numerical methods for nonlinear partial differential...

Numerical methods for nonlinear partial differential equationsand inequalities arising from option valuation undertransaction costsLesmana, D. (2014). Numerical methods for nonlinear partial differential equations and inequalities arisingfrom option valuation under transaction costs

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Numerical Methods for Nonlinear

Partial Differential Equations and

Inequalities Arising from Option

Valuation under Transaction Costs

Donny Citra Lesmana

This thesis is presented for the degree of

Doctor of Philosophy

of The University of Western Australia

Department of Mathematics & Statistics

June 2014

“An approximate answer to the right problem is worth a good deal more than an

exact answer to an approximate problem.”

John Tukey

Abstract

This thesis develops the numerical methods and their mathematical analysis for

solving nonlinear partial and integral-partial differential equations and inequalities

arising from the valuation of European and American option with transaction costs.

The models can hardly be solvable analytically. Therefore, in practice, approximate

solutions to such a model are always sought. In this thesis, we discuss two models

for the asset price movements: the geometric Brownian motion and jump diffusion

process. For the valuation of European options with transaction costs when the un-

derlying asset price follows a geometric Brownian motion, the classical Black-Scholes

model becomes a nonlinear partial differential equation. To approximately solve this,

we use an upwind finite difference scheme for the spatial discretization and a fully

implicit time-stepping scheme. We prove that the system matrix from this scheme

is an M -matrix and that the approximate solution converges unconditionally to the

exact one by proving that the scheme is consistent, monotone and unconditionally

stable. The discretized nonlinear system is then solved using a Newton iterative

algorithm.

For the valuation of American options with transaction costs when the underlying

asset follows geometric Brownian motion, we propose a power penalty method for

a finite-dimensional Nonlinear Complementarity Problem (NCP) arising from the

discretization of the continuous American option pricing model. We show that the

mapping involved in the system is continuous and strongly monotone. Thus, the

unique solvability of both the NCP and the penalty equation and the exponential

convergence of the solution to the penalty equation to that of the NCP are guaran-

teed by an existing theory.

In the presence of transaction costs and when the underlying asset price follows a

jump diffusion process, the problem becomes a nonlinear partial integro-differential

equation (PIDE). Since exact solutions can hardly be found, numerical approxima-

tions to the nonlinear PIDE are always sought. This is challenging as the PIDE

involves a nonlocal integration term. The method we propose is based on an upwind

finite difference scheme for the spatial discretization and a fully implicit time step-

ping scheme. The fully discretized system is solved by a Newton iterative method

coupled with a Fast Fourier Transform (FFT) for the computation of the discretized

integral term. The constraint in the American option model is imposed by adding

a penalty term to the original partial integro-differential complementarity problem.

We also perform some numerical experiments to illustrate the usefulness and accu-

racy of the method.

Acknowledgements

First and foremost, I would like to express my sincere gratitude to my supervisor

Prof. Song Wang for his continuous support of my Ph.D study and research, for

his patience, motivation, enthusiasm, and immense knowledge. His guidance helped

me in all the time of research and writing of this thesis. I could not have imagined

having a better supervisor for my Ph.D study.

My sincere thanks also go to all members of the School of Mathematics and Statistics

of the University of Western Australia. In one way or another and at different times

during my study, they have contributed towards the completion of my studies.

From a financial point of view, I would like to express my profound gratitude to

Government of Indonesia, Directorate General of Higher Education, for granting me

a full scholarship for my Ph.D study.

Finally, I am deeply thankful and extremely grateful to Allah, who made all the

things possible.

vi

Contents

Abstract iv

Acknowledgements vi

List of Figures ix

List of Tables xi

Abbreviations xiii

Symbols xv

1 Introduction 1

1.1 Formulation of the Mathematical Model . . . . . . . . . . . . . . . . 5

1.1.1 Nonlinear Black-Scholes Equation . . . . . . . . . . . . . . . . 5

1.1.2 Jump Diffusion Model . . . . . . . . . . . . . . . . . . . . . . 9

1.1.3 Viscosity Solution . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.1 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.2 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.4 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.5 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 15

2.1 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 The continuous problem . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Convergence of the numerical scheme . . . . . . . . . . . . . . . . . . 25

2.6 Solution of the nonlinear system (2.24) . . . . . . . . . . . . . . . . . 30

vii

Contents

2.7 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 A Penalty Approach for American Put Option Valuation UnderTransaction Costs 45

3.1 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 The discretized problem and penalty formulation . . . . . . . . . . . 48

3.4 Convergence of the penalty method . . . . . . . . . . . . . . . . . . . 53

3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Numerical scheme for pricing option with transaction costs underjump diffusion processes 69

4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 The continuous model . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Discretization of the PIDE . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4.1 Discretization of the integral . . . . . . . . . . . . . . . . . . . 75

4.4.2 Full discretization . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5 Convergence of the numerical scheme . . . . . . . . . . . . . . . . . . 82

4.6 Solution of the nonlinear system . . . . . . . . . . . . . . . . . . . . . 86

4.6.1 The European Case . . . . . . . . . . . . . . . . . . . . . . . . 86

4.6.2 The American case . . . . . . . . . . . . . . . . . . . . . . . . 93

4.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Conclusion 101

A Proof of the monotonicity of σ2(S, z)z 103

Bibliography 105

List of Figures

1.1 Payoff for Call Option with K = 40 at t = T . . . . . . . . . . . . . . 3

1.2 Payoff for Put Option with K = 40 at t = T . . . . . . . . . . . . . . 3

1.3 Payoff for Butterfly Spread Option with K1 = 20, k2 = 40, andK3 = 60 at t = T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Payoff for Cash-or-Nothing Option with K = 40 and B = 1 at t = T . 4

2.1 Price of the European call option with a = 0.05. . . . . . . . . . . . 34

2.2 Call Option Prices of Barles-Soner and HWW short position Models. 35

2.3 Prices of the European call option for different transaction costs. . . . 35

2.4 The call option prices for different values of σ0 . . . . . . . . . . . . . 36

2.5 Price of the European put option. . . . . . . . . . . . . . . . . . . . . 37

2.6 The Comparison of Put Option Price Between Barles-Soner Modeland HWW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7 Prices of the European put option for different transaction costs. . . . 38

2.8 The put option price for low and high volatility . . . . . . . . . . . . 39

2.9 Price of the Butterfly Spread Option. . . . . . . . . . . . . . . . . . . 40

2.10 The Comparison of Butterfly Spread Option Price Between Barles-Soner Model and HWW Model . . . . . . . . . . . . . . . . . . . . . 40

2.11 Prices of the butterfly spread option for different transaction costs. . 41

2.12 The butterfly spread option price for low and high volatility . . . . . 41

2.13 Price of the Cash or Nothing option. . . . . . . . . . . . . . . . . . . 42

2.14 The Comparison of Cash or Nothing Option Price Between Barles-Soner Model and HWW Model . . . . . . . . . . . . . . . . . . . . . 42

2.15 Prices of the Cash or Nothing option for different transaction costs. . 43

2.16 The cash or nothing option price for low and high volatility . . . . . . 43

3.1 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 V − V ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Prices of the American and European put options with a = 0.02 att = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Prices of the American and European put option with a = 0.02. . . . 67

4.1 Price of the European call option with a = 0.01 and b = 0.07. . . . . 94

4.2 Price of the European put option with a = 0.01 and b = 0.07. . . . . 95

ix

List of Figures

4.3 Price of the European call option for different transaction cost pa-rameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4 Price of the European put option for different transaction cost pa-rameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.5 American and European Put Option Price under Jump Diffusion Pro-cess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.6 American Put Option Price for Different Transaction Cost Parameterunder Jump Diffusion Process . . . . . . . . . . . . . . . . . . . . . . 98

List of Tables

2.1 Computed rates of convergence for the call option with a = 0.02 . . . 37

2.2 Computed results for put options with a = 0.02 . . . . . . . . . . . . 39

3.1 Computed rates of convergence in ϑ when k = 1 and a = 0.02 . . . . 64

3.2 Computed rates of convergence in ϑ when k = 2, 3 and a = 0.02 . . . 64

3.3 Computed rates of convergence in k when ϑ = 20 . . . . . . . . . . . 64

4.1 Computed rates of convergence for the call option with a = 0.01 andb = 0.07 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Computed rates of convergence for the put option with a = 0.01 andb = 0.07 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

xi

Abbreviations

BSE Black-Scholes Equation

CoN Cash or Nothing

FFT Fast Fourier Transform

HWW Hoggard Whalley Wilmott

LHS Left Hand Side

NCP Nonlinear Complementarity Problem

PDE Partial Differential Equation

PIDE Partial Integro Differential Equation

RHS Right Hand Side

xiii

Symbols

Vt partial derivatives of V with respect to time t

VS partial derivatives of V with respect to space variable S

| · | absolute value

‖ · ‖∞ l∞-norm

‖ · ‖2 l2-norm

xv

Dedicated to my lovely wife, Sukma Dini Miradani, for

her love and support. And to our newest addition to

Lesmana’s family: Zahra Alisha Lesmana.

xvii

Chapter 1

Introduction

Financial options are derivative instruments which can usually be traded in a se-

condary financial market. An option on a stock is a contract which gives its holder

the right, not obligation, to sell (put option) or buy (call option) a certain number

of the shares at a prescribed price. The time when the contract ends is known as the

expiry date or the maturity date of the option. The price at which the asset may be

purchased or sold is called the strike/exercise price. There are two major types of

option: European option and American option. The former can only be exercised

on the expiry date while the latter can be exercised on or before the expiry date.

The options of European type are also known as plain vanilla options.

Since an option is tradable on a financial market, a natural question is how to

determine the value of the option at any time before the expiry date. The fast

development in option valuation theory started with the publication of two seminal

papers by Black and Scholes [1] and Merton [2] respectively. In [1], the authors

introduced a continuous time model for pricing options on a stock in a complete

friction-free market of which the price follows a geometric Brownian motion. This

model is now known as the Black-Scholes model in which the value of an option

is equal to the value of a self-financing replicating portfolio comprising a risk-less

security and a risky stock and can be determined by a partial differential equation

of the form

1

2 Chapter 1

Vt +1

2σ2

0S2VSS + rSVS − rV = 0 (1.1)

for (S, t) ∈ (0,∞)×(0, T ] with a set of appropriate boundary and terminal conditions

depending on the type of an option, where V denotes the option value, S denotes

the underlying stock price, t is time, T > 0 is the expiry time (or maturity) of the

option, r ≥ 0 denotes a constant riskless interest rate and σ0 is a constant volatility.

The derivation of the closed form solution to the Black-Scholes equation is straight

forward and can be found in many literatures in finance and economics, for example

in [1, 3].

The most common types of plain vanilla options are call option, put option, butterfly

spread option, and cash-or-nothing (CoN) option. The payoff for the plain vanilla

option is given in this following function:

payoff =

max(S −K, 0) for a call,

max(K − S, 0) for a put,

max(S −K1, 0)− 2 max(S −K2, 0) + max(S −K3, 0) for butterfly,

B ×H(S −K) for a CoN,

where K,K1, K2 and K3 denote the strike prices of the options, B is a constant

and H is the Heaviside function. Typical payoff graphs for a call, a put, a butterfly

spread, and a cash-or-nothing option are given in Figures 1.1–1.4.

Black-Scholes model is very effective for pricing options in a complete market without

costs on transactions of risky and riskless securities. However, when trading in

the bond or/and stock involves transaction cost, the Black-Scholes option pricing

model does not hold anymore. To overcome this difficulty, various models have been

proposed to price European options under transaction costs [4–8]. All these models

give rise to a nonlinear Black-Scholes equation. There are also utility-maximization

based models for determining the so-called reservation prices of European and Ame-

rican options under transaction costs [9–12]. These models are of the form of a set

Chapter 1. Introduction 3

0 10 20 30 40 50 60 70 80−5

0

5

10

15

20

25

30

35

40

Stock Price

Pay

off

Payoff for call option with exercise price K = 40 at t = T

Figure 1.1: Payoff for Call Option with K = 40 at t = T .

0 10 20 30 40 50 60 70 80−5

0

5

10

15

20

25

30

35

40

Stock Price

Pay

off

Payoff for put option with exercise price K = 40 at t = T

Figure 1.2: Payoff for Put Option with K = 40 at t = T .

of Hamilton-Jacobi-Bellman equations for both European and American options.

In the presence of transaction costs, the classical Black-Scholes equation becomes

the following nonlinear Black-Scholes equation [13]:

Vt +1

2σ2(t, S, VS, VSS)S2VSS + rSVS − rV = 0, (1.2)

where σ is the modified volatility as a function of t, S, VS, and VSS.

Black and Scholes [1] assumed that the underlying asset price follows a geometric

4 Chapter 1

0 10 20 30 40 50 60 70 80−2

0

2

4

6

8

10

12

14

16

18

20

Stock Price

Pay

off

Payoff for butterfly spread with K1 = 20, K2 = 40, and K3 = 60, at t = T

Figure 1.3: Payoff for Butterfly Spread Option with K1 = 20, k2 = 40, andK3 = 60 at t = T .

0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1

Stock Price

Pay

off

Payoff for cash−or−nothing option with K = 40 and B = 1 at t = T

Figure 1.4: Payoff for Cash-or-Nothing Option with K = 40 and B = 1 att = T .

Brownian motion with constant volatility which implies a continuous market evolu-

tion. However, from the empirical study on financial data, it is clear that the price

process can jump (see for example [14]). More general models for the stochastic

dynamics have been proposed to handle this problem. The most recent ones are

stochastic volatility models ([15–17]), jump diffusion models ([18, 19]), and general

singular Levy model ([20]). For simplicity, we will focus on the jump diffusion model

in this thesis.


1.1 Formulation of the Mathematical Model

1.1.1 Nonlinear Black-Scholes Equation

To derive the nonlinear Black-Scholes model, we first give the property of Wiener

process. A Wiener process, W , is generally characterized by the following properties:

1. W (0) = 0,

2. for t ≤ s ≤ f , W (s)−W (t) and W (f)−W (s) are independent,

3. if t < s, then W (s)−W (t) ∈ N (0,√s− t), where N is a normal distribution,

4. W has continuous trajectories.

Assume that the financial market consists of one money market and one stock whose

price is governed by the dynamics

dS = µS dt+ σ0S dW, (1.3)

dB = rB dt, (1.4)

where µ is the drift rate, W is a standard Brownian motion and B is the bond price.

Let us consider a hedging portfolio that has x stocks and y bonds. In the case of

continuous time, the value of the option at any time t for all t ≤ T is

x(t)S(t) + y(t)B(t),

where S(t) is the stock price at time t and B(t) is the bond price at time t. If we

denote the value of the option at time t by V (t, S(t)), then we are seeking

V (t, S(t)) = x(t)S(t) + y(t)B(t). (1.5)

6 Chapter 1

Consequently, both sides should have the same dynamics:

dV (t, S(t)) = d(x(t)S(t) + y(t)B(t)

),

where d is an infinitesimal change. For a self-financing portfolio, we have

d(x(t)S(t) + y(t)B(t)

)= x(t) dS(t) + y(t) dB(t). (1.6)

From (1.3) and (1.4), Equation (1.6) becomes

x(t)(µS dt+ σ0Sφ√

dt) + y(t)rB dt = (xµS + yrB) dt+ xσ0Sφ√

dt.

Thus,

dV = (xµS + yrB) dt+ xσ0Sφ√

dt. (1.7)

From Ito’s lemma,

dV =∂V

∂tdt+

∂V

∂S(µS dt+ σ0Sφ

√dt) +

1

2σ2

0S2φ2∂

2V

∂S2dt

=

(∂V

∂t+ µS

∂V

∂S+

1

2σ2

0S2φ2∂

2V

∂S2

)dt+ σ0Sφ

∂V

∂S

√dt. (1.8)

If we invoke the assumption about the existence of transaction costs, then Equation

(1.8) becomes

dV =

(∂V

∂t+ µS

∂V

∂S+

1

2σ2

0S2φ2∂

2V

∂S2

)dt+ σ0Sφ

∂V

∂S

√dt− k|N |S, (1.9)

where k is the transaction cost parameter and N is the number of asset bought or

sold. From (1.7) and (1.9), we have

(∂V

∂t+ µS

∂V

∂S+

1

2σ2

0S2φ2∂

2V

∂S2

)dt+ σ0Sφ

∂V

∂S

√dt− k|N |S

= (xµS + yrB) dt+ xσ0Sφ√

dt.


Now we investigate the number of assets bought or sold. Given that x is evaluated

at the asset value S and time t, and following the same hedging strategy, we have

x =∂V

∂S(S, t).

Rebalancing or hedging after a finite time δt leads to a change in the value of assets

held as below∂V

∂S(S + δS, t+ δt).

The number N of assets bought or sold at the new time are

N =∂V

∂S(S + δS, t+ δt)− ∂V

∂S(S, t). (1.10)

Expressing δS = µSδt+ σ0Sφ√δt in the form δS = O(δt) + σ0Sφ

√δt and applying

Taylor expansion to the first term of (1.10), we have

∂V

∂S(S + δS, t+ δt) =

∂V

∂S(S, t) +

∂2V

∂S2δS +

1

2

∂3V

∂S3(δS)2 + . . . .

In this derivation, the third partial derivative of the option price does not play any

role and we remove it. Hence, it becomes

∂V

∂S(S + δS, t+ δt) =

∂V

∂S(S, t) +

∂2V

∂S2δS.

Thus, substituting for δS we have

N ≈ ∂2V

∂S2δS ≈ ∂2V

∂S2σ0Sφ

√δt. (1.11)

Hence the expected transaction cost in a finite time-step, E[k|N |S], is given by

E[k|N |S] = kSE[|N |]

= kSE

[∣∣∂2V

∂S2σ0Sφ

√δt∣∣]

= kσ0S2∣∣∂2V

∂S2

∣∣E[|φ|]√δt.

8 Chapter 1

Also,

E[|φ|] =1√2π

∫ ∞−∞

φe−0.5φ2 dφ

= 21√2π

∫ ∞0

φe−0.5φ2 dφ

=

√2

π

∫ ∞0

φe−0.5φ2 dφ

=

√2

π.

From the above, Equation (1.9) becomes

δV =

(∂V

∂t+ µS

∂V

∂S+

1

2σ2

0S2φ2∂

2V

∂S2

)δt+ σ0Sφ

∂V

∂S

√δt

−√

2

π

kσ0S2

√δt

∣∣∣∣∂2V

∂S2

∣∣∣∣ δt=

(∂V

∂t+ µS

∂V

∂S+

1

2σ2

0S2φ2∂

2V

∂S2−√

2

π

kσ0S2

√δt

∣∣∣∣∂2V

∂S2

∣∣∣∣)δt

+σ0Sφ∂V

∂S

√δt.

It follows that

E[δV ] =

(∂V

∂t+ µS

∂V

∂S+

1

2σ2

0S2∂

2V

∂S2−√

2

π

kσ0S2

√δt

∣∣∣∣∂2V

∂S2

∣∣∣∣)δt

since E[φ] = 0 and E[φ2] = 1. Recalling δV = (xµS + yrB)δt+ xσ0Sφ√δt, implies

that x = ∂V∂S

and

xµS + yrB =

(∂V

∂t+ µS

∂V

∂S+

1

2σ2

0S2∂

2V

∂S2−√

2

π

kσ0S2

√δt

∣∣∣∣∂2V

∂S2

∣∣∣∣).

From (1.5), we have yB = V − xS = V − S ∂V∂S

. Now, plugging yB in the above

equation yields

yrB =

(∂V

∂t+

1

2σ2

0S2∂

2V

∂S2−√

2

π

kσ0S2

√δt

∣∣∣∣∂2V

∂S2

∣∣∣∣).


Since yrB = rV − rS ∂V∂S

, we have

rV − rS ∂V∂S

=

(∂V

∂t+

1

2σ2

0S2∂

2V

∂S2−√

2

π

kσ0S2

√δt

∣∣∣∣∂2V

∂S2

∣∣∣∣).

Thus,

∂V

∂t+ rS

∂V

∂S+

1

2σ2

0S2∂

2V

∂S2−√

2

π

kσ0S2

√δt

∣∣∣∣∂2V

∂S2

∣∣∣∣− rV = 0.

This is one of the nonlinear Black-Scholes equations that will be discussed in this

thesis.

1.1.2 Jump Diffusion Model

To derive the model when the underlying stock price follows jump diffusion process,

we follow the work in [21] and give these following definitions.

Definition 1.1. A random process X(t) is said to be a counting process if X(t)

represents the total number of events that have occurred in the time interval (0, t).

A counting process must satisfy the following conditions:

1. X(t) ≥ 0, X(0) = 0

2. X(t) is integer valued

3. X(s) < X(t) if s < t

4. X(t) − X(s) equals the number of events that have occurred in the interval

(s, t).

Definition 1.2. A Poisson process is a counting process with intensity λ > 0 if

1. X(0) = 0

2. X(t) has independent and stationary increments

3. P [X(t+ dt)−X(t) = 1] = λdt+ o(dt) where P is the probability

10 Chapter 1

4. P [X(t+ dt)−X(t) ≥ 2] = o(dt) where

limdt→0

o(dt)

dt= 0

In this thesis, we can define the Poisson process dq as follows

dq =

0 with probability1− λdt

1 with probabilityλdt

where λ is the Poisson arrival intensity. Hence, if stock price follows a combination

of Brownian motion and rare jump events, then change in stock price is given by

dS = µSdt+ σ0SdW︸︷︷︸+ (η − 1)Sdq︸︷︷︸ .The first part of the equation is due to Brownian motion and the second part is

due to jump. Assume that the jump size has some known probability density g(η).

Given that a jump occurs, the probability of a jump in [η, η+dη] is g(η)dη. We also

have∫∞−∞ g(η)dη =

∫∞0g(η)dη = 1. If f = f(η), then the expected value of f is

E(f) =

∫ ∞0

f(η)g(η)dη. (1.12)

Suppose we have one option worth V and ∆ shares at price S. If Π is the value of

the portfolio, then Π = V −∆S. Consider the change in the value of portfolio

[dΠ]total = [dΠ]Brownian + [dΠ]jump. (1.13)

From Ito’s lemma

[dΠ]Brownian =

[Vt + µSVS +

σ20S

2

2VSS −∆µS

]dt+ σ0S[VS −∆]dW.


Noting that jump is of finite size, we have

[dΠ]jump =[V (ηS, t)− V (S, t)

]dq −∆(η − 1)Sdq.

If the Brownian motion risk is hedged by choosing ∆ = VS, then Equation (1.13)

becomes

[dΠ] =

[Vt +

σ20S

2

2VSS

]dt+

[V (ηS, t)− V (S, t)

]dq − VS(η − 1)Sdq. (1.14)

The change in the value of the portfolio still has a random component dq which

cannot be hedged away. Thus, we take the expectation on both sides of (1.14) to

get

E[dΠ] = E

([Vt +

σ20S

2

2VSS

])dt+ E

[V (ηS, t)− V (S, t)

]E[dq]

−VSSE[(η − 1)]E[dq]. (1.15)

We have assumed that the probability of jump and probability of jump sizes are

independent. Define E[η − 1] = κ, then Euqation (1.14) becomes

E[dΠ] = E

([Vt +

σ20S

2

2VSS

])dt+ E

[V (ηS, t)− V (S, t)

]λdt

−VSSκλdt. (1.16)

Assume that investor holds a diversified portfolio of hedging portfolios for different

stocks. We assume that the jumps for these portfolios are uncorrelated and the

variance of the portfolio is small. Then, the expected return should be

E[dΠ] = rΠdt. (1.17)

From (1.16) and (1.17), we have

Vt +σ2

0S2

2VSS + (rS − Sκλ)VS − (r + λ)V + E[V (ηS, t)λ = 0. (1.18)

12 Chapter 1

From (1.12) and (1.18), we get

Vt +σ2

0S2

2VSS + (r − κλ)SVS − (r + λ)V + λ

∫ ∞0

g(η)V (ηS, t)dη = 0. (1.19)

This is a partial integro-differential equation (PIDE) that models jump diffusion in

option pricing.

1.1.3 Viscosity Solution

Consider a second order partial differential equation of the form

F (x, u,Du,D2u) = 0, x ∈ Ω. (1.20)

Definition 1.3. Let Ω ⊂ Rn be an open set and u continuous in Ω;

• We say that u is a viscosity subsolution of (1.20) at a point x0 ∈ Ω, if and

only if, for any test function ϕ ∈ C2(Ω) such that u−ϕ has a local maximum

at x0, then

F (x0, u(x0), Dϕ(x0), D2ϕ(x0)) ≤ 0; (1.21)

• We say that u is a viscosity supersolution of (1.20) at a point x0 ∈ Ω, if and

only if, for any test function ϕ ∈ C2(Ω) such that u− ϕ has a local minimum

at x0, then

F (x0, u(x0), Dϕ(x0), D2ϕ(x0)) ≥ 0; (1.22)

• We say that u is a viscosity solution in the open set Ω if u is a viscosity

subsolution and a viscosity supersolution, at any point x0 ∈ Ω.


1.2 Thesis outline

The main contribution of this thesis is to develop and analyse numerical methods

to solving nonlinear partial differential equations and inequalities arising from the

valuation of European and American options with transaction costs. The first one is

for pricing European and American option with transaction costs when the underly-

ing asset price follows geometric Brownian motion and the second one is for pricing

European and American option with transaction costs when the underlying asset

price follows jump diffusion processes. The organization of the thesis is as follows.

1.2.1 Chapter 1

The general problem formulation and payoff functions are given. An introductory

discussion of options and assumption are also given in this chapter.

1.2.2 Chapter 2

A numerical method for solving European option with transaction costs when the

underlying asset price follows geometric Brownian motion is proposed here. We

also prove the convergence of the method by showing that the scheme is consistent,

monotone and unconditionally stable.

1.2.3 Chapter 3

In this chapter, we propose a penalty method for a finite-dimensional Nonlinear

Complementarity Problem (NCP) arising from the discretization of the infinite-

dimensional free boundary problem governing the valuation of American options

under transaction costs. The NCP is approximated by a penalty equation. We

show that the mapping involved in the system is continuous and strongly monotone

14 Chapter 1

and thus it is uniquely solvable. These also prove that the solution to the penalty

equation converges to that of the NCP.

1.2.4 Chapter 4

In this chapter, we develop a numerical method for a nonlinear partial integro-

differential equation (PIDE) and a partial integro-differential complementarity prob-

lem arising from European and American option valuations with transaction costs

when the underlying assets follow jump diffusion processes. The method is based

on an upwind finite difference scheme for the spatial discretization and a fully im-

plicit time stepping scheme. We prove that the system matrix from this scheme is

an M -matrix and that the approximate solution converges unconditionally to the

viscosity solution to the PIDE by showing that the scheme is consistent, mono-

tone, and unconditionally stable. We also add a penalty term to the original partial

integro-differential complementarity problem to impose the constraint in the Amer-

ican option model.

1.2.5 Chapter 5

Conclusion is presented in this chapter.

Chapter 2

An Upwind Finite Difference

Method for Pricing European

Options Under Transaction Costs

2.1 summary

This chapter develops a numerical method for a nonlinear partial differential equa-

tion arising from pricing European options under transaction costs. The method is

based on an upwind finite difference scheme for the spatial discretization and a fully

implicit time-stepping scheme. We prove that the system matrix from this scheme is

an M -matrix and that the approximate solution converges unconditionally to that

of the viscosity solution to the equation by proving that the scheme is consistent,

monotone and unconditionally stable. A Newton iterative algorithm is proposed

for solving the discretized nonlinear system of which the Jacobian matrix is shown

to be also an M -matrix. Numerical experiments are performed to demonstrate the

accuracy and robustness of the method.

The results from this chapter have been published in [22].

15

16 Chapter 2

2.2 Introduction

As mentioned in Chapter 1, Black-Scholes model is very effective for pricing options

in a complete market without costs on transactions of risky and riskless securities.

However, in the presence of transaction costs on trading in the riskless security

or stock, the Black-Scholes option pricing methodology is no longer valid, since

perfect hedging is impossible. The constant re-balancing used in the Black-Scholes

framework will be infinitely costly, no matter how small the transaction costs are,

since the geometric Brownian motion has infinite variations.

In recent years, different models have been proposed to accommodate transaction

costs arising in the hedging strategy, see, for example, [4–7]. Due to transaction costs

in trading, the classical model results in nonlinear equations in which the volatility

can depend on time, the underlying stock price, and/or a derivative of the option

price itself. In this case, the classical linear Black-Scholes equation becomes the

following nonlinear Black-Scholes equation:

Vt +1

2σ2(t, S, VS, VSS)S2VSS + rSVS − rV = 0, (2.1)

where σ is the modified volatility as a function of t, S, VS, and VSS.

Assuming that the transaction cost is proportional to the monetary value of an asset

bought or sold, Leland [4] argued that the option price is the solution to (2.1) with

the modified volatility

σ2 = σ20

(1 + Le sign(VSS)

), (2.2)

where Le is the Leland number given by

Le =

√2

π

(κ

σ0

√δt

),

where δt denotes the transaction frequency and κ denotes transaction cost measure.

Boyle and Vorst [5] derived from the binomial model that as the time step δt and

Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 17

the transaction cost κ tend to zero, the discrete option price converges to the Black-

Scholes price with the modified volatility of the form

σ2 = σ20

(1 + Le

√π

2sign(VSS)

). (2.3)

For both of the choices of σ2 in (2.2) and (2.3), the parameter Le has to be such

that σ2 > 0.

In [8] the authors derived the following nonlinear volatility

σ2 = σ20

(1−K sign(VSS)

)and σ2 = σ2

0

(1 +K sign(VSS)

)(2.4)

for long and short positions in an option respectively, where K = kσ0

√8π dt

, dt is fixed

trading frequency, and k is the transaction cost parameter. Equation (2.1) together

with nonlinear volatility (2.4) is known as HWW transaction costs model.

In [23] the authors derived the following risk-adjusted nonlinear volatility

σ2 = σ20

(1− 3

(C2R

2πSVSS

) 13

)(2.5)

under a rather unrealistic condition

SVSS <π

32C2R, (2.6)

where R ≥ 0 is the risk premium measure and C ≥ 0 is the transaction cost para-

meter.

A more comprehensive and robust model has been proposed by Barles and Soner [7]

based on the assumption that investor’s preferences are characterized by an expo-

nential utility function. In their model the nonlinear volatility is given by

σ2 = σ20

(1 + Ψ

[er(T−t)a2S2VSS

]), (2.7)

18 Chapter 2

where a = κ√γN with κ being the transaction cost parameter, γ a risk aversion

factor and N the number of options to be sold. The function Ψ is the solution of

the following nonlinear initial value problem:

Ψ′(z) =Ψ(z) + 1

2√zΨ(z)− z

for z 6= 0 and Ψ(0) = 0. (2.8)

This volatility model does not require any unrealistic conditions on the parameters

involved.

2.3 The continuous problem

In this chapter we present a numerical scheme with its analysis for the numerical

solution of the nonlinear Black-Scholes equation. For clarity, we only consider the

nonlinear model proposed by Barles and Soner [7], i.e. Eq.(2.1) with the nonlinear

volatility (2.7)–(2.8). All the theoretical results are also applicable to the nonlinear

volatility models in (2.2)–(2.5) and the results will be given in Appendix A.

Under the transformation τ = T − t, (2.1) can be written as

Uτ =1

2σ2(USS)S2USS + rSUS − rU (2.9)

for S > 0 and 0 < τ ≤ T , where

σ2(USS) = σ20

(1 + Ψ

[erτa2S2USS

])(2.10)

by (2.7) with Ψ determined by (2.8).

For the purpose of computations it is necessary to restrict the underlying stock price

S in a finite region I = (0, Smax), where Smax denotes a sufficiently large positive

number to ensure accuracy of the solution (cf., for example, [24]). Then, we define


the initial and boundary conditions for (2.9) on this finite region as follows

U(S, 0) = g1(S), S ∈ (0, Smax), (2.11)

U(0, τ) = g2(τ), τ ∈ (0, T ], (2.12)

U(Smax, τ) = g3(τ), τ ∈ (0, T ] (2.13)

satisfying the compatibility conditions that g1(0) = g2(0) and g1(Smax) = g3(0),

where g1, g2 and g3 are given functions.

The choices of g1, g2 and g3 depend on the type of an option. Popular European

options are vanilla, butterfly spread and cash-or-nothing options for which the initial

and boundary data are given by

g1 =

max(S −K, 0) for a vanilla call,

max(K − S, 0) for a vanilla put,

max(S −K1, 0)− 2 max(S −K2, 0) + max(S −K3, 0) for a butterfly,

B ×H(S −K) for a CoN,

g2 =

0 for a vanilla call,

Ke−rτ for a vanilla put,

0 for a butterfly spread,

0 for a cash or nothing,

g3 =

Smax −Ke−rτ for a vanilla call,

0 for a vanilla put,

0 for a butterfly spread,

Be−rτ for a cash or nothing,

where K,K1, K2 and K3 denote the strike prices of the options, B is a constant and

H is the Heaviside function.

20 Chapter 2

An implicit exact solution to (2.8) is derived in [25] as follows

√z =

−sinh−1√

Ψ√Ψ + 1

+√

Ψ for z > 0, Ψ(z) > 0, (2.14)

√−z =

sin−1√−Ψ√

Ψ + 1−√−Ψ for z < 0, −1 < Ψ(z) < 0. (2.15)

It has also been shown in [25] that

− 1 < Ψ(z) <∞, z ∈ R and Ψ′(z) > 0 for z 6= 0. (2.16)

Therefore, Ψ is strictly increasing in z.

There are limited studies on the numerical solution of (2.9) with the nonlinear volatil-

ity (2.10) in the open literature though the numerical solution of the linear Black-

Scholes equation has been discussed extensively (cf., for example, [26–29]). In [7]

the authors propose an explicit finite difference scheme which requires a restrictive

stability condition on the time and spatial mesh sizes. Ankudinova and Ehrhardt

[13] use a Crank-Nicolson method combined with a high order compact difference

scheme developed in [30] to construct a numerical scheme for the linearized Black-

Scholes equation using frozen values of nonlinear volatility. In [31] the authors

propose a high order finite difference scheme for the transformed equation of (2.9)

under the transformation x = ln(S/K). This transformation transforms (0, Smax)

to (−∞, ln(Smax/K)) and in computation, this infinite domain has to be truncated

which is essentially to omit the degeneracy of (2.9) at S = 0. In [32] the authors also

propose a fourth-order compact scheme by treating the nonlinear volatility explic-

itly. A more proper treatment of the nonlinear volatility was proposed by Company,

Jodar and Pintos [33] based on a semi-discretization technique (or the method of

lines) which approximate (2.9) with a system of ordinary differential equations and

solve the system using backward Euler scheme. A smoothing technique for the pay-

off condition is also used in [33] in order that the high order scheme works, which

essentially changes the nature of the pricing problem. Usually, a high order method

requires that the solution to the PDE is sufficiently smooth in order to achieve the


expected order of convergence. However, it is known that (2.9) with the initial and

boundary conditions (2.11)–(2.13) does not usually have any classic smooth solu-

tions, but only has the so-called viscosity solution. Therefore, a numerical solution

to (2.9)–(2.13) by a high order numerical scheme is not necessarily more accurate

than that from a first-order discretization scheme, mainly due to the non-smoothness

of the given data and the exact solution.

While most of the above works do not contain rigorous mathematical analysis, a

convergence analysis for the scheme in [31] is performed in [34] in which the authors

also show that the system matrix from the discretization is an M -matrix. Another

notable work is [35] in which the author applies a central difference scheme to the

transformed equation of (2.9) using x = ln(S/K) and shows that the approximate

solution converges to the exact viscosity solution. The convergence and other re-

sults in both [34] and [35] are established under restrictive and rather unrealistic

conditions on either the coefficients of (2.9) or the discretization mesh sizes, or both.

Also, the transformation x = ln(S/K) used in these works transforms 0 < S ≤ 1

into (−∞ < x ≤ 0) and thus solving the transformed equation is computationally

more expensive than solving (2.9), particularly when uniform meshes are used as in

[34] and [35].

In this chapter, we will examine the use of simple and popular spatial discretiza-

tion numerical schemes, central and the upwind finite differences, along with the

backward Euler (the fully implicit time stepping) scheme for (2.9)–(2.13). We prove

theoretically that the numerical method is unconditionally stable, the system ma-

trix of the discretized equation is an M -matrix and the solution from the method

converges unconditionally to the viscosity solution to (2.11)–(2.13). We will show

numerically that the rate of convergence of the numerical solutions from our method

to the exact one is roughly of 2nd order in a discrete L2-norm. We will also compare

the results from Barles and Soner’s model to those from HWW model as well as

other volatility models.

22 Chapter 2

2.4 Discretization

We now present a finite difference scheme for the discretization of (2.9) based

on an upwind finite differencing in space and the backward Euler’s time stepping

scheme. Upwind finite differencing techniques have long been used as stable nume-

rical schemes for convection-dominated diffusion equations [36] and more recently

for HJB equations [37] as the system matrix of a discretized system from such a

scheme is usually an M -matrix. We start this discussion by defining a mesh for

(0, Smax)× (0, T ).

Let I := (0, Smax) be divided into M sub-intervals

Ii = (Si, Si+1), i = 0, 1, . . . ,M − 1

satisfying 0 = S0 < S1 < . . . < SM = Smax. Similarly, we divide (0, T ) into N

sub-intervals with mesh nodes τnNn=0 satisfying 0 = τ0 < τ1 < . . . < τN = T . For

any i = 0, 1, . . . ,M−1 and n = 0, 1, ..., N−1, let hi = Si+1−Si and ∆τn = τn+1−τn.

For any vectors W n = (W n0 ,W

n1 , ...,W

nM)> and Wi = (W 0

i ,W1i , ...,W

Ni )> for i =

0, 1, ...,M and n = 0, 1, ..., N , we define the following finite difference operators on

the mesh defined above:

(δτWi)(n) =W n+1i −W n

i

∆τn,

(δ+SW

n)(i) =W ni+1 −W n

i

hi, (δ−SW

n)(i) =W ni −W n

i−1

hi−1

,

(δSSWn)(i) =

(δ+SW

n)(i)− (δ−SWn)(i)

(hi−1 + hi)/2= pi−1W

ni−1 − piW n

i + pi+1Wni+1,

where

pi−1 =2

hi−1(hi−1 + hi), pi+1 =

2

hi(hi−1 + hi), pi = pi−1 + pi+1 =

2

hi−1hi. (2.17)


Using these operators, we approximate (2.9) by the following finite difference system

(δτUi)(n)− 1

2σ2((δSSU

n+1)(i))S2i (δSSU

n+1)(i)−rSi(δ+SU

n+1)(i)+rUn+1i = 0 (2.18)

for i = 1, 2, ...,M − 1 and n = 0, 1, ..., N − 1,

where Un+1 = (Un+10 , Un+1

1 , ..., Un+1M )> and Ui = (U0

i , U1i , ..., U

Ni )> with Un

i being

an approximation to U(Si, τn) for any feasible index pair (i, n). Note that in (2.18)

we used the upwind technique to discretize the term rSUS in (2.9) which turns out

to be the forward finite differencing. Also, the time discretization is based on the

Backward Euler’s scheme and thus the above is a fully implicit numerical scheme.

For any n = 0, 1, ..., N − 1, (2.18) defines a nonlinear system in Un+1. Using the

definitions of the finite difference operators it is easy to show that (2.18) can also

be written as

αn+1i (Un+1)Un+1

i−1 + βn+1i (Un+1)Un+1

i + γn+1i (Un+1)Un+1

i+1 =1

∆τnUni (2.19)

for i = 1, 2, ...,M − 1 and n = 0, 1, ..., N − 1, where

αn+1i (Un+1) = −1

2σ2((δSSU

n+1)(i))S2i pi−1, (2.20)

βn+1i (Un+1) =

1

∆τn+

1

2σ2((δSSU

n+1)(i))S2i pi +

rSihi

+ r, (2.21)

γn+1i (Un+1) = −1

2σ2((δSSU

n+1)(i))S2i pi+1 −

rSihi. (2.22)

Using (2.11)–(2.13) we define the following initial and boundary conditions for (2.19):

U0i = g1(Si), Un

0 = g2(τn), UnM = g3(τn) (2.23)

for i = 0, 1, ...,M and n = 1, 2, ..., N . Equation (2.19), along with the above bound-

ary conditions, can further be written in the following matrix form:

An+1(Un+1)Un+1 =1

∆τnUn +Bn+1 (2.24)

24 Chapter 2

for n = 0, 1, ..., N − 1 with the discrete initial condition defined above, where

An+1(Un+1) =

βn+11 γn+1

1 0 . . . 0 0 0

αn+12 βn+1

2 γn+12 . . . 0 0 0

0 αn+13 βn+1

3 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . βn+1M−3 γn+1

M−3 0

0 0 0 . . . αn+1M−2 βn+1

M−2 γn+1M−2

0 0 0 . . . 0 αn+1M−1 βn+1

M−1

,

Uk =(Uk

1 , Uk2 , ..., U

kM−1

)>for k = n, n+ 1,

Bn+1 =(−αn+1

1 Un+10 , 0, . . . , 0,−γn+1

M−1Un+1N

)>.

Clearly, (2.24) is a nonlinear system in Un+1 of which the nonlinear system matrix

has the following properties.

Theorem 2.1. For any n = 0, 1, ..., N , the matrix An = (Anij) is an M-matrix for

any given Un.

Proof. To prove this theorem, it suffices to show that

αni < 0, βni > 0, γni < 0, (2.25)

βni ≥ |αn+1i |+ |γni |+

1

∆τn(2.26)

for i = 1, 2, ...,M − 1.

From (2.20)–(2.22) and (2.17) we see that (2.25) is obviously true and that

βni ≥ |αn+1i |+ |γn+1

i |+ r +1

∆τn≥ |αn+1

i |+ |γn+1i |+ 1

∆τn,

since r ≥ 0. From these and the definition of An we have that

Anij ≤ 0, i 6= j, Anii > 0, Anii >M−1∑j=1

|Anij|.


Also, it is obvious that An is irreducible. Therefore, by [38], An is an M -matrix for

any given Un.

Remark 2.2. It is possible to use Crank-Nicolson scheme for the time discretization

of (2.9) and the resulting system matrix is still an M -matrix. However, Crank-

Nicolson scheme cannot be used for at least the first time step. This is because

the initial condition (2.11) is non-smooth so that the discretization of USS becomes

unbounded. To remedy this, a combination of the above fully implicit scheme and

Crank-Nicolson scheme needs to be used. For clarity, we concentrate on the fully

implicit scheme and will discuss the use of Crank-Nicolson scheme in a future work.

2.5 Convergence of the numerical scheme

In [7] the authors show the existence and uniqueness of the viscosity solution to

(2.9). In this section we will prove that the solution to (2.24) converges to the

viscosity solution to (2.9). It has been shown in [39] that the convergence of the

fully discretized system (2.24) to the viscosity solution of a full nonlinear 2nd-order

PDE is guaranteed if the discretization is consistent, stable and monotone. Thus,

in rest of this section we will prove the convergence of our numerical scheme by

showing that it satisfies these properties.

For i = 1, 2, ...,M − 1 and n = 0, 1, ..., N − 1, introduce a functional Hn+1i defined

by

Hn+1i

(Un+1i , Un+1

i+1 , Un+1i−1 , U

ni

):= ηiU

n+1i+1 + λiU

n+1i − 1

∆τnUni

− 1

2S2i σ

2((δSSUn+1)(i))(δSSU

n+1)(i) (2.27)

where

ηi = −rSihi

and λi =1

∆τn+rSihi

+ r.

26 Chapter 2

Then, it is easy to see that (2.19) becomes

Hn+1i

(Un+1i , Un+1

i+1 , Un+1i−1 , U

ni

)= 0

for all feasible i and n. For this discretization scheme, we have the following lemma.

Lemma 2.3 (Monotonicity). The discretization (2.19) is monotone, i.e. for any

ε > 0 and i = 1, 2, . . . ,M − 1,

Hn+1i

(Un+1i , Un+1

i+1 + ε, Un+1i−1 + ε, Un

i + ε)≤

Hn+1i

(Un+1i , Un+1

i+1 , Un+1i−1 , U

ni

), (2.28)

Hn+1i

(Un+1i + ε, Un+1

i+1 , Un+1i−1 , U

ni

)≥

Hn+1i

(Un+1i , Un+1

i+1 , Un+1i−1 , U

ni

). (2.29)

Proof. Since ηi ≤ 0, 1∆τn

> 0, and λi > 0, the first three (linear) terms on the RHS

of (2.27) are respectively non-increasing in Un+1i−1 , increasing in Un+1

i , and decreasing

in Uni .

Let Ek = (0, 0, ..., 1︸︷︷︸kth

, 0, ..., 0)> be the (M − 1) × 1 column vector. From the

definition of δSS and (2.17) we have

(δSS(Un+1 + εEi−1 + εEi+1))(i) = pi−1(Un+1i−1 + ε)− piUn+1

i + pi+1(Un+1i+1 + ε)

= (δSSUn+1)(i) + piε, (2.30)

(δSS(Un+1 + εEi))(i) = pi−1Un+1i−1 − pi(Un+1

i + ε) + pi+1Un+1i+1

= (δSSUn+1)(i)− piε. (2.31)

Let us consider the nonlinear term on the RHS of (2.27). From (2.10) we see that

it is of the form

1

2S2i σ

2(un+1i )un+1

i =1

2σ2

0S2i

[1 + Ψ(Kn+1

i un+1i )

]un+1i


where

un+1i = (δSSU

n+1)(i) and Kn+1i = erτn+1a2S2

i > 0 (2.32)

for i = 1, 2, ...,M−1. For any K > 0, independent of u, Differentiating (1+Ψ(Ku))u

with respect to u and using (2.8) we have

d

du[(1 + Ψ(Ku))u] = Ψ′(Ku)(Ku) + (1 + Ψ(Ku))

=1 + Ψ(z)

2√zΨ(z)− z

z + (1 + Ψ(z))

= (1 + Ψ(z))2√zΨ(z)

2√zΨ(z)− z

= 2√zΨ(z)Ψ′(z)

≥ 0 (2.33)

by (2.16), where z = Ku. Therefore, (1 + Ψ(Ku))u is an increasing function of u.

Using the above notation and combining the monotonicity of (1 + Ψ(Ku))u, the

properties of the linear terms on the RHS of (2.27) and (2.30) we have

Hn+1i

(Un+1i , Un+1

i+1 + ε, Un+1i−1 + ε, Un

i + ε)

= ηi(Un+1i+1 + ε) + λiU

n+1i − 1

∆τn(Un

i + ε)

−1

2σ2

0S2i

[1 + Ψ

(Kn+1i (un+1

i + piε))] (

un+1i + piε

)≤ Hn+1

i

(Un+1i , Un+1

i+1 , Un+1i−1 , U

ni

).

This is (2.28). Similarly, using the monotonicity of (1 + Ψ(Ku))u and (2.31) it

is easy to show that (2.29) also holds true. Hence, the discretization scheme is

monotone.

The stability of the method is established in the following Lemma.

Lemma 2.4 (Stability). For n = 0, 1, 2, . . . ,M − 1,

let Un+1 = (Un+10 , (Un+1)>, Un+1

M )> where Un+1 is the solution to (2.24).

28 Chapter 2

Then, Un+1 satisfies

‖Un+1‖∞ ≤ max‖g1‖∞, ||g2||∞, ||g3||∞, (2.34)

where g1, g2 and g3 are the initial and boundary conditions defined in (2.11)–(2.13)

and ‖ · ‖∞ denotes the l∞-norm.

Proof. For any n = 0, 1, ..., N − 1, from (2.19) we have

βn+1i Un+1

i = −αn+1i Un+1

i−1 − γn+1i Un+1

i+1 +1

∆τnUni

for i = 1, 2, ...,M − 1. Recall αn+1i ≤ 0, γn+1

i ≤ 0 and βn+1i > 0.

From the above we get

βn+1i |Un+1

i | ≤ −αn+1i |Un+1

i−1 | − γn+1i |Un+1

i+1 |+1

∆τn|Un

i |

≤ −αn+1i ‖Un+1‖∞ − γn+1

i ‖Un+1‖∞ +1

∆τn‖Un‖∞

for i = 1, 2, ...,M − 1. We now consider the following two cases.

Case I: ‖Un+1‖∞ = |Un+1k | for an index k ∈ 1, 2, ...,M − 1.

In this case, the above estimate with i = k becomes

(αn+1k + βn+1

k + γn+1k )‖Un+1‖∞ ≤

1

∆τn‖Un‖∞.

Therefore, using (2.26) we obtain from the above inequality

‖Un+1‖∞ ≤1/∆τn

(αn+1i + βn+1

i + γn+1i )‖Un‖∞ ≤ ‖Un‖∞

≤ ‖Un−1‖∞ ≤ · · · ≤ ‖U0‖∞ ≤ ‖g1‖∞.

Case II: ‖Un+1‖∞ = |Un+10 | or ‖Un+1‖∞ = |Un+1

M |.


In this case, from (2.23), (2.12) and (2.13) it is easy to see that

‖Un+1‖∞ ≤ max|Un+10 |, |Un+1

M | ≤ max||g2||∞, ||g3||∞.

Combining the above two cases we have (2.34).

The consistency of the numerical scheme is given in the following lemma:

Lemma 2.5 (Consistency). The discretization scheme (2.18) is consistent.

The proof is standard since both of the time and spatial discretization schemes

are standard and have been used extensively in the literature for 2nd-order partial

differential equations. Therefore, we omit the proof of this lemma. Combining the

above three lemmas we have the following convergence result.

Theorem 2.6. The solution to (2.24) converges to the viscosity solution to (2.9)–

(2.13) as (h,∆τ)→ (0+, 0+), where h = max0≤i≤M−1 hi and ∆τ = max0≤n≤N−1 ∆τn.

Proof. In [39] the authors show that if a discretization scheme for a fully nonlinear

2nd order PDE is monotone, stable and consistent, then the solution to the fully

discretized system converges to the viscosity solution to the PDE. Therefore, this

theorem is just a consequence of Lemmas 2.3, 2.4 and 2.5.

Remark 2.7. Note that the monotonicity and stability of the numerical scheme

established above are unconditional, while the convergence results in [34] and [35]

are obtained under some restrictive and rather unrealistic conditions on the mesh

sizes of the schemes and some of the coefficients of the problem. Also, in [35]

the author uses the observation that monotonicity (or maximum principle) implies

stability to prove the stability of the scheme. However, this observation may not

be true as l∞-stability of a numerical scheme is equivalent to monotonicity only for

mappings which are invariant under translation by a constant (see, for example, [40]

in which an example is also given to demonstrate that a monotone scheme is not

stable).

30 Chapter 2

Remark 2.8. We also comment that though the above monotonicity, stability and

convergence results have been obtained for the volatility σ2 defined in (2.7), the

results are also true for the other choices of σ given in (2.2) and (2.3). As a matter

of fact, it is easy to see that the proofs of Lemmas 2.3 and 2.4 only need the

assumptions that σ2(t, S, VSS) > 0 and σ2(t, S, z)z is monotonically increasing in

z. It is easy to show (even graphically) that these conditions are satisfied by the

nonlinear models defined in (2.2) and (2.3). It is also easy to show that σ2(t, S, z)z

is monotonically increasing for the volatility defined in (2.5) when (2.6) is satisfied.

The proofs are given in Appendix A.

2.6 Solution of the nonlinear system (2.24)

In this section, we propose a Newton iterative method for the nonlinear system

(2.24) at each time step. To achieve this, we first write (2.24) in the following form

F n+1(Un+1) := An+1(Un+1)Un+1 −GUn −Bn+1 = 0.

Let

F n+1(Un+1) = (fn+11 (Un+1), fn+1

2 (Un+1), . . . , fn+1M−1(Un+1))>.

Then, from (2.19) it is easy to see that the ith component of F n+1(Un+1) is

fn+1i (Un+1) = αn+1

i Un+1i−1 + βn+1

i Un+1i + γn+1

i Un+1i+1 −

1

4τnUni ,


where Un+10 and Un+1

M are defined in (2.23). The Jacobian matrix of F n+1(Un+1),

denoted by Jn+1(Un+1), is given by

Jn+1(Un+1) =

Jn+111 Jn+1

12 0 . . . 0 0 0

Jn+121 Jn+1

22 Jn+123 . . . 0 0 0

0 Jn+132 Jn+1

33 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . Jn+1(M−3)(M−3) Jn+1

(M−3)(M−2) 0

0 0 0 . . . Jn+1(M−2)(M−3) Jn+1

(M−2)(M−2) Jn+1(M−2)(M−1)

0 0 0 . . . 0 Jn+1(M−1)(M−2) Jn+1

(M−1)(M−1)

,

where Jn+1ij :=

∂fn+1i

∂Un+1j

for all feasible i and j. Using (2.20)–(2.22), (2.7) and (2.8) and

following the notation used in the proof of Lemma 2.3, we derive explicit expressions

for the derivatives as follows:

Jn+1i,i−1 = αn+1

i + Un+1i−1

∂αn+1i

∂Un+1i−1

+ Un+1i

∂βn+1i

∂Un+1i−1

+ Un+1i+1

∂γn+1i

∂Un+1i−1

= αn+1i − S2

i σ20

2(pi−1U

n+1i−1 − piUn+1

i + pi+1Un+1i+1 )

∂

∂Un+1i−1

[(1 + Ψ(Kn+1

i un+1i ))

]= αn+1

i − S2i σ

20

2un+1i Ψ′(Kn+1

i un+1i )Kn+1

i pi−1

= αn+1i − S2

i σ20pi−1

2Ψ′(zn+1

i )zn+1i , (2.35)

where Kn+1i and un+1

i are defined in (2.32), zn+1i = Kn+1

i un+1i and Ψ′ is defined in

(2.8). Similarly, we have

Ji,i = βn+1i +

1

2S2i σ

20piΨ

′(zn+1i )zn+1

i ,

Ji,i+1 = γn+1i − 1

2S2i σ

20pi+1Ψ′(zn+1

i )zn+1i .

Using the Jacobian of F n+1, we propose the following Newton algorithm for (2.24):

Algorithm N

32 Chapter 2

1. Choose a tolerance ε > 0. Let n = 0 and evaluate the discrete initial condition

U0 = (U01 , ..., U

0M−1)> using (2.23).

2. Set l = 0 and W l = Un.

3. Solve

Jn+1(W l)δW = −F n+1(W l)

for δW and set

W l+1 = W l + δW.

4. If ‖δW‖∞ ≥ ε, set l := l + 1 and go to Step 3. Otherwise, continue.

5. Set Un+1 = W l+1. If n < N − 1, let n := n + 1 and go to Step 2. Otherwise,

stop.

Remark 2.9. The initial condition in Step 1 is calculated using Equation (2.23).

To efficiently solve linear systems in Step 3, we can use Thomas algorithm since it

is stable as the system matrix is diagonally dominant.

For the Jacobian Jn+1, we have the following results.

Theorem 2.10. For any given Un+1, Jn+1(Un+1) is an M-matrix.

Proof. For simplicity of notation, we omit the superscript n+ 1 in the proof of this

theorem.

To show that J is an M -matrix, from [38] we see that it suffices to prove that Jii > 0,

Ji,i−1, Ji,i+1 ≤ 0, Jii ≥ |Ji,i−1| + |Ji,i+1| and Jii > |Ji,i−1| + |Ji,i+1| for at least one

index i. Let us first consider Ji,i−1. Using the definition of αn+1i in (2.20) and (2.10)


we have from (2.35)

Ji,i−1 = −1

2σ2

0(1 + Ψ(zi))S2i pi−1 −

1

2σ2

0Ψ′(zi)ziS2i pi−1

= −1

2σ2

0S2i pi−1 [1 + Ψ(zi) + Ψ′(zi)zi]

= −σ20S

2i pi−1

√ziΨ(zi)Ψ

′(zi) (by (2.33))

≤ 0. (by (2.33))

Similarly it can be shown that

Ji,i = σ20S

2i pi√ziΨ(zi)Ψ

′(zi) +rSihi

+ r +1

∆τn> 0

Ji,i+1 = −σ20S

2i pi+1

√ziΨ(zi)Ψ

′(zi)−rSihi≤ 0.

From these expressions we see that

Ji,i = |Ji,i−1|+ |Ji,i+1|+ r +1

∆τn> |Ji,i−1|+ |Ji,i+1|

for any i = 1, 2, ...,M − 1 with the convention that J1,0 = 0 = JM−1,M . Therefore,

J is an M -matrix by [38].

Note that the linear system at Step 3 of Algorithm N is usually large-scale and the

above theorem ensures that the system has a unique solution and the solution of the

system by an LU decomposition or an iterative method is numerically stable.

2.7 Numerical experiments

To show the efficiency and accuracy of the discretization method, numerical experi-

ments on four model problems have been performed. All the numerical results were

computed in double precision using Matlab on a PC running Windows XP.

34 Chapter 2

020

4060

80

00.2

0.40.6

0.810

10

20

30

40

50

Stock Price

Price of European Call Option, a = 5%

Time

Opt

ion

Pric

e

Figure 2.1: Price of the European call option with a = 0.05.

Test 1: European Vanilla Call Option with r = 0.1, σ0 = 0.2, K = 40, T = 1 and

Smax = 80.

The problem in the independent variables (S, τ) is solved for various values of a on

a number of uniform meshes by the numerical method presented in the previous

sections. The numerical solutions are then transformed back in (S, t). The value of

the option for a = 0.05 on the uniform mesh with M = 20 (h = 4) and N = 10

(∆τ = 0.1) is depicted in Figure 2.1. From the figure we see that numerical solution

is stable. A comparison of the European call option prices from Barles-Soner model

and HWW short position model is given in Figure 2.2. As can be seen, the two

models produce almost the same result in this case. To see the influence of the

transaction cost parameter a on the option price, we plot the values of the option at

t = 0 (or τ = T ) for three different values of a on the interval [0, 50] in Figure 2.3 in

which the curve for a = 0 is the price from the standard Black-Scholes Model. From

this figure we see that the price of the option increases as the transaction parameter

a increases as expected in practice.

The call option prices for three different values of σ0 at t = 0 are plotted in Figure 2.4.

From the figure we see that the value of the call option increases as σ0 increases.

This is true in practice because the owner of a call option has higher chance to


0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

45

Stock Price

Cal

l Opt

ion

Pric

e

Call Option Price for Barles−Soner (a = 2%) & HWW (k = 2%, dt = 1/12)

Barles−Soner ModelHWW Model

Figure 2.2: Call Option Prices of Barles-Soner and HWW short position Models.

0 5 10 15 20 25 30 35 40 45 500

5

10

15

Stock Price

Cal

l Opt

ion

Pric

e

Call Option Price for Different Transaction Cost Parameters

: a = 0: a = 0.02: a = 0.05

Figure 2.3: Prices of the European call option for different transaction costs.

benefit from price increases due to higher volatility but has limited downside risk in

the event of price decreases.

We now investigate numerically rates of convergence of our method. To determine

these rates, we choose a sequence of meshes generated by successively halving the

mesh sizes of the previous ones, starting from a given coarse mesh. Since the exact

solution to the test problem is unknown, we use the numerical solution on the

uniform mesh with M = 2560 and N = 1280 as the “exact” or reference solution

Vexact. Using this reference solution we then calculate the following ratios of the

36 Chapter 2

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

45

Stock Price

Eur

opea

n C

all P

rice

Call Option Price with Low & High Volatility

: volatility = 5%: volatility = 20%: volatility = 70%

Figure 2.4: The call option prices for different values of σ0

numerical solutions from two consecutive meshes:

Ratio(‖ · ‖h,∞) =‖V ∆τ

h − Vexact‖h,∞‖V ∆τ/2

h/2 − Vexact‖h,∞,

Ratio(‖ · ‖h,2) =‖V ∆τ

h − Vexact‖h,2‖V ∆τ/2

h/2 − Vexact‖h,2,

where V βα denotes the computed solution on the mesh with spatial mesh size α and

time mesh size β and || · ||h,∞ and || · ||h,2 are discrete maximum norm and L2-norm

defined respectively by

‖V ∆τh − Vexact‖h,∞ := max

1≤i≤M ;1≤n≤N|V ni − Vexact(Si, τn)|,

‖V ∆τh − Vexact‖h,2 :=

( ∑1≤i≤M

∑1≤n≤N

|V ni − Vexact(Si, τn)|2h4τ

)1/2

.

The computed ratios for the chosen meshes in the two discrete norms are listed in

Table 2.1. From the table we see that the rates of convergence of our method are

respectively about 1.6 in || · ||h,∞ and 2 in || · ||h,2.

Test 2: European Put Option with r = 0.1, σ0 = 0.2, K = 40, T = 1 and Smax = 80.


M N ‖ · ‖h,∞ Ratio(‖ · ‖h,∞) ‖ · ‖h,2 Ratio(‖ · ‖h,2)21 11 3.60174e-1 5.82355e-241 21 2.04226e-1 1.76 3.02330e-2 1.9381 41 1.20698e-1 1.69 1.54699e-2 1.95

161 81 7.54435e-2 1.60 7.58883e-3 2.04321 161 4.80820e-2 1.57 3.53741e-3 2.15641 321 2.89570e-2 1.66 1.50513e-3 2.35

1281 641 1.36280e-2 2.12 4.97529e-4 3.03

Table 2.1: Computed rates of convergence for the call option with a = 0.02

0 10 20 30 40 50 60 70 80 0

0.5

1

0

5

10

15

20

25

30

35

40

Time

Stock Price

Price of European Put Option, a = 5%

Opt

ion

Pric

e

Figure 2.5: Price of the European put option.

The value of this option computed by our method on the uniform mesh with h = 4

and ∆τ = 0.1 is plotted in Figure 2.5. The cross sections at t = 0 of the computed

option prices from Barles-Soner and HWW short position models are displayed in

Figure 2.6 from which we see that the prices of the two models are qualitatively the

same. The values of the option corresponding to the three different values of a at

t = 0 are depicted in Figure 2.7. As can be seen from the figure, the option price

is also an increasing function of a as expected in practice. The value function of

the put option for three different values of σ0 is depicted in Figure 2.8. As in Test

1, it is also an increasing function in σ0, i.e., the higher the volatility is, the more

expensive the put option is.

As in the case of the European call option, we compute the ratios of the errors in

38 Chapter 2

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

Stock Price

Put

Opt

ion

Pric

e

Put Option Price for Barles−Soner (a = 2%) & HWW (k = 2%, dt = 1/12)


Figure 2.6: The Comparison of Put Option Price Between Barles-Soner Modeland HWW Model

30 35 40 45 50 55 60 65 70 75 800

1

2

3

4

5

6

7

8

Stock Price

Put

Opt

ion

Pric

e

Put Option Price for Different Transaction Cost Parameter

: a = 0: a = 0.02: a = 0.05

Figure 2.7: Prices of the European put option for different transaction costs.

the two discrete norms from two consecutive meshes and list them in Table 2.2.

Clearly, these ratios show that the orders of convergence of the proposed discretiza-

tion scheme are respectively around 1.6 and 2 in || · ||h,∞ and || · ||h,2.

Test 3: Butterfly Spread Option with r = 0.1, σ = 0.2, K1 = 30, K2 = 40, K3 = 50,

T = 1 and Smax = 80.


and ∆τ = 0.05 is plotted in Figure 2.9. The cross-sections of the value functions


0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

Stock Price

Eur

opea

n P

ut O

ptio

n P

rice

Put Option Price with Low & High Volatility


Figure 2.8: The put option price for low and high volatility

M N ‖ · ‖h,∞ Ratio(‖ · ‖h,∞) ‖ · ‖h,2 Ratio(‖ · ‖h,2)21 11 3.58222e-1 6.37343e-241 21 2.03738e-1 1.76 3.32999e-2 1.9181 41 1.20578e-1 1.69 1.70071e-2 1.96

161 81 7.54142e-2 1.60 8.33053e-3 2.04321 161 4.80752e-2 1.57 3.88013e-3 2.15641 321 2.89555e-2 1.66 1.64999e-3 2.35

1281 641 1.36277e-2 2.12 5.45038e-4 3.03

Table 2.2: Computed results for put options with a = 0.02

at t = 0 from Barles-Soner and HWW short position models are plotted in Figure

2.10 from which we see that the price from HWW model is slightly higher than that

from Barles-Soner model near S = K. The values of the option corresponding to the

three different values of a at t = 0 are depicted in Figure 2.11. As can be seen from

the figure, the option price is also an increasing function of a. The value function

of the butterfly spread option at t = 0 for different values of σ0 is given in Figure

2.12. From the figure we see that it is a decreasing function of σ0 when the option

is ‘in-the-money’, i.e. K1 < S < K3 in which the payoff function is positive. It is an

increasing function of σ0 when it is ‘out-of-the-money’, when S < K1 or S > K3, in

which the payoff function is of no value.

40 Chapter 2

020

4060

80

00.2

0.40.6

0.810

2

4

6

8

10

Stock Price

Butterfly Spread Price, a = 5%

Time

Opt

ion

Pric

e

Figure 2.9: Price of the Butterfly Spread Option.

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

Stock Price

But

terfl

y S

prea

d O

ptio

n P

rice

Butterfly Spread Price for Barles−Soner (a = 2%) & HWW (k = 2%, dt = 1/12)


Figure 2.10: The Comparison of Butterfly Spread Option Price Between Barles-Soner Model and HWW Model

Test 4: Cash or Nothing Option with r = 0.1, σ = 0.2, K = 40, T = 1, B = 1 and

Smax = 80.


and ∆τ = 0.05 is plotted in Figure 2.13. The comparison of the cash or nothing

option price at t = 0 between Barles-Soner model and HWW model is given in Figure

2.14. Again, from the figure we see that the value from HWW model is slightly higher

than that from Barles-Soner model. The values of the option corresponding to the


0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

Stock Price

But

terfl

y S

prea

d O

ptio

n P

rice

Butterfly Spread Option Price for Different Transaction Cost Parameter

: a = 0: a = 0.02: a = 0.05

Figure 2.11: Prices of the butterfly spread option for different transaction costs.

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

7

Stock Price

But

terfl

y O

ptio

n P

rice

Butterfly Option Price with Low & High Volatility


Figure 2.12: The butterfly spread option price for low and high volatility

three different values of a at t = 0 are depicted in Figure 2.15. As expected, the

option price is also an increasing function of a. We also computed the prices of the

cash or nothing option for three different values of σ0 and the results at t = 0 are

plotted in Figure 2.16. When the volatility is larger, it is more likely for the stock

price to increase or decrease, and hence the option price will be cheaper when it is

in-the-money and will be more expensive when it is out-of-the-money. Clearly, the

results in Figure 2.16 display this phenomenon.

42 Chapter 2

020

4060

80

0

0.5

10

0.2

0.4

0.6

0.8

1

Stock Price

Price of Cash or Nothing Option, a = 5%

Time

Opt

ion

Pric

e

Figure 2.13: Price of the Cash or Nothing option.

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stock Price

Cas

h or

Not

hing

Opt

ion

Pric

e

Cash or Nothing for Barles−Soner (a = 2%) & HWW (k = 2%, dt = 1/12)


Figure 2.14: The Comparison of Cash or Nothing Option Price Between Barles-Soner Model and HWW Model

2.8 Conclusion

In this chapter we proposed an upwind finite difference method for the nonlinear

Black-Scholes equation governing option pricing under transaction costs. Numerical

experiments, performed to demonstrate the accuracy and usefulness of the method,

show that the orders of convergence of the method are about 1.6 and 2 in respectively

the discrete L∞- and L2-norms. The results also show that the price of a European

option is an increasing function of the transaction cost parameter a.


0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stock Price

Cas

h or

Not

hing

Opt

ion

Pric

e

Cash or Nothing Option Price for Different Transaction Cost Parameter

: a = 0: a = 0.02: a = 0.05

Figure 2.15: Prices of the Cash or Nothing option for different transaction costs.

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stock Price

Cas

h or

Not

hing

Opt

ion

Pric

e

Cash or Nothing Option Price with Low & High Volatility


Figure 2.16: The cash or nothing option price for low and high volatility

Chapter 3

A Penalty Approach for American

Put Option Valuation Under

Transaction Costs

3.1 summary

We propose a power penalty method for a finite-dimensional Nonlinear Complemen-

tarity Problem (NCP) arising from the discretization of the infinite-dimensional free

boundary problem governing the valuation of American options under transaction

costs. The NCP is approximated by a penalty equation containing a penalty term.

We show that the mapping involved in the system is continuous and strongly mono-

tone. Thus, the unique solvability of both the NCP and the penalty equation and

the exponential convergence of the solution to the penalty equation to that of the

NCP are guaranteed by an existing theory. Numerical results will be presented to

demonstrate the convergence rates and usefulness of this penalty method.

The results from this chapter have been submitted for publication in [41].

45

46 Chapter 3

3.2 Introduction

In the previous chapter, we discussed the numerical solution for nonlinear PDE

arising from pricing European option with transaction cost. In a complete market

without transaction costs, Black-Scholes [1] proposed a partial differential equation

pricing model for European options. However, when trading in the bond or/and

stock involves transaction cost, the Black-Scholes option pricing model does not

hold anymore. To overcome this difficulty, various models have been proposed to

price European options under transaction costs [4–8]. All these models give rise

to a nonlinear Black-Scholes equation. There are also utility-maximization based

models for determining the so-called reservation prices of European and American

options under transaction costs [9–12]. These models are of the form of a set of

Hamilton-Jacobi-Bellman equations even for European options.

The aforementioned models can hardly be solvable analytically. In practice, approx-

imate solutions to such a model are always sought. In [22] we proposed an upwind

finite difference method for the nonlinear Black-Scholes equation governing Euro-

pean option pricing proposed in [7] and proved the convergence of the numerical

scheme. However, how to price American put options of this type still remains an

open problem. In this work, we will present a numerical technique for solving the

infinite-dimensional nonlinear complementarity problem (NCP), or equivalently a

nonlinear variational inequality, governing American option valuation under trans-

action costs involving the nonlinear Black-Scholes operator developed for European

option valuation in [7]. The infinite dimensional NCP is first discretized by the

numerical scheme proposed and analyzed in [22] for the nonlinear Black-Scholes

equation. A power penalty method is then proposed for the NCP in infinite dimen-

sions arising from the discretization. We show that the mapping defining the NCP is

locally Lipschitz continuous and strongly monotone so that an existing convergence

theory applies to our problem. Numerical results will then be presented to demon-

strate the convergence rates and usefulness of the method. Although we use the

Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 47

nonlinear Black-Scholes operator in [7], the principle developed in this work applies

to other nonlinear Black-Scholes’ operators such as those in [4–6, 8].

For an American vanilla put option without transaction cost in a complete market,

its value v(S, t) as a function of its underlying asset/stock price S and time t is

governed by the following linear complementarity problem (cf., [26]) involving the

Black-Scholes differential operator:

L0v(S, t) := −∂v∂t− 1

2σ2

0S2 ∂

2v

∂S2− rS ∂v

∂S+ rv ≥ 0, (3.1)

v(S, t)− u∗(S) ≥ 0, (3.2)

L0v(S, t) · [v(S, t)− u∗(S)] = 0, (3.3)

for (S, t) ∈ (0, Smax)× [0, T ) satisfying the payoff/terminal and boundary conditions

v(S, T ) = u∗(S), S ∈ (0, Smax),

v(0, t) = u∗(0) = K, t ∈ (0, T ],

v(Smax, t) = u∗(Smax) = 0, t ∈ (0, T ],

where Smax K is a positive constant defining a computational upper bound on S,

K is the strike price of the option, σ0 is a constant volatility of the asset, r > 0 is a

constant risk-free interest rate, and u∗(S) is the payoff function given by

u∗(S) := max(K − S, 0). (3.4)

In the presence of transaction costs and under the transformation τ = T−t, the value

of an American Vanilla put option, u(S, τ) is governed by the following nonlinear

complementarity problem:

L(u)(S, τ) :=∂u

∂τ− 1

2σ2

(τ, S,

∂u

∂S,∂2u

∂S2

)S2 ∂

2u

∂S2− rS ∂u

∂S+ ru ≥ 0, (3.5)

u(S, τ)− u∗(S) ≥ 0, (3.6)

L(u)(S, τ) · [u(S, τ)− u∗(S)] = 0, (3.7)

48 Chapter 3

for (S, τ) ∈ (0, Smax) × [0, T ), where σ is the modified volatility as a nonlinear

function of τ, S, ∂u∂S

and ∂2u∂S2 , and u∗(S) is the payoff function defined in (3.4). The

initial and boundary conditions become

u(S, 0) = u∗(S), S ∈ (0, Smax), (3.8)

u(0, τ) = K, τ ∈ (0, T ], (3.9)

u(Smax, τ) = 0, τ ∈ (0, T ]. (3.10)

We comment that different types of American options have different payoff functions.

The most common one is the payoff function for Vanilla American put options defined

in (3.4). For clarity, we only consider this type of payoff functions in this work and

the results to be presented can also be used for other types of payoff functions as

well.

As discussed in previous chapter, various models for the nonlinear volatility have

been proposed, for example [4, 5, 7, 8, 23]. In this work we will mainly consider the

nonlinear volatility (2.7) proposed by Barles and Soner [7]. Unlike other models of

nonlinear volatility, this volatility model does not require any unrealistic conditions

on the parameters involved. Thus, in the present work we will focus on this model,

though the developed theory applies to other models as well.

3.3 The discretized problem and penalty formu-

lation

The nonlinear complementarity problem (3.5)–(3.7) with the initial and boundary

conditions (3.8)–(3.10) are in infinite-dimensions. It cannot be usually solved ana-

lytically except for some trivial cases. Therefore, it is necessary to discretize the

problem. Various discretization schemes have been developed for (3.1)–(3.3). An


upwind finite difference scheme is recently proposed in [22] for the nonlinear Black-

Scholes equation arising in pricing European option with transaction costs and a

convergence theory for this scheme is also established in the work. In what follows,

we will apply this scheme to (3.5)–(3.7).

To discretize (3.5)–(3.7), we first define a mesh for (0, Smax) × (0, T ). For a given

positive integer M , let (0, Smax) be divided into M sub-intervals

Ii = (Si, Si+1), i = 0, 1, . . . ,M − 1

satisfying

0 = S0 < S1 < . . . < SM = Smax.

Similarly, we divide (0, T ) into N sub-intervals with mesh nodes τnNn=0 satisfying

0 = τ0 < τ1 < . . . < τN = T.

For any i = 0, 1, . . . ,M−1 and n = 0, 1, ..., N−1, let hi = Si+1−Si and ∆τn = τn+1−

τn. For simplicity, we assume that the spatial mesh is uniform, i.e., hi = h = Smax/M

for i = 0, 1, ...,M − 1.

Given any matrix W n = (W n0 ,W

n1 , ...,W

nM)> and Wi = (W 0

i ,W1i , ...,W

Ni )> for i =

0, 1, ...,M and n = 0, 1, ..., N, we define the following finite difference operators on

the mesh defined above:


i

∆τn,

(δ+SW

n)(i) =W ni+1 −W n

i

h,

(δSSWn)(i) =

W ni−1 − 2W n

i +W ni+1

h2.

50 Chapter 3

Using these operators, we approximate (3.5)–(3.10) by the following finite difference

inequality system:

Ln+1i (U) := (δτ Ui)(n)− 1

2σ2(kn+1

i (δSSUn+1)(i))S2

i (δSSUn+1)(i)

−rSi(δ+S U

n+1)(i) + rUn+1i ≥ 0, (3.11)

Un+1i − U∗i ≥ 0, (3.12)

Ln+1i (U)(Un+1

i − U∗i ) = 0, (3.13)

for i = 1, 2, ...,M − 1 and n = 0, 1, ..., N − 1, where

kn+1i = erτn+1a2S2

i ,

U∗i = u∗(Si),

Un+1 = (Un+10 , Un+1

1 , ..., Un+1M )>,

Ui = (U0i , U

1i , ..., U

Ni )>,

U = (U0, U1, ..., UN)>,

and Uni denotes an approximation to u(Si, τn) to be determined for any feasible index

pair (i, n). Using (3.8)–(3.10), we define the initial and boundary conditions for the

above system as follows

U0i = U∗i , i = 0, 1, ...,M, (3.14)

Un0 = K, n = 0, 1, ..., N, (3.15)

UnM = 0, n = 0, 1, ..., N. (3.16)

For a detailed discussion of the above discretization scheme and its convergence for

the nonlinear Black-Scholes equation, we refer the reader to Chapter 2.

The finite-difference inequality system (3.11)–(3.24) and the boundary and initial

conditions (3.14)–(3.16) form a recursive system that determines the unknown vector


Un+1 = (Un+11 , ..., Un+1

M−1)> for n = 0, 1, ..., N − 1. This system can be re-written as

the following complementarity problem:

Problem 3.1. For n = 0, 1, . . . , N − 1, find Un+1 ∈ RM−1 such that

F n+1(Un+1) ≥ 0, (3.17)

Un+1 − U∗ ≥ 0, (3.18)

F n+1(Un+1)>(Un+1 − U∗) = 0, (3.19)

where F n+1 : RM−1 7→ RM−1 is defined by

F n+1(Un+1) = An+1(Un+1)Un+1 − 1

4τnUn −Bn+1, (3.20)

An+1(Un+1) =

βn+11 γn+1

1 0 . . . 0 0 0

αn+12 βn+1

2 γn+12 . . . 0 0 0

0 αn+13 βn+1

3 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . βn+1M−3 γn+1

M−3 0

0 0 0 . . . αn+1M−2 βn+1

M−2 γn+1M−2

0 0 0 . . . 0 αn+1M−1 βn+1

M−1

,

Uk =(Uk

1 , Uk2 , ..., U

kM−1

)>for k = n, n+ 1,

U∗ = (U∗1 , U∗2 , ..., U

∗M−1)>,

Bn+1 =(−αn+1

1 Un+10 , 0, . . . , 0,−γn+1

M−1Un+1M

)>.

The entries of An+1 are defined by

αn+1i (Un+1) = −1

2

σ2(kn+1i (δSSU

n+1)(i))S2i

h2, (3.21)

βn+1i (Un+1) =

1

∆τn+σ2(kn+1

i (δSSUn+1)(i))S2

i

h2+rSih

+ r, (3.22)

γn+1i (Un+1) = −1

2

σ2(kn+1i (δSSU

n+1)(i))S2i

h2− rSi

h. (3.23)

Let K = V ∈ RM−1 : V ≥ 0. It is easy to verify that K is a convex, closed, and

52 Chapter 3

self-dual cone in RM−1. Using this K, we define the following variational inequality

problem corresponding to Problem 3.1.

Problem 3.2. For n = 0, 1, . . . , N − 1, find Un+1 ∈ RM−1 such that Un+1−U∗ ∈ K

and (V − (Un+1 − U∗)

)>F n+1(Un+1) ≥ 0

for all V ∈ K.

The equivalence of Problem 3.1 and Problem 3.2 is given in the following proposition

Proposition 3.1. A vector Un+1 ∈ RM−1 is a solution to Problem 3.1 if and only

if it is a solution to Problem 3.2.

A proof to Proposition 3.1 can be found in [42]. Problem 3.2 is a finite-dimensional

nonlinear variational inequality. We can also prove that Problem 3.2 has a unique

solution by showing that F n+1(·) is strongly monotone and continuous. This discus-

sion is deferred to the next section.

Various numerical methods for solving this problem have been developed and for

an overview of these methods we refer the reader to the monograph [42]. Recently,

penalty methods have become increasingly popular as efficient computational tools

for solving complementarity problems arising in pricing American options in both

infinite and finite dimensions [43–47]. One of these method is the power penalty

[37, 48, 49] which has the merits that it has an exponential convergence rate and it

does not require introduction of extra unknowns. In this work, we apply the penalty

method to Problem 3.1 or 3.2.

Consider the following problem:

Problem 3.3. For n = 0, 1, . . . , N − 1, find Un+1ϑ ∈ RM−1 such that

F n+1(Un+1ϑ )− ϑ[U∗ − Un+1

ϑ ]1/k+ = 0, (3.24)

where ϑ > 1 and k ≥ 1 are constants, [z]+ = max(z, 0) and yp = (yp1, yp2, ..., y

pM−1)>

for any y ∈ RM−1 and p > 0.


This is a penalty equation which approximates (3.17)–(3.19). In this formulation, the

penalty term ϑ[U∗ − Un+1ϑ ]

1/k+ penalizes the positive part of (U∗ − Un+1

ϑ ). Loosely

speaking, when (3.18) is violated, (3.24) yields [U∗ − Un+1ϑ ]+ = ϑ−kF n+1(Un+1

ϑ ).

Therefore, if F n+1(Un+1ϑ ) is bounded, [U∗−Un+1

ϑ ]+ approaches zero as either ϑ or k

approaches positive infinity, so that (3.18) is satisfied within a tolerance depending

on the values of ϑ and k. In fact, it is proved in [37] that the error between the

solutions to Problems 3.1 (or 3.2) and 3.3 is of the order O(1/ϑk) when F n+1 is

continuous and strongly monotone. Therefore, in the next section we shall prove

that both of these conditions are satisfied by F n+1 so that the convergence result

applies to the solution to Problem 3.3.

3.4 Convergence of the penalty method

In this section we establish an upper bound for the distance between the solutions to

Problem 3.1 and Problem 3.3. While a convergence theory is established in [37, 49]

for a general nonlinear continuous and ξ-monotone function, our main development

in this section is to prove that the mapping involved in Problem 3.3 is strongly mono-

tone and locally Litpschitz continuous so that the exponential convergence result in

[37, 49] applies to Problem 3.3. We start this discussion with the monotonicity of

the nonlinear volatility.

Lemma 3.2. Let σ(k(τ, S)γ) = σ2(k(τ, S)γ)γ, where σ2 is the nonlinear volatility

defined in (2.7) and k(τ, S) = erτa2S2. Then, we have the following results:

1. For any γ ∈ (−∞,∞),d

dγσ(k(τ, S)γ) > 0. (3.25)

2. The derivative ddγσ(k(τ, S)γ) is continuous for all γ ∈ (−∞,∞), if we define

ddγσ(0) = 1.

54 Chapter 3

3. When γ 1, there exists a constant ρ > 0 such that

d

dγσ(k(τ, S)γ) ≤ ργ. (3.26)

Proof. We prove Items 1 and 3 simultaneously. From [22] we have

d

dγσ(kγ) = (1 + Ψ(z))

2√zΨ(z)

2√zΨ(z)− z

= 2√zΨ(z)Ψ′(z) (3.27)

where z = kγ. Since zΨ(z) > 0 and Ψ′(z) > 0 by (2.16), we have from the above

ddγσ(kγ) > 0 when z 6= 0 (or γ 6= 0 since k > 0).

We now need to look into limγ→0d

dγσ(kγ). When γ > 0, z > 0 and thus from (3.27)

and (2.14) we have

d

dγσ(kγ) = (1 + Ψ(z))

2√

Ψ(z)

2√

Ψ(z)−√z

=2(1 + Ψ(z))3/2√

1 + Ψ(z) +sinh−1

√Ψ(z)√

Ψ(z)

.

But when z → 0+, Ψ(z)→ 0+. Therefore,

limγ→0+

d

dγσ(kγ) = lim

Ψ→0+

2(1 + Ψ(z))3/2√1 + Ψ(z) +

sinh−1√

Ψ(z)√Ψ(z)

= 1.

Similarly, when γ < 0, we have from (3.27) and (2.15)

d

dγσ(kγ) =

2(1 + Ψ(z))3/2√1 + Ψ(z) +

sin−1√−Ψ(z)√

−Ψ(z)

.

Therefore,

limγ→0−

d

dγσ(kγ) = lim

Ψ→0−

2(1 + Ψ(z))3/2√1 + Ψ(z) +

sin−1√−Ψ(z)√

−Ψ(z)

= 1.

Therefore, ddγσ(kγ) > 0 for any γ ∈ (−∞,∞). This is (3.25).

Note that when γ 6= 0, from (3.27) and (2.8) we see that ddγσ is continuous. From the

above limits we see that ddγσ(kγ) is also continuous at γ = 0 if we define d

dγσ(0) = 1.


We now prove (3.26). Again from (3.27) we have, when z > 0,

d

dγσ(kγ) = (1 + Ψ(z))

2√

Ψ(z)

2√

Ψ(z)−√z

= (1 + Ψ(z))1

1−√z

2√

Ψ(z)

. (3.28)

From the definition of sinh−1(·) we have

limx→+∞

sinh−1√x√x+ 1

= limx→+∞

ln(√x+√x+ 1)√

x+ 1= 0.

Therefore, from (2.14) we see that when z → +∞, Ψ(z)→ +∞ and

limz→+∞

√z√Ψ

= limz→+∞

sinh−1√

Ψ√Ψ(Ψ + 1)

= 0.

Combining this with (3.28) we see that, when z = kγ 1,

d

dγσ(kγ) ≤ (1 + Ψ(z))

[1 +O

(√z√Ψ

)]≤ ρ(1 + k(τ, S)γ) ≤ ργ

as k is bounded on the bounded domain I×(0, T ), where ρ denotes a generic positive

constant, independent of γ. Since k > 0, kγ >> 1 implies γ >> 1. Thus, the above

estimate implies (3.26).

We comment that it is easily seen by a graph that ddγσ(kγ) is an increasing function

of γ, though a rigorous proof for this is difficult. Using Lemma 3.2, we establish the

continuity and strongly monotonicity of F n+1 as given below.

Theorem 3.3 (Continuity). For any n = 0, 1, ..., N − 1 and fixed h and ∆τn, the

mapping F n+1 is locally Lipschitz continuous on RM−1, i.e., there exists a constant

L > 0 such that

‖F n+1(V )− F n+1(W )‖2 ≤ L (1 + ||V ||2 + ||W ||2) ‖V −W‖2, ∀V,W ∈ RM−1.

(3.29)

56 Chapter 3

Proof. For any V ∈ RM−1, we let V n+1 = (Un+10 , V >, Un+1

M )>, where Un+10 and Un+1

M

are the boundary values defined in (3.15) and (3.16). The reason for the introduction

of V n+1 is because the boundary conditions Un+10 and Un+1

M are needed in the central

differences (δSSVn+1)(1) and (δSSV

n+1)(M − 1). From (3.15) and (3.16) we see that

both Un+10 and Un+1

M are independent of n and thus, in what follows, we write V n+1

as V .

Let Γ : RM+1 7→ RM−1 be defined by

Γ(V ) = ((δSSV )(1), (δSSV )(2), ..., (δSSV )(M − 1))> =: QV , (3.30)

where δSS is the central difference scheme defined in the previous section and

Q =1

h2

1 −2 1 . . . 0 0 0 0

0 1 −2 1 . . . 0 0 0...

......

. . ....

......

......

......

.... . .

......

...

0 0 0 . . . 1 −2 1 0

0 0 0 . . . 0 1 −2 1

(M−1)×(M+1)

.

From (3.20) and (3.21)–(3.23) we see that F n+1 can also be written as

F n+1(V ) = P nV − 1

2SΣ(Kn+1,Γ(V ))Γ(V )− 1

4τnV n − Bn+1, (3.31)


where

P n =

bn1 cn1 0 . . . 0 0 0

0 bn2 cn2 . . . 0 0 0

0 0 bn3 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . bnM−3 cnM−3 0

0 0 0 . . . 0 bnM−2 cnM−2

0 0 0 . . . 0 0 bnM−1

,

S = diag(S2

1 , S22 , ..., S

2M−1

),

Σ(Kn+1,Γ(V )) = diag(σ2

1(kn+11 Γ1), σ2

2(kn+12 Γ2), ..., σ2

M−1(kn+1M−1ΓM−1)

)with Γi = (δSSV )(i), i = 1, 2, ...,M − 1,

bni =1

4τn+rSih

+ r > 0,

cni = −rSih

< 0,

Kn+1 = (kn+11 , kn+1

2 , ..., kn+1M−1)>,

Bn+1 = (0, 0, . . . , 0,−cnM−1Un+1M )>.

(Note that Σ is a matrix, not a sum.) Therefore, we have, using the mean value

theorem,

F n+1(V )− F n+1(W ) = P n(V −W )

−1

2S[Σ(Kn+1,Γ(V ))Γ(V )− Σ(Kn+1,Γ(W ))Γ(W )

]= P n(V −W )− 1

2S

d

dU

[Σ(Kn+1,Γ(U))Γ(U)

]U=ξ

(V −W )

= P n(V −W )− 1

2S

d

dΓ

[Σ(Kn+1,Γ)Γ

]Γ(ξ)

dΓ(U)

dU

∣∣∣∣ξ

×(V −W )

= P n(V −W )− 1

2S

d

dΓ

[Σ(Kn+1,Γ)Γ

]Γ(ξ)

Q(V −W ),

58 Chapter 3

where ξ = (Un+10 , ξ>, Un+1

M )>, ξ is a vector given by ξ = θV + (1 − θ)W for an

unknown constant θ ∈ (0, 1), and

Q =dΓ(U)

dU=

1

h2

−2 1 0 . . . . . . 0

1 −2 1 . . . . . . 0...

. . . . . . . . ....

......

.... . . . . . . . .

...

0 0 . . . 1 −2 1

0 0 . . . 0 1 −2

(M−1)×(M−1)

by (3.30) and the definition of Q. Using Item 3 of Lemma 3.2 and Cauchy-Schwarz

inequality we have from the above

‖F n+1(V )− F n+1(W )‖2 ≤ ‖P n‖2‖V −W‖2

+1

2‖S‖2

∥∥∥∥ d

dΓ

[Σ(Kn+1,Γ)Γ

]Γ(ξ)

∥∥∥∥2

‖Q‖2‖V −W‖2

≤ C1‖V −W‖2 + C2||Γ(ξ)||2‖V −W‖2

≤(C1 + C2||Q||2||ξ||2

)‖V −W‖2

≤ L(1 + ||V ||2 + ||W ||2)||V −W ||2

for some positive constants C1, C2 and L, where || · ||2 denotes either the l2-norm

or subordinate matrix norm associated with the l2-norm on RM−1 or RM+1. Hence,

F n+1 is locally Lipschitz continuous on RM−1.

We comment that although L is a constant, it does depend on h and ∆τn, as P n, Q

and Q depend on h and ∆τn. We now show that F n+1 is strongly monotone in the

following theorem.

Theorem 3.4 (Monotonicity). The mapping F n+1 is strongly monotone on RM−1,

i.e., there exists a constant α > 0 such that

(V −W )>(F n+1(V )− F n+1(W )) ≥ α‖V −W‖22, ∀V,W ∈ RM−1, (3.32)


for n = 0, 1, . . . , N − 1.

Proof. In this proof we follow the notation used in Theorem 3.3. For any V,W ∈

RM−1, from (3.31) we have

(V −W )>(F n+1(V )− F n+1(W ))

= (V −W )>(P n(V −W )− 1

2S[Σ(Kn+1,Γ(V ))Γ(V )− Σ(Kn+1,Γ(W ))Γ(W )

])= (V −W )>P n(V −W ) +

1

2

(V −W

)>S

d

dΓ

(Σ(Kn+1,Γ)Γ

)Γ(ξ)

(−Q)(V −W ),

(3.33)

where ξ = θV + (1− θ)W for some θ ∈ [0, 1]. In the last step of the above, we used

the mean value theorem and the chain rule as in the proof of Theorem 3.3. Let us

consider the first term on RHS of (3.33). From the definition of P n in the proof of

Theorem 3.3 we see that it can be decomposed into:

(V −W )>P n(V −W ) = (V −W )>((

1

4τn+ r

)I +

r

hΦ

)(V −W ),

where I is the (M − 1) × (M − 1) identity matrix and Φ the the following upper

triangular matrix:

Φ =

S1 −S1 0 . . . 0 0 0

0 S2 −S2 . . . 0 0 0

0 0 S3 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . SM−3 −SM−3 0

0 0 0 . . . 0 SM−2 −SM−2

0 0 0 . . . 0 0 SM−1

.

60 Chapter 3

Clearly, Φ is positive-definite, since Si > 0 for i = 1, 2, ...,M−1. Therefore, we have

(V −W )>P n(V −W ) =

(1

4τn+ r

)(V −W )>I(V −W )

+r

h(V −W )>Φ(V −W )︸︷︷︸

≥0

≥(

1

4τn+ r

)‖V −W‖2

2

≥ C1‖V −W‖22 (3.34)

for some constant C1 > 0.

Now, let us consider the second term on the RHS of (3.33). Let

Rn+1(ξ) := Sd

dΓ

(Σ(Kn+1,Γ)Γ

)Γ(ξ)

(−Q)

=1

h2× diag(an+1

1 , an+12 , ..., an+1

M−1)

2 −1 0 . . . 0 0 0

−1 2 −1 . . . 0 0 0

0 −1 2 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . 2 −1 0

0 0 0 . . . −1 2 −1

0 0 0 . . . 0 −1 2

,

(3.35)

where

an+1i =

d

dΓ

[σ(kn+1

i Γ)Γ]

Γ(ξ)S2i > 0

for i = 1, 2...,M − 1 and any V,W ∈ RM−1 by Lemma 3.2. Clearly, Rn+1k is a

tridiagonal symmetric matrix. We now show that it is positive-definite.


For any k = 1, 2, ...,M − 1, it is easy to show that the kth-order leading principal

minor of the last matrix in (3.35) is

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

2 −1 0 . . . 0 0 0

−1 2 −1 . . . 0 0 0

0 −1 2 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . 2 −1 0

0 0 0 . . . −1 2 −1

0 0 0 . . . 0 −1 2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣k×k

= k + 1.

Hence, from this and (3.35) we see that the kth-order leading principal minor of

Rn+1 is given by

|Rn+1k | =

( 1

h2

)k ∣∣diag(an+11 , an+1

2 , ..., an+1k )

∣∣

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

2 −1 0 . . . 0 0 0

−1 2 −1 . . . 0 0 0

0 −1 2 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . 2 −1 0

0 0 0 . . . −1 2 −1

0 0 0 . . . 0 −1 2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣k×k

.

=( 1

h2

)k× an+1

1 × an+12 × . . .× an+1

k × (k + 1)

> 0

for all k = 1, 2, . . . ,M − 1. Therefore, using Sylvester’s criterion [50] we have that

Rn+1 is a positive definite matrix, and thus the last term in (3.33) satisfies

1

2(V −W )>Rn+1(ξ)(V −W ) ≥ 0 (3.36)

for any V,W ∈ RM−1.

62 Chapter 3

Using (3.34) and (3.36), we finally have from (3.33)

(V −W )>(F n+1(V )− F n+1(V )) ≥ α‖V −W‖22

for some constant α > 0. Hence F n+1 is strongly monotone.

Remark 3.5. Under (3.29) and (3.32), it is easy to show that Problem 3.2 has a

unique solution (cf. [42]). Also, Problem 3.1 is uniquely solvable by Proposition 3.1.

Moreover, because ϑ[·]1/k+ is also monotone for any ϑ > 1 and k ≥ 1, the variational

problem corresponding to Problem 3.3 also has a unique solution and hence (3.24)

is uniquely solvable. The convergence of the solution to Problem 3.3 to that to

Problem 3.1 (or equivalently Problem 3.2) is given in the following theorem.

Theorem 3.6. For given mesh sizes h and 4τn, let Un+1 and Un+1ϑ be the solutions

to Problem 3.1 and Problem 3.3, respectively, for n = 0, 1, ..., N − 1. There exists a

constant C > 0, independent of ϑ, such that

∥∥Un+1 − Un+1ϑ

∥∥2≤ C

ϑk(3.37)

for any ϑ > 1, k ≥ 1 and n = 0, 1, ..., N − 1.

Proof. For any given h and 4τn, Theorems 3.3 and 3.4 hold. Therefore, (3.37) is a

consequence of Theorem 3.1 of [37].

Remark 3.7. We comment that (3.24) is a non-smooth nonlinear system in Un+1ϑ for

each n = 0, 1, ..., N−1, because of the penalty term. In fact, when k > 1, the penalty

term in (3.24) is continuous, but not Lipschitz continuous. In the case k = 1, i.e.,

the linear penalty method, a non-smooth Newton’s method can be used for (3.24).

When k > 1, we may use a smoothing technique proposed in [37, 45] to locally

smooth out the penalty term near zero and use a Newton-like algorithm to solve the

resulting system. An analytic expression for the Jacobian matrix of F n+1(Un+1) is

derived in [22] and it is trivial to derive the Jacobian matrix corresponding to the

penalty term.


Remark 3.8. We also remark that although the convergence of O(ϑ−k)-order for

Un+1ϑ is established in Theorem 3.6, this rate is not uniform in the spatial dimensions

as the constant C in (3.37) depends inversely on h, as can be seen from the proofs

of Theorems 3.3 and 3.4. In fact, it is shown in [45] that in the limiting case that

discretized system (3.11)–(3.24) becomes continuous and that σ2 is a constant, the

rate of convergence is O(ϑ−k/2), rather than O(ϑ−k). The reason for the difference

between finite and infinite dimensional cases is because in finite dimensions, all the

norms are equivalent which is not true in infinite dimensions.

3.5 Numerical Results

In this section we demonstrate the rates of convergence and numerical performance

of our method using the following test problem

Test Problem: American Vanilla Put Option under transaction costs with system

parameters: r = 0.1, σ0 = 0.2, K = 40, T = 1, Smax = 80 and a = 0.02.

Let us first investigate the rate of convergence computationally, which requires an

exact or reference solution. Since the exact solution to this problem is unknown,

we use the numerical solution to (3.24) with ϑ = 106 on the uniform mesh with

M = 20480 (h = 1256

) and N = 10240 (4τn = 110240

) as an “exact” or reference

solution.

The problem is then solved on the uniform mesh with M = 320 (h = 14) and

N = 160 (4τn = 1160

) using ϑn = 10 × 2n for n = 0, 1, 2, 3, 4 when k = 1. The

distances between the reference and the numerical solutions in Euclidean norm are

computed and listed in Table 3.1 in which we also list the ratios of the errors from

two consecutive values of ϑ. From Theorem 3.6 we see that, theoretically, the ratio

is ϑn+1/ϑn = 2(n+1)k/2nk = 2k. Thus, the computed ratios in Table 3.1 match the

theoretical one well for k = 1. The computed errors and ratios of the numerical

solutions corresponding to k = 2 and 3 are given in Table 3.2. Again, the ratios are

64 Chapter 3

ϑ 10 20 40 80 160Error 0.23371 0.12326 0.06330 0.03213 0.01632Ratio – 1.90 1.95 1.97 1.97

Table 3.1: Computed rates of convergence in ϑ when k = 1 and a = 0.02

ϑ 5 10 20 40 80k = 2 Error 0.30726 0.09584 0.02535 0.00648 0.00203

Ratio - 3.21 3.78 3.91 3.19k = 3 Error 0.29554 0.03936 0.00508 0.00090 0.00022

Ratio - 7.51 7.75 5.64 4.09

Table 3.2: Computed rates of convergence in ϑ when k = 2, 3 and a = 0.02

k 1 2 3Error 0.12326 0.02535 0.00508Ratio - 4.86 4.99

Table 3.3: Computed rates of convergence in k when ϑ = 20

close to the theoretical value 4 and 8 respectively for k = 2 and 3, except for the last

two numbers when k = 3. The reason that the computed ratios are smaller than

the respective theoretical ones is mainly because when ϑ is large, the discretization

errors between the reference and the numerical solutions dominate the errors due

to the penalty parameters ϑ and k. From Remark 3.3 we also see that the rate of

convergence may be smaller than that in (3.37) when the number of mesh points is

large.

To show the convergence in k of the method, we choose ϑ = 20 and calculate the

errors in numerical solution for k = 1, 2, 3. The computed errors and ratios of errors

from two consecutive values of k are listed in Table 3.3. From Theorem 3.6 we see

that the ratio is theoretically a constant Cϑ, since C depends on k. Therefore, from

Table 3.3 we see that the computed ratios are close to a constant, coinciding with

the theory.

To further demonstrate the performance we plot in Figure 3.1, Figure 3.2 and Figure

3.3 the Greeks, ∆ = ∂V∂S

and Γ = ∂2V∂S2 , of the original option price V (S, t) and the


020

4060

80

00.2

0.40.6

0.81

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Time

Delta of V for k = 3, lambda = 20

Stock Price

Del

ta V

Figure 3.1: ∆

020

4060

800

0.5

1

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time

Gamma of V for k = 3, lambda = 20

Stock Price

Gam

ma

V

Figure 3.2: Γ

constraint V − V ∗ computed using k = 3 and ϑ = 20. From the plots of ∆ and Γ,

we can see the free boundary or the optimal exercise curve is very well captured by

our method. Also from the graphs we can see that the constraint is always (up to a

tolerance) satisfied.

To see the difference between the European and American options, we plot the

computed values of both of the options at t = 0 (or τ = T ) in Figure 3.4. From the

figure it is clear that the American put option is more valuable than the European

put option as expected. From Figure 3.4 we also see that the price of the American

66 Chapter 3

020

4060

80

0

0.5

1−0.5

0

0.5

1

1.5

2

2.5

3

Stock Price

The value of V−V* for k = 3, lambda = 20

Time

V −

V*

Figure 3.3: V − V ∗

option touches the lower bound V ∗ when in the sub-interval from 0 to the point on

the optimal exercise curve at the cross-section t = 0.

Finally, let us investigate the influence of the transaction cost parameter a on the

option price. To see this, we plot the values of the option at t = 0 (or τ = T ) for

three different values of a on the interval [30, 60] in Figure 3.5 in which the curve for

a = 0 is the price of the American put option without transaction cost. From this

figure we see that the price of the option increases as the transaction parameter a

increases as expected in practice.

3.6 Conclusion

In this chapter we proposed and analyzed a nonlinear penalty method for the solution

of finite-dimensional nonlinear complementarity problem. We have shown that the

system is continuous and strongly monotone and also have shown that the solution to

the penalty equation converges to that of NCP at an arbitrary rate depending on the

choice of parameters in the penalty term. Numerical experiments were performed

to confirm the theoretical results. The numerical results showed agreement with the

theoretical ones.


0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

Stock Price

Put

Opt

ion

Pric

e

American Put Option v.s. European Put Option

: American Option (lambda = 20): European Option (lambda = 0)

Figure 3.4: Prices of the American and European put options with a = 0.02 att = 0.

30 35 40 45 50 55 600

1

2

3

4

5

6

7

8

9

10

Stock Price

Put

Opt

ion

Pric

e

American Put Option Price for Different Transaction Cost Parameter

: a = 0: a = 0.02: a = 0.05

Figure 3.5: Prices of the American and European put option with a = 0.02.

Chapter 4

Numerical scheme for pricing

option with transaction costs

under jump diffusion processes

4.1 Summary

In this chapter we develop a numerical method for a nonlinear partial integro-

differential equation and a partial integro-differential complementarity problem aris-

ing from European and American option valuations respectively with transaction

costs when the underlying assets follow a jump diffusion process. The method is

based on an upwind finite difference scheme for the spatial discretization and a fully

implicit time stepping scheme. The fully discretized system is solved by a Newton it-

erative method coupled with an FFT for the computation of the discretized integral

term. The constraint in the American option model is imposed by adding a penalty

term to the original partial integro-differential complementarity problem. We prove

that the system matrix from this scheme is an M -matrix and that the approximate

solution converges unconditionally to the viscosity solution to the PIDE by showing

69

70 Chapter 4

that the scheme is consistent, monotone, and unconditionally stable. Numerical re-

sults will be presented to demonstrate the convergence rates and usefulness of this

method.

4.2 Introduction

In a complete market without transaction costs, Black-Scholes [1] proposed a partial

differential equation pricing model for European options. This model is based on

the assumptions that the price of the underlying stock price follows a geometric

Brownian motion with a log-normal diffusion and constant volatility. However, there

is evidence that these assumptions are not consistent with that of market price

movements in practice, which is often called the volatility skew or smile [51]. A

number of models have been proposed to remedy this problem. These improved

models can be categorized into three classes: stochastic volatility model [16, 17], the

deterministic volatility function model [52], and the jump diffusion model [18, 28,

53, 54]. Among them, the jump diffusion model is very popular.

It has been shown that the price of an option under a jump diffusion process without

transaction costs satisfies a linear partial integro-differential equation (PIDE). Since

exact solutions can hardly be found, numerical approximations to the linear PIDE is

always sought in practice in order to determine the price of such an option. This is

challenging as the PIDE involves a nonlocal integration term. In [20, 54], the authors

treat the integral term explicitly and the remaining terms implicitly. However, the

method is only conditionally stable. Operator splitting method coupled with an

FFT for the evaluation of the integral term has been used in [55]. This method is

unconditionally stable and two order accurate. There are various other methods for

the linear PIDE such as [56–58].

In [59, 60] the authors showed that, in the presence of transaction costs, the price of

a European option whose underlying asset price satisfies a jump diffusion process is

governed by a nonlinear PIDE. Although some theoretical properties of the nonlinear

Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 71

model are established in [59, 60], to our best knowledge, no numerical methods

for this nonlinear PIDE in their original asset variable can be found in the open

literature. In this work, we propose a numerical method based on some conventional

discretization schemes for the time and spatial derivatives as well as the integral

term. We show that the system matrix of the fully discretized equation is an M -

matrix and establish a convergence theory for the numerical method. To evaluate

the discretized integral term, we use the FFT technique, and a Newton iterative

method coupled with a regular operator splitting is used to avoid the inversion of

the dense matrix arising from the non-local term. The rest of this work is organized

as follows.

In the next section, we will give a brief account of the PIDE model. In Section 4.4,

we will present some discretization schemes for the PIDE. A convergence analysis

is given in Section 4.5 in which we show that the numerical scheme is consistent,

unconditionally stable and monotone. In Section 4.6, we propose a Newton iterative

algorithm for solving the nonlinear algebraic system and show it is convergent. In

Section 4.7, we present some numerical experimental results to demonstrate the

rates of convergence and usefulness of the numerical method for solving a number

of model problems.

4.3 The continuous model

Consider a European option with strike price K and expiry time T on an asset whose

price S follows the following stochastic differential equation

dS

S= (ν − λκ) dt+ σ0 dZ + (η − 1) dq,

where dZ is an increment of the standard Gauss-Wiener process and dq is the

independent Poisson process with deterministic jump intensity λ. Here, σ0 is the

volatility, ν is the drift rate, η − 1 is an impulse function producing a jump from S

to Sη, and κ = E(η − 1) with E an expectation operator. Following [61], it is easy

72 Chapter 4

to show that the value V (S, t) of the European option at any t ∈ [0, T ) and S > 0

when transactions do not incur any costs satisfies the following linear PIDE:

Vτ =1

2σ2

0S2VSS + (r − λκ)SVS − (r + λ)V + λ

∫ ∞0

V (Sη)g(η) dη,

for (S, τ) ∈ [0,+∞) × [0, T ), where r is the risk-free interest rate, τ = T − t,

and g(η) is the probability density function of the jump amplitude η, satisfying∫∞0g(η) dη = 1. Clearly, various choices of g are available, but for clarity, we only

consider, in this work, the following lognormal density function in Merton’s model:

g(η) =1√

2πσJηexp

(−(ln η − µ)2

2σ2J

),

in which case, κ = E(η − 1) = exp(µ + σ2J/2) − 1, where µ and σJ determine the

mean and variance of the jumps.

When transactions of selling and buying the asset involve costs, the value of a

European option, V (S, τ) is governed by the following nonlinear PIDE:

Vτ =1

2σ2 (τ, S, VS, VSS)S2VSS + (r − λκ)SVS (4.1)

−(r + λ)V + λ

∫ ∞0

V (Sη)g(η) dη,

where σ is the modified volatility as a function of τ, S, VS and VSS. Assuming that

the transaction cost is proportional to the value of the transaction and the portfolio

is revised every ∆t units of time, where ∆t denotes a small, but non-infinitesimal,

fixed time interval, Mocioalca [59] showed that the option price is the solution to

(4.1) with the modified volatility models

σ2 = σ20 +

2

∆tρE

(∣∣∣∆SS

∣∣∣) and σ2 = σ20 −

2

∆tρE

(∣∣∣∆SS

∣∣∣)

for, respectively, long and short positions in the option, where ρ is the transaction

cost parameter, measured as a fraction of the volume of transactions. It is shown in


[4] that these can be written as

σ2(S, VSS) = σ20

(1 + Le sign(VSS)

), (4.2)

where Le is the Leland number given by

Le =

√2

π

(ρ

σ0

√∆t

).

Clearly, Le has to be such that σ2(VSS) > 0, that is 0 ≤ Le < 1.

In [60], the author assumes that the transaction costs behave like a non-increasing

positive linear function, h(x) = a − bx, where a, b ≥ 0 are two transaction cost

parameters. Using this cost function, the author derived the following modified

volatility

σ2(S, VSS) = σ20 (1− k1sign(VSS) + k2SVSS) (4.3)

with

k1 =a

σ0

√2

π∆t> 0, k2 = 2b > 0.

We assume that k1, k2 and VSS satisfy the following inequalities

1− k1 > 0 and 1− k1sign(VSS) + 2k2SVSS > 0 ∀S ∈ (0, Smax). (4.4)

Under these conditions, we can easily see from (4.3) that σ2(S, VSS) > 0. Note that

VSS ≥ 0 is usually satisfied for conventional European and American vanilla options

as used in [60] and the 2nd condition in (4.4) extend this condition to the case that

VSS can be negative.

If we define

LV = Vτ −1

2σ2 (τ, S, VS, VSS)S2VSS − (r − λκ)SVS

+(r + λ)V − λ∫ ∞

0

V (Sη)g(η) dη,

74 Chapter 4

then the American option pricing problem can be stated as the following nonlinear

partial integro-differential complementarity problem (PIDC):

LV ≥ 0, (4.5)

V − V ∗ ≥ 0, (4.6)

LV · (V − V ∗) = 0, (4.7)

where V ∗ denotes a payoff function.

To solve (4.1) and (4.5)–(4.7), boundary and payoff conditions need to be defined.

There are various types of boundary and initial conditions and payoff functions

depending on the type of contingent claims. For European vanilla call and put

options, they are given by

V (S, τ = 0) = g1(S) :=

max(S −K, 0) for a call

max(K − S, 0) for a put,S ∈ (0, Smax), (4.8)

V (0, τ) = g2(τ) :=

0 for a call

Ke−rτ for a put,, τ ∈ (0, T ], (4.9)

V (Smax, τ) = g3(τ) :=

Smax −Ke−rτ for a call

0 for a put,, τ ∈ (0, T ], (4.10)

where K denotes the strike price of the option and Smax K is a positive constant

defining a computational upper bound on S. For American put options, they are

given by

V (S, τ = 0) = V ∗(S) = g1(S) := max(K − S, 0), (4.11)

V (0, τ) = g2(τ) := K, (4.12)

V (Smax, τ) = g3(τ) := 0. (4.13)


4.4 Discretization of the PIDE

We divide discussion into two parts: the discretization of the integral term and that

of the differential terms.

4.4.1 Discretization of the integral

We follow the idea in [57] to approximate the integral in (4.1) numerically. First, we

use a logarithmic transformation to transform the integral into a correlation integral

so that it can be computed by the FFT algorithm. Let x = lnS and y = ln η. Then

Q(S) :=

∫ ∞0

V (Sη)g(η) dη =

∫ ∞−∞

V (x+ y)f(y)dy =: Q(x),

where f(y) = g(ey)ey and V (z, τ) = V (ez, τ). Let R be partitioned uniformly into

xi = i∆x for i = 0,±1,±2, . . . where ∆x > 0 is a constant step length. Let yi = xi

for all i = 0,±1,±2, . . . and ∆y = ∆x. Note that the density function g(η) → 0

exponentially as |η| → ∞. So does f(y). Therefore, we approximate the correlation

integral at xi by the following finite sum

Q(xi) ≈ Qi :=

J2∑

j=−J2

+1

Vi+jfj∆y (4.14)

for all feasible i, where J >> 0 is a positive even integer, Vk denotes a nodal

approximation of V (k∆x) for any k and

fj =1

∆y

∫ (j+ 12

)∆y

(j− 12

)∆y

f(y) dy.

Eq. (4.14) defines nodal approximations to Q(x) at the mesh nodes of the trans-

formed region using nodal values of V . However, as can be seen in the next sub-

section, we will discretize (4.1) in the original solution domain (0, Smax). Therefore,

76 Chapter 4

it is necessary to define nodal approximations to V (S) using those of V (x) de-

fined in (4.14) which requires interpolating V using nodal values of V and Q using

nodal values of Q. Given a set of nodes S1, S2, . . . , Sn, . . . on the S-axis satisfy-

ing 0 = S1 < S2 < . . . < Sn < . . ., let Vi be a nodal approximation to V (Si) for

i = 1, 2, .... We use the following steps to approximate Q(Si).

1. Approximation of Vj by the linear interpolation of Vi.

For any integer j, let p(j) be the index such that

Sp(j) ≤ exj ≤ Sp(j)+1.

Then, we define the following approximation to Vj

Vj = φp(j)Vp(j) + (1− φp(j))Vp(j)+1, (4.15)

where

φp(j) =exj − Sp(j)

Sp(j)+1 − Sp(j).

2. Approximation of Qi by the linear interpolation of Qj.

For any i = 1, 2, . . ., let q(i) be the integer satisfying xq(i) ≤ lnSi ≤ xq(i)+1. Then,

Qi is defined as

Qi = ψq(i)Qq(i) + (1− ψq(i))Qq(i)+1, (4.16)

where

ψq(i) =lnSi − xq(i)xq(i)+1 − xq(i)

.

From the definition, it is clear that 0 ≤ φp(j) ≤ 1 and 0 ≤ ψq(i) ≤ 1. Using (4.14),

(4.15) and (4.16), it is easy to verify that Qi is defined by

Qi =

J2∑

j=−J2

+1

Πij(V )fj∆y, (4.17)


where

Πij(V ) = ψq(i)[φp(q(i)+j)Vp(q(i)+j) + (1− φp(q(i)+j))Vp(q(i)+j)+1] + (1− ψq(i))

×[φp(q(i)+1+j)Vp(q(i)+1+j) + (1− φp(q(i)+1+j))Vp(q(i)+1+j)+1]. (4.18)

The operator Πji (V ) is linear in V . This approximation will be used in the spatial

discretization of the problem as discussed below.

4.4.2 Full discretization

To discretize (4.1) with nonlinear volatility (4.3) and boundary and initial conditions

(4.8)–(4.10), we first define a mesh for (0, Smax) × (0, T ). For a given positive even

integer M , let (0, Smax) be divided uniformly into M sub-intervals with mesh nodes

Si = (i− 1)h, i = 1, . . . ,M + 1,

where h = Smax/M . Similarly, we divide (0, T ) into N sub-intervals with mesh nodes

τnN+1n=1 satisfying

0 = τ1 < τ2 < . . . < τN+1 = T.

We put ∆τn = τn+1 − τn.

Given any matrices W n = (W n1 ,W

n2 , ...,W

nM+1)> and Wi = (W 1

i ,W2i , ...,W

N+1i )>

for i = 1, ...,M + 1 and n = 1, ..., N + 1, we define the following finite difference

operators on the mesh defined above:


i

∆τn,

(δ+SW

n)(i) =W ni+1 −W n

i

h, (δ−SW

n)(i) =W ni −W n

i−1

h,

(δSSWn)(i) =

W ni−1 − 2W n

i +W ni+1

h2.

78 Chapter 4

Using these operators and Qi defined in (4.17) for a sufficiently large even integer

J , we approximate (4.1) by the following finite difference equation:

(δτVi)(n)− 1

2σ2((δSSV

n+1)(i))S2i (δSSV

n+1)(i)−(

1 + sign(r)

2

)rSi(δ

+S V

n+1)(i)

−(

1− sign(r)

2

)rSi(δ

−S V

n+1)(i) + (r + λ)V n+1i − λQi = 0

(4.19)

for i = 2, ...,M and n = 1, ..., N , where

r = r − λκ,

V n+1 = (V n+11 , V n+1

2 , ..., V n+1M+1)>,

Vi = (V 1i , V

2i , ..., V

N+1i )>,

and V ni denotes an approximation to V (Si, τn) to be determined for any feasible

index pair (i, n). The discretization of VS is based on the so-called upwind finite

difference scheme. For a detailed discussion of the above discretization scheme and

its convergence for the nonlinear Black-Scholes equation without any jump, we refer

the reader to [22]. Using (4.8)–(4.10), we define the initial and boundary conditions

for the above system as follows

V 1i = g1(Si), i = 1, ...,M + 1, (4.20)

V n1 = g2(τn), n = 1, ..., N + 1, (4.21)

V nM+1 = g3(τn), n = 1, ..., N + 1. (4.22)

The system (4.19) can be rewritten as

αn+1i (V n+1)V n+1

i−1 + βn+1i (V n+1)V n+1

i + γn+1i V n+1

i+1 − λ∆y∑l

Πil(V

n+1)fl =1

∆τnV ni ,

(4.23)


where

αn+1i (V n+1) = −1

2

σ2((δSSVn+1)(i))S2

i

h2+(1− sign(r)

2

) rSih, (4.24)

βn+1i (V n+1) =

1

∆τn+σ2((δSSV

n+1)(i))S2i

h2+|r|Sih

+ r + λ, (4.25)

γn+1i (V n+1) = −1

2

σ2((δSSVn+1)(i))S2

i

h2−(1 + sign(r)

2

) rSih. (4.26)

In (4.23), the range = J/2 + 1 ≤ l ≤ J/2 of the sum for a sufficient large even

integer J is omitted for notation simplicity. The finite-difference system (4.23) along

with (4.20)–(4.22) form a recursive system that determines the unknown vector

V n+1 = (V n+12 , . . . , V n+1

M )> for n = 1, ..., N . From (4.17) and (4.18) we see that the

sum on the LHS of (4.23) is linear in V . Let D = (dij) be the (M − 1) × (M − 1)

matrix such that [DV n

]i

=M∑j=2

dijVnj =

∑l

Πil(V

n)fl∆y. (4.27)

Then, (4.23) can be written in a matrix form as follows

[An+1(V n+1)− λD

]V n+1 =

1

4τnV n +Bn+1, (4.28)

80 Chapter 4

where

An+1(V n+1) =

βn+12 γn+1

2 0 . . . 0 0 0

αn+13 βn+1

3 γn+13 . . . 0 0 0

0 αn+14 βn+1

4 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . βn+1M−2 γn+1

M−2 0

0 0 0 . . . αn+1M−1 βn+1

M−1 γn+1M−1

0 0 0 . . . 0 αn+1M βn+1

M

,

Bn+1 =

−αn+12 (V n+1)V n+1

1 + λ∆y(f−1Vn+1

1 + fM−1Vn+1M+1)

λ∆y(f−2Vn+1

1 + fM−2Vn+1M+1)

...

λ∆y(f2−MVn+1

1 + f2Vn+1M+1)

−γn+1M (V n+1)V n+1

M+1 + λ∆y(f1−MVn+1

1 + f1Vn+1M+1)

.

Clearly, Bn+1 contains contributions from the boundary conditions in (4.21)–(4.22).

For the system matrix of (4.28) we have the following results.

Theorem 4.1. For any n = 1, ..., N + 1, both An(V n) and An(V n) − λD are M-

matrices for any given V n.

Proof. To prove An defined above is an M -matrix, it suffices to show that

αni < 0, βni > 0, γni < 0, (4.29)

βni > |αni |+ |γni | (4.30)

for i = 2, ...,M .

From (4.24)–(4.26) we see that (4.29) is obviously true and that

βni = |αni |+ |γni |+1

4τn+ r + λ > |αni |+ |γni |,


since r ≥ 0, 14τn > 0, and λ ≥ 0. Therefore, (4.30) is also satisfied. Also, it is

obvious that An is irreducible and thus it is irreducibly diagonally dominant with

positive diagonal and non-positive off-diagonal entries. By [38], An is an M -matrix

for any given V n.

Now we investigate dij defined in by (4.27). From (4.17) and (4.18) we see that when

Vi = 1 for all i = 1, 2, ...,M + 1,

M∑j=2

dij =∑l

fl∆y. (4.31)

Since f is a probability density function, we have

∫ ∞−∞

f(y) dy =

∫ ∞0

g(η) dη = 1, f(y) = g(ey)ey ≥ 0. (4.32)

Therefore, it follows from (4.32) that

∑l

fl∆y ≤ 1 and fl ≥ 0.

From (4.31), we have

0 ≤ dij ≤ 1 and 0 ≤∑j

dij ≤ 1.

Using these inequalities we can conclude that all the off-diagonal entries of An(V n)−

λD are non-positive and

αni + βni + γni − λM∑j=2

dij =1

4τn+ r + λ− λ

M∑j=2

dij

=1

4τn+ r + λ

(1−

M∑j=2

dij

)

≥ 1

4τn+ r > 0. (4.33)

82 Chapter 4

This implies

βni − λdii =1

4τn+ r +

(1−

∑j

dij

)λ+ |αni |+ |γni |+ λ

∑j 6=i

dij

=1

4τn+ r + (1− dii)λ+ |αni |+ |γni |

> |αni |+ |γni |. (4.34)

Therefore, all the diagonal entries of An(V n) − λD are positive and the matrix is

strictly diagonally dominant. Hence, it is an M -matrix.

4.5 Convergence of the numerical scheme

In [60], the author shows the existence and uniqueness of the viscosity solution to

(4.1). In this section we will prove that the solution to (4.28) converges to the

viscosity solution to (4.1). It has been shown in [39] that the convergence of the

fully discretized system (4.28) to the viscosity solution of a full nonlinear 2nd-order

PDE is guaranteed if the discretization is consistent, stable and monotone. Thus, in

rest of this section we will prove the convergence of our numerical scheme by showing

that it satisfies these properties. For i = 2, 3, ...,M and n = 1, ..., N , introduce a

functional Hn+1i defined by

Hn+1i

(V n+1i , V n+1

i+1 , Vn+1i−1 , V

n+1j , V n

i

):= ηiV

n+1i−1 + ξiV

n+1i + ζiV

n+1i+1 − λ

∑j 6=i

dijVj

− 1

4τnV ni −

1

2S2i σ

2(Si, (δSSVn+1)(i))(δSSV

n+1)(i)

(4.35)

where

ηi =

(1− sign(r)

2

)rSih, ξi =

1

4τn+|r|Sih

+ r + (1− dii)λ

and ζi = −(

1 + sign(r)

2

)rSih.


Then, it is easy to see that (4.23) becomes

Hn+1i

(V n+1i , V n+1

i+1 , Vn+1i−1 , V

n+1j , V n

i

)= 0

for all feasible i and n. For this discretization scheme, we have the following lemma.

Lemma 4.2 (Monotonicity). The discretization (4.23) with σ defined in (4.2) is

monotone. Furthermore, (4.23) with σ defined in (4.3) is also monotone for all V n

such that (4.4) is satisfied with VSS replaced with (δSSVn+1).

Proof. To prove this theorem, we show that, for any ε > 0 and i = 2, 3, . . . ,M,

Hn+1i

(V n+1i , V n+1

i+1 + ε, V n+1i−1 + ε, V n+1

j + ε, V ni + ε

)≤

Hn+1i

(V n+1i , V n+1

i+1 , Vn+1i−1 , V

n+1j , V n

i

), (4.36)

Hn+1i

(V n+1i + ε, V n+1

i+1 , Vn+1i−1 , V

n+1j , V n

i

)≥

Hn+1i

(V n+1i , V n+1

i+1 , Vn+1i−1 , V

n+1j , V n

i

). (4.37)

Since ηi ≤ 0, ξi > 0, ζi ≤ 0, 0 ≤∑

j 6=i dij ≤ 1, λ ≥ 0, and 14τn > 0, the first five

terms on the RHS of (4.35) are respectively non-increasing in V n+1i−1 , increasing in

V n+1i , non-increasing in V n+1

i+1 , non-increasing in V n+1j , and decreasing in V n

i .

Let Ek = (0, 0, ..., 1︸︷︷︸kth

, 0, ..., 0)> be the (M − 1) × 1 column vector. From the

definition of δSS and (4.3) we have

(δSS(V n+1 + εEi−1 + εEi+1))(i) =(V n+1

i−1 + ε)− 2V n+1i + (V n+1

i+1 + ε)

h2

= (δSSVn+1)(i) +

2ε

h2< (δSSV

n+1)(i),(4.38)

(δSS(V n+1 + εEi))(i) =V n+1i−1 − 2(V n+1

i + ε) + V n+1i+1

h2

= (δSSVn+1)(i)− 2ε

h2> (δSSV

n+1)(i).(4.39)

Let us consider the nonlinear term on the RHS of (4.35). We discuss this term in

the following two cases corresponding to the choices of σ defined in (4.2) and (4.3).

84 Chapter 4

When σ is defined in (4.2), we have, for any S, z1 and z2,

1

σ20

[σ2(S, z1)z1 − σ2(S, z2)z2] = [(z1 − z2) + Le(sign(z1)z1 − sign(z2)z2)]

= [1 + Lesign(z1)](z1 − z2) + Le[sign(z1)− sign(z2)]z2.

=

C1(z1 − z2) z1 · z2 > 0,

(1 + Le)z1 + (Le− 1)z2 > 0 z1 > 0 > z2,

(1− Le)(z1 − z2)− 2Lez2 < 0 z1 < 0 < z2,

where C1 = (1 + Lesign(z1)) > 0, since 0 ≤ Le < 1.

Therefore, σ2(S, z)z defined in (4.2) is an increasing function in z.

When σ2(S, z) is by (4.3), we have

d

dz[(1− k1sign(z) + k2Sz)z] = (1− k1sign(z)) + 2k2Sz > 0

when z satisfies (4.4). Therefore, σ2(S, z)z is an increasing function in z on the set

defined by the 2nd inequality in (4.4).

Combining the monotonicity of σ2(S, z)z and the properties of the linear terms on

the RHS of (4.35) and (4.38) we have

Hn+1i

(V n+1i , V n+1

i+1 + ε, V n+1i−1 + ε, V n+1

j + ε, V ni + ε

)= ηi(V

n+1i−1 + ε) + ξiV

n+1i + ζi(V

n+1i+1 + ε)− λ

∑j 6=i

dij(Vn+1j + ε)

− 1

4τn(V n

i + ε)− 1

2S2i σ

2

(Si, (δSSV

n+1)(i) +2ε

h2

)((δSSV

n+1)(i) +2ε

h2

)≤ Hn+1

i

(V n+1i , V n+1

i+1 , Vn+1i−1 , V

n+1j , V n

i

).

This is (4.36). Similarly, using the monotonicity of σ2(S, z)z and (4.39) it is easy to

show that (4.37) also holds true. Hence, the discretization scheme is monotone.

The stability of the method is given in the following lemma.


Lemma 4.3 (Stability). For n = 1, 2, . . . ,M , let V n+1 = (V n+11 , (V n+1)>, V n+1

M+1)>,

where V n+1 is the solution to (4.28). Then, V n+1 satisfies

‖V n+1‖∞ ≤ max‖g1‖∞, ||g2||∞, ||g3||∞, (4.40)

where g1, g2 and g3 are the initial and boundary conditions defined in (4.20)–(4.22)

and ‖ · ‖∞ denotes the l∞-norm.

Proof. For any n = 1, ..., N , from (4.23) and (4.34) we have

(βn+1i − λdii)V n+1

i = −αn+1i V n+1

i−1 − γn+1i V n+1

i+1 + λΣj 6=idijVn+1j +

1

4τnV ni

for i = 2, ...,M . Recall that αn+1i ≤ 0, γn+1

i ≤ 0, βn+1i > 0, 0 ≤ dij ≤ 1, and

0 ≤ Σjdij ≤ 1.

From the above we get

(βn+1i − λdii)|V n+1

i | ≤ −αn+1i |V n+1

i−1 | − γn+1i |V n+1

i+1 |+ λΣj 6=idij|V n+1j |+ 1

∆τn|V ni |

≤ −αn+1i ‖V n+1‖∞ − γn+1

i ‖V n+1‖∞ + λΣj 6=idij‖V n+1‖∞

+1

∆τn‖V n‖∞

for i = 2, ...,M . We now consider the following two cases.

Case I: ‖V n+1‖∞ = |V n+1k | for an index k ∈ 2, ...,M.

In this case, the above estimate with i = k becomes

(αn+1k + βn+1

k + γn+1k − λΣjdkj)‖V n+1‖∞ ≤

1

4τn‖V n‖∞.

Therefore, using (4.33) we obtain from the above inequality

‖V n+1‖∞ ≤1/4τn

(αn+1k + βn+1

k + γn+1k − λΣjdkj)

‖V n‖∞ ≤ ‖V n‖∞

≤ ‖V n−1‖∞ ≤ · · · ≤ ‖V 1‖∞ ≤ ‖g1‖∞.

86 Chapter 4

Case II: ‖V n+1‖∞ = |V n+11 | or ‖V n+1‖∞ = |V n+1

M+1|.

In this case, from (4.20), (4.21) and (4.22) it is easy to see that

‖V n+1‖∞ ≤ max|V n+11 |, |V n+1

M+1| ≤ max‖g2‖∞, ‖g3‖∞.

Combining the above two cases we have (4.40).

The consistency of the numerical scheme is given in the following lemma:

Lemma 4.4 (Consistency). The discretization scheme (4.23) is consistent.

The proof is standard since both of the time and spatial discretization schemes

are standard and have been used extensively in the literature for 2nd-order partial

differential equations. Therefore, we omit the proof of this lemma. Combining the

above three lemmas we have the following convergence result.

Theorem 4.5. The solution to (4.28) converges to the viscosity solution to (4.1) as

(h,∆τ)→ (0+, 0+), where ∆τ = max1≤n≤N 4τn.

Proof. In [60] the authors show that if a discretization scheme for a fully nonlinear

2nd order PDE is monotone, stable and consistent, then the solution to the fully

discretized system converges to the viscosity solution to the PDE. Therefore, this

theorem is just a consequence of Lemmas 4.2, 4.3 and 4.4.

4.6 Solution of the nonlinear system

4.6.1 The European Case

Note that D arising from discretization of correlation product term is a dense ma-

trix. Therefore, the solution of (4.28) is computationally expensive because of the

inversion of D. To solve this, we use a Newton iterative method coupled with a


regular splitting technique used in [62]. To achieve this, we first write (4.28) in the

following form

Hn+1(V n+1) :=[An+1(V n+1)− λD

]V n+1 − 1

4τnV n −Bn+1 = 0.

Let

Hn+1(V n+1) = (hn+12 (V n+1), hn+1

3 (V n+1), . . . , hn+1M (V n+1))>.

Then, from (4.23) it is easy to see that the ith component of Hn+1(V n+1) is

hn+1i (V n+1) = αn+1

i V n+1i−1 + βn+1

i V n+1i + γn+1

i V n+1i+1 − λ

M∑j=2

dijVn+1j − 1

4τnV ni ,

where V n+11 and V n+1

M+1 are defined in (4.21) and (4.22). The Jacobian matrix of

Hn+1(V n+1), denoted by Jn+1(V n+1)− λD, is given by

Jn+1(V n+1) =

Jn+122 Jn+1

23 0 . . . 0 0 0

Jn+132 Jn+1

33 Jn+134 . . . 0 0 0

0 Jn+143 Jn+1

44 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . Jn+1(M−2)(M−2) Jn+1

(M−2)(M−1) 0

0 0 0 . . . Jn+1(M−1)(M−2) Jn+1

(M−1)(M−1) Jn+1(M−1)(M)

0 0 0 . . . 0 Jn+1(M)(M−1) Jn+1

(M)(M)

,

where Jn+1ij := ∂(An+1(V n+1))(i)

∂V n+1j

for all feasible i and j. D is as defined before.

For the Jacobian Jn+1(V n+1)− λD, we have the following results.

Theorem 4.6. For any given V n+1, both Jn+1(V n+1) and Jn+1(V n+1) − λD are

M-matrices.

Proof. For simplicity of notation, we omit the superscript n+ 1 in the proof of this

theorem.

88 Chapter 4

To show that J is an M -matrix, from [38] we see that it suffices to prove that Jii > 0,

Ji,i−1, Ji,i+1 ≤ 0, Jii ≥ |Ji,i−1|+|Ji,i+1| and Jii > |Ji,i−1|+|Ji,i+1| for at least one index

i. For volatility given by (4.2), it is easy to check that Jn+1(V n+1) = An+1(V n+1).

Thus, from Teorema 4.1, both J and J − λD are M -matrices.

Now we will prove that J and J − λD are M -matrices for volatility given by (4.3).

Let us first consider Ji,i−1. Using the definition of αn+1i in (4.24) and (4.3) we have

Ji,i−1 = αn+1i + V n+1

i−1

∂αn+1i

∂V n+1i−1

+ V n+1i

∂βn+1i

∂V n+1i−1

+ V n+1i+1

∂γn+1i

∂V n+1i−1

= αn+1i − σ2

0k2S3i

2h2

(δSSV

)(i)

= −1

2

σ20

(1− k1sign

(δSSV

)(i) + 2k2Si

(δSSV

)(i))S2i

h2+

(1− sign(r)

2

)rSih

≤ 0. (by (4.4))

Similarly it can be shown that

Ji,i =σ2

0

(1− k1sign

(δSSV

)(i) + 2k2Si

(δSSV

)(i))S2i

h2+

1

∆τn+|r|Sih

+ r + λ > 0

Ji,i+1 = −1

2

σ20

(1− k1sign

(δSSV

)(i) + 2k2Si

(δSSV

)(i))S2i

h2−(

1 + sign(r)

2

)rSih≤ 0.

From these expressions we see that

Ji,i = |Ji,i−1|+ |Ji,i+1|+1

∆τn+ r + λ > |Ji,i−1|+ |Ji,i+1|

for any i = 2, 3, ...,M with the convention that J2,1 = 0 = JM,M+1. Therefore, J is

an M -matrix by [38].


Now, let us consider J − λD. From the definition of D, it is clear that all the

off-diagonal entries of J − λD are non-positive and

Ji,i−1 + Ji,i + Ji,i+1 − λM∑j=2

dij =1

4τn+ r + λ− λ

M∑j=2

dij

=1

4τn+ r + λ

(1−

M∑j=2

dij

)

≥ 1

4τn+ r > 0. (4.41)

This implies

Ji,i − λdii =1

4τn+ r +

(1−

∑j

dij

)λ+ |Jni,i−1|+ |Jni,i+1|+ λ

∑j 6=i

dij

=1

4τn+ r + (1− dii)λ+ |αni |+ |γni |

> |αni |+ |γni |. (4.42)

Therefore, all the diagonal entries of Jn+1(V n+1)− λD are positive and the matrix

is strictly diagonally dominant. Hence, it is an M -matrix.

Using the Jacobian of Hn+1, we propose the following Newton algorithm for (4.28):

Algorithm N1

1. Choose a tolerance ε1 > 0. Let n = 1 and evaluate the discrete initial condition

V 1 = (V 12 , ..., V

1M)> using (4.20).

2. Set l = 1 and W l = V n.

3. Solve

[Jn+1(W l)− λD]δW = −Hn+1(W l) (4.43)

for δW and set

W l+1 = W l + δW.

90 Chapter 4

4. If ‖δW‖∞ ≥ ε1, set l := l + 1 and go to Step 3. Otherwise, continue.

5. Set V n+1 = W l+1. If n < N − 1, let n := n + 1 and go to Step 2. Otherwise,

stop.

Note that D is a dense matrix. Therefore, the solution of (4.43) is computationally

expensive because of the inversion of D. To solve this, we use a regular splitting

technique. Let

M = Jn+1(V n+1)− λD.

We split M into

M = Jn+1(V n+1)− (λD) =: P −R. (4.44)

Now, we introduce this following definition

Definition 4.7. A splitting M = P −R is said to be a regular splitting if P−1 ≥ 0

and R ≥ 0 (cf. [63])

From Theorem 4.6 we have Jn+1(V n+1) is an M -matrix for any given V n+1 and hence

P−1 ≥ 0. Also from (4.23) and (4.28) we have D > 0 and thus R ≥ 0. Therefore,

we have a regular splitting. Using this splitting, we define an iterative scheme for

(4.43) as follows.

P (δW )k+1 = R(δW )k −H. (4.45)

The following lemma establishes the convergence of the iterative method (4.45).

Lemma 4.8. The iterative scheme (4.45) associated with the regular splitting (4.44)

is convergent.

Proof. From Theorem 4.6, we know that M = Jn+1(V n+1) − λD is an M -matrix.

Hence we have M−1 ≥ 0. By the result in [63], we have (4.45) is convergent.

Recall that matrix D is a dense matrix, hence it is computationally expensive to eval-

uate the multiplication in (4.45) directly, because the computational cost is O(N2).


Thus, we present a fast algorithm to evaluate this matrix. This algorithm, based on

Fast Fourier Transform (FFT), has a computational cost of order O(N lnN).

From (4.14) and (4.44) we know that R is a Toeplitz matrix. Applying FFT to

R(δW )k and RV n+1 will produce wrap-round pollution. Hence, we embed the

Toeplitz matrix R into circulant matrix C (cf. [64]) as follow

C =

f0 f1 f2 . . . fM−3 fM−2 f2−M f3−M . . . f−2 f−1

f−1 f0 f1 . . . fM−4 fM−3 fM−2 f2−M . . . f−3 f−2

f−2 f−1 f0 . . . fM−5 fM−4 fM−3 fM−2 . . . f−4 f−3

......

.... . .

......

......

. . ....

...

f2−M f3−M f4−M . . . f−1 f0 f1 f2 . . . fM−3 fM−2

fM−2 f2−M f3−M . . . f−2 f−1 f0 f1 . . . fM−4 fM−3

fM−3 fM−2 f2−M . . . f−3 f−2 f−1 f0 . . . fM−5 fM−4

fM−4 fM−3 fM−2 . . . f−4 f−3 f−2 f−1 . . . fM−6 fM−5

......

.... . .

......

......

. . ....

...

f1 f2 f3 . . . fM−2 f2−M f3−M f4−M . . . f−1 f0

.

Define

δUk = [(δW )k2, . . . , (δW )kM , 0, . . . , 0︸︷︷︸M−2

]>,

U l = [V l2 , . . . , V

lM , 0, . . . , 0︸︷︷︸

M−2

]>.

Then, matrix-vector products R(δW )k and RV l are realized as the first M−1 entries

in C(δU)k and CU l.

The product R(δW )k is computed in the following two FFT operations. Let FFT (u)

denote the FFT of u and define the vector

F = (f0, f1, . . . , fM−2, f2−M , f3−M , . . . , f−1),

92 Chapter 4

which generates the row vector of C by permutation. First, we compute FFT (F )

and FFT ((δU)k). Then, we compute the inverse FFT of the product of FFT (F )

and FFT ((δU)k).

The numerical implementation for the system (4.28) can be summarized in this

following algorithm.

Algorithm N2

1. Choose tolerances ε1, ε2 > 0. Let n = 1 and evaluate the discrete initial

condition V 1 = (V 12 , ..., V

1M)> using (4.20).

2. Compute FFT (F ).

3. Set l = 1 and V l = V n.

4. Compute FFT (U l) and the inverse FFT on the product of FFT (F ) and

FFT (U l) which gives DV l in H.

5. Set k = 1 and (δW )1 = [0, . . . , 0︸︷︷︸M−1

]>.

6. Compute FFT ((δU)k) and the inverse FFT on the product of FFT (F ) and

FFT ((δU)k) which gives R(δW )k.

7. Solve

Jn+1(V l)(δW )k+1 = R(δW )k −Hn+1(V l)

for (δW )k+1.

8. If max (δW )k+1−(δW )k‖∞max(1,(δW )k+1‖∞)

< ε1 then stop. Otherwise, set k := k + 1, go to Step

6.

9. Set (δW )l = (δW )k+1. if ‖(δW )l‖∞ < ε2 then stop. Otherwise, compute

V l+1 = V l + (δW )l and set l = l + 1, go to Step 4.

10. Set V n+1 = V l+1. If n < N , let n := n+ 1 and go to Step 3. Otherwise, stop.


4.6.2 The American case

Now we consider the solution of (4.5)–(4.7). It is known that (4.5)–(4.7) can be

approximated by the following penalized equation

Vτ =1

2σ2 (τ, S, VS, VSS)S2VSS + (r − λκ)SVS (4.46)

−(r + λ)V + λ

∫ ∞0

V (Sη)g(η) dη + ϑ[V ∗ − V ]+,

where ϑ > 1 is a constant and [z]+ = max(z, 0).

This is a penalty equation which approximates (4.5)–(4.7). In this formulation, the

penalty term ϑ[V ∗− V ]+ penalizes the positive part of (V ∗− V ). Loosely speaking,

when (4.6) is violated, (4.46) yields [V ∗ − V ]+ = ϑ−1LV . Therefore, if LV is

bounded, [V ∗ − V ]+ approaches zero as ϑ approaches positive infinity, so that (4.6)

is satisfied within a tolerance depending on the value of ϑ.

Following the same argument in the previous section, it is easy to show that the full

discretization of (4.46) by the upwind finite difference method is of the form

Hn+1(V n+1ϑ )− ϑ[V ∗ − V n+1

ϑ ]+ = 0, (4.47)

where Hn+1(V n+1) = [An+1(V n+1)− λD] V n+1 − 14τn V

n −Bn+1.

Note that (4.47) is a non-smooth nonlinear system in V n+1ϑ for each n = 1, ..., N ,

because of the penalty term, hence we use a smoothing technique proposed in [37, 45]

to locally smooth out the penalty term near zero to solve this equation.

4.7 Numerical Results

In this section we demonstrate the rates of convergence and numerical performance

of our method using European call and put options as well as American put option.

94 Chapter 4

020

4060

80

00.2

0.40.6

0.810

10

20

30

40

50

Stock Price

European Call Option Price in Jump Diffusion with Transaction Costs

Time

Opt

ion

Pric

e

Figure 4.1: Price of the European call option with a = 0.01 and b = 0.07.

Test Problem 1: European vanilla call and put options with the system parame-

ters: r = 0.1, σ0 = 0.2, K = 40, T = 1, Smax = 80, µ = 0.01, σJ = 0.5, and λ = 0.1.

The payoff and boundary conditions are given in (4.8)–(4.10) for the call and put.

This test problem is solved using our method for the σ defined in (4.3) with the

transaction cost parameters a = 0.01 and b = 0.07. The computed European call

and put option prices are plotted in Figures 4.1 and 4.2. From the figures we see

that numerical solutions are stable. To see the effect of the transaction costs to the

price of the call and put options, we plot the values of the options at t = 0 (or

τ = T ) for a = 0 and three different values of b on the interval [30, 50] in Figure 4.3

and 4.4 for European call option and put option, respectively. From these figures

we see that the prices of the options increase as the transaction cost parameter b

increases, as expected in practice.

We now investigate the rate of convergence computationally, which requires an exact

or reference solution. Since the exact solution to this problem is unknown, we use

the numerical solution to (4.28) on the uniform mesh with M = 5120 (h = 164

) and

N = 2560 (4τn = 12560

) as an “exact” or reference solution. To determine these

rates, we choose a sequence of meshes generated by successively halving the mesh

sizes of the previous ones, starting from a given coarse mesh. Using the reference


0 10 20 30 40 50 60 70 80 0

0.5

1

0

5

10

15

20

25

30

35

40

Time

Stock Price

European Put Option Price in Jump Diffusion with Transaction Cost

Opt

ion

Pric

e

Figure 4.2: Price of the European put option with a = 0.01 and b = 0.07.

30 32 34 36 38 40 42 44 46 48 500

2

4

6

8

10

12

14

Stock Price

Opt

ion

Pric

e

European Call Option Price for Different Transaction Cost Parameter

: a = 0, b = 0: a = 0, b = 7%: a = 0, b = 20%

Figure 4.3: Price of the European call option for different transaction costparameter.

solution we then calculate the following ratios of the numerical solutions from two

consecutive meshes:

Ratio(‖ · ‖h,2) =‖V ∆τ

h − Vexact‖h,2‖V ∆τ/2

h/2 − Vexact‖h,2,

96 Chapter 4

30 32 34 36 38 40 42 44 46 48 502

3

4

5

6

7

8

Stock Price

Opt

ion

Pric

e

European Put Option for Different Transaction Cost Parameters

a = 0, b = 0a = 0, b = 7%a = 0, b = 20%

Figure 4.4: Price of the European put option for different transaction costparameter.

M N ‖ · ‖h,2 Ratio(‖ · ‖h,2)21 11 0.21568041 21 0.116543 1.8581 41 0.061550 1.89

161 81 0.031986 1.92321 161 0.016228 1.97641 321 0.007861 2.06

1281 641 0.003457 2.272561 1281 0.001170 2.96

Table 4.1: Computed rates of convergence for the call option with a = 0.01 andb = 0.07

where V βα denotes the computed solution on the mesh with spatial mesh size α and

time mesh size β and || · ||h,2 denotes the discrete L2-norm defined by

‖V ∆τh − Vexact‖h,2 :=

( ∑1≤n≤M

∑1≤i≤N

|V ni − Vexact(Si, τn)|2h4τ

)1/2

.

The computed errors in || · ||h,2 and the ratios two consecutive errors are listed in

Table 4.1 and 4.2 for the European call and put respectively. From the table we see

that the rates of convergence of our method is about 2, showing that our numerical

method is 2nd order accurate in the L2-norm.


M N ‖ · ‖h,2 Ratio(‖ · ‖h,2)21 11 0.45459641 21 0.438884 1.0481 41 0.390547 1.12

161 81 0.327934 1.19321 161 0.259319 1.26641 321 0.189478 1.37

1281 641 0.121703 1.562561 1281 0.058168 2.09

Table 4.2: Computed rates of convergence for the put option with a = 0.01 andb = 0.07

20 30 40 50 60 70 800

2

4

6

8

10

12

14

16

18

20

Stock Price

Put

Opt

ion

Pric

e

American and European Put Option Price Under Jump Diffusion Process

European put optionAmerican put option

Figure 4.5: American and European Put Option Price under Jump DiffusionProcess

Test Problem 2: American put options with the system parameters: r = 0.1,

σ0 = 0.2, K = 40, T = 1, Smax = 80, µ = 0.01, σJ = 0.5, λ = 0.1, and ϑ = 1000.

The payoff and boundary conditions are given in (4.11)–(4.13). To see the difference

between the European and American options, we plot the computed values of both

of the options at t = 0 (or τ = T ) in Figure 4.5. From the figure it is clear that the

American put option is more valuable than the European put option as expected.

Furthermore, we investigate the effect of the transaction cost parameters to the value

of the option. To see this, we plot the values of the option at t = 0 (or τ = T ) for

three different values of b on the interval [30, 50] in Figure 4.6 in which the curve

98 Chapter 4

30 32 34 36 38 40 42 44 46 48 502

3

4

5

6

7

8

9

10

Stock Price

Put

Opt

ion

Pric

e

American Put Option Price for Different Transaction Cost Parameter Under Jump Diffusion Process

a = 0, b = 5%a = 0, b = 10%a = 0, b = 20%

Figure 4.6: American Put Option Price for Different Transaction Cost Param-eter under Jump Diffusion Process

for a = 0 and b = 0 is the price of the American put option without transaction

cost. From this figure we see that the price of the option increases as the transaction

parameter b increases as expected in practice.

4.8 Conclusion

In this chapter we proposed and analyzed an upwind finite difference scheme to

approximate a nonlinear partial integro-differential equation and a partial integro-

differential complementarity problem arising from European and American option

pricing with transaction costs under jump diffusion model. The convergence of the

solution to discretized system to the viscosity solution of the continuous problem

has been proven.

To solve the dense system, which is resulted from the fully implicit scheme for

the nonlinear PIDE, we develop a fast iterative method coupled with a regular

splitting technique. To speed up the computation of the integral term, we used

FFT algorithm. We add a penalty term to the original partial integro-differential

complementarity problem to impose the constraint in the American option model.


Numerical experiments were performed to confirm the theoretical results. The orders

of convergence of the method is about 2 in L2-norms. The results also show that

the prices of a European and an American option are increasing functions of the

transaction cost parameter a and b.

Chapter 5

Conclusion

In this thesis we have developed three numerical algorithms for obtaining approxi-

mate solutions to the valuation of European and American options with transaction

costs under geometric Brownian motion and jump diffusion process. These algo-

rithms were based on an upwind finite difference scheme for the spatial discretization

and a fully implicit time-stepping scheme.

The first algorithm was built to analytically solving European option with trans-

action costs under geometric Brownian motion. We proved that the system matrix

from the discretization scheme is an M -matrix. We also proved the convergence

of the solution to discretized system to the viscosity solution of the continuous

problem. Furthermore, a Newton’s iterative method was proposed for solving the

resulting nonlinear algebraic system and it was shown that the Jacobian matrix of

the nonlinear system is also an M -matrix.

The second algorithm was constructed to solve the American put option problem

with transaction cost when the underlying asset price follows geometric Brownian

motion. The problem can be written as an infinite-dimensional nonlinear comple-

mentarity problem (NCP) or equivalently a nonlinear variational inequality. We pro-

posed and analyzed a nonlinear penalty method for the solution of finite-dimensional

NCP. We have shown that the system is continuous and strongly monotone and also

101

102 Chapter 5

have shown that the solution to the penalty equation converges to that of NCP at

an arbitrary rate depending on the choice of parameters in the penalty term.

The third algorithm was proposed to approximate a nonlinear partial integro-differential

equation (PIDE) arising from European option pricing with transaction costs un-

der jump diffusion model. We showed that the scheme converges to the viscosity

solution of the continuous problem. To solve the dense system, which was resulted

from the fully implicit scheme for the nonlinear PIDE, we developed a fast iterative

method coupled with a regular splitting technique. To speed up the computation of

the integral term, we used FFT algorithm.

Appendix A

Proof of the monotonicity of

σ2(S, z)z

Given σ2(S, z) as in Leland Model (Equation (2.2)), Boyle & Vorst Model (Equation

(2.3)), HWW Model (Equation (2.4)), and Jandacka & Sevcovic Model (Equation

(2.5)), σ2(S, z)z is an increasing function in z.

Proof. First, we will prove the monotonicity of σ2(S, z)z for volatilities in Equation

(2.2), (2.3), and (2.4).

Note that Equations (2.2), (2.3), and (2.4) can be written in a general form as follow

σ2 = σ20

(1 + C sign(VSS)

)(A.1)

where

C = Le =

√2

π

(κ

σ0

√δt

)for Eqn. (2.2),

C = Le

√π

2for Eqn. (2.3),

C =k

σ0

√8

π dtfor Eqn. (2.4),

103

104 Appendix A. Proof of the monotonicity of σ2(S, z)z

and 0 ≤ C < 1.

We have, for any S, z1 and z2,

1

σ20

[σ2(S, z1)z1 − σ2(S, z2)z2] = [(z1 − z2) + C(sign(z1)z1 − sign(z2)z2)]

= [1 + Csign(z1)](z1 − z2) + C[sign(z1)− sign(z2)]z2

=

C1(z1 − z2) z1 · z2 > 0,

(1 + C)z1 + (C − 1)z2 > 0 z1 > 0 > z2,

(1− C)(z1 − z2)− 2Cz2 < 0 z1 < 0 < z2,

where C1 = (1 + Csign(z1)) > 0, since 0 ≤ C < 1.

Therefore, σ2(S, z)z is an increasing function in z.

Now, when σ2(S, z) is by (2.5), we have

σ2(S, z)z = σ20

(1− 3

(C2R

2πSz

) 13

)z.

Furthermore,

d

dz

[σ2(S, z)z

]= σ2

0

(1− 3

(C2R

2πSz

) 13

)

−σ20

(C2R

2πSz

)−23(C2R

2πSz

)= σ2

0

(1− 4

(C2R

2πSz

) 13

)> 0 (by (2.6))

Therefore, σ2(S, z)z is an increasing function in z on the set defined by the inequality

in (2.6).

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