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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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Download date: 04. Jun. 2018
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.
Chapter 1. Introduction 5
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.
Chapter 1. Introduction 7
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
∣∣∣∣).
Chapter 1. Introduction 9
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.
Chapter 1. Introduction 11
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.
Chapter 1. Introduction 13
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
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 19
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
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 21
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)
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 23
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|.
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 25
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
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 27
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 |.
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 29
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 ,
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 31
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)
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 33
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
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 35
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.
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 37
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)
Barles−Soner ModelHWW Model
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.
The value of this option computed by our method on the uniform mesh with h = 2
and ∆τ = 0.05 is plotted in Figure 2.9. The cross-sections of the value functions
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 39
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
: volatility = 5%: volatility = 20%: volatility = 70%
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)
Barles−Soner ModelHWW Model
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.
The value of this option computed by our method on the uniform mesh with h = 2
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
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 41
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
: volatility = 5%: volatility = 20%: volatility = 70%
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)
Barles−Soner ModelHWW Model
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.
Chapter 2. An Upwind Finite Difference Method for Pricing European OptionsUnder Transaction Costs 43
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
: volatility = 5%: volatility = 20%: volatility = 70%
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
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 49
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:
(δτWi)(n) =W n+1i −W n
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
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 51
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.
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 53
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.
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 55
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)
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 57
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)
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 59
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.
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 61
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.
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 63
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
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 65
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.
Chapter 3. A Penalty Approach for American Put Option Valuation UnderTransaction Costs 67
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
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 73
[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)
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 75
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)
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 77
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:
(δτWi)(n) =W n+1i −W n
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)
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 79
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 |,
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 81
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.
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 83
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.
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 85
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
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 87
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].
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 89
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).
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 91
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.
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 93
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
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 95
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.
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 97
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.
Chapter 4. Numerical scheme for pricing option with transaction costs under jumpdiffusion processes 99
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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