Examination of Impact from Different Boundary Conditions on the 2D Black-Scholes Model

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Examination of Impact from Different

Boundary Conditions on the 2D

Black-Scholes Model

Evaluating Pricing of European Call Options

Tomas Sundvall

David Trång

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Teknisk- naturvetenskaplig fakultet UTH-enheten

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Abstract

Examination of Impact from Different Boundary

Conditions on the 2D Black-Scholes Model

Tomas Sundvall & David Trång

This paper examines different combinations of close-field and far-field boundary conditions for solving the 2D Black-Scholes model using finite difference methods in space. The combinations were also tested for different parameter settings. The research showed that in the area close to the strike price, the error was not particularly affected by the boundary conditions but rather by the characteristics of the problem itself. The main differences in error for the combinations of conditions are located close to the boundaries. However, if the computational domain for some reason has to be reduced, e.g. to save computational time, the boundary conditions will play an important role on the error in the area close to the strike price. Based on the findings presented in this report, Dirichlet boundary condition on the far- field boundary together with no boundary condition on the close-field is the best

combination. If any of those are not applicable, the linearity condition should be used on that boundary instead.

Examinator: Martin Sjödin Ämnesgranskare: Petter Tammela Handledare: Lina Von Sydow

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en specifik tillgång, kallad för den underliggande tillgången, till ett specifikt pris kallat för lösenpriset, vid den tidpunkt då kontraktet löper ut. Om marknadsvärdet på den underliggande tillgången är högre än lösenpriset när optionen löper ut har ägaren till optionen möjligheten att göra en vinst genom att köpa den underliggande tillgången till lösenpriset och sedan direkt sälja den till marknadspriset. Om lösenpriset istället skulle vara lägre än marknadspriset då optionen löper ut behöver ägaren inte exercera optionen, och riskerar därmed inte att göra en större förlust än den premie som betalats för optionen.

Eftersom ägandet av optioner ger en möjlighet utan motsvarande skyldighet kräver utfärdaren av optioner en premie av köparen. Problemet med att bestämma options premier är av stort intresse inom finansbranschen, och en vanlig metod som ofta används är att lösa Black-Scholes ekvation. En option kan ha flera underliggande tillgångar, och i detta arbete har olika randvilkor utvärderats när den två dimensionella Black-Scholes ekvationen löses med Finita Diﬀerens Metoder.

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CONTENTS

1 Introduction

A European call option is a financial contract that gives the holder the right but not the obligation to buy a particular asset for an agreed price K (the strike price) at a certain time T [7, p.28,44]. The asset is usually referred to as the underlying asset, and if the price of the underlying asset is higher than the strike price at time T , the owner of the option can buy the asset to the strike price and then immediately sell it to the higher market price, and thus make a profit. If however, the strike price is lower than the price of the underlying asset at time T , the option does not have to be exercised, which means that the owner of the option can suﬀer no other loss than the cost of the option itself. If the price of the underlying asset is denoted s, then the payoﬀ function for the option contract is given by

(s) =

(s K s K,

0 s < K. (1)

Because the ownership of an option gives an opportunity without a corresponding obligation, the issuer of the option will require a payment for the option. This payment is called the premium of the option. The issuer must determine what premium to demand for a certain option at a time t before the time of maturity T . The problem of determining option premiums are of great interest in the financial industry. A common approach for solving the problem which is frequently used by practitioners, is to solve the Black-Scholes model [3, p.216], which is given by

Ft(t, s) + rsFs(t, s) + s^{2 2}

2 Fss(t, s) rF (t, s) = 0, F (T, s) = (s).

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Here s is the market price of the underlying asset at time t, r is the risk free interest rate and is the volatility. The Black-Scholes model was developed by Fischer Black and Myron Scholes [2]. The partial diﬀerential equation (PDE) (2) can be solved with finite diﬀerence methods (FDM), but in order to do that, the model has to be bounded to a finite spatial domain. Because the price of an underlying asset cannot be less than zero, the spatial domain has a lower bound here called the close-field boundary, with the boundary condition

F (t, 0) = 0. (3)

There is no upper bound for the price of the underlying asset, but because the area of interest is usually close to the strike price, a rule of thumb is to set the upper limit of the computational domain to 4K multiplied with the number of spatial dimensions.

This boundary is called the far-field boundary. When using central finite diﬀerences to approximate the derivatives both the discretized points vi 1 and vi+1 are used to approximate the derivate for the point vi. Thus we have to treat the boundary points diﬀerently since the point vi 1 are outside the domain for the close-field, and

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2 THE 2D BLACK-SCHOLES MODEL

correspondingly vi+1 is outside of the domain for the far-field boundary. There are several methods to deal with the boundaries, and the aim of this paper is to examine how the accuracy of the solution is aﬀected by diﬀerent boundary condition to handle the boundaries when solving the Black-Scholes model with two spatial dimensions.

2 The 2D Black-Scholes Model

An option is not bound to have only one underlying asset, but can have many. In this paper options with two underlying assets are considered, which gives two spatial dimensions. The 2D Black-Scholes model is given by

Ft= rxFx ryFy

1

2x^{2 2}(1, 1)Fxx

1

2y^{2 2}_y(2, 2)Fyy xy ²(1, 2)Fxy + rF, F (T, x, y) = (x, y).

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Here x and y are the prices of the underlying assets at time t and is the correlation matrix for the volatility of the two underlying assets, and are given by

=

✓

xx yx

xy yy

◆ ,

xy = _yx,

xx = yy.

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The initial condition (x, y) is, as for the one-dimensional problem, the payoﬀ function for the option at time of maturity T , which in two dimensions is given by

(x, y) =

✓x + y

2 K

◆+

. (6)

We now introduce the operator L given by L = rx @

@x ry @

@y 1

2x^{2 2}(1, 1) @²

@x² 1

2y^{2 2}(2, 2) @²

@y² xy ²(1, 2) @²

@x@y + r. (7) The PDE (4) can then be written as

@F

@t =LF. (8)

The main part of this paper will hereinafter be about the discretization of L since that’s where the spatial domains boundaries are dealt with. The discretization in time will also be treated, but only briefly.

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3 Discretization in Space

3.1 Central Diﬀerences

The discretization of the operator L is conducted with central diﬀerences of second order accuracy on a equidistant grid. The 2D Black-Scholes model have a first and second order derivative with respect to x and y respectively and a mixed derivative.

This gives a total number of five diﬀerent derivatives. A representation of the grid can be seen in figure 1 below. The point vi,j approximates F (xi, y_j) and the gray and black points are the points needed for each approximation. The stars are the far-field boundary, the diamonds are the close-field boundary and the triangles are the corner points which belongs to both the far-field and close-field boundary.

0 1 2

. . . ...

i-1 j-1

i j

i+1 j+1

. . . ...

N-2 N-2

N-1 N-1

N N

Figure 1 : The black points are used for the approximation of the first and second order derivatives of x and y, while the gray points are used for approximating the mixed derivative. The far-field boundary is marked with stars and the close-field boundary is marked with diamonds, while the corner points which belongs to both the far-field and close-field boundary are marked with triangles.

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3 DISCRETIZATION IN SPACE

The central diﬀerences of second order accuracy is given by

✓@F

@x

◆

i,j

⇡ vi+1,j vi 1,j

2h_x ,

✓@²F

@x²

◆

i,j

⇡ vi+1,j 2vi,j+ vi 1,j

h²_x ,

✓@F

@y

◆

i,j

⇡ vi,j+1 vi,j 1

2h_y ,

✓@²F

@y²

◆

i,j

⇡ vi,j+1 2vi,j+ vi,j 1

h²_y ,

✓ @²F

@x@y

◆

i,j

⇡ vi+1,j+1 vi+1,j 1 vi 1,j+1+ vi 1,j 1

4h_xh_y .

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Here hx and hy are the step lengths in x and y direction. Given these approximations of the spatial derivatives, the semi-discrete 2D Black-Scholes model is given by

✓@F

@t

◆

i,j

= rxvi+1,j vi 1,j

2hx

ryvi,j+1 vi,j 1

2hy 2(1, 1)x²_i

2

vi+1,j 2vi,j+ vi 1,j

h²_x

2(2, 2)y²_j 2

vi,j+1 2vi,j + vi,j 1

h²_y

2(1, 2)xyv_i+1,j+1 v_{i+1,j 1} v_{i 1,j+1}+ v_{i 1,j 1} 4hxhy

+ rvi,j.

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This discretization in space is well-defined for all interior points in the domain but not on the boundaries, since the boundary points will have to use points outside of the domain with this discretization.

There are several diﬀerent boundary conditions to overcome this problem, and the ones that will be examined in this paper are the following

Boundary condition Boundary

One-sided diﬀerences Far-field

Linearity condition Far-field

Dirichlet boundary condition Far-field

One-sided diﬀerences Close-field

Linearity condition Close-field

No boundary condition Close-field

The three boundary conditions for the far-field will each be examined together with the three boundary conditions for the close-field, which gives nine diﬀerent cases.

The parameters and r will also be varied since they to great extent aﬀects the characteristics of the solution.

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3.2 Far-field Boundaries

What distinguishes the far-field boundaries from the close-field boundaries is that the price of the underlying assets has no absolute upper limit, and thus the far-field boundaries can be located at an arbitrary price of the underlying assets. The rule of thumb is to limit the domain by setting the upper limit to 4K times the number of spatial dimensions, since the stiﬀest part of the Black-Scholes model is close to the strike price. Using 8K or 12K should yield about the same result (but requiring more computational power due to the larger grids). In this paper 4K is used to restrict the computational domain. Below follows an explanation of the diﬀerent methods to handle the far-field boundary that will be examined.

3.2.1 Dirichlet Boundary Condition

Using a Dirichlet boundary conditions means that the values of the unknown is prescribed on the boundaries, eliminating the need for finite diﬀerence approximations in those points. For the Black-Scholes model the diﬀerence between the options price F (t, s) and the price of the underlying asset s becomes

slim!1

⇣F (t, s) s⌘

= Ke ^{r(T t)} (11)

when s approaches infinity [6, s.121]. Even though the far-field boundary cannot be located at infinity, it can be assumed that (11) is a good approximation when the price of the underlying asset is suﬃciently larger than K. By using this, the far-field boundary points can be approximated by

F (t, xmax, y)⇡ xmax+ y

2 Ke ^{r(T t)},

F (t, x, ymax)⇡ x + y_max

2 Ke ^{r(T t)}.

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The Dirichlet boundary condition can then be used on the far-field with values calculated by (12).

3.2.2 Linearity Boundary Condition

The linearity boundary condition is based on the assumption that F is approximately linear for large s. Based on (11) it can be concluded that F is linear when s ! 1, and thus F is approximately linear when s is large compared to K. This means, in terms of options, that the option price F is nearly linear with respect to the spot price s at the far-field boundaries [5]. If F is linear, then the second order derivative of F is equal to zero. Then, in one dimension

@²F (t, s)

@s² = 0, vn+1 2vn+ vn 1

h² = 0, vn+1 = 2vn vn 1. (13) The problem with using the original semi-discretization (10) on the far-field is that the

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3 DISCRETIZATION IN SPACE

points vi,j+1, vi+1,j and vi+1,j+1 lies outside of the computational domain when i = N and/or j = N where N is the size of the grid. However, according to (13), these points can be substituted by

v_i,j+1= 2v_i,j v_{i,j 1}, vi+1,j = 2vi,j vi 1,j,

vi+1,j+1 = 4vi,j 2vi,j 1 2vi 1,j + vi 1,j 1.

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Consequently, the central diﬀerences at the far-field boundary can be rewritten as

✓@F

@x

◆

N,j

⇡ vN,j vN 1,j

hx

,

✓@²F

@x²

◆

N,j

⇡ 0,

✓@F

@y

◆

i,N

⇡ vi,N vi,N 1

hy

,

✓@²F

@y²

◆

i,N

⇡ 0,

✓ @²F

@x@y

◆

N,j

⇡ vN,j+1 vN,j 1 vN 1,j+1+ vN 1,j 1

2hxhy

,

✓ @²F

@x@y

◆

i,N

⇡ vi+1,N vi+1,N 1 vi 1,N + vi 1,N 1

2hxhy

.

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By combining these diﬀerences with the diﬀerences (9), all points at the far-field boundary can be computed.

3.2.3 One-sided Diﬀerence Boundary Condition

The problem with central differences on the boundaries is that points from both sides of the point being approximated are needed. This problem can be circumvented by using one-sided differences. One-sided differences only uses points from one side of the approximated point, and can thus be used on the boundaries. The one-sided differences are simply a method for handling boundaries and has no physical interpretation. They are given by

✓@F

@x

◆

N,j

⇡ vN,j vN 1,j

h_x ,

✓@²F

@x²

◆

N,j

⇡ vN,j 2vN 1,j+ vN 2,j

h²_x ,

✓@F

@y

◆

i,N

⇡ vi,N vi,N 1

h_y ,

✓@²F

@y²

◆

i,N

⇡ vi,N 2vi,N 1+ vi,N 2

h²_y ,

✓ @²F

@x@y

◆

N,j

⇡ vN,j+1 vN,j 1 vN 1,j+1+ vN 1,j 1

2h_xh_y ,

✓ @²F

@x@y

◆

i,N

⇡ vi+1,N vi+1,N 1 vi 1,N + vi 1,N 1

2hxhy

,

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(11)

and have one less order of accuracy compared to the corresponding central diﬀerences.

This is however not an issue since the far-field boundary is just a fraction of the computational domain as a whole [1]. These one-sided diﬀerences can together with the diﬀerences in (9) be used to compute all the far-field points.

3.3 Close-field Boundaries

The points in the domain closest to the x and y axes form the close-field boundary.

The main diﬀerence between the close-field and the far-field boundary is that equation (11) does not hold for the close-field boundary.

3.3.1 No Boundary Condition

At the origin, which is part of the close-field boundary, the 2D Black-Scholes model (4) becomes

Ft(t, 0, 0) = rF (t, 0, 0),

Ft(0, 0, 0) = 0, (17)

since all terms containing either the variable x or y are zero. The solution to Equation (17) yields that

F (t, 0, 0) = 0. (18)

Along the x-axis, the variable y is zero, and therefore all the terms containing the variable y vanish along the x-axis. The equation along the x-axis therefore becomes

Ft= rxFx

1

2x^{2 2}(1, 1)Fxx+ rF. (19)

In the same manner, the equation along the y-axis becomes Ft= ryFy

1

2y^{2 2}(2, 2)Fyy+ rF. (20)

Since the central diﬀerences only uses points in the same direction as the derivative, the semi-discrete equivalents to equation (19) and (20) will only use points in the same direction as the axes. Therefore, the central diﬀerences (9) can be used without modification since no points outside of the domain will be present.

As described above, if the close-field is set on the border of the domain, along an axis where the spot price of the underlying assets is zero, the 2D Black-Scholes equation becomes one-dimensional along the close-field boundary and only points within the domain are needed. However, in some cases it might be of interest to use a domain

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4 DISCRETIZATION IN TIME

that does not contain the borders, and thus the boundaries have to be dealt with.

The Dirichlet boundary condition cannot be used on the close-field boundary since it is close to the strike price. However, both the linearity condition and one-sided diﬀerences can be used. If the linearity condition is applied in the same way as for the far-field, then the diﬀerences one-point in form the close-field boundary is given by

✓@F

@x

◆

1,j

⇡ v1,j v2,j

hx

,

✓@²F

@x²

◆

1,j

⇡ 0,

✓@F

@y

◆

i,1

⇡ vi,1 vi,2

hy

,

✓@²F

@y²

◆

i,1

⇡ 0,

✓ @²F

@x@y

◆

1,j

⇡ v1,j+1 v1,j 1 v2,j+1+ v2,j 1

2hxhy

,

✓ @²F

@x@y

◆

i,1

⇡ vi+1,1 vi+1,2 vi 1,1+ vi 1,2

2hxhy

.

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3.3.3 One-sided Boundary Condition

The one-sided diﬀerences can be used one-point in from the close-field boundary in the same manner as for the far-field boundary. The one-sided diﬀerences one-point in from the close-field are given by

✓@F

@x

◆

1,j

⇡ v1,j v2,j

h_x ,

✓@²F

@x²

◆

1,j

⇡ v1,j 2v2,j + v3,j

h²_x ,

✓@F

@y

◆

i,1

⇡ vi,1 vi,2

h_y ,

✓@²F

@y²

◆

i,1

⇡ vi,1 2vi,2+ vi,3

h²_y ,

✓ @²F

@x@y

◆

1,j

⇡ v1,j+1 v1,j 1 v2,j+1+ v2,j 1

2h_xh_y ,

✓ @²F

@x@y

◆

1,j

⇡ v1,j+1 v1,j 1 v2,j+1+ v2,j 1

2h_xh_y .

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4 Discretization in Time

When implementing the boundary condition in the discretized system, one can either alter the diﬀerence matrix A or add a factor b to the equation. The general form is therefore vt= Av + b where either A is altered and b is zero (e.g. when using linearity boundary conditions), or A is unaltered and b is set to some value, which might depend on time (e.g. when using Dirichlet boundary conditions).

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The time is discretized using a backward diﬀerentiation formula with second order of accuracy (BDF-2). When using BDF-2 the step length in time does only aﬀect the accuracy of the solution, not the stability. The BDF-2 for this problem is derived as follows:

Writing vⁿ_t using Taylor series and backward diﬀerences for the first and second time derivatives as

vⁿ_t = vⁿ v^{n 1}

k + k

2

vⁿ 2v^{n 1}+ v^{n 2}

k² + O(k²). (23)

The problem can then be written with second order accuracy in time as

vⁿ_t ⇡ vⁿ v^{n 1}

k +k

2

vⁿ 2v^{n 1}+ v^{n 2}

k² = Avⁿ+ b. (24)

Solving for vⁿ then yields P vⁿ = 4

3v^{n 1} 1

3v^{n 2}+ 2k

3 b where P = (1 2kA

3 ). (25)

In order to use BDF-2 it is not enough to only use the initial value since it needs two previous time steps. Therefore, the second time step is approximated with backward euler (BDF-1). The variable b is used for boundary conditions where it cannot be included in the operator e.g. for Dirichlet boundary conditions.

5 Numerical Results

Each boundary setup was examined for three diﬀerent problems with diﬀerent parameter setups (see table 5). As gets smaller compared to r, the problem gets more convection characteristics and thus more error prone. Below follows a summary of the parameter setups.

Parameter setup Problem type

r = 0.1, (1, 1) = 0.3, (1, 2) = 0.05 Problem 1 r = 0.01, (1, 1) = 0.3, (1, 2) = 0.05 Problem 2 r = 0.1, (1, 1) = 0.03, (1, 2) = 0.005 Problem 3

Since there is no analytical solution to the 2D Black-Scholes model, the evaluations of the test cases was carried out by comparing the results with a reference solution computed on a very fine mesh. The reference solution used the Dirichlet boundary condition at the far-field boundary and no boundary condition on the close-field

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5 NUMERICAL RESULTS

boundary since these methods are known to work [4]. Three reference solutions was computed, one for each problem type. The most important part of the domain when analysing the result are the area close to the strike price. Therefore the error has been measured in the square around the strike price defined by

8>

><

>>

: K

3  x  5K 3 , K

3  y  5K 3 ,

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which will hereinafter be referred to as the closeup region. The error plots that shows the error in the closeup region will be referred to as closeup error plots, and the error plots that shows the entire computed domain will be referred to as global error plots.

The numerical result for problem 1 and problem 2 are similar and are therefore presented together, while the result from the more error prone problem 3 deviates from the other two and will therefore be presented separately. The closeup and global error plots are presented in order to show how the error behaves both in the closeup region and along the boundaries. In order to compare how well the diﬀerent boundary conditions perform compared to each other, the summed error in the closeup region was used for the evaluation.

5.1 Problem 1 and Problem 2

5.1.1 Dirichlet and No Boundary Condition

Using the Dirichlet boundary condition on the far-field did not introduce any error along the boundary. Since the Dirichlet boundary condition approximates the far-field using analytical values calculated for x, y ! 1, the far-field will be identical with it’s counterpart in the reference solution. When using no boundary condition on the close-field the error will also be smaller than for the linearity condition and one-sided diﬀerences since the reference solution uses no boundary condition.

Since using the Dirichlet boundary condition on the far-field and no boundary condition on the close-field does not introduce large errors along the boundaries, the closeup error plot when imposing those conditions can be used as a comparison to the other closeup error plots. For problem 2 and problem 3, the closeup error plots for all other boundary condition setups showed the same characteristics as Figure 2, and will thus not be presented explicitly. This also implies that the error behaves as expected in the closeup region for all boundary condition setups imposed on problem 1 and problem 2.

The global error plots will be presented in order to show how the diﬀerent boundary conditions aﬀects the error along the boundaries, and the final evaluation of the boundary conditions will be carried out by comparing the error in

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the closeup region for diﬀerent number of grid points.

4 6 8 10 12 14 16

x-axis

y-axis

2· 10 ⁴ 4· 10 ⁴ 6· 10 ⁴ 8· 10 ⁴ 1· 10 ³

Error

Figure 2 : Closeup error plot with the far-field Dirichlet boundary condition and no boundary on the close-field.

Because the linearity boundary condition is based upon the assumption that the problem is linear on the boundaries, it can be assumed to give a better result for the far-field boundary than for the close-field boundary. Since the global error plots for problem 1 and problem 2 exhibited the same characteristics, only one plot will be presented for each boundary condition setup.

From Figure 3 it can be seen that the linearity condition applied to the close-field boundaries have the greatest error between 20 and 30 on both the x-axis and y-axis. The large error on the close-field boundaries does not spread into the closeup region to any great extent, and the corresponding closeup plot to Figure 3 exhibits the same characteristics as Figure 2. From Figure 4 it can be observed that the linearity condition imposed on the far-field boundaries yields the largest errors where the far-field boundary meets the close-field boundary. By comparing Figure 3 and Figure 4 it can be concluded that the linearity condition on the close-field boundary results in a large error close to and above the strike price, while the linearity condition imposed to the far-field yields a large error in the corner where the far-field meets the close-field.

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5 NUMERICAL RESULTS

5 10 15 20 25 30 35 40

10 20 30 40

5 10 15 20 25 30 35 40

10 20 30 40

x-axis

y-axis

0 2· 10 ² 4· 10 ² 6· 10 ² 8· 10 ² 0.1 0.12 0.14 0.16

Error

Figure 3 : Global error with linearity condition on the close-field and the Dirichlet boundary condition on the far-field for problem 1 and problem 2.

0 5 10 15 20 25 30 35 40

0 10 20 30 40

0 5 10 15 20 25 30 35 40

0 10 20 30 40

x-axis

y-axis

0 2· 10 ² 4· 10 ² 6· 10 ² 8· 10 ² 0.1 0.12 0.14 0.16

Error

Figure 4 : Global error with linearity condition on the far-field and no boundary condition on the close-field for problem 1 and problem 2.

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5 10 15 20 25 30 35 40 10

20 30 40

5 10 15 20 25 30 35 40

10 20 30 40

x-axis

y-axis

0 2· 10 ² 4· 10 ² 6· 10 ² 8· 10 ² 0.1 0.12 0.14 0.16

Error

Figure 5 : Global error with linearity condition on both the far-field and the close-field boundary for problem 1 and problem 2.

When the linearity condition is applied to both the far-field and the close-field boundary, the errors from the corners where the far-field boundary meets the close-field boundary magnifies due to the errors on the close-field boundary, as can be seen from Figure 5. However, the errors only propagate along the upper part of the close-field boundary. The result in Figure 3 can be explained by the linearity condition being a bad approximation close to the strike price at the close-field boundaries, and the result in Figure 4 may be due to the corner points being hard to approximate, resulting in a propagating error that smooths out futher away from the boundaries.

5.1.3 One-sided Diﬀerences

The one-sided differences do not assume the solution to be linear along the boundaries as for the linearity condition. However, both the one-sided differences and linearity condition only provides differences with first order accuracy. Using first order accuracy approximations to some extent does however not ruin the stability.

As can be observed in Figure 6, 7 and 8, the error on the boundaries using one-sided differences behaves in the same way as for the linearity condition. The errors for the one-sided differences are however of greater magnitude and more spread out. However, this difference between the one-sided differences and linearity condition might still be of considerable interest since the smaller and less spread out errors when using the linearity condition allows smaller computational domain without having a negative impact on the area of interest.

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5 NUMERICAL RESULTS

0 5 10 15 20 25 30 35 40

0 10 20 30 40

0 5 10 15 20 25 30 35 40

0 10 20 30 40

x-axis

y-axis

0 2· 10 ² 4· 10 ² 6· 10 ² 8· 10 ² 0.1 0.12 0.14 0.16

Error

Figure 6 : Global error with no boundary conditions on the close-field and one-sided diﬀerences on the far-field for problem 1 and problem 2.

5 10 15 20 25 30 35 40

10 20 30 40

5 10 15 20 25 30 35 40

10 20 30 40

x-axis

y-axis

0 2· 10 ² 4· 10 ² 6· 10 ² 8· 10 ² 0.1 0.12 0.14 0.16

Error

Figure 7 : Global error with one-sided diﬀerences on the close-field and the Dirichlet boundary condition on the far-field for problem 1 and problem 2.

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5 10 15 20 25 30 35 40 10

20 30 40

5 10 15 20 25 30 35 40

10 20 30 40

x-axis

y-axis

0 2· 10 ² 4· 10 ² 6· 10 ² 8· 10 ² 0.1 0.12 0.14 0.16

Error

Figure 8 : Global error with one-sided diﬀerences on both the close-field and the far-field for problem 1 and problem 2.

5.1.4 Linearity and one-sided diﬀerences boundary conditions

5 10 15 20 25 30 35 40

10 20 30 40

5 10 15 20 25 30 35 40

10 20 30 40

x-axis

y-axis

0 2· 10 ² 4· 10 ² 6· 10 ² 8· 10 ² 0.1 0.12 0.14 0.16

Error

Figure 9 : Global error with linearity condition on the close-field and one-sided diﬀerences on the far-field for problem 1 and problem 2.

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5 NUMERICAL RESULTS

5 10 15 20 25 30 35 40

10 20 30 40

5 10 15 20 25 30 35 40

10 20 30 40

x-axis

y-axis

0 2· 10 ² 4· 10 ² 6· 10 ² 8· 10 ² 0.1 0.12 0.14 0.16

Error

Figure 10 : Global error with one-sided diﬀerences on the close-field and the linearity boundary condition on the far-field for problem 1 and problem 2.

In Figure 9 large errors are visible in the corners and on the close-field. The errors in the corners are not visible in Figure 10. This might be due to that the linearity condition is better used on the far-field than on the close-field. The corresponding closeup plot looks like Figure 2

5.2 Problem 3

For a problem with a comparably small volatility, errors are not so easily damped out which makes those problems more error prone, and thus the larger error is what distinguishes problem 3 from problem 1 and problem 2. The most critical area is where the solution and it’s derivatives are changing most rapidly, which for the Black-Scholes model is close to the strike price.

As can be seen in Figures 11 and Figure 12, there are an error, as expected, parallel to the strike price. All combinations of close-fields and far-fields results in the same type of figures which indicates that changing boundary conditions will not aﬀect the solution. The reason why all the global error plots looks the same is probably because the error propagates with so little damping along the upper part (above the strike price) of the close-field boundary so that it does not matter if the linearity condition or the one sided diﬀerences are used as boundary condition.

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5 10 15 20 25 30 35 40 10

20 30 40

5 10 15 20 25 30 35 40

10 20 30 40

x-axis

y-axis

0 2· 10 ² 4· 10 ² 6· 10 ² 8· 10 ² 0.1 0.12 0.14 0.16

Error

Figure 11 : Global error for all combinations of far-field and close-field for problem 3

4 6 8 10 12 14 16

x-axis

y-axis

2· 10 ⁴ 4· 10 ⁴ 6· 10 ⁴ 8· 10 ⁴ 1· 10 ³

Error

Figure 12 : Closeup error for all combinations of far-field and close-field for problem 3

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5 NUMERICAL RESULTS

5.3 Convergence

The error for each solution was measured by

e = vu ut 1

N XN

i=1

(di ri)² (27)

where N is the number of points in the solution, di is a point in the problem solution and ri is the corresponding point in the reference solution. The points used to calculate e are only taken from the closeup region in each solution.

10^1.4 10^1.6 10^1.8 10² 10^2.2 10^2.4 10 ⁴

10 ³ 10 ²

Number of grid points per dimension

Error

CF: no boundary condition; FF: All CF: Linearity, One-sided; FF: All

Figure 13 : The error plotted against the number of grid points per dimension for problem 1.

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10^1.4 10^1.6 10^1.8 10² 10^2.2 10^2.4 10 ⁴

10 ³ 10 ²

Error

CF: Linearity; FF: All CF: no boundary condition, One-sided; FF: All

10^1.4 10^1.6 10^1.8 10² 10^2.2 10^2.4 10 ³

10 ² 10 ¹

Error

CF: All; FF: All

The accuracy is of second order in space for all solutions as it can be seen from

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6 CONCLUSIONS

the slope of the lines in figure 13, 14 and 15. This was expected since the FDM scheme used was of second order in space. It is interesting that the error lines are so close to each other for all combinations of boundary conditions. This leads to the conclusion that the absolute error, and the rate of error convergence is not particularly aﬀected by which boundary condition was used, but rather on the characteristics of the problem itself.

It is clear that the error converges for all problems as the number of grid points increases. Problem 3 has a much larger error than the other problems which probably is due to the characteristics of the equation and could therefor not be resolved by using other boundary conditions. The error distribution can be seen in Figure 12. The large error for small N in Figure 13 when using one-sided diﬀerences or linearity condition might be due to that the closeup region includes points which are directly neighbours to the close-field, which therefor can be aﬀected by the large error on the boundary. However, using very few nodes is never a good idea if a solution with high accuracy is needed, therefor the increased error for small N is not considered a problem for in real life applications since such small N would not be used.

6 Conclusions

The main conclusion drawn from this investigation of boundary conditions for the 2D Black-Scholes PDE is that all variants tested gave a reasonably good result. In fact the error in the most important part of the domain, i.e. close to the strike price K, was not affected by the boundary conditions used but rather of the characteristics of the problem itself. However, there are some advantages and disadvantages among the different methods. The far-field boundary was in general less error prone than the close-field boundary, and in the closeup region there were no significant difference in the magnitude of the error depending on the condition used on the far-field boundary. By inspection of the error plots it could however be seen that both the one-sided differences and linearity condition resulted in large errors in the corners of the computational domain where the far-field boundary meets the close-field boundary. The error for the one-sided differences were somewhat larger than the errors using the linearity condition. This is of interest if the computational domain for some reason have to be reduced and the distance between the boundaries and the area of interest therefore are smaller. The Dirichlet boundary condition is therefore the best alternative, followed by the linearity condition, to handle the far-field boundary.

On the close-field boundary, the method that gave the overall best results were using no boundary conditions. The diﬀerence in convergence between this method and the alternatives where mainly very small. However, in Figure 13 it can be seen that when the number of grid points are small, using no boundary condition gives a smaller error. Apart from this, by inspection of the error plots it is seen that the unaltered boundaries have the smallest errors on the boundaries as well.

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However, the strategy of using no boundary conditions can only be used where the x-axis and y-axis respectively are zero. Therefore the comparison between one-sided diﬀerences and linearity condition one point in from the boundary is of great interest. From an error convergence perspective, the methods are equally good. But just as for the far-field boundary, the linearity condition gives less error on the boundaries, and are therefore preferable over the one-sided diﬀerences. The recommendation that can be made based on this experimental study is therefore to use the Dirichlet boundary condition on the far-field boundary if possible, and no boundary condition on the close-field if possible. If any of these for some reason are not applicable, the linearity condition should be used.

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REFERENCES

References

[1] Heinz-Otto Kreiss Bertil Gustafsson and Arne Sundström. Stability theory of diﬀerence approximations for mixed initial boundary value problems. ii.

Mathematics of Computation Vol. 26, No. 119 (Jul., 1972), pp. 649-686.

[2] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities.

Journal of Political Economy, 81(3):pp. 637–654, 1973.

[3] Zvi Bodie, Robert C. Merton, and David L. Cleeton. Financial Economics.

Pearson Education, Inc., Upper Saddle River, New Jersey, 07458., second edition edition, 2006.

[4] Slobodan Milovanovic. Personal communication.

[5] Jonas Persson and Lina von Sydow. Pricing European multi-asset options using a space-time adaptive fd-method.

[6] Domingo Tavella and Curt Randall. Pricing Financial Instruments the Finite Diﬀerence Method. John Wiley & Sons, Ltd, 2000.

[7] Paul Wilmott. Paul Wilmott Introduces Quantitative Finance. John Wiley & Sons, Ltd, second edition edition, 2007.

Examination of Impact from Different Boundary Conditions on the 2D Black-Scholes Model

Examination of Impact from Different

Boundary Conditions on the 2D

Black-Scholes Model

Evaluating Pricing of European Call Options

Tomas Sundvall

David Trång

Abstract

Examination of Impact from Different Boundary

Conditions on the 2D Black-Scholes Model

Contents

1 Introduction

2 The 2D Black-Scholes Model

3 Discretization in Space

3.1 Central Diﬀerences

3.2 Far-field Boundaries

3.3 Close-field Boundaries

4 Discretization in Time

5 Numerical Results

5.1 Problem 1 and Problem 2

5.2 Problem 3

5.3 Convergence

6 Conclusions

References