time setting, and proposed a modied replicating strategy, which depends uponthe level of transaction costs, the revision interval, the option to be replicated and theenvironment. Subsequently, several authors proposed new models, but all in discrete time[2, 19]. However, these models based on the diusion process known as geometric Brownianmotion (GBM) have severe shortcomings, for example, long-range correlations, heavy-tailedand skewed marginal distributions, lack of scale invariance and periods of constant values, toenumerate these only. Recently, Wang [9] obtained a European call option pricing formulausing a mean self-nancing delta hedging argument in a discrete time setting and then showedhow scaling and long-range dependence impact dramatically on the pricing of options usingthe fractional and multifractional Black-Scholes model with transaction costs.

In continuous time a lot of eort has been done in order to alleviate the problem ofinnite amount of transaction costs over any trading interval when the asset process followsa geometric Brownian motion. Magdziarz [20] et al. introduced a subdiusive geometricBrownian process which captures the subdiusive characteristics of nancial markets. Theyshowed that their model is arbitrage free. The same idea was used later by Wang et al. [8],Hui and Yun-Xiu [21] to obtain a Black-Scholes equation with transaction costs in subd-iusive fractional Brownian motion regime. However, closed-form solutions of these PDEsin nance are generally rare. Therefore, one has to consider numerical methods to obtainsolutions.

In this paper, we are concerned with the application of the barycentric Jacobi interpola-tion to value options with transaction costs under fractional [9], mixed Brownian-fractional[10] and subdiusive fractional [8] Black-Scholes models. Barycentric spectral methods wereintroduced by Baltensperger et al. [22] to solve boundary value problems. Recently thesemethods have emerged in the eld of nance as a promising tool to solve option pricingproblems. Pindza and Patidar [3] proposed an accurate method, namely the barycentricChebyshev spectral method to price options in illiquid markets. Ngounda et al. [6] used thebarycentric Chebyshev domain decomposition method to provide fast and accurate resultsfor pricing European options with jumps, which was later extended and applied to Heston'svolatility model (see [7]). Most of the work on barycentric spectral methods have been basedon the use of uniform or Chebyshev grids. Recently, Wang et al. [15] computed explicitbarycentric weights for Jacobi polynomial interpolation in the roots or extrema of classicalorthogonal polynomials in terms of the nodes and weights of the corresponding Gaussianquadrature rule. Hence, we investigate the utility of this new barycentric interpolation inthe eld of nance. The semi-discretization of the PDE in time is realized using a 7-stage13th-order fully implicit Runge-Kutta Radau IIA method with adaptive time stepping [18].

This paper is structured as follows. Section 2 reviews the option pricing formulation withtransaction costs under fractional, mixed Brownian-fractional and subdiusive fractionalprocesses. In Section 3, we introduce the Jacobi barycentric spectral method, semi-discretizethe PDE in the asset space and propose a conformal map in order to improve the accuracyof our method. In Section 4, we perform numerous experiments in order to advocate theutility of the barycentric spectral method. Finally, Section 5 gives a summary and scope forfurther research.

2

2 Pricing with transaction costs under fractional, mixed

Brownian-fractional and subdiusive fractional pro-

cesses

Let M = ;F ;P; (F )t0 ; (St)t0 be a complete probability space carrying a fractionalBrownian motion (BH(t))t2R with Hurst exponent H 2 (0; 1), where is the set of allpossible outcomes of the experiment known as the sample space, F is the set of all events, Pis a real world probality, Ft is a natural ltration, St a risky underlying asset price process.Assume that the price St of the underlying stock at time t satises a fractional Black-Scholesmodel

St = S0 exp(t+ BH(t)); (2.1)

where ,H 2 (0; 1) and S0 > 0 are constants. Assume that the portfolio is revised every smallxed time step t; transaction costs are proportional to the value of the transaction in theunderlying. Let k denote the round trip transaction cost per unit dollar of transaction. Thenunder all the assumptions given by Wang [9], the Black-Scholes equation with transactioncosts assuming factional Brownian motion can be written as

@u

@t+ rS

@u

@S+

~2

2S2

@2u

@S2 ru = 0; (2.2)

where the modied volatility is given by

~2 = 2(t)2H1 + Le(H)sign()

: (2.3)

Here Le(H) is the fractional Leland number [9].If we assume that the price St of the underlying stock at time t satises a mixed Brownian-

fractional Brownian model

St = S0 exp(t+ B(t) + HBH(t)); (2.4)

then under all the assumptions given by Wang [10], the Black-Scholes equation with trans-action costs in mixed Brownian-fractional Brownian motion can be written as

@u

@t+ rS

@u

@S+

~2

2S2

@2u

@S2 ru = 0; (2.5)

where the modied volatility is given by

~ =

2 + 2H(t)

2H1 + k

s2

2

t+ 2H(t)

2H2sign()

! 12

; (2.6)

with 2H =

2k2

+

q2k2

2+ 8k

2

2t

2; t =

2

12Hk

1H

, and =@2u

@S2.

A subdiusive fractional Brownian motion is describeb by

dS(t) = S(t)(dt)2H + S(t)dB(t): (2.7)

3

Wang et al. [8] obtained the modied volatility corresponding to the continuous Black-Scholes equation with transaction cost in subdiusive regime as

@u

@t+ rS

@u

@S+

~2H2S2

@2u

@S2 ru = 0; (2.8)

where the modied volatility is given by

~2H = 2Ht2H1

t1

()

2H 1 +

k

2

22 1 sign()2: (2.9)

Here () represents the gamma function evaluated at and sign() is the second derivativeof the option value with respect to the asset.

The dierence between American and European call and put options is made by theinitial and boundary conditions. In this work we focus exclusively on the European calloption. Such options have the following initial and boundary conditions

u(0; t) = 0; u(S; 0) = maxfS K; 0g; u(S; t) = S Kert; as S !1; (2.10)

where S 2 [0;1] and t 2 [0; T ]. We want to solve the three PDEs (2.2), (2.5) and (2.8)subjected to modied volatilities (2.3), (2.6) and (2.9), respectively. Before moving to the ap-plications of the barycentric Jacobi spectral method to solve these problems, it is worthwhileto consider some preliminaries of this method.

3 Barycentric Jacobi spectral collocation method

In this section, we turn our attention to the problem of barycentric Jacobi interpolationand present a fast algorithm for the ecient computation of the interpolation formula usingJacobi-Gauss-Lobatto (JGL) points. This interpolation is realized by a class of the Lagrangeform of the interpolating polynomial, as follows. Let xj, j = 0; 1; : : : ; N be a set of distinctnodes. Then the polynomial of degree N that interpolates the function u(x) at these pointsis given by [11]

pN(x) =NXj=0

uj`j(x); `j(x) =NY

k=0;1;k 6=j

x xkxj xk ; (3.1)

where the Lagrange polynomial `j corresponding to the node xj has the property

`j(xk) =

(1; j = k

0; otherwise,(3.2)

with j; k = 0; 1; : : : N: Generally, the Lagrange form of the interpolating polynomial (3.1)is not advocated for numerical computations. In particular, it requires O(N2) additionsand multiplications for each evaluation of pN(x) and every time a node xj is modied oradded, all Lagrange fundamental polynomials have to be recalculated. However, with slightmodications, the Lagrange formula is indeed of great practical use. This has been noted

4

by several authors, including Henrici [12] and Werner [13]. Berrut and Trefethen [14] mod-ied the Lagrange polynomial through barycentric interpolation and proposed an improvedLagrange formula. Following [14], we dene `(x), the numerator of `j in (3.1) divided byx xj, as

`(x) =NYj=0

(x xj): (3.3)

In addition, if we dene the barycentric weight by

j =1

NYk=0;k 6=j

(xj xk); j = 0; 1; : : : ; N; (3.4)

i.e., j = 1=`0(xj), then `j in (3.1) becomes

`j(x) = `(x)j

x xj : (3.5)

Consequently, the Lagrange formula (3.1) becomes

pN(x) = `(x)NXj=0

jx xj uj: (3.6)

In particular, if u(x) = 1, we obtain

1 =NXj=0

`j(x) = `(x)NXj=0

jx xj : (3.7)

Dividing (3.6) by (3.7), we get the barycentric formula for pN as

pN(x) =

NXj=0

jx xj uj

NXj=0

jx xj

: (3.8)

This is the most used form of Lagrange interpolation in practice and admitsO(N) operations.In order to obtain good approximations via interpolation, the choice of interpolation nodesand barycentric weights is particularly important. For certain particular sets of points,such as equidistant points as well as Chebyshev points, the barycentric weights j can becomputed analytically [14]. Recently, Wang et al. [15] computed explicit barycentric weightsfor polynomial interpolation in the roots or extrema of classical orthogonal polynomials interms of the nodes and weights of the corresponding Gaussian quadrature rule.

5

In this paper, we are interested in the barycentric Jacobi interpolation. The Jacobi-Gauss-Lobatto quadrature rule is dened byZ 1

1u(x) =

NXj=0

wju(xj); (3.9)

where the Jacobi-Gauss-Lobatto points fxjgNj=0 are the zeros of (1x2)P (;)N1 (x) and P (;)N (x)is the Jacobi polynomial of degree N . The following result gives an analytical formula ofbarycentric weights for the Jacobi-Gauss-Lobatto points.

Theorem 3.1 ([15]). Let xj; j = 0; ; N , be the roots of (1 x2)P (;)N1 (x) and let wjbe the corresponding weights of the interpolatory quadrature rule with weight function (1 x)(1 + x). Then the simplied barycentric weights are For Jacobi-Gauss-Lobatto points,the simplied barycentric weights j are given by

j = (1)jpjwj; j =

8>: + 1; j = 0

+ 1; j = N

1; otherwise,

(3.10)

The proof of Theorem 3.1 can be found in [15].The computation of entries of the rst and second order dierentiation matrices D(1) and

D(2) whereD

(1)ij = `

0j(xi) and D

(2)ij = `

00j (xi); (3.11)

is given as in [22] by

`0j(xi) =

8>>>>>:j=ixi xj if i 6= j,

NXi 6=j

`0j(xi); if i = j,(3.12)

`00j (xi) =

8>>>>>>>>>:2 j=i

xi xj

NX

k=0;k 6=i

k=ixi xk

1

xi xj

!if i 6= j,

NXi6=j

`00j (xi); if i = j,

(3.13)

where i; j = 0; 1; : : : ; N .

3.1 Conformal mappings for high resolution of non-smooth initialconditions

It is well-known that the solution of Equation (2.8) is very sensitive to localization errorswhen S is in the vicinity of K, because the second derivative of the payo does not exist at

6

this point. Therefore, to increase accuracy it would be reasonable to use an adaptive meshwith high concentration of the mesh points around S = K, while a rareed mesh could beused far away from this area. If we assume that S 2 [a; b], where both a and b are nite, thenwithout further loss of generality we may assume that the interval of integration is [1; 1],as the linear transformation

2S = (b a)x (a+ b): (3.14)In this paper, we use the conformal map g given in [17] by

x = g(y) = ~ +1

~sinh

(y ) ; (3.15)

where =

+

2; =

+

; (3.16)

with

= sinh1[~(1 + ~)]; = sinh1[~(1 ~)]; (3.17)

where ~ and ~ determine the magnitude of the region of rapid change and the location,respectively. Here, y represents the Jacobi-Gauss-Lobatto points. In Figure 1, we plot the

0 50 100 150 200 2501

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

S(x)

y

Figure 1: Asset direction grids S obtained from the mapped Jacobi grids x.

new grid obtained from the original Jacobi y grid by using transformations (3.15) and (3.14)together. The new grid in S(x) contains 100 nodes distributed from a = 0 to b = 250.Value of parameters used in this example are: ~ = 104 and ~ = (2K (b a))=(b a),where K is the strike price. In addition, we show the transformation function around K for~ = 10 (black), ~ = 102 (green), ~ = 103 (purple) and ~ = 104 (red). We observe thatwhen ~ decreases the grid approaches a Jacobi-Gauss-Lobatto grid distribution one. When~ increases we accommodate more grid points around the strike price.

A signicant advantage of the barycentric Jacobi spectral method is that it eliminatestedious computations of transformed derivatives using the chain rule as is usually the casein other spectral collocation methods.

7

3.2 Application to the Black-Scholes PDE

We discretize the Black-Scholes PDEs (2.2), (2.5) and (2.8) in the asset (space) directionby means of barycentric Jacobi spectral method. Let x = g(y) be the transformed Jacobi-Gauss-Lobatto points, the rst step is to transform x 2 [1; 1] into S 2 [0; Smax] that bettersuits the option at hand using x = (2S Smax)=Smax: Now writing u(S; t) = u(x; t), thePDE (2.8) together with its initial and boundary conditions yields8>:

ut = p(x)uxx + q(x)ux + ru;

u(x; 0) = u0; 1 x 1; 0 S Smax;u(1; t) = u0; u(1; t) = uN ; 0 t T;

(3.18)

where

p(x) =1

22S2

2

Smax

2; q(x) = (r )S

2

Smax

:

Substituting (3.1) into (3.18) yields the following system of nonlinear ODEs8>: ut(x; t) = p(x)NXk=0

uk(t)`00k(x) + q(x)

NXj=0

uk(t)`0k(x) + r

NXk=0

uk(t)`k(x);

u0 = u(1; t); uN = u(1; t):(3.19)

In order to write (3.19) in matrix form, we introduce the following matrix and vector nota-tions 8>>>>>>>:

u = [u1; u2 : : : ; uN1]T ;

D(1) =D

(1)ij

; D

(1)ij = `

0j(xi); i; j = 1; : : : ; N 1;

D(2) =D

(2)ij

; D

(2)ij = `

00j (xi); i; j = 1; : : : ; N 1;

P = diag(p(xi)); Q = diag(q(xi)); i = 1; : : : ; N 1;

(3.20)

where I is an (N 1) (N 1) identity matrix. Consequently, (3.19) can be expressed asan initial value problem of the form

du

dt= g(t;u); u(0) = u0: (3.21)

where

g(u; t) =

8

4 Results and discussions

We apply the barycentric Jacobi spectral method to value Black-Scholes equations withtransaction costs under fractional (FBS), mixed Brownian-fractional (MBS) and subdiusivefractional (SBS) processes. To show the eciency of the present method we report the rootmean square norm error (L2) of the solution computed with N grid points, given by

L2 =

vuut NXj=1

(u(Sj) uexact(Sj))2; (4.1)

and the maximal norm error (L1)

L1 = maxj=1;:::;N

ju(Sj) uexact(Sj)j; (4.2)

where u(Sj) and uexact(Sj) are the benchmark and computed values of the solution u at Sj.The analytic solutions of Black-Scholes equations with transaction costs under FBS, MBS

and SBS regime are possible if we assume that the Greek is always positive for the abovementioned models as in the case of standard Black-Scholes model, then for European calloptions under SBS [9] regime is known, and expressed as

VSBS(S; ) = SN(d^1())KerN(d^2()); (4.3)where

d^1() =ln (S=E) +

r + 1

2^2H;()

^H;()p

; d^2() =ln (S=E) +

r 1

2^2H;()

^H;()p

; (4.4)

here ^2H;() =R T ~

2H;(t)dt

is given by

^2H; =2

(())2

1 +

k

2

22 1T 2H (T )2H

; (4.5)

= T t and N() is the cumulative probability distribution function for a standardizednormal variable

N(y) =1p2

Z y1

ex2

2 dx: (4.6)

If we assume the same sign behaviour for in the MBS model, then the analytic solution isgiven by

VMBS(S; ) = SeN(d1) EerN(d2); (4.7)

where

d1 =ln (S=K) +

r + ~

2

2

~p

; d2 = d1 ~p ; (4.8)

with the modied volatility given by

~ =

2 + 2H(t)

2H1 + k

s2

2

t+ 2H(t)

2H2! 12

: (4.9)

9

Here 2H =

2k2

+

q2k2

2+ 8k

2

2t

2and t =

2

12Hk

1H

. In the case of FBS model,

the closed-form solution is given as

VFBS(S; ) = SeN(d1) EerN(d2); (4.10)

where

d1 =ln (S=K) +

r + ~

2

2

~p

; d2 = d1 ~p ; (4.11)

with~2 = 2

(t)2H1 + Le(H)

: (4.12)

In order to test the accuracy of the method, we present a comparison against the above

100 150 200 250 300 350 400 450 5001011

1010

109

108

107

Smax

L

no

rm

SBSFBSMBS

100 150 200 250 300 350 400 450 5001010

109

108

107

106

Smax

L 2

no

rm

FBSSBSMBS

Figure 2: Eect of the truncation domain on the L2-norm and L1-norm in computation ofthe three models with parameters K = 100, = 0:2, r = 0:03, T = 1, N = 200, Smin = 0,Smax = 250, H = 0:6, = 0:2, k = 1 and = 0:7

exact solutions. In Figure 3 (top left) we plot together numerical and exact solutions forcomparison purposes. We select numerical values of the parameters to be K = 100, = 0:2,r = 0:03, T = 1, N = 200, Smin = 0, Smax = 250, H = 0:6, = 0:2, k = 1, = 0:7. Wechoose the tolerance of the 7-stage 13th-order fully implicit Runge-Kutta Radau IIA method[18] to be tol = 1010, so that the error is dominated by the spatial error. Clearly, it isobserved that all numerical solutions are in good agreement with exact ones. To see howgood our numerical approach approximates exact solutions, we plot the absolute error, i.e.,absolute distance between the exact solution and the numerical approximation for all threemodels. This shows very good accuracy for our method.

4.1 Eect of changes in N and Smax

To determine the convergence of the discretization scheme, we solve the problem by keepingsome parameters xed and varying others. We st investigate the eect of the truncationdomain on the errors by varying Smax between 120 and 500 and keeping other parameters

10

0 50 100 150 200 25020

0

20

40

60

80

100

120

140

160

180

S

u

FBS NumericalPayoffFBS ExactMBS numericalMBS ExactSBS NumericalSBS Exact

0 50 100 150 200 2501013

1012

1011

1010

109

108

107

106

S

Abso

lute

erro

r

FBSMBSSBS

0 50 100 150 200 2500.005

0

0.005

0.01

0.015

0.02

0.025

0.03

S

FBSSBSMBS

0 50 100 150 200 2500.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

S

FBSSBSMBS

Figure 3: Solutions of the European call options with transaction costs under the threemodels (top left), absolute error (top right), the Delta (bottom left) and Gamma (bottomright) of the three models with K = 100, = 0:2, r = 0:03, T = 1, N = 200, Smin = 0,Smax = 250, H = 0:6, = 0:2, k = 1 and = 0:7.

xed as K = 100, = 0:2, r = 0:03, T = 1, N = 200, Smin = 0, Smax = 250, H = 0:6, = 0:2, k = 1 and = 0:7. Figure 2 show the L1-norm and L2-norm errors, respectively.We notice that the errors remain bounded and do not vary signicantly in term of thetruncation domain. This is an important result, since we can accurately solve the optionpricing problem on a small truncated domain, which will result in better eciency.

In the next experiment, we investigate the convergence of the barycentric Jacobi spectralmethod. We vary the number of collocation points N between 20 and 300, with parametersK = 100, = 0:2, r = 0:03, T = 1, N = 200, Smin = 0, Smax = 250, H = 0:6, = 0:2,k = 1, = 0:7; and we plot the results in Figure 4 (top left and right).

For all three models, our method converges rapidly to the exact solutions. This is usuallyknown as spectral or exponential convergence. The eciency of our approach is measuredin Figure 4 (bottom left and right) by the CPU time. The results on the eciency of ourmethod are very satisfactory for all three methods. An accuracy of 108 can be obtained inless than a second.

11

0 50 100 150 200 250 3001010

108

106

104

102

100

N

L

no

rm

FBSMBSSBS

0 50 100 150 200 250 3001010

108

106

104

102

100

L 2

no

rm

N

FBSMBSSBS

0 0.5 1 1.5 2 2.51010

108

106

104

102

100

CPUtime

L

no

rm

FBSSBSMBS

0 0.5 1 1.5 2 2.51010

108

106

104

102

100

CPUtime

L

no

rm

FBSSBSMBS

Figure 4: Convergence of L1 and L2-norms and eciency of the barycentric Jacobi spectralmethod with parametersK = 100, = 0:2, r = 0:03, T = 1, N = 200, Smin = 0, Smax = 250,H = 0:6, = 0:2, k = 1 and = 0:7.

4.2 Convergence of the method

We explore the eect of changes in time, as well as grid-stretching on the accuracy of themodel, keeping N = 200, E = 100; = 0:2; r = 0:03; T = 1; Smin = 0; Smax = 250; H =0:6; = 0:2; k = 1; = 1; = 1 xed and allowing ~ to vary between 102, 103 and 104,we compute the L2- and L1-norms for t = 0:5, t = 1 and t = 2. The results are shownin Table 4.2. We notice that when the grid stretching parameter ~ increases, the accuracyof our method is improved. By accommodating more grid points around the strike priceK, we can overcome the poor convergence of naive application of numerical methods whenpricing options. We also observe that our method is highly accurate (even for long maturityoptions).

We also explore the eect of changes in time, as well as Jacobi parameters and on theaccuracy of the model, keeping N = 200, E = 100; = 0:2; r = 0:03; T = 1; Smin = 0; Smax =250H = 0:6; = 0:2; k = 1; ~ = 104 xed and allowing and to vary between 0:01, 0:5 and1, and to vary between 0:07, 1 and 2, we compute the L2- and L1-norms for t = 0:5, t = 1and t = 2. The results are shown in Table 2. Once again the barycentric Jacobi spectral

12

Table 1: Norm innity and norm relative of errors for the European call options with transactioncosts under subdiusive regime (SBS), fractional Brownian (FBS) and multifractional Brownian(MBS) motions .

Models ~L2 L1t = 0:5 t = 1 t = 2 t = 0:5 t = 1 t = 2

SBS102 6.5320 (-5) 4.6050 (-5) 2.8437 (-5) 6.3396 (-6) 4.2439 (-6) 2.5463 (-6)103 3.0229 (-8) 2.0966 (-8) 1.2747 (-8) 2.7323 (-9) 1.8275 (-9) 1.0850 (-9)104 6.6854 (-9) 1.7416 (-8) 3.9353 (-8) 6.8067 (-10) 1.6954 (-9) 3.6965 (-9)

FBS102 9.2345 (-5) 6.5861 (-5) 4.4286 (-6) 9.7058 (-6) 6.5277 (-6) 4.1726 (-6)103 4.3900 (-8) 3.0709 (-8) 2.0257 (-8) 4.1927 (-9) 2.8170 (-9) 1.7942 (-9)104 1.0145 (-9) 5.9468 (-9) 1.6573 (-8) 1.6911 (-10) 6.1812 (-10) 1.6266 (-9)

MBS102 2.2002 (-5) 9.8368 (-6) 1.1066 (-5) 1.9491 (-6) 8.6613 (-7) 9.9914 (-5)103 9.6097 (-9) 1.5761 (-7) 2.7000 (-5) 8.1278 (-10) 1.0433 (-7) 2.6269 (-5)104 5.2222 (-8) 5.2571 (-7) 5.1523 (-5) 4.7722 (-9) 3.4097 (-7) 5.0791 (-5)

Table 2: Norm innity and norm relative of errors for the European call options with transaction costsunder subdiusive regime (SBS), fractional Brownian (FBS) and multifractional Brownian (MBS) motions.

Models L2 L1t = 0:5 t = 1 t = 2 t = 0:5 t = 1 t = 2

SBS0:01 0.07 1.1292 (-8) 1.8273 (-8) 3.3933 (-8) 1.0863 (-9) 1.7419 (-9) 3.1798 (-9)0:5 1 1.3716 (-8) 5.1996 (-9) 2.7702 (-8) 1.3623 (-9) 5.2704 (-10) 2.6757 (-9)1 2 2.2325 (-8) 1.3673 (-8) 5.6170 (-9) 1.9366 (-9) 1.1587 (-9) 5.5163 (-10)

FBS0:01 0.07 8.7690 (-9) 1.1102 (-8) 1.8003 (-8) 8.4659 (-10) 1.0787 (-9) 1.7388 (-9)0:5 1 2.8309 (-8) 1.4683 (-8) 5.2317 (-9) 2.5996 (-9) 1.4726 (-9) 6.1682 (-10)1 2 3.3504 (-8) 2.2705 (-8) 1.4448 (-8) 3.0405 (-9) 2.0069 (-9) 1.2536 (-9)

MBS0.01 0.07 4.7082(-8) 2.3641 (-7) 1.9080(-5) 4.3052 (-9) 1.3894 (-7) 1.3665 (-5)0.5 1 4.2324 (-8) 3.2348 (-7) 5.1813 (-5) 3.8964 (-9) 3.4502 (-7) 5.1095 (-5)1 2 7.5002 (-9) 1.4137 (-6) 1.1420 (-4) 5.9867 (-9) 1.3157 (-6) 1.1316 (-4)

method achieves very good accuracy for all three models (even for long term maturity). TheJacobi parameters chosen here do not impact the accuracy of our methodology signicantly.It would be useful to investigate how to choose these parameters in an optimal way, however,this is beyond the scope of this paper.

5 Conclusion

The work of Leland [5] on the discrete option pricing model and replication with transactioncost has paved the way to develop the continuous version. We exploited the continuousversion by Wang et al. [8] to construct spectral-based solutions to the model. In practice,option pricing problems are solved numerically since analytical solutions rarely exist. Weacknowledge that many studies have been conducted in the domain of nance and bothnumerical and analytical solutions have been investigated. However, the barycentric Jacobispectral method has just been proven to have a better accuracy and has not been studiedin the eld of PDEs in nance. Furthermore, this method obtains solutions with greateraccuracy than the usual well-known and studied numerical schemes. The barycentric Jacobispectral method has full dierentiation matrices, which require more computational eortthan sparse dierentiation matrix methods. In order to improve the eciency of the rational

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Jacobi spectral method we are currently investigating the domain decomposition algorithmwhich yields block diagonal matrices and we expect to have greater accuracy, and at leastless computational time and memory.

Acknowledgement

This work is supported by the Faculty of Natural and Agricultural Science of the Universityof Pretoria. B F Nteumagne and E Pindza would like to thank Mr Brad Welch for hisnancial support in this research.

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