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Renato Alencar Adelino da Costa

Risk Neutral Option Pricing under some special

GARCH models

Tese de Doutorado

Thesis presented to the Postgraduate Program in Engenharia

El´etrica of the Departamento de Engenharia El´etrica, PUC–Rio

as partial fulﬁllment of the requirements for the degree of Doutor

em Engenharia El´etrica

Advisor : Prof.

Alvaro de Lima Veiga Filho

Co–Advisor: Prof. Tak Kuen Siu

Rio de Janeiro

August 2010

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Renato Alencar Adelino da Costa

Risk Neutral Option Pricing under some special

GARCH models

Thesis presented to the Postgraduate Program in Engenharia

El´etrica of the Departamento de Engenharia El´etrica, PUC–Rio

as partial fulﬁllment of the requirements for the degree of Doutor

em Engenharia El´etrica. Approved by the following commission:

Prof.

Alvaro de Lima Veiga Filho

Advisor

Departamento de Engenharia El´etrica –PUC-Rio

Prof. Tak Kuen Siu

Co–Advisor

Department of Actuarial Studies, Faculty of Business and

Economics, Macquarie University

Prof. Caio Ibsen de Almeida

EPGE–FGV

Prof. Adrian Heringer Pizzinga

Departamento de Estat´ıstica – UFF

Prof. Paulo Henrique Soto Costa

Faculdade de Ciˆencias Econˆomicas, Departamento de An´alise

Quantitativa – UERJ

Prof. Marcelo Cunha Medeiros

Departamento de Economia – PUC-Rio

Prof. Joel Maur´ıcio Corrˆea da Rosa

Departamento de Estat´ıstica – UFF

Prof. Carlos Kubrusly

Departamento de Engenharia El´etrica – PUC-Rio

Prof. Jos´e Eugenio Leal

Coordinator of the Centro T´ecnico Cient´ıﬁco — PUC–Rio

Rio de Janeiro — August 17, 2010

ads:

Renato Alencar Adelino da Costa

Renato Costa is a Master in Mathematics, graduated from

the Pontiﬁcal Catholic University of Rio de Janeiro (Rio

de Janeiro, Brazil). There he studied Applied mathematics

with the intent of using it in Finance. He recently spent 9

months at the Department of Mathematics and Statistics of

Curtin University of Technology where he did research for his

PhD thesis as a exchange/sandwich student sponsored by the

National Council for Scientiﬁc and technological Development

(CNPq)

Bibliographic data

Costa, Renato Alencar Adelino da

Risk Neutral Option Pricing under some special GARCH

models / Renato Alencar Adelino da Costa ; advisor:

Alvaro

de Lima Veiga Filho; co–advisor: Tak Kuen Siu. — 2010.

103 f. : il. ; 30 cm

Tese (Doutorado em Engenharia El´etrica)-Pontif´ıcia Uni-

versidade Cat´olica do Rio de Janeiro, Rio de Janeiro, 2010.

Inclui bibliograﬁa

1. Engenharia El´etrica – Teses. 2. Apre¸camento de op¸c˜oes.

3. Mudan¸ca de medida. 4. GARCH. 5. FC-GARCH. 6. Mistura

de GARCHs. 7. Transformada de Esscher condicional. I. Veiga,

Alvaro de Lima . II. Siu, Tak Kuen. III. Pontif´ıcia Universidade

Cat´olica do Rio de Janeiro. Departamento de Engenharia

El´etrica. IV. T´ıtulo.

CDD: 510

Acknowledgments

Firstly, I’d like to thank my advisors

Alvaro de Lima Veiga Filho and

Tak Kuen Siu for the patience and help provided.

I also would like to thank Professor Carlos Kubrusly for the knowledge

and advice given during some of my doctoral courses, Professors Cristiano

Fernandes, Marcos da Silveira and Tara Baydia for the initial guidances in my

ﬁrst year as a PhD student and Thomas Lewiner for helping me ﬁxing my

thesis tex ﬁle.

My life relies strongly in the support given by my family, not only

ﬁnancially but also emotionally, so I must thank a lot my wife Cristiane Gomes

Silva da Costa, for her forbearance and understanding when I couldn’t be

with her to study, my father Jos´e Adelino da Costa and my mother Regina

Maria Alencar da Costa for everything they’ve done for me, my sister Roberta

Alencar Adelino da Costa Lopes Correia, my nephew J´ulio C´esar, my aunt

Alcina da Concei¸c˜ao Coutinho, my un cle Ricardo Borges Alencar and my

grandparents Neemias Alencar and Maria Aparecida Borges Alencar and also

my grandmother Celeste Costa Adelino that unfourtunately is not among us

anymore.

I also would like to thank my personal friends for their friendship, in

particular to Anderson Mazzoli Lisboa, D´ecio Marques de Paiva, Rafael Brasil,

Rafael Ferr˜ao, S´ergio Adriano and Viviane Martins for the support given as

true friends.

I thank PUC-Rio as well as the secretaries for the support provided and

the National Council of Research (CNPq) for providing me the scholarship not

only in Brazil but also during my research at Curtin University of Technology

as a sandwich/exchange student where I learned a lot and made many personal

and professional important contacts.

My thanks to my friends and colleagues at Curtin and PUC-Rio: Adrian

Pizzinga, Andr´e Cunha, Aldo Ferreira, Alexandre Santos, Alexandre Street,

Bernardo Pagnoncelli, Caio Azevedo, Camila Epprecht, Cristina Vidigal, Davi

Michel, Deepak Candeep, Ling Wang, Luiz Oz´orio, M´arcio Candeias, Mariana

Paix˜ao, Marlene Marchena, Masnita Misirian, Nattakorn Phewchean (Mick),

Regina Fukuda, Reinaldo Marques, Rodrigo Atherino, Rodrigo Moreira Topin,

Rodrigo Sechin, Rodrigo Silva Mello, Waseem AlShanti and Wilson Freitas.

Abstract

Costa, Renato Alencar Adelino da; Veiga,

Alvaro de Lima ; Siu, Tak

Kuen. Risk Neutral Option Pricing under some special GARCH

models. Rio de Janeiro, 2010. 103p. Tese de Doutorado — Departamento

de Engenharia El´etrica, Pontif´ıcia Universidade Cat´olica do Rio de

Janeiro.

Option pricing is a very imp ortant issue nowadays. The use of probabilis-

tic methods is required for risk neutral pricing. Here we apply the method of

Siu et al. for two classes of GARCHs, viz., the FC-GARCH and the Mixture

of GARCHs.

In both models we derive the risk neutral version of the mo d el which is

essential for pricing contracts, in two diﬀerent cases, when the noise is normal

as well as when it is shifted gamma.

We also performed simulations with both models and compared to the

benchmark Black Scholes model, checked for the smile eﬀect and made some

sensibility analysis in the parameters.

Keywords

Option Pricing. Change of measure. GARCH. FC-GARCH. Mix-

ture of GARCHs. Conditional Esscher transform.

Resumo

Costa, Renato Alencar Adelino da; Veiga,

Alvaro de Lima ; Siu, Tak

Kuen. Apre¸camento neutro ao risco de opc˜oes sob modelos

GARCH especiais. Rio de Janeiro, 2010. 103p. Tese de Doutorado —

Departamento de Engenharia El´etrica, Pontif´ıcia Universidade Cat´olica

do Rio de Janeiro.

O apre¸camento de op¸c˜oes ´e um assunto muito importante nos dias de

hoje. M´etodos probabilisticos s˜ao necess´arios para fazer o apre¸camento neutro

ao risco. Usaremos o m´etodo de Siu et al. para duas classes de GARCHs, o

FC-GARCH e a mistura de GARCHs

Em amb os os modelos n´os encontramos a vers˜ao neutra ao risco do

modelo que ´e necess´aria para a preciﬁca¸c˜ao de contratos, em dois diferentes

casos, quando o ru´ıdo ´e normal e quando ´e shifted gamma.

Fizemos tamb´em simula¸c˜oes para ilustrar e comparamos os resultados

com o valor de Black Scholes, veriﬁcamos a existˆencia de smile e ﬁzemos uma

an´alise de sensibilidade nos parˆametros

Palavras–chave

Apre¸camento de op¸c˜oes. Mudan¸ca de medida. GARCH. FC-

GARCH. Mistura de GARCHs. Transformada de Esscher condicional.

Contents

1 Introduction 8

2 Continuous Time Finance 12

2.1 Initial Deﬁnitions 12

2.2 Equivalences among Processes and other Deﬁnitions 13

2.3 Martingales 14

2.4 Stochastic Integrals 16

2.5 Itˆo Processes, Diﬀerentiation Rule and Solution of a SDE 17

2.6 Examples of Itˆo Formula 20

2.7 Change of Measure 24

3 GARCH Models 26

3.1 ARCH Models 26

3.2 Flexible Coeﬃcient Generalized Autoregressive Conditional Het-

eroskedastic (FC-GARCH) 27

4 Option Pricing 29

4.1 Pricing Formula in the Risk-Neutral Measure 29

4.2 Black-Scholes-Merton Formula 31

4.3 Put-Call Parity 33

4.4 Incomplete Markets 34

4.5 Duan’s breakthrough 37

5 Option Pricing under a Nonlinear and Nonnormal GARCH 40

5.1 Flexible Coeﬃcient Generalized Autoregressive Conditional Het-

eroskedastic (FC-GARCH) models for Asset Returns 40

5.2 The Conditional Esscher Transform 42

5.3 Some Parametric Cases 46

5.4 Simulation Studies 53

5.5 Sensitivity Analysis 64

5.6 Discussion of the results 65

5.7 Conclusions 65

6 Option Valuation under Mixture GARCH models 67

6.1 Asset Price Dynamics and Pricing Model 67

6.2 Parametric Cases 76

6.3 Empirical Results and Discussion 82

6.4 Sensitivity Analysis 94

6.5 Summary and discussion 95

7 Future works and ﬁnak remarks 96

Bibliography 97

A Appendix 101

Introduction

Option valuation is one of the most important topics in ﬁnancial eco-

nomics. The path-breaking work by Black and Scholes (1973)(6) and Merton

(1973)(37) provides the cornerstone for an explosive growth in the literature

describing the theory and p ractice of option pricing models. The key economic

insights behind the Black-Scholes-Merton model are the concept of perfect

hedging of an option, by constructing a replicating portfolio via trading the

underlying assets continuously, and pricing by the no-arbitrage principle. Cox,

Ross and Rubinstein (1979) (12) were the ﬁrst to establish the relationship be-

tween the risk-neutral valuation and the no-arbitrage principle. Harrison and

Kreps (1979)(33), and Harrison and Pliska (1981, 1983)(34) (35) established

a solid mathematical foundation for the relationship between the no-arbitrage

principle and the notion of risk-neutral valuation using the language of proba-

bility theory. They also provide a solid theoretical foundation to the concept of

market incompleteness. If the securities market is complete, there is a unique

equivalent martingale measure, i.e., a unique risk neutral measure and , hence,

the unique price of any contingent claim is given by its expected discounted

payoﬀ at expiry under the martingale measure. However, in an incomplete mar-

ket, there are inﬁnitely many equivalent martingale measures and, so, a range

of no-arbitrage prices for a contingent claim can be found, and this complicates

the pricing and hedging issues.

F¨ollmer and Sondermann (1986)(29), F¨ollmer and Schweizer (1991)(28)

and Schweizer (1996)(40) determined the equivalent martingale pricing mea-

sure by minimizing the quadratic utility of the losses due to imperfect hedg-

ing. Davis (1997)(14) adopted a traditional valuation approach in economics,

namely the marginal rate of substitut ion, to determine a pricing measure by

solving a utility maximization problem. The seminal work by Gerber and Shiu

(1994)(31) provided a pertinent solution to the option pricing problem in an

incomplete market by using the Esscher transform, a time-honored tool in

actuarial science introduced by Esscher (1932)(25). Their model provided a

convenient and ﬂexible way to price options under diﬀerent parametric as-

sumptions on the stock returns within the class of inﬁnitely divisible distribu-

Risk Neutral Option Pricing under some special GARCH models 9

tions. They can justify the pricing result by considering a utility maximization

problem with respect to a power utility function. Their signiﬁcant contribu-

tions highlighted the interplay between the ﬁnancial and insurance pricing

in incomplete markets and its importantance is mentioned in B¨uh lmann et

al. (1996) (9) and Embrechts (2001) (24). B¨uhlmann et al. (1996) (9) devel-

oped the conditional Esscher transform by generalizing the classical Esscher

transform to stochastic processes in order to incorporate the richer theory of

semi-martingales under the no-arbitrage condition in the Gerber-Shiu option-

pricing model. B¨uhlmann et al. (1998) (10) investigated the use of Esscher

transforms in discrete ﬁnance models and established a solid foundation for its

use based on economic arguments. Siu, Tong and Yang (2001) (42) introduced

the concept of a random Esscher transform with a random Esscher parameter

and adopted the random Esscher transform to incorporate the uncertainty of

the probability measures for r isk measurement. Elliott, Chan and Siu (2004)

(22) provided a modiﬁcation of the random Esscher transform and developed

the regime switching random Esscher transform to identify a pricing measure

for option valuation und er a Markov-mod ulated Geometric Brownian Motion

(MMGBM). Yao (2002) (46) adopted the Esscher Transform to specify the

forward-risk-adjusted measure and provided a general and consistent frame-

work for pricing derivatives on stocks, interest rates and currency rates.

Autoregressive conditional heteroskedastic (ARCH) models were pro-

posed by the Nobel Laureate Robert Engle as a tool to describe time-varying

volatility dynamics and other stylised empirical facts of many ﬁnancial time

series. Bollerslev (1986) (7) and Taylor (1986) (44) generalized the idea of the

ARCH models and developed the generalized ARCH (GARCH) models inde-

pendently by assuming that the current level of the conditional variance not

only depends on the past values of the innovations but also the past values

of the conditional variances. For an excellent overview of ARCH-type models,

see Bollerslev, Chou and Kroner (1992)(8).

There has been a considerable interest in option valuation under GARCH

models in the ﬁnance literature. The seminal work by Duan (1995) (17) was

the pioneer to provide a solid theoretical foundation for option valuation in

the context of GARCH models. He generalized the concept of risk-neutral val-

uation and introduced the notion of locally risk-neutral valuation relationship

(LRNVR) which provides a sou nd economic argument to choose a particu-

lar equivalent martingale measure in the GARCH mod el with a conditionally

normal stock innovation. Under the preference assumptions and distributional

assumptions, Duan (1995) provided a rigorous theoretical foundation and eco-

nomic justiﬁcation of the validity of LRNVR. Duan, Popova and Ritcken (2002)

Risk Neutral Option Pricing under some special GARCH models 10

(19) developed a family of option pricing models where the underlying stock

price d ynamics are modelled by a regime switching process in which the prices

stay in one volatility regime for a random amount of time before switching

over into a new regime. Barone- Adesi, Engle and Mancini (2008)(3) proposed

a new approach to compute option prices under the GARCH models in an in-

complete market framework. Their model allows the actual volatility of asset

returns to be diﬀerent from the volatility of asset returns under the pricing

probability measure. Siu, Tong and Yang (2004) (43) proposed an alternative

method to price options under the GARCH models with inﬁnitely divisible in-

novations by using the conditional Esscher transform proposed by B¨uhlmann

et al. (1996). Elliott, Siu and Chan (2004) introduced the use of a modiﬁed

version of the conditional Esscher transform, namely the Markov switching

conditional Esscher transform (MS CET), to determine an equivalent martin-

gale pricing measure under a Markov switching GARCH model. They justiﬁed

their pricing results by considering the stochastic power utility function with

Markov switching risk-aversion parameters.

The thesis extends the literature in two main points. First, we deal with

more general processes to perform option pricing than in Duan(1995) (17)

and Siu et al.(2004) (43), viz. the Flexible Coeﬃcient GARCH model (FC-

GARCH) which is a nonlinear model and nests several well-known GARCH

speciﬁcations in the literature and th e mixture of GARCHs. Second, we can

treat those two models in a variety of innovation distributions, not being

restricted to the normal case as in Duan(1995) (17). In particular in this

thesis we perform calculations and simulations for the Normal and Shifted-

Gamma cases. A minor contribution is that we include the possibility of a

negative innovation in the Shifted-Gamma case. This allow us to mimic the

small and negative skewness usually found in the empirical literature. This

contribution give the practitioners ﬂexibility in choosing among the Normal,

the positive and negative Shifted-Gamma innovation cases according to the

sign and magnitude of the skewness.

Although the non-linearity of the FC-GARCH aﬀects little the prediction

of the volatility, it considerably aﬀects the option prices. We notice both in

the FC-GARCH and in the Mixture of GARCHs that the option prices vary

a lot in our experiments. We also noticed that th e choice of the innovation

distribution is important for better describing the skewness of the series. We

deal with negative shifted-Gamma innovations to treat negative skewness data.

When using a diﬀerent innovation, a signiﬁcant diﬀerence in the option price

is also detected.

The document is structured as follows. First we make a review on the

Risk Neutral Option Pricing under some special GARCH models 11

mathematical background knowledge, then in chapter three we talk about the

simplest ARCH models. In chapter 4, we brieﬂy discuss the option pricing

methodologies. Chapters 5 and 6 contain the main results of the research. They

consist of two risk n eutral option pricing papers, one assuming the FC-GARCH

as the underlying log-return process and the other a mixture of GARCHs. In

chapter 5, we develop the methodology in Siu et al.(2004) (43) to the FC-

GARCH. In chapter 6, we perform a similar methodology to the Mixture

of GARCHs. The theoretical results we achieved are the Theorems 35, 36,

41 and 42. In both these chapters we discuss the model, the methodology,

the calculations and simulation experiments. We ﬁnish the thesis with the

conclusions, future intent of work and a small app endix.

Continuous Time Finance

This Chapter will cover the basics of Probability Theory used in Finance

and then the use of it in Finance. We deal, in particular, with the topics

concerning Continuous Time Finance.

The work here is done with discrete time models but as the continuous

case account for the discrete case also as its particular case we will do all

discussion in continuous time for mental exercising and generality.

Most of what is here was made initially as beamer slides presented to my

co-advisor as a weekly seminar based from the Elliot and Kopp book (21) but

I also did a mixing of contents from other books such as the Lecture Notes of

Evans version 1.2 (26), Billingsley (4), Oksendal (38), Shreve (41) and Bingham

and Kiesel (5). We refer to these for some proofs and additional details.

2.1

Initial Deﬁnitions

This subsection deals with continuous time stochastic processes, so we

are going to consider that t lies in T , where T is one of the following sets:

[0, T ], [0, ∞) or [0, ∞]. And we will be always considering a probability space

(Ω, F, P).

Deﬁnition 1 (Filtration) F = {F

}

t∈T

is called a ﬁltration if it is a increasing

family of sub-σ-algebras, i.e., if it is a family of σ-algebras such that for s ≤ t:

⊂ F, F

⊂ F

We also suppose that the ﬁltration is complete and is right continuous in

the folowing sense: (Information now comes continuously.)



s>t

, t ∈ T

Deﬁnition 2 (Stochastic Processes) A continuous-time stochastic process X

taking values in a measurable space (E, E) is a family of r.v’s {X

} deﬁned in

(Ω, F, P), indexed by t taking values in (E, E). Notice that:

Risk Neutral Option Pricing under some special GARCH models 13

• X

(·) is a random variable.

• X

(ω) is a path of the process X.

2.2

Equivalences among Processes and other Deﬁnitions

We are going to deﬁne diﬀerent ways of comparing sto chastic processes as

well as clarify the d iﬀerences among those concepts. Besides that, we are going

to give some common deﬁnitions concerning the measurability of stochastic

processes.

Deﬁnition 3 (Equivalent Processes) Let

,...,t

(A) = P({ω ∈ Ω : (X

(ω), X

(ω), ..., X

(ω)) ∈ A})

be a measure in R

. X and Y are equivalent if t heir families of ﬁnite

distributions coincides and we denote by X ∼ Y .

Deﬁnition 4 (Modiﬁcation of a process) Suppose (X

)

t≥0

and (Y )

t≥0

two

proce sses deﬁned on the same probability space (Ω, F, P) and taking values

in (E, E). The process {Y

} is said to be a modiﬁcation of {X

} if

= Y

a.s. ∀t ∈ T ;

i.e.,

P(X

= Y

) = 1 ∀t ∈ T .

Remark: Note that for each t, it is possible to have a null set associated to it.

Deﬁnition 5 (Indistinguishable) Suppose (X

)

t≥0

and (Y )

t≥0

two processes

deﬁned on the same probability space (Ω, F, P) and taking values in (E, E).

The process {Y

} is said to be indistinguishable from {X

} if for almost every

ω ∈ Ω,

(ω) = Y

(ω) ∀t ∈ T ;

i.e.,

P({ω ∈ Ω : X

(ω) = Y

(ω), ∀t ∈ T }) = 1.

Remark: Note that in this case, there is only one null set.

Now we are going to present a concept concerning sets but that is related

to indistinguishability:

Risk Neutral Option Pricing under some special GARCH models 14

Deﬁnition 6 (Evanescent) A ⊂ [0, ∞] × Ω is evanescent if

(t, ω) =



1, if (t, ω) ∈ A

0, if (t, ω) /∈ A

(2-1)

is indistinguishable from the zero process, i.e.,

P({ω ∈ Ω : 1

(t, ω) = 0, ∀t ∈ T }) = 1. (2-2)

what is equivalent to

P({ω ∈ Ω : ∃t ∈ T with (t, ω) ∈ A}) = 0. (2-3)

which means that the probability of having an omega that allows the existence

of a t such that the pair (t, ω) is in A is null.

Deﬁnition 7 (Adapted Process) X is said to be adapted to (F

)

t≥0

if X

-measurable ∀t ∈ T .

It means that information comes according to time. No future information is

known. Here we can for each t, check the measurability of X as a function only

of omega.

Deﬁnition 8 (Progressively Measurable) The process X deﬁned in ([0, T ] ×

Ω, B([0, T ] × F)) to a measurable space (E, E) is said to be “progressively

measurable” (or simply “progressive”) if, for every time t ∈ [0, T ], the map

X : [0, T ] × Ω → E

(t, ω) → X

(ω)

is B([0, T ]) ⊗ F

-Measurable function. This implies that X is F

-adapted.

This property asks a jointly measurability condition that concerns not only

the space but also time.

2.3

Martingales

One of the most important concepts in Finance is the process property

of being a martingale. Here we are going to deﬁne, give examples and see the

main related results shortly.

Risk Neutral Option Pricing under some special GARCH models 15

2.3.1

Deﬁnition

Deﬁnition 9 (Martingale) A real valued adapted process (M

) is said to be a

martingale with respect to the ﬁltration {F

}

t∈T

if E|M

| < ∞ ∀t and ∀s ≤ t:

E[M

] = M

. a.s.

If the equality is replaced by ≤ then (M

) is said to be a supermartingale.

If the equality is replaced by ≥ then (M

) is said to be a submartingale.

The martingale condition can be regarded as E[X

] being a version of

the process X



E[X

]dP =



dP A ∈ F

(2-4)

but by the deﬁnition of conditional expectation we have:



E[X

]dP =



dP A ∈ F

(2-5)

so that for s ≤ t:



dP =



dP A ∈ F

. (2-6)

The martingale can be seen as a model for “fair games” or a “pure random

process” because given the information available ab out the process until now,

the expected value is the present value.

Remark: A martingale is a pr ocess “constant in mean”, in the sense

that

E[M

] = E[M

] ∀t ≥ 0

Indeed,

E[M

] = M

a.s. s ≤ t

implies

E[E[M

]] = E[M

]

so that by the iterated expectation property:

E[M

] = E[M

] ∀s ≤ t.

Theorem 10 (Levy) Let (B

)

t≥0

be a standard Brownian Motion with respect

to the ﬁltration (F

)

t≥0

. Then:

Risk Neutral Option Pricing under some special GARCH models 16

a) (B

)

t≥0

is an F

-martingale.

b) (B

− t)

t≥0

is an F

-martingale.

c) (e

σB

−

)

t≥0

is an F

-martingale.

Also, the converse holds true (stated as Characterization of a Brownian

Motion in section 3.6). Besides that, there is a curious property of the

Brownian Motion, viz., its paths although continuous a.s. are non-diﬀerentiable

almost everywhere.

For the proof of this curiosity and the Theorem see Evans (26) and Elliot (21)

respectivelly.

2.4

Stochastic Integrals

Here we build the deﬁnition of the stochastic integral starting from simple

functions and ﬁnishing with a wider range of functions.

Deﬁnition 11 (Simple processes)Consider (W

) a (F

)-Brownian motion de-

ﬁned on (Ω, F, P) . A real-valued simple process on [0, T ] is a function H for

which

a) There is a partition 0 = t

< t

< ... < t

= T ; and

b) H

= H

(ω) and H

= H

(ω) for t ∈ (t

, t

i+1

], where H

(·) is F

measurable and square integrable. That is,

= H

(ω) +

n−1



i=0

(ω)1

i+1

]

, t ∈ [0, T ].

Deﬁnition 12 (Stochastic Integral of a simple process) If H is a simple

proce ss, the stochastic integral of H with respect to the Brownian Motion (W

)

is the process deﬁned for t ∈ (t

, t

k+1

], by



k−1



i=0

i+1

− W

) + H

− W

This can be written as a martingale transform:





i=0

i+1

∧t

− W

∧t

Theorem 13 Suppose H is a simple process. Then:

Risk Neutral Option Pricing under some special GARCH models 17



is a continuous F

-martingale.

b) E



(



)



= E







(Itˆo Isometry).

c) E



sup

0≤t≤T





≤ 4E







Lemma 14 Let H be the space of processes adapted to (F

) that satisfy



ds] < ∞. Suppose H

∈ H. Then there is a sequence {H

} of simple

proce sses such that

lim

n→∞





− H



= 0.

i.e., we can now deﬁne the stochastic integral to a broader space taking

this special L

limit of simple processes. Simple processes are dense in H if we

consider this convergence.

There is a broader class of processes that the integral can be deﬁned keeping

the properties in the Theorem above. Please refer to Elliott (21) for this

generalization.

2.5

Itˆo Processes, Diﬀerentiation Rule and Solution of a SDE

In this section we are going to present the Itˆo Processes, th e so called

Itˆo Lemma and see through some examples the usual uses of it in solving

Stochastic Diﬀerential Equations.

2.5.1

Initial Deﬁnitions

Deﬁnition 15 (Itˆo Processes) Suppose (Ω, F, P) is a probability space with a

ﬁltration (F

)

t≥0

and (W

) is a standard (F

)-Brownian Motion. A real valued

Itˆo process (X

)

t≥0

is a process of the form:

= X



ds +



where:

(a) X

is F

-measurable,

(b) K and H are adapted to F

(c)



|ds < ∞ a.s. and



ds < ∞ a.s.

Risk Neutral Option Pricing under some special GARCH models 18

Deﬁnition 16 (Quadratic Variation) Given a partition 0 = t

< t

< ... <

= t of the interval [0, t] and writing |π| = max

i+1

− t

), the quadrati c

variation of a continuous martingale (M

)

t≥0

, denoted by M

, is deﬁned by:

M

= lim

|π|→0



i=0

i+1

− M

)

For the Brownian Motion, W 

= t

2.5.2

Itˆo Formula and SDEs

Let’s start with the main result in Stochastic Calculus:

Theorem 17 (Itˆo Formula) Suppose {X

}

t≥0

is an Itˆo process of the form

= X



ds +



Suppose f twice diﬀerentiable. Then,

f(X

) = f(X

) +



′

)dX



′′

)dX

Here, by deﬁnition, X



ds ; that is the (predictable) quadratic vari-

ation of X is the quadratic variation of the martingale component



Also,



′

)dX



′

ds +



′

Deﬁnition 18 (Solution of a SDE)

A process X

, 0 ≤ t ≤ T is a solution of the stochastic diﬀerential

equation

= f(X

, t)dt + σ(X

, t)dW

with initial condition X

= ξ if for all t the integrals



f(X

, s)ds and



σ(X

, s)dW

are well deﬁned and

= ξ +



f(X

, s)ds +



σ(X

, s)dW

a.s.

Risk Neutral Option Pricing under some special GARCH models 19

Note that the symbols “dt”, “dW ” have no meaning alone they just

make sense when they are in a equation. Even then the meaning is that of a

notation for the integral equation of its solution. Now that we have introduced

the diﬀerential notation, for pencil and paper compuations, we prefer to write

the result in Itˆo Lemma as:

df(X

) = f

′

)dX

′′

)dX

together with the fact that X



ds we have ultimately:

df(X

) = f

′

)dX

′′

ds (2-7)

= f

′

dt + f

′

′′

dt (2-8)



′

′′



dt + f

′

(2-9)

It is worth writing the more general result although later we are going to show

a suﬃcient general particular case.

Theorem 19 (Multidimensional Itˆo Lemma)

Suppose X

= (X

, ..., X

) is a n-dimensional Itˆo process with

= K

dt +



j=1

, (2-10)

and suppose f : [0, T ] × R

→ R is C

1,2

. Then

df(t, X

, ..., X

) =

∂f

∂t

(t, X

, ..., X

)dt +



i=1

∂f

∂x

(t, X

, ..., X

)dX



i,j=1

∂f

∂x

(t, X

, ..., X

)





r=1

i,r

j,r



Note that if m = 1 we have:

df(t, X

, ..., X

) =

∂f

∂t

(t, X

, ..., X

)dt +



i=1

∂f

∂x

(t, X

, ..., X

)dX



i,j

∂f

∂x

(t, X

, ..., X

Risk Neutral Option Pricing under some special GARCH models 20

2.5.3

Existence and Uniqueness

The result we are going to present assure us about the existence and

uniquiness of a solution in some Stochastic Diﬀerential Equations.

Theorem 20 (Existence and Uniq ueness of solutions) Suppose the usual as-

sumptions and that ξ,f and σ satisfy:

|f(x, t) − f(x

′

, t)| + |σ(x, t) − σ(x

′

, t)| ≤ K|x − x

′

|, (2-11)

|f(x, t)|

+ |σ(x, t)|

≤ K

(1 + |x|

), (2-12)

E[|ξ|

] < ∞ (2-13)

for all 0 ≤ t ≤ T, x, ˆx ∈ R

and some constant K. Then there is a unique

solution X (up to indistinguishability) given by

= ξ +



f(X

, s)ds +



σ(X

, s)dW

a.s.

such that



sup

0≤t≤T



< C



1 + E[|ξ|

]



Proof :

Please refer to (21) and (26).

2.6

Examples of Itˆo Formula

Example 21 Lets Calculate the integral of W

w.r.t. W

, i.e.,



For that, we consider dX

= dW

so that K

= 0 and H

= 1 and the

fuction y = x

By the Itˆo formula,

X

= W 



1dv = u

Risk Neutral Option Pricing under some special GARCH models 21

and

d(W

) = mW

m−1

m(m − 1)W

m−2

du.

or equivalently

− W

= m



m−1



m(m − 1)W

m−2

du.

Substituting m = 2, we have

− W

= 2



2du

so that

− W

= 2



Then

− W

−

(t − s)



instead of the expected



− W

that we used to have in calculus.

Lemma 22 (Two-dimensional formula)

For,

= K

dt + H

= K

dt + H

We have

f(t, X

, Y

) = f(0, X

, Y

) +



∂f

∂s

(s, x, y)ds +



∂f

∂x

(s, x, y)dX +



∂f

∂y

(s, x, y)dY +



∂f

∂x∂y

(s, x, y)H



∂f

∂x

(s, x, y)H

ds +



∂f

∂y

(s, x, y)H

Example 23 (Itˆo Product Rule) Suppose

= K

dt + H

Risk Neutral Option Pricing under some special GARCH models 22

= K

dt + H

The Itˆo Formula applied to f(x, y) = x.y we have:

f(t, X, Y ) = f(0, X

, Y

) +



Y dX +



XdY



d(X

) = Y

+ X

+ H

Example 24 (Solution of a Lognormal SDE)

Suppose

= µdt + σdW

, S

= X

or equivalently

= X



µds +



σdW

This is a Itˆo Process if we think K

= S

µ and H

= S

σ satisfying the

usual conditions.

Then X



ds.

If S

> 0, by the Itˆo Formula with f(x) = log(x) we have

logS

= logX



−

ds (2-14)

= logX





µ −



ds +



σdW

(2-15)

= logX



µ −



t + σW

(2-16)

so that

= X



µ−



t+σW



Example 25 (Testing a Solution of a Lognormal SDE) Consider

F (t, x) = X



µ−



t+σx



Show that S

= F (t, W

) satisﬁes the lognormal SDE of the example using Itˆo

Formula.

Risk Neutral Option Pricing under some special GARCH models 23

By Itˆo Formula,

F (t, W

) = F (0, W

) +



∂F

∂s

(s, W

)ds



∂F

∂W

(s, W

)dW



∂

∂W

(s, W

)d W 

Equivalently



µ−



t+σW



= X





µ −





µ−



s+σW





σX



µ−



s+σW







µ−



s+σW



Then



µ−



t+σW



= X



µX



µ−



s+σW





σX



µ−



s+σW



Now, by the deﬁnition of S

= X



µS

ds +



σS



Example 26 (Quotient Rule) Suppose B a Brownian Motion and

= X

(µ

dt + σ

)

= Y

(µ

dt + σ

)

Deﬁne Z by Z

= Y

. Show that Z is lognormal with dyn amics

= Z

(µ

dt + σ

)

and determine µ

and σ

in terms of the coeﬃcients of X and Y .

Risk Neutral Option Pricing under some special GARCH models 24

The Itˆo Formula applied to f(x, y) = y/x gives us:





−Y

dt −

−Y

dt +

−Y

dt +

dt −

(µ

− µ

+ σ

− σ

)dt +

(σ

− σ

)dB

so that





(µ

− µ

+ σ

− σ

)dt +

(σ

− σ

)dB

means by the deﬁnition of Z

that

= Z

(µ

dt + σ

)

with

= (µ

− µ

+ σ

− σ

)

= (σ

− σ

)



2.7

Change of Measure

Let Q be a second probability measure on (Ω, F) t hat is absolutely

continuous with respect to P (Q << P) such that



, and Q(A) =



dP, for A ∈ F

Lemma : (X

) is a martingale under P iﬀ X

is a martingale under

Proof : Suppose s ≤ t and A ∈ F

. Then



dQ =



dP =



dP =



dQ.

Theorem 27 (Characterization of a Brownian Motion) Suppose (W

)

t≥0

a continuous (scalar) martingale on the ﬁltered space (Ω, F, P, F), such that

− t)

t≥0

is a martingale. Then (W

) is a Brownian Motion.

Risk Neutral Option Pricing under some special GARCH models 25

So this Theorem implies that a real process (B

)

t≥0

is a standard Brown-

ian Motion if:

a) t → B

(ω) is continuous a.s.,

b) B

is a martingale

c) B

− t is a martingale.

Deﬁnition 28 (History of the Brownian Motion Process) Write F

= σ(B

s ≤ t) for the σ-algebra on Ω generated by the history of the Brownian Motion

up to time t. Then, (F

)

t≥0

will denote the right-continuous complete ﬁltration

generated by the F

The idea of the Girsanov’s Theorem is to show how (B

) behaves under

a change of measure.

Theorem 29 (Girsanov’s Theorem) Suppose (θ

)

0≤t≤T

is an adapted, mea-

surable process such that



ds < ∞ a.s. and also so that the process

= e

{

−



−



}

(2-17)

is an (F

, P) martingale. Deﬁne a new measure Q

on F



= Λ

(2-18)

Then the process

= B



is a standard Brownian motion on (F

, Q

Proof : By Itˆo diﬀerentiation rule and by the deﬁnition of Λ. For details please

refer to (21).

GARCH Models

We will here, deﬁne the basic econometric models that will be the basis for

our work. In chapters 5 and 6 we are going to adapt the methodoly developed

in Siu et al. (43) to the FC-GARCH model of

Alvaro and Medeiros (36) and

for the Mixture of GARCHs.

To start, we should ﬁrst go to the basics and deﬁne the ARCH and

GARCH models.

3.1

ARCH Models

When we want to express the volatility in terms of past returns, we should

use an Autoregresive Conditional Heterokesdastic model (henceforth ARCH)

model, which is deﬁned by a model with:

= µ

+ h

1/2

(3-1)

where

= α



i=1

t−i

(3-2)

that can be rewritten as

= α



i=1

′ǫ

t−i

(3-3)

where ǫ

is a noise usually a N(0, 1) random variable and µ

is the conditional

mean that we will be always considering conditionally deterministic, or more

precisely, F

t−1

measurable, sometimes even being considered to be zero for

stationarity purposes.

When we want to model the conditional volatility taking in account not

only the past errors but also the past volatilities we should use a General

Autoregresive Conditional Heterokesdastic model (henceforth GARCH) which

Risk Neutral Option Pricing under some special GARCH models 27

is modeled as below:

= µ

+ h

1/2

(3-4)

with

= α



i=1

t−i



j=1

t−j

(3-5)

Considering the same observations for µ

and ǫ

as above. For simplicity let´s

suppose it zero.

Another way of writing the same thing (but note the diﬀerent alpha) is:

= α



i=1

′ǫ

t−i



j=1

t−j

(3-6)

In the next subsection we are going to show a more general GARCH-like

model, viz., the FC-GARCH by Medeiros and

Alvaro, that poorly speaking,

gives a GARCH to each regime of the economy.

3.2

Flexible Coeﬃcient Generalized Autoregressive Conditional Heteroskedas-

tic (FC-GARCH)

First, we consider a discrete-time ﬁnancial model consisting of one risk-

free bond B and one risky stock S. We consider a complete probability space

(Ω, F, P) with P being the r eal-world probability measure. Let T denote the

time index set {0, 1, 2, . . . , T } of the ﬁnancial model. Let r denote a constant

continuously compounded risk-free rate of interest or force of interest. Under

P, the dynamics of the bond-price process {B

}

t∈T

satisfy, for each t ∈ T \{0}:

= B

t−1

, B

= 1 . (3-7)

Let ǫ = {ǫ

}

t∈T

denote the innovations of the returns from the risky stock S,

where ǫ

= 0. We assume that {ǫ

}

t∈T

are i.i.d. with distribution D(0, 1), where

D(0, 1) represents a general distribution with mean zero and unit variance.

Let S := {S

}

t∈T

denote the price process of the risky asset S. Let

:= ln(S

t−1

), which represents the continuously compounded rate of

return from the risky asset S over the time horizon [0, t]. We suppose t hat the

return process Y := {Y

}

t∈T

follows a ﬁrst-order ﬂexible coeﬃcient Generalized

Risk Neutral Option Pricing under some special GARCH models 28

Autoregressive Conditional Heteroskedastic model with m = H + 1 limiting

regimes, henceforth, FC-GARCH(m, 1, 1) as below:

= µ

+ h

1/2

= G(w

; ψ) . (3-8)

Here G(w

; ψ) is a nonlinear function of a vector of variables w

t−1

, h

t−1

, s

]

(i.e. “T

′′

represents the transpose of a vector or a matrix) de-

ﬁned by:

G(w

; ψ) := α

+ β

t−1

+ λ

t−1



i=1

[α

+ β

t−1

+ λ

t−1

]f(s

; γ

, c

), t = 1, ..., T.

where

1. for each i = 1, 2, . . . , H, the logistic function

f(s

, γ

, c

) :=

1 + e

−γ

−c

)

;

2. The vector of parameters

ψ := [α

, β

, λ

, α

, ..., α

, β

, ..., β

, λ

, ..., λ

, γ

, ..., γ

, c

, ..., c

]

∈ R

3+5H

;

3. The parameter γ

, i = 1, ..., H is called the slop e parameter. Wh en

→ ∞, the function becomes a step function. Here, we consider the

case when s

= Y

t−1

The rationale of introducing the FC-GARCH model is to provide a useful

and practical way to incorporate the asymmetric eﬀect of the sign and the

size of the previous return Y

t−1

on the current variance level h

. It can also

capture the heavy-tailedness of return’s distribution and the slow decaying of

the autocorrelation of the squared returns process {Y

}

t∈T

. The FC-GARCH

model can also incorporate another important stylized empirical feature of

returns data, namely, the Taylor eﬀect, ﬁrst documented by Taylor (1986)(44).

The Taylor eﬀect refers to th e strong autocorrelation of absolute daily returns

data. This also relates to the long-memory eﬀ ect of volatility; that is, the decay

of the autocorrelations of volatility is too slow to be described by any short

memory autoregressive moving average time series models. We are going to get

back to this model in Chapter 5.

Option Pricing

This Chapter explains basically how should we price an option.

In a ﬁrst Finance Mathematics course we are thaught the concept of

present value, which consists in discounting by a constant risk free interest

rate some value in the future so that we can ﬁnd the price of it in the present.

It is correct if we think of loans and bank accounts with ﬁxed interest

rates because we know the evolution of these values, so we can track it and

use it in a reverse way to ﬁnd the present value, but not in general. For risky

assets we don’t know the evolution of the price so we cannot discount by a

proper rate.

On the other hand Probability Theory have some answers to solve that

problem. Under some assumptions, if we change properly the measure in which

we take expectations we can think of any discounted portfolio as a risk neutral

one in t hat measure. That’s why such measures are more usually called Risk

Neutral Measures than Equivalent Martingale Measures. The latter name is

because under that new equivalent measure, the discounted portfolio becomes

a martingale.

4.1

Pricing Formula in the Risk-Neutral Measure

For this section, we follow the chapter 5 in Shreve (41) for the background

and to show the Black-Scholes example.

Let D

= e

−



r(u)du

be a discount factor process and X

be a self-ﬁnancing

portfolio made of ﬁxed interest investment and risky assets where the amount

invested into risky assets in time t is ∆

. It means that changes in the value

of the portfolio are due to changes in the price of the risk asset and changes

in the safe investment and not for putting more money into this portfolio.

Mathematically we ask that

= ∆

+ r

− ∆

)dt

Note that according to th e Itˆo product rule, asking for being self-ﬁnancing

is very restrictive because we lose some terms in the equation above. Suppose

Risk Neutral Option Pricing under some special GARCH models 30

also that the asset follows

= α

dt + σ

(4-1)

or as in example 25 of the use of Itˆo’s Lemma:

= S



σd





(

r−

)

(4-2)

By the Itˆo Lemma, we get ﬁrst that

= e

−



r(u)du

r(t)dt = −D

Then

= ∆

+ r

− ∆

)dt (4-3)

= ∆

dt + ∆

+ r

− ∆

)dt (4-4)

= (∆

dt + r

− r

∆

)dt + ∆

(4-5)

= (r

+ (α

− r

)∆

)dt + ∆

(4-6)

= r

dt + σ

∆

(θ

dt + dW

) (4-7)

= r

dt + σ

∆



. (4-8)

where



is a Brownian Motion by Girsanov’s Theorem and θ

(α

−r

)

Proposition 30 D

is a martingale under Q

Proof : We will proceed as Shreve (41). By the product rule (example 23 of

the use of Itˆo’s Lemma) we have

d(D

) = X

+ D

+ dD

(4-9)

= X

(−D

)dt + D

+ 0 (4-10)

= −X

dt + D

dt + σ

∆



) (4-11)

= D

∆



. (4-12)

The G irsanov’s Theorem garantee us that



is a Brownian Motion and

so, as a stochastic integral with respect to a Brownian Motion is a martingale,

is a martingale. 

Let t he value of some derivative V

be F

-measurable. We want to know

what is the initial money X

and which process ∆

are required to be in a

portfolio so that an investor can hedge a position in this derivative, i.e., such

that

= V

Risk Neutral Option Pricing under some special GARCH models 31

If we can do that, given that D

is a martingale under Q we have

= E

] = E

]

Then, D

is the amount required in t to make the hedge of this

derivative with payoﬀ V

. We can then say that the price of the derivative

in t is V

where

= E

] 0 ≤ t ≤ T (4-13)

= E

−



r(u)du

] 0 ≤ t ≤ T (4-14)



4.2

Black-Scholes-Merton Formula

The most famous ﬁnance formula is the Black-Scholes formula. It is easy

to ﬁnd in many books the proof that consists in deriving the Black-Scholes

PDE and then turning it in a heat equation problem. Here we present it under

the risk-neutral framework as in Shreve (41).

The Black-Scholes-Merton Model supposes that the interest rate and the

volatility are ﬁxed. For an European Call option, we have the payoﬀ

= (S

− K)

(4-15)

Using the pricing formula just discussed we need to calculate

= E

−r(T −t)

− K)

] (4-16)

Let’s think of the formula above as a function of t and S

C(t, S

) = E

−r(T −t)

− K)

] (4-17)

The equation

= S



σd





(

r−

)

(4-18)

Risk Neutral Option Pricing under some special GARCH models 32

can be written as

= S



σd





(

r−

)

(4-19)

= S

σ(



−



(

r−

)

(4-20)

where τ = T −t. If we further deﬁne Y = −



−



√

T −t

we can write

= S

−σ

√

τY +

(

r−

)

(4-21)

Then, as S

is F

-measurable and e

−σ

√

τY +

(

r−

)

is independent of F

C(t, x) = E

−rτ

− K)

] (4-22)

√

2π



∞

−∞

−rτ

(xe

−σ

√

τY +

(

r−

)

− K)

−

dy (4-23)

Now, note that (xe

−σ

√

τY +

(

r−

)

− K)

is positive if and only if

y < d

−

≡

√









r −





(4-24)

Indeed,

−σ

√

τy+

(

r−

)

> K/x ⇐⇒ e

−σ

√

τy

> (K/x)e

−

(

r−

)

⇐⇒ −σ

√

τy > ln(K/x) −



r −



∗(−1)

⇐⇒ y <

√









r −





We don’t have to consider the integral when it is zero. This leads us to:

C(t, x) =

√

2π



−

−∞

−rτ

(xe

−σ

√

τy+

(

r−

)

− K)e

−

√

2π



−

−∞

−

−σ

√

τy+

(

−

)

dy −



−

−∞

−rτ

−

√

2π



−

−∞

−

(y+σ

√

τ )

dy − Φ(d

−

−rτ

z=y+σ

√

= x

√

2π



−

+σ

√

−∞

−

(z)

dz − Φ(d

−

−rτ

= xΦ(d

) − Φ(d

−

−rτ

Risk Neutral Option Pricing under some special GARCH models 33

where

= d

−

+ σ

√









r −





+ σ

√







r −



τ + σ

√







r +



√



4.3

Put-Call Parity

Now, we will see that who can solve the problem of pricing the European

Call option, can also ﬁnd the price of an Europena Put Option.

Build a portfolio made of:

• Buy a stock.

• Buy a put option with this stock bought as the underlying.

• Sell a call option also based in the stock bought.

Then, if we call this portfolio Π, we can write:

= S

+ P

− C

But then at time T we will have:

payoﬀ

(Π) = S

+ (K −S

)

− (S

− K)



+ K −S

, if S

≤ K

− (S

− K) , if S

> K

= K.

This portfolio always gives a return K, the exercise price. Then, because

it is a deterministic portfolio, we can think of its present value being that of

K discounted:

= Ke

−r(T −t)

Risk Neutral Option Pricing under some special GARCH models 34

Then we have:

−r(T −t)

= S

+ P

− C

⇒ P

= Ke

−r(T −t)

+ C

− S



4.4

Incomplete Markets

Here we are going to show two well known techniques for pricing in

incomplete markets, viz. the Davis method and the Esscher method.

When we can not ﬁnd a hedge for some asset in the economy we say

that the market is incomplete and t hen it is known that there are inﬁnite

equivalent martingale measures. The risk neutral way of pricing do not involve

requirements concerning the risk preferences of the investors, but in order to

justify the choice of one particular risk neutral measure among many, we will

have to think about a reasonable way to select it.

Besides the two methodologies that we are going to discuss brieﬂy here,

there are other approaches to hedging and pricing in incomplete markets, such

as the quadratic hedging approach. They are mainly interested in minimizing

risk, Maximizing expected utility or minimize loss. For a comprehensible

reading see Bingham and Kiesel (2004)(5).

4.4.1

Davis Method

We start by the method of Davis. Let Φ

be the set of all self-ﬁnancing

strategies (i.e. in which you can not inject money in the portfolio. For details

see (21)and (26)) and write

U(x) = sup

ϕ∈Φ

E[U(V

ϕ,x

(T ))]

for the maximum expected utility of an investor and suppose p is the price of

the asset X.

Let

W (δ, x, p) = sup

ϕ∈Φ

E[U(V

ϕ,x−δ

(T ) +

X)]

be a disturb created for being able to sense the eﬀect of changing the strategy.

If the maximum utility does not get aﬀected by this “tilt”, ˆp is the fair price

for the option.

Under some conditions, ˆp is the unique solution of

Risk Neutral Option Pricing under some special GARCH models 35

∂W

∂δ

W (0, p, x) = 0

which is the fair price of the option in t = 0. Furthermore, this value is

given by:

ˆp =

E[U

′

∗

(T ))X]

′

(x)

For more details see Davis(1994)(13) and Bingham and Kiesel(2004) (5).

4.4.2

Esscher Method

Now let’s see the Esscher method. Let S

= S

be the model for the

asset prices where X

has independent and stationary increments. Assume that

M(h, t) = M(h, 1)

= E[e

] (4-25)

Then, having that in mind, let’s deﬁne

E[e

]

= e

M(h, 1)

−t

E[S

]

; t ≥ 0 (4-26)

that is a positive martingale and will be used for the change of measure.

Now, by the Baye’s rule,

[Ψ(Y )] = E[Ψ(Y ); h] =

E[Ψ(Y )e

]

E[e

]

(4-27)

= E[Λ

Ψ(Y )] (4-28)

where Y is a random variable and Ψ is a measurable function. For

example Ψ(Y ) could be the logreturn e

We will call Q the Esscher measure of parameter h = h

∗

if {e

−rt

}

t≥0

a martingale.

By the condition on the asset prices we have the following equivalences:

Risk Neutral Option Pricing under some special GARCH models 36

E[e

−rt

, h

∗

] = S

⇐⇒ E



, h

∗



= e

(4-29)

⇐⇒ E[e

, h

∗

] = e

(4-30)

⇐⇒ E



∗

M(h

∗

, 1)



= e

(4-31)

⇐⇒

E[e

∗

+1)

]

M(h

∗

, 1)

= e

(4-32)

⇐⇒

M(h

∗

+ 1, 1)

M(h

∗

, 1)

= e

(4-33)

⇐⇒



M(h

∗

+ 1, 1)

M(h

∗

, 1)



= e

(4-34)

⇐⇒

M(h

∗

+ 1, 1)

M(h

∗

, 1)

= e

(4-35)

(4-36)

The equation

M(h

∗

+ 1, 1)

M(h

∗

, 1)

= e

(4-37)

determines the parameter h

∗

uniquely. We will be interested in this

parameter for ﬁnding the risk neutral measure to be used in the option pricing.

There is a useful result that we should show here. It is known as the

factorization formula:

Theorem 31 Let g be a measurable function and h, k and t be real values

with t ≥ 0, then

E[S

g(S

); h] = E[S

; h]E[g(S

); k + h]. (4-38)

Proof :

E[S

g(S

); h] = E[S

g(S

M(h, 1)

−t

] (4-39)

= E



g(S

)

E[S

]



(4-40)

E[S

k+h

]E[S

k+h

g(S

)]

E[S

]E[S

k+h

]

(4-41)

= E[S

; h]E[g(S

); k + h] (4-42)

Risk Neutral Option Pricing under some special GARCH models 37



For more details see Gerber and Shiu (1994)(31) and Bingham and

Kiesel(2004)(5). Some of the points discussed above will be revisited in the

conditional Esscher Tr ansform section.

4.5

Duan’s breakthrough

In 1995, Duan developped a method for pricing options under GARCH

processes. Although there are economic restrictions and is valid only for

normal noises, it was a milestone in Finance. He also developped some papers

concerning the relationship between the diﬀusions and the econometric models

and their risk neutral versions. This discussion doesn’t help us in this thesis

and it can b e found in Duan(1997)(18). Here we show the main results from

the paper published in 1995 (17).

Consider the model:

t−1

= r + λ



−

+ ξ

where

|φ

t−1

∼ N(0, h

) under mesure P

= α



i=1

t−i



j=1

t−j

Basically the paper raises the question: What would the risk-neutral

GARCH be?

For answering this question, he deﬁnes the concept of Local Risk Neutral

Valuation Relationship hereafter LRNVR, as below:

A measure Q satisﬁes the LRNVR if:

1 Q is mutually absolutely continuous with respect to measure

P.(Equivalents)

t−1

is lognormally distributed (und er Q)

3 E

[

t−1

|φ

t−1

] = e

4 V ar

(ln

t−1

|φ

t−1

) = V ar

(ln

t−1

|φ

t−1

) a.s.

Having in mind this concept we can state the main Theorem of the paper:

Theorem 32 Supposing the LRNVR, under Q we have:

t−1

= r −

+ ǫ

Risk Neutral Option Pricing under some special GARCH models 38

where:

|φ

t−1

∼ N(0, h

)

= α



i=1

(ǫ

t−i

− λ



t−i

)



j=1

t−j

Proof : It follows exactly as in Duan (17). Since

t−1

|φ

t−1

lognormally dis-

tributed under measure Q, it can be written as:

t−1

= v

+ ǫ

where v

is the conditional mean and ǫ

is a Q-normal random variable. The

conditional mean of ǫ

is zero and its conditional variance is to be determined.

The proof is in two parts.

First we prove that v

= r −

. Indeed,



t−1

|φ

t−1



= E

+ǫ

|φ

t−1

] = e

where h

= V ar

[

t−1

|φ

t−1

] = V ar

[

t−1

|φ

t−1

] by the LRNVR.

Since E

[

t−1

|φ

t−1

] = e

also by the LRNVR, it follows that v

= r −

Now we prove the second part. It remains to prove that h

can indeed

be expressed as stated in th e Theorem. By the preceding result and the model

under P:

t−1

= r + λ



−

+ ξ

, (4-43)

we have, comparing the logs:

r + λ



−

+ ξ

= r −

+ ǫ

. (4-44)

so that ξ

= ǫ

− λ

√

. Substituting ǫ

into the conditional variance

equation yields the desired result:

= α



i=1

(ǫ

t−i

− λ



t−i

)



j=1

t−j



The result just proved can be found as a particular case of the method

described in the next section. This veriﬁcation can be found in section 3.1

Risk Neutral Option Pricing under some special GARCH models 39

in the paper by Siu et al. (2004)(43). The method below doesn’t require the

noises to be normal, they just have to have a moment generation function.

Option Pricing under a Nonlinear and Nonnormal GARCH

This chapter is based in a pap er written together with both my advisor

and my co-advisor, Professors

Alvaro Veiga and Ken Siu respectively. We

investigate the pricing of options in a class of discrete-time Flexible Coeﬃcient

Generalized Autoregressive Conditional Heteroskedastic (FC-GARCH) models

with non-normal innovations. A conditional Esscher transform was used to

select a price kernel for valuation in the incomplete market. This choice of the

pricing kernel can be justiﬁed by an economic equilibrium argument based on

maximizing the expected power utility. We provide a numerical study on the

pricing results when the GARCH innovations have a normal distribution or

a shifted-Gamma distribution and identify some key features of the pricing

results.

The conditional Esscher transform provides a convenient and ﬂexible way

to determine a price kernel under nonlinear time series models. Here we exploit

this important tool in actuarial science to determine a price kernel for option

valuation.

5.1

Flexible Coeﬃcient Generalized Autoregressive Conditional Heteroskedas-

tic (FC-GARCH) models for Asset Returns

We consider a discrete-time economy with a bond B and a share S. Let T de-

note the time index set {0, 1, 2, . . . , T } of the economy. To model uncertainty,

we ﬁx a complete probability space (Ω, F, P) where P is a real-world prob-

ability measure. To simplify our analysis, we assume that the continuously

compounded rate of interest from the bond is a constant, say r per period.

Consequently, th e bond-price process {B

|t ∈ T } evolves over time as:

= B

t−1

, B

= 1 . (5-1)

Let ǫ = {ǫ

}

t∈T

be the return innovations of the share S, where

we take ǫ

= 0 by convention. Suppose {ǫ

|t ∈ T \{0}} are independent

and identically distributed, (i.i.d.), with common distribution D(0, 1), where

D(0, 1) represents a general distribution with zero mean and unit variance.

Risk Neutral Option Pricing under some special GARCH models 41

Let S := {S

}

t∈T

be the price process of the share S. Let Y

ln(S

t−1

), which is the continuously compounded rate of return from the

share S from time t − 1 and time t. Then we assume that the return

process Y := {Y

|t ∈ T } follows a ﬁrst-order Flexible Coeﬃcient Generalized

Autoregressive Conditional Heteroscedastic model with m = H + 1 limiting

regimes, henceforth, FC-GARCH (m, 1, 1):

= µ

+ h

1/2

= G(w

; ψ) . (5-2)

Here G(w

; ψ) is a nonlinear function of a vector of variables w

t−1

, h

t−1

, s

)

′

, (i.e. “′” represents the transpose of a matrix, or in particu-

lar a vector), deﬁned by:

G(w

; ψ) := α

+ β

t−1

+ λ

t−1



i=1

[α

+ β

t−1

+ λ

t−1

]f(s

; γ

, c

) ,

where

1. for each i = 1, 2, . . . , H, the logistic function

f(s

, γ

, c

) :=

1 + e

−γ

−c

)

;

2. the vector of parameters

ψ := (α

, β

, λ

, α

, ··· , α

, β

, ··· , β

, λ

, ··· , λ

, γ

, ··· , γ

, c

, ··· , c

)

′

∈ R

3+5H

;

3. for each i = 1, 2, ··· , N, the parameter γ

is the slope parameter and

is considered positive. When γ

→ ∞, the function becomes a step

function. Please refer to Medeiros and Veiga(2009)(36), for a complete

list of assumptions made on the parameters. Note that their restrictions

are for a simpler FC-GARCH in which the conditional mean is constant.

Here, we consider a simple case that s

= Y

t−1

The class of FC-GARCH models provides the ﬂexibility in incorporating

the asymmetric eﬀect of the sign and the size of the previous return Y

t−1

on the current variance level h

. It can also capture the heavy-tailedness of

return’s distribution and the slow decay of the autocorrelation of the squared

returns process {Y

|t ∈ T }. In addition, the FC-GARCH model can capture

another important stylized empirical feature of returns data, namely, the

Taylor eﬀect, ﬁrst documented by Taylor (1986)(44). The Taylor eﬀect refers

Risk Neutral Option Pricing under some special GARCH models 42

to the strong autocorrelation of absolute daily returns data. This also relates to

the long-memory eﬀect of volatility; that is, the decay of th e autocorrelations

of volatility is too slow to be described by any short memory autoregressive

moving average time series models. In the empirical studies by Ding et al.

(1993) (16), it has been documented that the realized volatility decays in a

hyperparabolic rate.

When γ

= 0, or α

, β

, λ

= 0, i = 1, 2, ··· , H, the FC-GARCH model

reduces to the GARCH(1,1) model. The FC-GARCH model also nests other

important ARCH-type models in the literature. Some examples include the

LST-GARCH(1,1) model, the GJR-GARCH(1,1) model, the VS-GARCH(1,1)

model, the ANST-GARCH(1,1) model, the DT-ARCH(1,1) model, the DT-

GARCH(1,1) model, and others. For detail, interested readers may refer to

Medeiros and Veiga(2009)(36).

5.2

The Conditional Esscher Transform

In this section, we recall the method of the conditional Esscher transform

described in Siu et al. (2004) (43) to determine a pricing kernel for option

valuation. The method applies to determine a price kernel for a general FC-

GARCH model in the next section.

For each t ∈ T , write F

for the P-completed, σ-ﬁeld generated by the

share price process up to and including time t and write also F := {F

|t ∈ T }.

We assume that under P,

= µ

+ ξ

where ξ

is an i.i.d innovation pro cess having distribution D(0, h

) and µ

and

are F

t−1

-measurable.

We noticed while p reparing that paper that the methodology is applied

not only to the FC-GARCH but also for any model that has a µ

and h

structure being F

t−1

−measurable and noises given by a inﬁnitely divisible

distribution having a moment generation function. As many of the GARCH

especiﬁcations has those properties, the Siu et al. methodology can be used

very broadly.

We now deﬁne the conditional Esscher transform. Let {θ

|t ∈ T \{0}} be

an F -predictable, real-valued, process on (Ω, F, P). It means we know in time

t − 1 its value in time t. Denote, for each t ∈ T \{0}, the moment generating

function of Y

given F

t−1

under P evaluated at z ∈ ℜ by M

(t, z); that is,

(t, z) := E[e

t−1

] .

Risk Neutral Option Pricing under some special GARCH models 43

Here E is expectation under P. Assume that, for each t ∈ T \{0} and z ∈ ℜ,

(t, z) exists and consider an F-adapted process {Λ

|t ∈ T } on (Ω, F, P)

with Λ

= 1, P-a.s., deﬁned by:



k=1

(k, θ

)

, t ∈ T \{0} .

Then, it is easy to check that {Λ

}

t∈τ

is an (F, P)-martingale. So, E[Λ

] = 1.

Now we deﬁne a new probability measure P

equivalent to P on F

setting



:= Λ

. (5-3)

We call P

the conditional Esscher transform associated with θ.

Let M

(t, z) be the moment generating function of the return Y

given

t−1

under the new measure P

. Write E

[·] for expectation under P

. Then,

by the Bayes’ rule, it is easy to check that

(t, z) =

(t, θ

+ z)

(t, θ

)

. (5-4)

Indeed, by the Bayes’ rule (Theorem 45 in the Appendix),

(t, z) := E

t−1

]

E[e

t−1

]

E[Λ

t−1

]

= E



t−1



E[e

(z+θ

t−1

]

(t, θ

)

(t, θ

+ z)

(t, θ

)

. (5-5)

According to the fundamental theorem of asset pricing (see Harrsion

and Kreps(1979)(33) and Harrsion and Pliska (1981, 1983))(34) and (35), the

absence of arbitrage opportunities is “essentially” equivalent to the existence

of an equivalent martingale measure under which discounted price processes

are martingales. We call the latter a martingale condition. Please refer to Siu

et al.(43) for the economic argument to select the measure. A similar argument

will be shown in chapter 6.

Now we write

:= e

−rt

, which is the discounted asset price at time t,

Risk Neutral Option Pricing under some special GARCH models 44

for each t ∈ T . Then in our case, the martingale condition is:

= E

[

] , for all u, t ∈ T with u ≤ t . (5-6)

Here E

is expectation under P

The following proposition gives the necessary and suﬃcient condition for

the martingale condition. It is in Siu et al. (2004)(43).

Proposition 33 The martingale condition is satisﬁed if and only if there

exists an F -predictable process {θ

|t ∈ T \{0}} such that

r = ln M

(t, θ

+ 1) − ln M

(t, θ

) . (5-7)

Proof :

Let Y

= ln



t−1



, such that e

t−1

(⇐) First we prove that

t−1

= E

[

t−1

]

Indeed, if r = ln



(t,θ

+1)

(t,θ

)



, then

(t, θ

+ 1)

(t, θ

)

= M

(t, 1) = E

t−1

]. (5-8)

Then,

−rt

t−1

] = S

t−1

−rt

t−1

] (5-9)

= S

t−1

−rt

t−1

] (5-10)

= S

t−1

−rt

E[e

(θ

+1)

t−1

]

E[e

t−1

]

(5-11)

= S

t−1

−rt

(t, θ

+ 1)

(t, θ

)

(5-12)

= S

t−1

−r(t− 1)

(5-13)

Now, we will show that for any u, t ∈ τ with u < t,

−rt

t−1

] = e

−ru

, a.s. with respect to P.

Risk Neutral Option Pricing under some special GARCH models 45

−rt

] = E

−rt

t−1

] (5-14)

= E

−rt

t−1

]|F

] (5-15)

= E

−r(t− 1)

t−1

−r

t−1

]|F

] (5-16)

= E

−r(t− 1)

t−1

] (5-17)

= ... (5-18)

= E

−r(u +1)

u+1

] (5-19)

= e

−ru

(5-20)

almost surely with respect to P as desired.

(⇒) Now we are going to prove that if

= E

[

] ∀u ≤ t (5-21)

then

r = ln



(t, θ

+ 1)

(t, θ

)



In particular, the hypothesis is true at u = t − 1. Then

t−1

−r(t− 1)

= E

−rt

t−1

]

⇒

−r(t− 1)

= E

[

t−1

]

⇒

= E

t−1

]

⇒

r = ln E

t−1

]

⇒

r = ln



E[e

(θ

+1)

t−1

]

E[e

t−1

]



⇒

r = ln



(t, θ

+ 1)

(t, θ

)



The existence and uniqueness of the pr ocess θ can be established using

some standard arguments.



Risk Neutral Option Pricing under some special GARCH models 46

Consider an European-style option with payoﬀ V (S

) at maturity T .

Then, a conditional price of the option at time t given F

is determined as:

= e

−r(T −t)

[V (S

)|F

] . (5-22)

The expected value of V (S

) is calculated via Monte Carlo simulation.

To obtain the results we also use some variance reduction techniques: control

variate (the Black and Scholes option price as the benchmark) and antithetic

variables (Normal case only).

5.3

Some Parametric Cases

In this section, we consider some parametric cases of our model when the

GARCH innovations have a normal distribution and a shifted gamma distri-

bution. The development in this section follows that of Siu et al. (2004)(43).

In this section two of the main theoretical results of our research will appear,

viz., Theorems 35 and 36.

5.3.1

Normal innovations

Firstly, under P, consider some F

t−1

−measurable conditional mean µ

an the model below.

= µ

+ ξ

t−1

= N(0, h

)

= α

+ β

t−1

+ λ

t−1



i=1

(α

+ β

t−1

+ λ

t−1

)f(s

; γ

, c

where

f(s

, γ

, c

) :=

1 + e

−γ

−c

)

Then, under P, Y

t−1

∼ N(µ

, h

), as µ

depends only on information

available in F

t−1

In order to ﬁnd the Esscher parameter:

Risk Neutral Option Pricing under some special GARCH models 47

r = ln(M

t−1

(1, θ

)) = ln



t−1

(1 + θ

)

t−1

(θ

)



(5-23)

= ln





(1+θ

)





= ln(e

) = µ

+ h

(5-24)

i.e.,

r = µ

+ h

r −µ

−

= h

which gives us θ

r−µ

−

. (The Esscher parameter)

Using that θ

r−

(

)

in the relation

t−1

(z, θ

) =

t−1

(z + θ

)

t−1

(θ

)

we have

t−1

(z, θ

) =

(z+θ

)

(5-25)

= e

z+zθ

(5-26)

= e

z+z

(

r−

(

))

(5-27)

= e

(

r−

)

, (5-28)

which is the mgf of a normal, i.e.,

t−1

∼ N



r −

, h



under P

. By the dynamics

= µ

+ ξ

; ξ

= h

1/2

we have

[ξ

t−1

] = E

t−1

] − µ

= r − µ

−

Risk Neutral Option Pricing under some special GARCH models 48

V ar

[ξ

t−1

] = V ar

t−1

] = h

= V ar

[ξ

t−1

Note that the variance does not change but the mean does and we want a

zero mean variable. So make ǫ

:= ξ

−r+µ

such that ǫ

t−1

∼ N

(0, h

Then, we can write the model under measure P

= r −

+ ǫ

, (5-29)

where

= α

+ β

t−1

+ λ



t−1

+ r − µ

−

t−1



(5-30)



i=1



+ β

t−1

+ λ



t−1

+ r − µ

−

t−1





f(s

; γ

, c

)(5-31)

If we take µ

= r+λ

√

−

to be the conditional mean as Duan(1995),

we would have to consider

: = ξ

− r + r + λ



−

(5-32)

= ξ

+ λ



(5-33)

so t hat ǫ

t−1

∼ N

(0, h

). And then we would conclude for this particular

case that under P

= r −

+ ǫ

, (5-34)

where

= α

+ β

t−1

+ λ

(ǫ

t−1

− λ



)

(5-35)



i=1



+ β

t−1

+ λ

(ǫ

t−1

− λ



)



f(s

; γ

, c

) (5-36)

We summarize the discussion above in the following Theorem:

Risk Neutral Option Pricing under some special GARCH models 49

Theorem 34 Let

= r + λ



−

+ ξ

t−1

= N(0, h

)

= α

+ β

t−1

+ λ

t−1



i=1

(α

+ β

t−1

+ λ

t−1

)f(s

; γ

, c

) ,

be the model under P, where

f(s

, γ

, c

) :=

1 + e

−γ

−c

)

Then, under the risk neutral measure the model is

= r −

+ ǫ

t−1

∼ N

(0, h

)

= α

+ β

t−1

+ λ

(ǫ

t−1

− λ



)



i=1



+ β

t−1

+ λ

(ǫ

t−1

− λ



)



f(s

; γ

, c

)

5.3.2

Shifted-Gamma Innovations

Some authors (Gerber and Shiu(1994) (31) and Siu et al. (2004)(43) )

have been using shifted-gamma innovations to model log-returns in order to

handle the skewness that real ﬁnancial series usually exhibits. However, the

skewness of the Gamma distribution is strictly positive whilst ﬁnancial time

series can present both signs. In practice, before adopting the shifted-gamma

model, one may check if there is any skewness to be modeled. Otherwise,

the Normal model should suﬃce. Then, one should check for the sign of the

skewness so as to select an appropriate formulation of the shifted-gamma

innovations. The positive case is similar and for the GARCH case it has already

documented in Siu et al.(2004) (43). In the following, we develop a model to

incorporate negative skewness.

Suppose that for each t ∈ T \{0}, X

∼ Ga(a, b), where Ga(a, b)

represents a Gamma distribution with shape parameter a and scale parameter

b. We now suppose that under P the innovation at time t is given by:

:= −





− a/b



a/b



, (5-37)

Risk Neutral Option Pricing under some special GARCH models 50

so we write ξ

t−1

∼ −SGa(0, h

Then, under P,

= r + λ



−

+ ξ

, (5-38)

= α

+ β

t−1

+ λ

t−1



i=1

(α

+ β

t−1

+ λ

t−1

)f(s

; γ

, c

) ,(5-39)

where

f(s

, γ

, c

) :=

1 + e

−γ

−c

)

The return process Y can be expressed as:

= r + λ



−



− b



Note that b



∼ Ga(a,



), and that if W ∼ Ga(a, b) is a Gamma

random variable, the moment generation function of −W is M

−W

(t) =



b+θ



Then the moment generation function of Y

|F is given by

t−1

(θ

) =





+ θ



(r+λ

√

−

√

)θ

(5-40)

provided that θ



> 0.

Applying the formula

(z, θ

) =

(t, θ

+ z)

(t, θ

)

, (5-41)

we have

t−1

(z, θ

) =



√

+θ



(r+λ

√

−

√

)(θ

+z)



√

+θ



(r+λ

√

−

√

)θ

(5-42)





+ θ



+ θ

+ z



(r+λ

√

−

√

(5-43)

as long as z <



+ θ

By this formula and the relation

r = ln M

t−1

(1, θ

)

Risk Neutral Option Pricing under some special GARCH models 51

we have:

= e

r+λ

√

−

√





+ θ



+ θ

+ 1



(5-44)

or equivalently

1 = e

√

−

√





+ θ



+ θ

+ 1



(5-45)

Note that



+ θ



+ θ

+ 1

= 1 −



+ θ

+ 1

(5-46)

Then,

1 − e

√

−

√

= −

√

−

√



+ θ

+ 1

(5-47)

⇐⇒



+ θ

+ 1 = −

√

−

√

1 − e

√

−

√

(5-48)

⇐⇒

= −

√

−

√

1 − e

√

−

√

−



− 1 (5-49)

⇐⇒

= −

1 − e

√

−

√

−



(5-50)

⇐⇒

√

−

√

− 1

−



(5-51)

Consequently, th e martingale condition implies that

√

−

√

− 1

−



(5-52)

Risk Neutral Option Pricing under some special GARCH models 52

Now if we take b



and b

√

−

√

−1

, then

= θ

+ b

Let ∼ be “equal in distribution”. Under P

t−1

∼ −SGa(a, b

, −r −λ



−



) . (5-53)

Then, we can write

∼ r + λ



−



+ X

Here X

∼ −

Ga(a, 1)

, and

= α

+ β

t−1

+ λ

t−1



t−1

)



i=1

[α

+ β

t−1

+ λ

t−1



t−1

)

]f(s

; γ

, c

) .

From the discussion above we have achieved:

Theorem 35 Let the mode l under P be

= r + λ



−

+ ξ

t−1

∼ −SGa(0, h

= α

+ β

t−1

+ λ

t−1



i=1

(α

+ β

t−1

+ λ

t−1

)f(s

; γ

, c

where

f(s

, γ

, c

) :=

1 + e

−γ

−c

)

We write Gamma(a, 1) a Gamma random variable with shape parameter a and the

scalar parameter 1.

Risk Neutral Option Pricing under some special GARCH models 53

Then, under the risk neutral measure, the model is

∼ r + λ



−



+ X

∼ −

Ga(a, 1)

= α

+ β

t−1

+ λ

t−1



t−1

)



i=1

[α

+ β

t−1

+ λ

t−1



t−1

)

]f(s

; γ

, c

) .

5.4

Simulation Studies

In this section we conduct simulation experiments and compare both Call

and Put option prices arising from diﬀerent processes for the log-returns of the

underlying asset.

We consider ﬁve pricing schemes for options with 90 days to maturity:

the classical Black and Scholes formulae assuming a GBM process and the

conditional Esscher transform method for GARCH and FC-GARCH processes,

each one with Normal and shifted-Gamma innovations.

In the Esscher transform approach, as noted in section 3, the expected

value in equation (3.20) is computed by Monte Carlo simulation. In simulation

experiments, we used a sample of size of 10000 for the Gamma models and

20000 for the Normal ones, due to the antithetic variables.

The ﬁve pricing schemes are applied to two artiﬁcial series produced

by a FC-GARCH model with Normal and Shifted-Gamma innovations with

3200 data points, obtained after a warm-up period of 1000 observations. The

FC-GARCH parameters used in the simulations are given in table 1. These

parameters, except for the a parameter and the risk-premium, are in Medeiros

and Veiga (2009)(36).

FC-GARCH Parameters

α [9.77 × 10

−16

, 5.14 × 10

−7

, 1.81 × 10

−5

]

β [1.21, −0.32, −0.25]

λ [0.06, −0.01, −0.04]

γ [2.52, 2.85]

c [−0.72, 1.56]

Risk Premium 0.0349

a (Gamma case) 100

Table 5.1: Parameters for the FC-GARCH including the values of α, β and λ

in the three diﬀerent regimes.

Risk Neutral Option Pricing under some special GARCH models 54

To evaluate the Black and Scholes prices we estimate the volatility

parameters by the sample variances, 4.0812 × 10

−5

, for the Normal data and

3.8244 × 10

−5

for the Gamma data.

For the estimation of the GARCH parameters we use an iterated two-

stage method. Initially, we suppose h

a constant equal to the sample variance.

Then, we estimate the risk premium by weighted least squares (WLS). Then

we ﬁt a GARCH(1,1) model to the residuals of the WLS by performing a Quasi

Maximum Likelihhod. We iterate these two steps until convergence is attained.

The estimated parameters are shown in table 2.

GARCH Parameters in Normal Case

α 7.4079 × 10

−7

β 0.9375

λ 0.0445

Risk Premium 0.0288

Table 5.2: Estimated GARCH Parameters in the Normal Case

We estimated the parameters using the two stage procedure as described

before and then for ﬁnding a we used the method of moments estimate as in

Siu et al. (2004) (43) as follows:

ˆa =





t=1

3/2



t=1



, (5-54)

which led us to the estimated parameters shown in table 3.

GARCH Parameters in Shifted-Gamma Case

α 5.4959 × 10

−7

β 0.9461

λ 0.0397

Risk Premium 0.0363

a 79.4022

Table 5.3: Estimated GARCH Parameters in the Gamma Case

As discussed in section 4.2, one must check for the presence and the sign

of the skewness before selecting the Normal or Shifted-Gamma approaches.

The resulting prices are presented in the tables that follows. IV stands

for the initial volatility, here the ratio between the initial variance used for

simulation and the sample variance. For the case where IV = 1.0, after each

option price table there is another table with the ratios between the (FC)-

GARCH prices and the Black-Scholes prices for each model. A graph of these

comparative tables is also shown.

Risk Neutral Option Pricing under some special GARCH models 55

Call Prices with IV=1.0 for Artiﬁcial Normal FCGARCH Series

K/S

BS FC-Normal GARCH-Normal FC-Gamma GARCH-Gamma

0.80 20.0002 20.0009 19.9978 20.0149 20.0103

0.85 15.0063 15.0110 15.0079 15.0292 15.0238

0.90 10.0957 10.0939 10.1022 10.1216 10.1227

0.95 5.6537 5.5895 5.6292 5.6255 5.6565

1.00 2.4175 2.2812 2.3714 2.2757 2.3762

1.05 0.7397 0.6704 0.7163 0.6449 0.6845

1.10 0.1575 0.1515 0.1667 0.1362 0.1440

1.15 0.0234 0.0298 0.0310 0.0223 0.0294

1.20 0.0025 0.0049 0.0045 0.0032 0.0050

Table 5.4: Call Prices with IV=1.0 for Artiﬁcial Normal FCGARCH Series and

T=90. The parameters used for the FCGARCH are in table 1, and the table

with GARCH parameters are in table 2

K/S

BS FC-Normal GARCH-Normal FC-Gamma GARCH-Gamma

0.8 1 1.0000 0.9999 1.0007 1.0005

0.85 1 1.0003 1.0001 1.0015 1.0012

0.9 1 0.9998 1.0006 1.0026 1.0027

0.95 1 0.9886 0.9957 0.9950 1.0005

1 1 0.9436 0.9809 0.9413 0.9829

1.05 1 0.9063 0.9684 0.8718 0.9254

1.1 1 0.9619 1.0584 0.8648 0.9143

1.15 1 1.2735 1.3248 0.9530 1.2564

1.2 1 1.9600 1.8000 1.2800 2.0000

Table 5.5: Call Prices ratios with IV=1.0 for Artiﬁcial Normal FCGARCH

Series and T=90

Call Prices with IV=1.2 for Artiﬁcial Normal FCGARCH Series

K/S

BS FC-Normal GARCH-Normal FC-Gamma GARCH-Gamma

0.80 20.0006 20.0044 20.0013 20.1102 20.0202

0.85 15.0144 15.0151 15.0164 15.1265 15.0386

0.90 10.1509 10.1177 10.1298 10.2467 10.1584

0.95 5.8159 5.6726 5.7121 5.7923 5.7428

1.00 2.6481 2.4260 2.4961 2.4920 2.5059

1.05 0.9156 0.7588 0.8015 0.7778 0.8111

1.10 0.2359 0.1828 0.2016 0.1688 0.2072

1.15 0.0454 0.0354 0.0478 0.0267 0.0367

1.20 0.0066 0.0053 0.0096 0.0043 0.0078

Table 5.6: Call Prices with IV=1.2 for Artiﬁcial Normal FCGARCH Series

and T=90. The parameters used are in t able 1, and the table with G ARCH

parameters are in table 2

Risk Neutral Option Pricing under some special GARCH models 56

Call Prices with IV=1.0 for Artiﬁcial Shifted-Gamma FCGARCH Series

K/S

BS FC-Gamma GARCH-Gamma FC-Normal GARCH-Normal

0.80 20.0001 20.0149 19.8948 20.0009 20.0079

0.85 15.0045 15.0292 14.9062 15.0110 15.0176

0.90 10.0803 10.1216 9.9955 10.0939 10.1031

0.95 5.6017 5.6255 5.5110 5.5895 5.6136

1.00 2.3402 2.2757 2.2413 2.2812 2.3280

1.05 0.6831 0.6449 0.6171 0.6704 0.6907

1.10 0.1349 0.1362 0.1209 0.1515 0.1537

1.15 0.0180 0.0223 0.0227 0.0298 0.0314

1.20 0.0017 0.0032 0.0041 0.0049 0.0049

Table 5.7: Call Prices with IV=1.0 for Artiﬁcial Shifted-Gamma FCGARCH

Series and T=90. The parameters used are in tab le 1, and the table with

GARCH parameters are in table 3

K/S

BS FC-Gamma GARCH-Gamma FC-Normal GARCH-Normal

0.80 1.0000 1.0007 0.9947 1.0000 1.0004

0.85 1.0000 1.0016 0.9934 1.0004 1.0009

0.90 1.0000 1.0041 0.9916 1.0013 1.0023

0.95 1.0000 1.0042 0.9838 0.9978 1.0021

1.00 1.0000 0.9724 0.9577 0.9748 0.9948

1.05 1.0000 0.9441 0.9034 0.9814 1.0111

1.10 1.0000 1.0096 0.8962 1.1231 1.1394

1.15 1.0000 1.2389 1.2611 1.6556 1.7444

1.20 1.0000 1.8824 2.4118 2.8824 2.8824

Table 5.8: Call Prices with IV=1.0 for Artiﬁcial Shifted-Gamma FCGARCH

Series

Call Prices with IV=1.2 for Artiﬁcial Shifted-Gamma FCGARCH Series

K/S

BS FC-Gamma GARCH-Gamma FC-Normal GARCH-Normal

0.80 20.0004 20.1102 20.0946 20.0044 20.0052

0.85 15.0109 15.1265 15.1095 15.0151 15.0163

0.90 10.1290 10.2467 10.2206 10.1177 10.1205

0.95 5.7553 5.7923 5.7780 5.6726 5.6910

1.00 2.5635 2.4920 2.4712 2.4260 2.4512

1.05 0.8499 0.7778 0.7554 0.7588 0.7742

1.10 0.2052 0.1688 0.1839 0.1828 0.1851

1.15 0.0362 0.0267 0.0375 0.0354 0.0340

1.20 0.0047 0.0043 0.0073 0.0053 0.0069

Table 5.9: Call Prices with IV=1.2 for Artiﬁcial Shifted-Gamma FCGARCH

Series and T=90. The parameters used are in tab le 1, and the table with

GARCH parameters are in table 3

Risk Neutral Option Pricing under some special GARCH models 57

Figure 5.1: Graphs of Call Prices ratios with IV=1.0 for Artiﬁcial

Normal FCGARCH Series

Figure 5.2: Graphs of Call Prices ratios with IV=1.0 for Artiﬁcial

Shifted-Gamma FCGARCH Series

Risk Neutral Option Pricing under some special GARCH models 58

Put Prices with IV=1.0 for Artiﬁcial Normal FCGARCH Series

K/S

BS FC-Normal GARCH-Normal FC-Gamma GARCH-Gamma

0.80 0.0002 0.0001 0.0008 0.0024 0.0010

0.85 0.0063 0.0083 0.0097 0.0169 0.0107

0.90 0.0957 0.0893 0.0940 0.1095 0.1041

0.95 0.6537 0.5753 0.6230 0.6088 0.6523

1.00 2.4175 2.2835 2.3685 2.3003 2.3666

1.05 5.7397 5.6568 5.7088 5.5981 5.6728

1.10 10.1575 10.1466 10.1593 10.0596 10.1118

1.15 15.0234 15.0281 15.0358 14.9438 14.9701

1.20 20.0025 20.0059 20.0135 19.9292 19.9468

Table 5.10: Put Prices with IV=1.0 for Artiﬁcial Normal FCGARCH Series

and T=90. The parameters used are in table 1, , and the table with GARCH

parameters are in table 2

K/S

BS FC-Normal GARCH-Normal FC-Gamma GARCH-Gamma

0.8 1 0.5 4 12 5

0.85 1 1.31746 1.539683 2.68254 1.698413

0.9 1 0.933124 0.982236 1.144201 1.087774

0.95 1 0.880067 0.953037 0.931314 0.997858

1 1 0.944571 0.979731 0.95152 0.978945

1.05 1 0.985557 0.994616 0.97533 0.988344

1.1 1 0.998927 1.000177 0.990362 0.995501

1.15 1 1.000313 1.000825 0.994702 0.996452

1.2 1 1.00017 1.00055 0.996335 0.997215

Table 5.11: Put Prices ratios with IV=1.0 for Artiﬁcial Normal FCGARCH

Series

Put Prices with IV=1.2 for Artiﬁcial Normal FCGARCH Series

K/S

BS FC-Normal GARCH-Normal FC-Gamma GARCH-Gamma

0.80 0.0006 0.0015 0.0015 0.0045 0.0016

0.85 0.0144 0.0147 0.0162 0.0248 0.0174

0.90 0.1509 0.1143 0.1325 0.1369 0.1444

0.95 0.8159 0.6534 0.7193 0.6856 0.7842

1.00 2.6481 2.4055 2.4884 2.4509 2.6057

1.05 5.9156 5.7375 5.8160 5.8001 5.9466

1.10 10.2359 10.1780 10.2118 10.2156 10.3430

1.15 15.0454 15.0418 15.0502 15.0689 15.1735

1.20 20.0066 20.0087 20.0134 20.0372 20.1402

Table 5.12: Put Prices with IV=1.0 for Artiﬁcial Normal FCGARCH Series

and T=90. The parameters used are in t able 1, and the table with G ARCH

parameters are in table 2

Risk Neutral Option Pricing under some special GARCH models 59

Put Prices with IV=1.0 for Artiﬁcial Shifted-Gamma FCGARCH Series

K/S

BS FC-Gamma GARCH-Gamma FC-Normal GARCH-Normal

0.80 0.0001 0.0024 0.0009 0.0001 0.0002

0.85 0.0045 0.0169 0.0099 0.0083 0.0085

0.90 0.0803 0.1095 0.0946 0.0893 0.0948

0.95 0.6017 0.6088 0.5760 0.5753 0.6149

1.00 2.3402 2.3003 2.2122 2.2835 2.3398

1.05 5.6831 5.5981 5.4999 5.6568 5.6952

1.10 10.1349 10.0596 9.9490 10.1466 10.1505

1.15 15.0180 14.9438 14.8426 15.0281 15.0213

1.20 20.0017 19.9292 19.8236 20.0059 19.9995

Table 5.13: Put Prices with IV=1.0 for Artiﬁcial Shifted-G amma FCGARCH

Series and T=90. The parameters used are in tab le 1, and the table with

GARCH parameters are in table 3

K/S

BS FC-Gamma GARCH-Gamma FC-Normal GARCH-Normal

0.8 1.0000 24.0000 9.0000 1.0000 2.0000

0.85 1.0000 3.7556 2.2000 1.8444 1.8889

0.9 1.0000 1.3636 1.1781 1.1121 1.1806

0.95 1.0000 1.0118 0.9573 0.9561 1.0219

1 1.0000 0.9830 0.9453 0.9758 0.9998

1.05 1.0000 0.9850 0.9678 0.9954 1.0021

1.1 1.0000 0.9926 0.9817 1.0012 1.0015

1.15 1.0000 0.9951 0.9883 1.0007 1.0002

1.2 1.0000 0.9964 0.9911 1.0002 0.9999

Table 5.14: Pu t Prices ratios with IV=1.0 for Artiﬁcial Shifted-Gamma FC-

GARCH Series

Put Prices with IV=1.2 for Artiﬁcial Shifted-Gamma FCGARCH Series

K/S

BS FC-Gamma GARCH-Gamma FC-Normal GARCH-Normal

0.80 0.0004 0.0045 0.0008 0.0015 0.0008

0.85 0.0109 0.0248 0.0184 0.0147 0.0121

0.90 0.1290 0.1369 0.1357 0.1143 0.1086

0.95 0.7553 0.6856 0.7007 0.6534 0.6632

1.00 2.5635 2.4509 2.4390 2.4055 2.4242

1.05 5.8499 5.8001 5.7436 5.7375 5.7507

1.10 10.2052 10.2156 10.1765 10.1780 10.1735

1.15 15.0362 15.0689 15.0415 15.0418 15.0329

1.20 20.0047 20.0372 20.0196 20.0087 20.0058

Table 5.15: Put Prices with IV=1.2 for Artiﬁcial Shifted-Gamma FCGARCH

Series and T=90. The parameters used are in tab le 1, and the table with

GARCH parameters are in table 3

Risk Neutral Option Pricing under some special GARCH models 60

Figure 5.3: Graphs of Put Prices ratios with IV=1.0 for Artiﬁcial

Normal FCGARCH Series

Figure 5.4: Graphs of Put Prices ratios with IV=1.0 for Artiﬁcial

Shifted-Gamma FCGARCH Series

Risk Neutral Option Pricing under some special GARCH models 61

We can see that Calls and Puts have a diﬀerent behavior. In tables 5.11

and 5.14, the put option price ratios have their largest values deep in the money.

The more pronounced eﬀect is in the FC-Gamma scheme. The values are larger

than those in other schemes. Tables 5.5 and 5.8, on the other hand, show their

largest values deep out the money, although its eﬀect is not as signiﬁcant as in

the pu t case. For call options, the Normal models overprice the other models

whilst in the put options, the Shifted-Gamma models do. This behavior may

be explained by the negative asymmetry we introduced changing the signs of

the innovations, in the Shifted-Gamma case.

To illustrate the changes in the option prices when we change the

measures, we simulated prices under both the physical and risk neutral

measures in all schemes. Then we checked for the proportions of scenarios

where the options were exercised, which then give an estimate of the real-

world probability of exercising an option. We chose S

= 100 and K = 100 to

perform this exercise. We notice that in all schemes presented in table 5.16,

the prices under the risk neutral measure are less likely to exceed the strike

price than the prices under the physical measure.

Table 5.16: Average rate of exercising

Model/rate Risk Neutral Measure Physical Measure

FCGARCH Normal 0.4881 0.6197

FCGARCH Gamma 0.5032 0.6198

Gamma GARCH 0.4940 0.6338

Normal Garch 0.4885 0.5939

GARCH-Gamma with Normal data 0.4805 0.5994

GARCH-Normal with Gamma data 0.4870 0.6238

Then we focus on the GARCH-Gamma case using parameters estimated

from artiﬁcial FC-GARCH Gamma data to produce the numerical results

presented in the table below. The choice of this scheme is justiﬁed by its largest

diﬀerence in the estimates ab ove between the two measures. We considered

= 100 and varied the strike price. Note that for every strike price, the risk

neutral prices have a smaller chance of exceeding the strike price than the

prices in the physical measure.

Risk Neutral Option Pricing under some special GARCH models 62

Average rate of exercising (Gamma GARCH)

K/S

Risk Neutral measure Physical measure

0.80 0.9991 1.0000

0.85 0.9932 0.9978

0.90 0.9491 0.9784

0.95 0.7922 0.8885

1.00 0.4908 0.6336

1.05 0.2004 0.2991

1.10 0.0552 0.0862

1.15 0.0095 0.0162

1.20 0.0015 0.0023

Table 5.17: Average rate of exercising for Artiﬁcial Shifted-Gamma GARCH

Series and T=90. The parameters used are in tab le 1, and the table with

GARCH parameters are in table 3

Risk Neutral Option Pricing under some special GARCH models 63

70 80 90 100 110 120 130

100

150

200

250

300

350

Figure 5.5: Histogram of S

in the physical measure

70 80 90 100 110 120 130

100

150

200

250

300

350

400

Figure 5.6: Histogram of S

in the risk neutral measure

Risk Neutral Option Pricing under some special GARCH models 64

5.5

Sensitivity Analysis

Now we are going to check how the option prices change when some

of the parameters are perturbed. We performed simulations imposing a small

variation around the values of the parameters.

In some cases, it is important to bear in mind the stationarity condition.

We note that putting the parameters so that the stationarity condition is close

to 2 makes the eﬀect on the option prices more pronounced. The stationarity

condition for the Normal case, what is going to be used as a benchmark, once

we don’t have a stability condition for the Gamma case, is given by:

2λ

+ 2β

+ β

+ λ

+ β

+ λ

< 2.

Proceeding with this exercise we capture the importance of each pa-

rameter in the option prices. We perform this analysis with the FC-GARCH

models having Normal and Shifted-Gamma innovations. Graphs are shown in

the appendix to illustrate such analysis.

5.5.1

Normal innovations

As we increase th e value β

by steps of 0.05, the option price also

increases. Note that a larger value of β

has a deeper impact than the others.

This is because the last β

are closer to the stationarity condition. Then we

increase the value of β

by the same increment, we have the same eﬀect. After

that, we notice that changing β

doesn’t aﬀect much the option price. When

we repeated the experiment with steps of 0.1, even then the graph was the

same, but we can see a slightly increasing pattern in the output numbers.

For the λ

and λ

graphs we can see a slightly increase in the option

prices. For the λ

, although a slight increase occurs it is not obviously shown

in the graph even when we performed with increments of 0.05. Run ning

γ ∗ σ = 1, 6, 11, 16 and 21, no changes are seen in both cases γ

and γ

For the eﬀect of c

, we noticed that the option value increases with an

increase in c

. On the other hand there is no clear eﬀect of changing c

The risk premium didn’t show any clear eﬀect. The graphic shows ﬁve

variations of the risk premium with steps of 0.05. It seems having an increasing

trend, but it is not clear even performing steps of 0.1.

Risk Neutral Option Pricing under some special GARCH models 65

5.5.2

Shifted Gamma innovations

For the Gamma case, we chose 0.05 to b e the increment of β

. The option

value increases with an increase in the value of β

as in the Normal case. Note

that in the last curve, when the stationarity condition exceeds 2, the increase

in option prices has a more pronounced eﬀect. Again, with steps of 0.05 we

perturbed β

and the option values were also increased. We can see that as β

is closer to 2 the bigger its impact on the option price. In the analysis of β

can not see the same behavior. It may be attributed to the insigniﬁcant eﬀect

of the last regime.

The a parameter doesn’t have a clear eﬀect on the option prices. It varied

from 10 to 190 in steps of 45.

An increase in λ

and λ

with increments of 0.01 increases the option

prices. We performed the analysis of λ

with increments 0.01,0.05 and 0.1. No

signiﬁcant eﬀect is noticed in any of the cases.

For the c

, with 0.1 increments, we see a slight increase of the price of the

option. The c

has no signiﬁcant eﬀect on prices. For the Gamma parameter,

running γ ∗σ = 1, 6, 11, 16 and 21 no changes are seen in both cases γ

and γ

The risk premium has no clear monotonic pattern inﬂuence on the option

prices. We simulated with increments of 0.05 and also steps of 0.1, that are

relatively large relative to the initial value 0.0363.

5.6

Discussion of the results

In the tables and graphs of section 5, we noticed that the prices obtained

from the FC-GARCH models are slightly higher than the Black Scholes prices,

but th e FC-GARCH option prices are lower than the GARCH option prices in

both the Normal and Gamma cases for the calls, but for the puts this eﬀect is

only noticed in the normal case.

In the sensitivity analysis, we noticed that the GARCH parameters for

the regime zero and the ﬁrst regime were the most sensitive to perturbations

of these model parameters while the GARCH parameters of the third regime,

the logistic parameter γ, and the risk premium λ have little or no impact on

option prices.

5.7

Conclusions

In this chapter we adopted the method of Siu et al. (2004) (43) to ﬁnd

a pricing kernel for the FC-GARCH models with two diﬀerent parametric

Risk Neutral Option Pricing under some special GARCH models 66

distributions for innovations, the Normal and the shifted-Gamma cases. We

also performed simulations and showed tables comparing the Black Scholes

prices and the GARCH prices to our simulation results of the FC-GARCH

models as well as we performed a sensitivity analysis to understand how

changes in some parameters aﬀect the option valuation results.

The FC-GARCH models can capture features that some other models

cannot, like the high kurtosis with low ﬁrst-order autocorrelation of the squared

observations, so that the option prices are more precise if calculated in the way

we did in this chapter. Here we performed simulations with 3 regimes but the

model can mimic an economy with many regimes. A further research would be

developing tests to ﬁnd the optimal numb er of regimes for each situation.

Option Valuation under Mixture GARCH models

This chapter is based in a working paper together with the Professors Ken Siu,

John Lau and

Alvaro Veiga.

It has been documented that normal mixture GARCH models can pro-

vide a better description for the leptokurtosis behavior in ﬁnancial returns

data compared with the GARCH models with normal innovations and the

student’s t-GARCH models (see Alexander and Lazar (2006) (1)). In this pa-

per, we shall consider the pricing of options under the class of discrete-time

mixture of GARCH models with innovations having a ﬁnite mixture of in-

ﬁnitely divisible distributions. The option valuation model can provide market

practitioners with a convenient and ﬂexible way to price options under various

forms of mixture GARCH models, which can incorporate diﬀerent degrees of

conditional skewness and conditional leptokurtosis of the distribution of the

asset returns. The market described by the discrete-time mixture GARCH

models is incomplete and, hence, there are inﬁnitely many equivalent mar-

tingale measures. We shall employ the doubly stochastic Esscher transform

to determine an equivalent martingale measure for pricing. The p ricing result

can be justiﬁed by a stochastic version of the power utility maximization. Em-

pirical results for comparing the call and put option prices obtained from the

mixture GARCH models with those from the standard Black-Scholes model

based on the recent 25-year S&P 500 data will be presented and discussed.

6.1

Asset Price Dynamics and Pricing Model

We consider a discrete-time ﬁnancial model consisting of one risk-free bond

B and one risky stock S. We assume that the dynamics of the risky stock

is governed by a mixture GARCH model with innovations having a ﬁnite

mixture of inﬁnitely divisible distributions. The mixture GARCH model

can incorporate various parametric forms of the mixture GARCH models,

such as the Normal-Mixture (NM) GARCH models in Alexander and Lazar

(2006)(1), the GARCH models with innovations having a ﬁnite mixture of

shifted gamma distributions and a ﬁnite mixture of shifted Inverse Gaussian

Risk Neutral Option Pricing under some special GARCH models 68

distributions. It p rovides market practitioners with a great deal of ﬂexibility

in modelling various empirical “stylised” behavior of asset price dynamics, like

the conditional skewed and leptokurtosis (or heavy-tailed) behaviors of the

asset returns. In the following, we present the setup of the mo del.

First, we describe the general mixture GARCH model. Let (Ω, F, P) be

a complete probability space, where P is a real-world probability. Let T be the

time index set {0, 1, 2, . . . , T } of the ﬁnancial model. Let S := {S

}

t∈T

denote

a stochastic process deﬁned on (Ω, F) with state space R

, where R

is the

set of non- negative real numbers. For each t ∈ T , S

represents the price of the

risky stock S at time t. Wr ite F

:= {F

}

t∈T

for the P-augmentation of the

natural ﬁltration generated by the process S. For each t ∈ T , F

represents

the observable information about the prices of the risky stock S up to and

including time t. Let B := {B

}

t∈T

denote the price process of the risk-free

bond B, which is assumed to be a deterministic process.

Let {ξ

}

t∈T

denote a stochastic process deﬁned on (Ω, F) taking values

on the real line R, with ξ

∼ D(0, h

) and ξ

= 0, which represents t he

random ﬂuctuations of the returns from the risky asset S. For each t ∈ T ,

we call ξ

the innovation of asset return at time t. For each k = 1, 2, . . . , K,

let h

:= {h

}

t∈T \{0}

denote a stochastic process on (Ω, F) with state space

, where R

is the set of positive real numb ers. For each k = 1, 2, . . . , K,

we assume that th e dynamics of h

is governed by the following GARCH(p,

q) structure:

= α



j=1

t−j



i=1



l=1

kil

i,t−1

, (6-1)

where p ≥ 1, q ≥ 1 and α

> 0, α

≥ 0, j ∈ {1, 2, . . . , q}, β

kil

≥ 0,

l ∈ {1, 2, . . . , p} in order to ensure the positivity of h

In the particular case of the GARCH(1, 1) that we are going to deal

with, for ensuring covariance stationarity of th e GARCH(1, 1) structure for

each k = 1, 2, . . . , K, we further impose the condition that the matrix

+ B

has all the eigenvalues smaller than 1, where α

= [α

, α

, ..., α

]

, p =

, p

..., p

]

and B = β

; k, i = 1, ..., K.

Write U := {U

}

t∈T

for a sequence of independent and identically

distributed (i.i.d.) K-dimensional random vectors, which take values from the

state space U := {e

, e

, . . . , e

}, where e

:= (0, 0, . . . , 1, . . . , 0, 0) ∈ R

a unit vector with one in the k

component and zero otherwise. We suppose

Risk Neutral Option Pricing under some special GARCH models 69

that the common probability distribution of U

is speciﬁed by:

P(U

= e

) = p

, k = 1, 2, . . . , K , (6-2)

where p

≥ 0 and



k=1

= 1.

Let H := {H

}

t∈T \{0}

denote a stochastic process on (Ω, F) with

state space H := {h

, h

, . . . , h

}. For each t ∈ T \{0}, let H

, h

, . . . , h

). Then, we can write H

as follows:

=< H

, U



k=1

< H

, e

>< U

, e

> . (6-3)

Write F

for the information set generated by the values of the pro cess U up

to and including time t. Write F

for the enlarged information set F

t−1

∨ F

generated by F

t−1

and F

. We shall specify the distributional structure of the

process {ξ

}

t∈T

F (x|0, H

) =



k=1

F (x|0, < H

, e

>) < U

, e



k=1

F (x|0, h

) < U

, e

> .(6-4)

Hence, the conditional distribution of ξ

given the observable information set

t−1

is given by the following ﬁnite mixture of inﬁnitely divisible distributions:

(x|F

t−1

) =



k=1

F (x|0, h

) , (6-5)

where F (···|0, h

) is the mixing kernel of the mixture distribution.

We write ξ

t−1

∼ MID(p

, . . . , p

; h

, . . . , h

), which represents the

ﬁnite mixture of inﬁnitely divisible distributions. We shall introduce the price

dynamics of the risk-free bond B and the underlying risky asset S. Let r

be the

continuously compounded risk-free interest rate of the bond B over the time

interval [t − 1, t], where t ∈ T \{0}; λ

the unit risk premium over the time

interval [t − 1, t]. We suppose that both r

and λ

are deterministic functions

of time t. Then, we assume that, under P, the dynamics of the bond-price

process {B

}

t∈T

and the stock-price process {S

}

t∈T

satisfy:

= B

t−1

, B

= 1 ,

= S

t−1

exp(r

+ λ



< H

, U

> −

< H

, U

> +ξ

) , S

= s ,(6-6)

for each t ∈ T \{0}.

For each t ∈ T \{0}, Y

denotes the continuously compounded one-p eriod

Risk Neutral Option Pricing under some special GARCH models 70

rate of return ln(

t−1

) of the stock S. Clearly, Y

is measurable with respect

to the observable information set F

. Then, under P, the dynamics of Y

governed by the following GARCH(p, q) model with innovations having a ﬁnite

mixture of inﬁnitely divisible distributions:

= r

+ λ



< H

, U

> −

< H

, U

> +ξ

t−1

∼ MID(p

, . . . , p

; h

, . . . , h

) ,

= α



j=1

t−j



i=1



l=1

kil

i,t−1

. (6-7)

The above model is called MID(K)-GARCH (p, q) model. It can in-

corporate the general form NM(K)-GARCH(p, q) speciﬁed by Haas, Mit-

tnik and Paolella ( 2002) (32) without explanatory variables and the NM(K)-

GARCH(1,1) examined by Alexander and Lazar (2006)(1). The model by

Haas, Mittnik and Paolella (2002)(32) can incorporate the cross dependence

of individual conditional variances by assuming the inter-dependent autore-

gressive evolution of the time series of conditional variances. Alexander and

Lazar (2006)(1) mentioned that the mixture GARCH model can be related to

other important GARCH models with non-normal innovations and the class

of Markov-Switching GARCH models. In fact, the MID(K)-GARCH (p, q)

model can be considered a particular case of a general class of regime-switch ing

GARCH(p, q) models with innovations having a ﬁnite mixture of inﬁnitely di-

visible distributions and the GARCH dynamics driven by a hidden Markov

chain model. The MID(K)-GARCH (p, q) model is diﬀerent from the one in

Alexander and Lazar (2006)(1) in two aspects. First, the proposed mixture

GARCH model here has a time-varying drift depending on the conditional

volatility of the return’s process while the mixture GARCH m odel in Alexan-

der and Lazar (2006)(1) has a constant drift. Second, the proposed mixture

GARCH model has a general mixing kernel, which is speciﬁed by an inﬁnite

divisible distribution with a ﬁnite moment generating function. Hence, it is

ﬂexible enough to incorporate the Normal-Mixture (NM) GARCH models in

Alexander and Lazar (2006)(1), the GARCH models with innovations having

a ﬁnite mixture of shifted gamma distributions and a ﬁnite mixture of shifted

Inverse Gaussian distributions, and others. This provides market practitioners

with a great deal of ﬂexibility in modelling diﬀerent empirical “stylised” behav-

iors of asset price dynamics, such the skewed behavior and the leptokurtosis

behavior of the distribution of asset’s returns.

We shall present the discrete-time doubly stochastic Esscher transform in

the sequel. The doubly stochastic Esscher transform is deﬁned by the product

Risk Neutral Option Pricing under some special GARCH models 71

of two stochastic processes. The idea is similar with the conditional Esscher

transform introduced in B¨uhlmann et al. (1996)(9). Elliott, Siu and Chan

(2006)(23) considered a similar type of doubly stochastic Esscher transform,

namely the regime-switching Esscher transform, in the context of a continuous-

time regime-switching Geometric Brownian Motion model. The discrete-time

version of the regime-switching Esscher transform has been adopted in Elliott,

Siu and Chan (2006)(23) for pricing options under a discrete-time Markov-

switching GARCH models.

First, we consider a real-valued stochastic process {Θ

}

t∈T

\{0} deﬁned

on (Ω, F). We assume th at for each t ∈ T \{0}, Θ

is measurable with respect

to F

Let M

(t, Θ

) denote the moment generating function of Y

given F

t−1

under P; that is,

Y |F

t−1

(t, Θ

) := E

t−1

) . (6-8)

We assume that E

t−1

) < ∞, for t ∈ T \{0}. As in B¨uhlmann et al.

(1996)(9), we deﬁne a sequence {Λ

}

t∈T

with Λ

= 1 as follows:



u=1

(u, Θ

)

, t ∈ T \{0} , (6-9)

where Λ

is speciﬁed by a product of two stochastic p rocesses {Θ

}

t∈T \{0}

and

}

t∈T \{0}

Lemma 36 {Λ

}

t∈T

is a (F, P)-martingale.

Proof : From its deﬁnition, Λ

is F

t+1

-measurable, for each t ∈ T and that



t−1



t−1



= E



t−1

)



t−1



= 1 , P −a.s. (6.1)

Hence, the result follows. 

Then, the doubly stochastic Esscher transform P

∼ P on F

with

respect to {Θ

, Θ

, . . . , Θ

t−1

} is given by:



:= Λ

t−1

, t = 2, 3, . . . , T + 1 . (6-2)

Risk Neutral Option Pricing under some special GARCH models 72

Let M

(z, t; Θ) be the moment generating function of Y

given F

under P

Then, we have the following lemma:

Lemma 37

(z, t; Θ) =

(t, z + Θ

)

(t, Θ

)

. (6-3)

Proof : The proof is adapted to the argument in Elliott, Siu and Chan

(2006)(23) in the case of mixture GARCH models. By Baye’s rule,

(z, t; Θ) = E

t−1

)

(Λ

t−1

)

(Λ

t−1

)

= E



t−1



t−1



(z+Θ

t−1

)

t−1

)

(t, z + Θ

)

(t, Θ

)

. (6-4)

Hence, the result follows. 

Harrison and Krep (1979) (33)and Harrison and Pliska (1981, 1983)(34,

35) provided a solid theoretical foundation to establish the relationship be-

tween the absence of arbitrage and the existence of an equivalent martingale

measure using the modern language of probability. They introduced the fun-

damental theorem of asset pricing which states that the absence of arbitrage is

equivalent to the existence of an equivalent martingale measure under which

the discounted asset price process is a martingale. The fundamental theorem of

asset pricing was then further extended by Dybyig and Ross (1987)(20), Back

and Pliska (1991)(2), Schachermayer (1992)(39) and Delbaen and Schacher-

mayer (1994)(15). Back and Pliska (1991)(2) has shown that the absence of

arbitrage is equivalent to the existence of an equivalent m artingale measure in

a discrete-time and an inﬁnite-state-space setting.

We shall determine an equivalent martingale measure using the doubly

stochastic Esscher transform P

. The suﬃcient condition on Θ

for P

to be

an equivalent martingale measure is presented in the following proposition.

Proposition 38 (Martingale Condition) Suppose Θ

satisﬁes the following

condition:

(t, Θ

+ 1)

(t, Θ

)

= e

. (6-5)

Risk Neutral Option Pricing under some special GARCH models 73

Then, the discounted price process {exp(−



u=1

}

t∈T

is a (F, P

martingale.

Proof :



−



u=1



t−1



= e

−



t−1

u=1

t−1

−r

t−1

)

= e

−



t−1

u=1

t−1



t−1



−r



t−1



= e

−



t−1

u=1

t−1

−r

(Θ

+1)Y

t−1

)

t−1

)

= e

−



t−1

u=1

t−1

−r

(t, Θ

+ 1)

(t, Θ

)

= e

−



t−1

u=1

t−1

, P −a.s. (6-6)

Hence, the result follows.



Note that the existence and uniqueness of Θ

satisfying the condition

in Proposition 2.3 can be proved by following the arguments in Chan and

van-der Hoek (2003) (11). The martingale condition with respect to the

enlarged ﬁltration {F

}

t∈T

is stronger than that with respect to the observable

information structure {F

}

t∈T

. In other words, if there exists a probability

measure P

satisfying the martingale condition with respect to {F

}

t∈T

, P

also satisﬁes the martingale condition with respect to {F

}

t∈T

. This can be

veriﬁed easily by the double expectation formula for conditional expectations.

Then, given F

:= F

t−1

∨ F

, the price of a European-style contingent

claim written on the underlying stock S with payoﬀ V (S

) at maturity T is

given by:

V (t − 1, T |F

) = E



exp



−



u=t



V (S

)





. (6-7)

Given the observable information F

t−1

, the price of the option can be deter-

mined as follows:

V (t − 1, T |F

t−1

) = E

[V (t − 1, T |F

)|F

t−1

]

= E



exp



−



u=t



V (S

)



t−1



In order to compute the option price V (t − 1, T |F

t−1

) under a given para-

Risk Neutral Option Pricing under some special GARCH models 74

metric distribution for the innovations, we ﬁrst need to estimate the unknown

parameters of the mixture GARCH model using the observed market price

data for S under P and then simulate the terminal stock prices from the mix-

ture GARCH model under P

for approximating V (t − 1, T |F

t−1

) via Monte

Carlo simulation coupled with its control variates technique. Due to the struc-

ture of the mixture GARCH model, its likelihood function can be determined

completely given the observed market price data F

t−1

. Hence, we do not need

to use information from the values of the hidden mixing process {U

}

t∈T \{0}

for the estimation part. However, in the Monte Carlo simulation, we need to

generate the values of the pro cess {U

}

t∈T \{0}

and the values of the innova-

tions process {ξ

}

t∈T \{0}

for simulating the terminal stock prices. In section 4,

we shall discuss in some detail the estimation and simulation procedures for

computing the prices of a European call option under two parametric mixture

GARCH models described in Section 3.

We shall justify the pricing result by a power utility maximization

problem with respect to F

. For each t ∈ T \{0}, let ζ

denote a random

variable measurable with respect to F

. Then, for each t ∈ T \{0}, we consider

a power utility function u

with the stochastic risk-averse parameter ζ

; that

is, for each t ∈ T \{0},

(x) =



1−ζ

if ζ

= 1,

ln x if ζ

= 1.

Equivalently, the power utility function can be written as:

(x) =

1−ζ

1 − ζ

{ζ

=1}

+ (ln x)I

{ζ

=1}

. (6-8)

We call the above power utility function a stochastic power utility function. We

assume that an economic agent can adjust the stochastic risk-averse parameter

based on F

. For each t ∈ T \{0}, following Gerber and Shiu (1994) (31),

we impose the following assumptions:

1. The economic agent has m

units of stock S and η

units of the option

from time t − 1 to time t, where m

, η

∈ F

V (t −1, T |F

) represents the economic agent’s equilibrium price at time

t − 1 of the option with maturity at time T so that it is optimal for the

agent not to buy or sell any units of the option at time t − 1.

For each t ∈ T \{0}, the conditional expected utility function U

on the

Risk Neutral Option Pricing under some special GARCH models 75

economic agent’s wealth at time t given F

under P is given by:

(η

)

= E





+ η



V (t, T |F

t+1

) − e

V (t − 1, T |F

)







, (6-9)

where η

is the choice variable.

The following proposition justiﬁes the pricing result by the doubly

stochastic Esscher transform.

Proposition 39 For each t ∈ T \{0},

V (t − 1, T |F

) =

V (t − 1, T |F

) , (6-10)

and

= −Θ

. (6-11)

Proof : We follow the argument in Gerber and Shiu (1994)(31). The optimal

condition of the conditional expected utility function on the economic agent’s

wealth is equivalent to that U

(η

) attains its maximum value when η

= 0, for

each t ∈ T \{0}. Mathematically, this can be translated to:

′

(η

= 0 , (6-12)

where U

′

is the derivative of U

with respect to η

For simplicity, we write

t−1

for

V (t − 1, T |F

). Then, the optimal

condition implies that

t−1

= e

−r

[

′

)|F

]

E[u

′

)|F

]

, (6-13)

where u

′

is the derivative of u

with respect to η

We notice that

′

(x) = x

−ζ

{ζ

=1}

+ (1/x)I

{ζ

=1}

. (6-14)

Then,

t−1

= e

−r



)

−ζ

]

E[(m

)

−ζ

]

{ζ

=1}

/(m

)|F

]

E[1/(m

)|F

]

{ζ

=1}



= e

−r

)

−ζ

]

E[(m

)

−ζ

]

. (6-15)

Risk Neutral Option Pricing under some special GARCH models 76

Since the above result applies for any option, it can be applied for the

underlying asset S. Hen ce,

t−1

= e

−r

E(S

1−ζ

)

E(S

−ζ

)

= e

−r

t−1

E(e

(1−ζ

)

E(e

−ζ

)

= e

−r

t−1

(t, 1 − ζ

)

(t, −ζ

)

. (6-16)

Then,

(t, 1 − ζ

)

(t, −ζ

)

= e

. (6-17)

Hence, by uniqueness of Θ

satisfying the martingale condition,

= −Θ

, (6-18)

and

t−1

= V

t−1

. (6-19)



6.2

Parametric Cases

In this section, we deal with some parametric cases, namely the Normal-

Mixture (NM) GARCH model and the mixture GARCH mo del with innova-

tions having a ﬁnite mixture of shifted gamma distributions. We consider K-

component ﬁrst-order mixture GARCH models in this section. The derivation

of the pricing result for the case when the innovations have a ﬁnite mixture

of shifted inverse Gaussian distributions is very similar to that of the ﬁnite

mixture of shifted gamma distributions. We are able to pr eserve the paramet-

ric forms of the distributions for the GARCH innovations under the change of

probability measures using the doubly stochastic Esscher tranform in Section

6.2.1

Normal-Mixture (NM) GARCH models

First, we assume that under P, the conditional distribution of ξ

given

t−1

is a normal distribution N(0, < H

, U

>) with mean 0 and conditional

Risk Neutral Option Pricing under some special GARCH models 77

variance < H

, U

>, for each t ∈ T \{0}. Then, the conditional distribution

of ξ

given F

t−1

under P is given by the following ﬁnite mixture of normal

distributions:

(x|F

t−1

) =



k=1

N(x|0, h

) ,



k=1

= 1 . (6-20)

The conditional distribution of Y

given F

t−1

under P is a normal distri-

bution N(r

+ λ

√

< H

, U

> −

< H

, U

>, < H

, U

>) with mean

+ λ

√

< H

, U

> −

< H

, U

> and variance < H

, U

>, where the distri-

bution of U

is given by:

P (U

= e

) = p

and the dynamics of h

is governed by:

= α

+ α

t−1



k=1

ki1

i,t−1

for each k = 1, 2, . . . , K. Hence we have

= r

+ λ



< H

, U

> −

< H

, U

> +ξ

t−1

∼ NM(p

, . . . , p

; h

, . . . , h

) ,

= α

+ α

t−1



i=1

ki1

i,t−1

Then,

t−1

(t, Θ

) = exp





+ λ



< H

, U

> −

< H

, U



< H

, U



.(6-21)

From the martingale condition proposition, the risk-neutralized stochastic

Esscher parameter Θ

is given by:

= −

√

< H

, U

= −λ



k=1

√

< U

, e

> , t ∈ T \{0} . (6-22)

The moment generating function M

t−1

(z, t; Θ) is given by:

t−1

(z, t; Θ) = exp[z(r −

< H

, U

>) +

< H

, U

>] .(6-23)

Hence, under P

, the conditional distribution of Y

given F

t−1

is a normal

Risk Neutral Option Pricing under some special GARCH models 78

distribution with conditional mean r

−

< H

, U

> and conditional variance

< H

, U

>, where

P (U

= e

) = p

, (6-24)

and

= α

+ α

t−1



i=1

ki1

i,t−1

. (6-25)

for each k = 1, 2, . . . , K.

Note that ξ

t−1

∼ N(−λ

√

< H

, U

>, < H

, U

>) under P

. Let ǫ

t−1

+ λ

t−1

√

< H

t−1

, U

t−1

>. Then, under P

, ǫ

t−1

∼ N(0, < H

, U

>). We

can write the dynamics of h

as follows:

= α

+ α

(ǫ

t−1

− λ

t−1



< H

t−1

, U

t−1



i=1

ki1

i,t−1

.(6-26)

We have just proved the following Theorem:

Theorem 40 Let

= r

+ λ



< H

, U

> −

< H

, U

> +ξ

t−1

∼ NM(p

, . . . , p

; h

, . . . , h

) ,

= α

+ α

t−1



i=1

ki1

i,t−1

be the model in the real world probability P. Then, the risk neutral version of

the model is:

= r

−

H

, U

 + ǫ

t−1

∼ N (0, H

, U

)

= α

+ α



t−1

− λ

t−1



H

t−1

, U

t−1







i=1

ki1

i,t−1

6.2.2

Shifted-Gamma-Mixture (SGM) GARCH models

We assume t hat the innovations for the general mixture GARCH model in

Section 2 follow a ﬁnite mixture of shifted gamma distributions. First, for each

k = 1, 2, . . . , K, we consider a sequence of i.i.d. random variables {X

(k)

}

t∈T \{0}

Risk Neutral Option Pricing under some special GARCH models 79

with common distribution being a gamma distribution Ga(a

, b) with shape

parameter a

and scale parameter b. For each t ∈ T \{0}, we deﬁne the random

variable ν

(k)

by standardizing the gamma random variable X

as follows:

(k)

− a



. (6-27)

Note that ν

(k)

follows a standard shifted gamma distribution SGa(·|0, 1) with

zero mean and unit variance, for each k = 1, 2, . . . , K. For each t ∈ T \{0}, let

:= (ν

, ν

, . . . , ν

) and ν

be deﬁned as follows:

:=< V

, U



k=1

< V

, e

>< U

, e



k=1

< U

, e

> . (6-28)

Then, ν

follows a standard shifted gamma distribution SGa(·|0, 1) with zero

mean and unit variance.

We assume that ξ

:= −

√

< H

, U

>ν

; then, ξ

= −



k=1

√

(k)

, e

>. Hence, the conditional distribution of ξ

given F

t−1

under P is

minus a shifted gamma distribution SGa(·|0, < H

, U

>) with zero mean and

variance < H

, U



k=1

< U

, e

>. Then, the conditional distribution

of ξ

given F

t−1

is given by the following ﬁnite mixture of shifted gamma

distributions:

(x|F

t−1

) =



k=1

SGa(x|0, h

) ,



k=1

= 1 , (6-29)

where SGa(·|0, h

) is the probability distribution of a shifted gamma distri-

bution with mean 0 and variance h

We write ξ

t−1

∼ −MSGa(p

, . . . , p

; h

, . . . , h

). Hence, under

P, the conditional distribution of Y

given F

t−1

is minus a shifted gamma

distribution SGa(·|r

+ λ

√

< H

, U

> −

< H

, U

>, < H

, U

>) with mean

+ λ

√

< H

, U

> −

< H

, U

> and variance < H

, U

>. Then, under

P, the dynamics of Y

is governed by the following K-component ﬁrst-order

mixture GARCH model with innovations having a ﬁnite mixture of shifted

gamma distributions:

= r

+ λ



< H

, U

> −

< H

, U

> +ξ

t−1

∼ −MSGa(p

, . . . , p

; h

, . . . , h

) ,

= α

+ α

t−1



i=1

ki1

i,t−1

. (6-30)

Risk Neutral Option Pricing under some special GARCH models 80

Let A := (a

, a

, . . . , a

) and A

be deﬁned as follows:

:=< A, U



k=1

< U

, e

> . (6-31)

We can also write the dynamics of {Y

}

t∈T \{0}

under P as follows:



k=1



+ λ



< H

, e

> −

< H

, e

> +



< A, e

>< H

, e

−b



< H

, e

< A, e

(k)



< U

, e



k=1



+ λ



−



− b



(k)



< U

, e

> .(6-32)

Deﬁne B



; that is,



k=1



< U

, e

> . (6-33)

Then, the conditional distribution of Y

given F

t−1

is minus a shifted gamma

distribution with shape parameter A

, scale parameter B

and shifted param-

eter −r

−λ

√

< H

, U

> +

< H

, U

> −

√

< A, U

>< H

, U

>. Hence, the

moment generating function of Y

given F

t−1

under P is given by:

Y |F

t−1

(t, Θ

) =



+ Θ



(

+λ

√

−

√

, (6-34)

where 0 < B

+ Θ

Hence, u nder P

, the moment generating fun ction of Y

given F

t−1

given by:

Y |F

t−1

(t, z; Θ

) =



+ Θ

− z



(

+λ

√

−

√

. (6-35)

From the martingale condition, the risk-neutralized stochastic Esscher param-

eter Θ

is given by:



√

−

√

− 1



−1

− B

(6-36)

Risk Neutral Option Pricing under some special GARCH models 81

Then, we deﬁne the parameter

as follows:

:= B

+ Θ



√

−

√

− 1



−1



k=1





√

−

√



− 1



−1

< U

, e

> , t ∈ T \{0} .

Hence, under P

, Y

t−1

follows a minus shifted gamma distribution with

shape parameter A

, scale parameter

and shifted parameter −r

−

√

< H

, U

> +

< H

, U

> −

√

< A, U

>< H

, U

>, where

P (U

= e

) = p

. (6-37)

Let X

denote a random variable such that X

t−1

∼ −

Ga(A

, 1). Then,

= α

+ α



t−1



t−1





i=1

ki1

i,t−1

where k = 1, 2, . . . , K.

From the discussion above we have:

Theorem 41 Let the mode l under P be:

= r

+ λ



< H

, U

> −

< H

, U

> +ξ

t−1

∼ −MSGa(p

, . . . , p

; h

, . . . , h

)

= α

+ α

t−1



i=1

ki1

i,t−1

Then, under the risk neutral measure the model becomes:

= r

+ λ



< H

, U

> −

< H

, U

> +



< A, U

>< H

, U

> + X

t−1

∼ −

Ga(A

, 1)

= α

+ α



t−1



t−1





i=1

ki1

i,t−1

Finally, we note that the scale parameter b does not appear in both the

real-world and the risk-neutral dynamics due to the structure of the mixture

Risk Neutral Option Pricing under some special GARCH models 82

shifted-gamma model for the innovations.

6.3

Empirical Results and Discussion

In this section we conduct a simulation exercise in order to compare the

option prices induced by assuming diﬀerent processes for the log-returns of th e

underlying asset.

We consider the two-component mixture GARCH(1, 1) models with a

ﬁnite mixture of normal innovations and a ﬁnite mixture of shifted gamma

innovations for illustration. We employ the daily close values of S&P500 index

from November 7, 1980, to November 4, 2005, 25 years of data with 6,130

observations. The data were obtained from the database in Yahoo Finance.

We estimate the parameters underlying the option pricing models for empirical

studies using the 25-year S&P500 index data. We assume that the risk-free

interest rate r is zero and that the unit risk premium is a constant λ throughout

this section as in Duan (1995)(17) and Siu et al. (2004)(43).

We shall discuss the estimation procedures and the estimation results for

the two-mixture GARCH models in the sequel. Tong (1990)(45) and Fan and

Yao (2003)(27) mentioned that the conditional maximum likelihood estimation

enjoys some desirable sampling and asymptotic properties. For estimating

the two-component mixture GARCH model with a ﬁnite mixture of normal

innovations, we employ the conditional maximum likelihoo d estimation. For

estimating the two-component mixture GARCH mo dels with a ﬁnite mixture of

shifted gamma distributions, we use a two-stage estimation procedure adopted

in Siu et al. (2004) (43). In the ﬁrst-stage, the quasi-maximum likelihood

estimation (QMLE) is used to estimate the mixture GARCH parameters,

namely p

, α

, β

k11

and β

k21

, where k = 1, 2. The QMLE is an

approximation to the exact MLE by assuming that the innovations of a time

series model follow a normal distribution even though the “true” innovations

may not be normally distributed. In the QMLE, the exact likelihood is replaced

by the normal likelihood. Franses and van Dijk (2000) (30) mentioned that if

it is not sure whether the speciﬁed parametric assumption for the GARCH

innovations is correct, the QMLE can be employed for estimation. Fan and

Yao (2003) (27) pointed out that one does not know the “true” distribution

for the innovations in practice and, hence, the QMLE can provide a practical

way to estimate the parameters in a GARCH model. In fact, the QMLE

method can provide market practitioners with a convenient way to estimate

non-normal GARCH models since many standard computing and statistical

packages have included the conditional MLE for GARCH models with n ormal

Risk Neutral Option Pricing under some special GARCH models 83

innovations. One undesirable feature of the QMLE is that the standard errors

of the estimates are larger than those obtained by the exact MLE in the

ﬁnite sample case. However, the QMLE also enjoys some d esirable asymptotic

properties and the standard errors of the est imates from the QMLE can be

reduced by estimating the model using a large data set. In the second-stage

of our estimation, we adopt the method of moments approach to estimate

the unknown parameters in the ﬁnite mixture of shifted gamma distributions.

Tong (1990) (45) mentioned that the method of moments is one of the common

approaches to estimate parametric non-linear time series models. Gerber and

Shiu (1994)(31) employed the method of moments approach to estimate the

unknown parameter in the shifted gamma distribution underlying th eir option

pricing model. Taylor (1986)(44) pioneered the use of the method of moments

approach for estimating stochastic volatility mod els. Here, given the observed

market data of the logarithmic returns {Y

, Y

, . . . , Y

} and the values of

the estimated parameters ˆp

, ˆα

k11

and

k21

(k = 1, 2), we can

employ the three realized time series {ξ

, ξ

, . . . , ξ

}, {h

, h

, . . . , h

} and

, h

, . . . , h

} to evaluate the method of moments estimators for the

shifted gamma parameters, namely ˆa

and ˆa

. By using the method of moments

approach for the mixture GARCH model and matching the theoretical and

the empirical third moments of the shifed-gamma innovations, we provide the

following formula for the method of moments estimators ˆa

and ˆa

ˆa



B+



1−p



A − B





and ˆa



B+



1−p



A − B





where

B =



t=1



t=1

3/2

and A =



t=1



t=1

−

and

= p



−





1,t

−

1,t



+ (1 − p

)



−





2,t

−

2,t



3/2

= p

3/2

1,t

+ (1 − p

) h

3/2

2,t

= p



−





1,t

−

1,t



+ (1 − p

)



−





2,t

−

2,t



= p

1,t

+ (1 − p

) h

2,t

The ﬁrst table below, displays th e estimation results for the mixture

GARCH parameters p

, α

, β

k11

and β

k21

(k = 1, 2) using the QMLE in

the ﬁrst-stage of the est imation procedure. Both the estimation results based

Risk Neutral Option Pricing under some special GARCH models 84

on the 25-year S&P 500 index data are presented. The estimation results were

obtained u sing R package “rgenoud” to search for an optimum of the log-

likelihood.

We consider ﬁve pricing schemes for options with 90 days to maturity: the

classical Black and Scholes formulae assuming a GBM process and the discrete-

time doubly stochastic Esscher transform method for GARCH and Mixture of

GARCHs (henceforth MGARCH) processes, each one with Normal and shifted-

Gamma innovations(henceforth NMGARCH and SGMGARCH respectively).

In th e Normal and Gamma codes we simulated 10000 times but as in the

Normal case we used the antithetic method we actually have 20000 prices. In

both cases the control variate technique was used.

The ﬁve pricing schemes are applied to two artiﬁcial series pro duced by a

Mixture of GARCH model with Normal and Shifted-Gamma innovations with

3200 data points, obtained after a warm-up period of 1000 observations. The

MGARCH parameters are given in the table 6.1.

Mixture of GARCHs Parameters

α [1.275531 × 10

−4

, 0.433127; 1.965270 × 10

−9

, 4.315862 × 10

−2

]

β [0.383459, 0.182873; 3.035471 × 10

−3

, 0.935921]

λ 7.715479 × 10

−2

p [0.068147, 0.9319]

a (Gamma case) [0.0964, 6.0256]

Table 6.1: Parameters for the Mixture of GARCHs. Each line in the matrices

α, β contains the parameters of a regime.

In order to ﬁnd the Black and Scholes price we only need th e volatility

estimated by the sample variance, 2.0186 × 10

−4

, for the Normal data and

6.0211 × 10

−5

for the Gamma data. because the drift is not required by the

Black Scholes formulae.

For the estimation of the GARCH parameters we use an iterated two-

stage method. Initially, we suppose h

a constant equal to the sample variance.

Then, we estimate the risk premium by weighted least squares(WLS). Next,

we ﬁt a GARCH(1,1) model to the residuals of the WLS by performing a Quasi

Maximum Likelihhod. We iterate these two steps until convergence is attained.

The estimated parameters are shown in table 6.2.

GARCH Parameters in Normal Case

α 5.1325 × 10

−6

β 0.9114

λ 0.0673

Risk Premium 0.1343

Table 6.2: Estimated GARCH Parameters in the Normal Case

Risk Neutral Option Pricing under some special GARCH models 85

We estimated the parameters using the two stage procedure as described

before and then for ﬁnding a we used the method of moments as in Siu et

al(2004)(43) to obtain the expression:

ˆa =





t=1

3/2



t=1



(6-38)

which led us to the parameters shown in table 6.3.

GARCH Parameters in Shifted-Gamma Case

α 3.7707 × 10

−6

β 0.8631

λ 0.0702

Risk Premium 0.0798

a 22.5253

Table 6.3: Estimated GARCH Parameters in the Gamma Case

Some authors (Gerber and Shiu(1994) (31) and Siu et al. (2004) (43)

) have been using shifted-gamma innovations to mo del log-returns in order

to handle the skewness that real ﬁnancial series usually exhibits as can be

seen in Medeiros and Veiga (2009)(36). However, the skewness of the Gamma

distribution is strictly positive whilst ﬁnancial time series can present both

signs. In practice, before adopting the shifted-gamma model, one may check

if there is any skewness to be modeled. The sign of the asymmetry in the

data, has to be taken into consideration too. Then, one should check for the

sign of the skewness so as to select an appropriate formulation of the shifted-

gamma innovations. In Section 4, we develop a model to incorporate negative

skewness. The positive case is similar, and for the GARCH case it has already

been documented in Siu et al.(2004) (43) but accounts for positive skewness

only. Here, in the Shifted-Gamma case, we perform an experiment consisting of

using a mixture of positive and negative noises. The ﬁ srt regime, will provide

positive innovations whilst the second negative ones.

The resulting prices are presented in tables that follows. We show both

Call and Put option prices. After each option price table there is another table

with the ratio between the price and the the Black S choles price for each model.

A graph of these comparative tables is also shown.

Risk Neutral Option Pricing under some special GARCH models 86

Artiﬁcial MGARCH Call Prices (Normal data)

K/S

BS NMGARCH GARCH-Normal SGMGARCH GARCH-Gamma

0.80 20.2451 20.0396 20.3704 20.6131 19.9422

0.85 15.6879 15.1600 15.8647 16.3017 15.0241

0.90 11.5852 10.5621 11.7831 12.4678 10.3457

0.95 8.1130 6.6096 8.3018 9.2143 6.3225

1.00 5.3731 3.6432 5.5584 6.6011 3.3357

1.05 3.3638 1.7641 3.5860 4.6039 1.5130

1.10 1.9930 0.7591 2.2336 3.1507 0.5988

1.15 1.1203 0.3048 1.3582 2.1351 0.2060

1.20 0.5994 0.1208 0.8043 1.4586 0.0720

Table 6.4: Artiﬁcial MGARCH Call Prices, and T=90. The parameters used

are in table 1

Table 6.5: Artiﬁcial MGARCH Call Price ratios (Normal data)

K/S

BS NMGARCH GARCH-Normal SGMGARCH GARCH-Gamma

0.8 1.0000 0.9898 1.0062 1.0182 0.9850

0.85 1.0000 0.9663 1.0113 1.0391 0.9577

0.9 1.0000 0.9117 1.0171 1.0762 0.8930

0.95 1.0000 0.8147 1.0233 1.1357 0.7793

1 1.0000 0.6780 1.0345 1.2285 0.6208

1.05 1.0000 0.5244 1.0661 1.3687 0.4498

1.1 1.0000 0.3809 1.1207 1.5809 0.3005

1.15 1.0000 0.2721 1.2124 1.9058 0.1839

1.2 1.0000 0.2015 1.3418 2.4334 0.1201

Call Option ratio (Normal data)

Figure 6.1: Graph of the ratio of Call options

Risk Neutral Option Pricing under some special GARCH models 87

Artiﬁcial MGARCH Call Prices (Gamma data)

K/S

BS SGMGARCH GARCH-Gamma NMGARCH GARCH-Normal

0.80 20.0023 20.6131 20.0617 20.0396 19.9935

0.85 15.0324 16.3017 15.0553 15.1600 15.0185

0.90 10.2388 12.4678 10.0617 10.5621 10.1955

0.95 6.0304 9.2143 5.3346 6.6096 5.9153

1.00 2.9361 6.6011 1.8991 3.6432 2.7815

1.05 1.1469 4.6039 0.4111 1.7641 1.0286

1.10 0.3553 3.1507 0.0464 0.7591 0.3053

1.15 0.0875 2.1351 0.0042 0.3048 0.0724

1.20 0.0174 1.4586 0.0010 0.1208 0.0157

Table 6.6: Artiﬁcial MGARCH Call Prices, and T=90. The parameters used

are in table 1

Table 6.7: Artiﬁcial MGARCH Call Price ratios (Gamma data)

K/S

BS SG MGARCH GARCH-Gamma NMGARCH GARCH-Normal

0.8 1.0000 1.0305 1.0030 1.0019 0.9996

0.85 1.0000 1.0844 1.0015 1.0085 0.9991

0.9 1.0000 1.2177 0.9827 1.0316 0.9958

0.95 1.0000 1.5280 0.8846 1.0960 0.9809

1 1.0000 2.2483 0.6468 1.2408 0.9473

1.05 1.0000 4.0142 0.3584 1.5381 0.8969

1.1 1.0000 8.8677 0.1306 2.1365 0.8593

1.15 1.0000 24.4011 0.0480 3.4834 0.8274

1.2 1.0000 83.8276 0.0575 6.9425 0.9023

Risk Neutral Option Pricing under some special GARCH models 88

Call Option ratio (Gamma data)

Figure 6.2: Graph of the ratio of Call options

Risk Neutral Option Pricing under some special GARCH models 89

Artiﬁcial MGARCH Put Prices (Normal data)

K/S

BS NMGARCH GARCH-Normal SGMGARCH GARCH-Gamma

0.80 0.2451 0.0487 0.3486 0.4922 0.0259

0.85 0.6879 0.1752 0.8335 1.0361 0.0844

0.90 1.5852 0.5728 1.7654 2.0291 0.3973

0.95 3.1130 1.5923 3.3021 3.6551 1.3615

1.00 5.3731 3.6200 5.5684 5.9864 3.3947

1.05 8.3638 6.7476 8.5754 8.9910 6.6014

1.10 11.9930 10.7743 12.2063 12.5913 10.7275

1.15 16.1203 15.3253 16.3190 16.6484 15.3697

1.20 20.5994 20.1390 20.7742 21.0324 20.2601

Table 6.8: Artiﬁcial MGARCH Put Prices, and T=90. The parameters used

are in table 1

Table 6.9: Artiﬁcial MGARCH Put Price ratios (Normal data)

K/S

BS NMGARCH GARCH-Normal SGMGARCH GARCH-Gamma

0.8 1.0000 0.1987 1.4223 2.0082 0.1057

0.85 1.0000 0.2547 1.2117 1.5062 0.1227

0.9 1.0000 0.3613 1.1137 1.2800 0.2506

0.95 1.0000 0.5115 1.0607 1.1741 0.4374

1 1.0000 0.6737 1.0363 1.1141 0.6318

1.05 1.0000 0.8068 1.0253 1.0750 0.7893

1.1 1.0000 0.8984 1.0178 1.0499 0.8945

1.15 1.0000 0.9507 1.0123 1.0328 0.9534

1.2 1.0000 0.9776 1.0085 1.0210 0.9835

Risk Neutral Option Pricing under some special GARCH models 90

Put Option ratio (Normal data)

Figure 6.3: Graph of the ratio of Put Options

Risk Neutral Option Pricing under some special GARCH models 91

Artiﬁcial MGARCH Put Prices (Gamma data)

K/S

BS SGMGARCH GARCH-Gamma NMGARCH GARCH-Normal

0.80 0.0023 0.4922 0.0097 0.0487 0.0030

0.85 0.0324 1.0361 0.0122 0.1752 0.0305

0.90 0.2388 2.0291 0.0489 0.5728 0.2198

0.95 1.0304 3.6551 0.4256 1.5923 0.9632

1.00 2.9361 5.9864 2.0676 3.6200 2.8462

1.05 6.1469 8.9910 5.5272 6.7476 6.0732

1.10 10.3553 12.5913 10.1236 10.7743 10.3267

1.15 15.0875 16.6484 15.0670 15.3253 15.0818

1.20 20.0174 21.0324 20.0585 20.1390 20.0158

Table 6.10: Artiﬁcial MGARCH Put Prices, and T=90. The parameters used

are in table 1

Table 6.11: Artiﬁcial MGARCH Put Price ratios (Gamma data)

K/S

BS SG MGARCH GARCH-Gamma NMGARCH GARCH-Normal

0.8 1.0000 214.0000 4.2174 21.1739 1.3043

0.85 1.0000 31.9784 0.3765 5.4074 0.9414

0.9 1.0000 8.4971 0.2048 2.3987 0.9204

0.95 1.0000 3.5473 0.4130 1.5453 0.9348

1 1.0000 2.0389 0.7042 1.2329 0.9694

1.05 1.0000 1.4627 0.8992 1.0977 0.9880

1.1 1.0000 1.2159 0.9776 1.0405 0.9972

1.15 1.0000 1.1035 0.9986 1.0158 0.9996

1.2 1.0000 1.0507 1.0021 1.0061 0.9999

Risk Neutral Option Pricing under some special GARCH models 92

Put Option ratios (Gamma data)

Figure 6.4: Graph of the ratio of Put options

Risk Neutral Option Pricing under some special GARCH models 93

We can see that Calls and Puts have a diﬀerent behavior. In tables 6.9 and

6.11, the put option price ratios have their largest values deep in the money.

The more pronounced eﬀect is in the SGMGARCH scheme. The values are

larger than those in other schemes. Tables 6.5 and 6.7, on the other hand,

show their largest values deep out the money. The more pronounced eﬀect is

again in the SGMGARCH scheme. The values of the SGMGARCH scheme are

much more drastic when the data is generated by Shifted-Gamma noises than

with Normal noises as we can see in the ﬁgures. The second largest values

alternates between the NMGARCH scheme for Shift-Gamma d ata and the

GARCH-Normal for Normal data. The smallest values appears in the GARCH-

Gamma scheme.

To illustrate the changes in the option prices when we change the

measures, we simulated prices under both the physical and risk neutral

measures in all schemes. Then we checked for the proportions of scenarios

where the options were exercised, which then give an estimate of the real-

world probability of exercising an option. We chose S

= 100 and K = 100 to

perform this exercise. We notice that in all schemes presented in table 6.12,

the prices under the risk neutral measure are less likely to exceed the strike

price than the prices under the physical measure. Note also that the diﬀerence

is more pronounced than in the FC-GARCH schemes in chapter 5.

Table 6.12: Average rate of exercising

Model/rate Risk Neutral Measure Physical Measure

NMGARCH 0.4824 0.7720

SGMGARCH 0.4665 0.7620

Gamma GARCH 0.4832 0.7652

Normal Garch 0.4692 0.8872

GARCH-Gamma with Normal data 0.4827 0.8813

GARCH-Normal with Gamma data 0.4856 0.7639

Risk Neutral Option Pricing under some special GARCH models 94

6.4

Sensitivity Analysis

Now we are going to check how the option prices change when some

of the parameters are disturbed. We performed simulations imposing a small

variation around the values of the parameters.

Proceeding with this exercise we capture the importance of each param-

eter in the option prices. We perform this analysis with the MGARCH models

having Normal and Shifted-Gamma innovations.

6.4.1

Normal innovations

If we increase the value of α

, by steps of 0.05 the option price does not

change signiﬁcantly. We made 5 variations in steps of 0.01 for α

and as we

increase its value, the option value also increases.

We made β

with increments of 0.05, and for β

in steps of 0.001 and

the option price slightly increased. For β

the eﬀect is not clear. On the other

hand for the β

with 0.05 increments, we can clearly see the increase eﬀect on

option prices.

The p parameter increases the value of the option as is shown in ﬁgure

11. We performed 5 varations from 0.0081 to 0.1281 in steps of 0.03.

The risk premium didn’t show any clear eﬀect. The graphics shows ﬁve

variations of the risk premium with steps of 0.05 and 0.1. It seems having an

increasing trend, but this increase is very slightly even performing steps of 0.1.

6.4.2

Shifted Gamma innovations

If we increase the value of α

and α

, by steps of 0.05 and 0.01

respectively the option price does not have a monotonic pattern.

We made for β

increments of 0.05, and the option price does not have

a monotonic pattern. For β

, β

and β

the eﬀect is not clear too.

We performed 5 varations of the p parameter from 0.0081 to 0.1281 in

steps of 0.03. No clear eﬀect was noticed.

The a

parameter doesn’t have a clear eﬀect on the option prices. a

was varied swith increments of 0.01 and 0.03. In any of the cases there was

a monotonic pattern. a

started from its value having increments of 0.03 and

also varied but without a monotonic clear eﬀect.

The risk premium has no clear monotonic pattern inﬂuence on the option

prices. We simulated with increments of 0.03 and also steps of 0.1,

Risk Neutral Option Pricing under some special GARCH models 95

In summary, in the Gamma case, maybe due to the mixture of signs

in the noises, even when the prices varied signiﬁcantly, no monotonic pattern

could be detected.

6.5

Summary and discussion

In this paper we adopted the discrete-time doubly stochastic Esscher

transform to ﬁnd a pricing kernel for the MGARCH models with two dif-

ferent p arametric distributions for innovations, the Normal and the shifted-

Gamma cases. We also performed simulations and showed tables comparing

the Black Scholes prices and the GARCH prices to our simulation results of the

MGARCH models as well as we performed a sensitivity analysis to understand

how changes in some parameters aﬀect the option valuation results.

In the tables and graphs of section 5, we noticed large values for the

SGMGARCH scheme and small values for the GARCH schemes. In the

sensitivity analysis, we noticed that the GARCH parameters β

and p were

the most sensitive to perturbations of these model parameters in the Normal

case, while the other parameters and the risk premium have little or no impact

on option prices. In the Gamma case, maybe due to the mixture of signs in the

noises, we could not measure the importance of any of the parameters. Even

when the prices varied signiﬁcantly, no monotonic pattern could be detected.

The MGARCH models can capture features that some other models

cannot like the high kurtosis, so the option prices are more precise if calculated

in the way we did in this paper. Here we performed simulations with a mixture

of 2 regimes but the model can mimic an economy with many regimes.

Future works and ﬁnak remarks

This work concerned option pricing of two special GARCH models, viz

the FC-GARCH and the Mixture of GARCHs. We found the theoretical risk

neutral version of those models and also simulated the option prices under

diﬀerent schemes. We also discuss the possibility of using negative noise to the

Shifted-Gamma case and perform the calculations in both models as well as

we performed simulation experiments. Then we made some sensibility analysis

to the paramters. The only theoretical contributions we achieve in these two

papers are the four theorems relating the econometric models in the original

P measure and in the risk neutral measure, viz., Theorems 35, 36, 41 and 42

showed in chapters 5 and 6.

We intend to continue in this direction, studying pricing methods and

applying them in more complex models and more soﬁsticated kind of contracts

so that practioners have a variety of possibilities of models to choose from.

According to our results, if the d ata comes from one of those two models

proposed, there is a signiﬁcant diﬀerence among the option prices in the

diﬀerent schemes. It shows that it is worth pricing using a reasonable model

and with the proper innovation distribution in order to obtain more precise

option prices.

We tried to extend the methodology in Siu et al.(43) to characteristic

functions, but the imaginary i brought us concern when dealing with the

martingale condition. It would be nice if we could ﬁnd a way in solving this

issue and generalize the methodology.

Another possible improvement is to create a trading scheme to compare

our results to the obtained by real options data. In such a paper we intend

to compare real data prices with both the FC-GARCH and the Mixture of

GARCHs option prices.

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Appendix

Deﬁnition 42 (Conditional Expectation). The conditional expectati on of a

nonnegative random variable ξ with respect to the σ-algebra G is a nonnegative

extended random variable, denoted by E[ξ|G] or E[ξ|G](ω), such that

(a) E[ξ|G] is G-measurable;

(b) For every A ∈ G,



ξdP =



E[ξ|G]dP. (A-1)

Theorem 43 Let G, H be σ-algebras such that G ⊂ H. Then

E[X|G] = E[E[X|H]|G] (A-2)

Proof : If G ∈ G then G ∈ H and therefore



E[X|H]dP =



XdP. (A-3)

Hence

E[E[X|H]|G] = E[X|G]. (A-4)



Corollary 44 (Iterated expectations)

E[E[X|H]] = E[X]. (A-5)

Risk Neutral Option Pricing under some special GARCH models 102

Proof : In particular take G = Ω and we will have



Ω

E[X|H]dP =



Ω

XdP (A-6)



Theorem 45 (Baye’s rule) Let µ and ν be two probability measures on a

measurable space (Ω, G) such that

dν(ω) = f (ω)dµ(ω)

for some f ∈ L

(µ). Let X be a random variable on (Ω, G) such that

[|X|] =



Ω

|X(ω)|f(ω)dµ(ω) < ∞ (X is ν − integrable) (A-7)

Let H be a σ-algebra, H ⊂ G. Then,

[X|H ].E

[f|H] = E

[fX|H] a.s.

[X|H ] =

[fX|H]

[f|H]

a.s. (A-8)

Proof : By the deﬁnition of conditional expectation we have that if H ∈ H

then



[X|H ]f dµ =



[X|H ]dν =



Xdν (A-9)



Xfdµ =



[fX|H]dµ. (A-10)

On the other hand, by iterated expectations we have



[X|H ]f dµ = E

[X|H ]f χ

] (A-11)

= E

[X|H ]f χ

|H]] (A-12)

= E

[χ

[X|H ]E

[f|H]] (A-13)



[X|H ]E

[f|H]dµ (A-14)

Combining (3.3) and (3.8) we get



[X|H ]E

[f|H]dµ =



[fX|H]dµ (A-15)

Risk Neutral Option Pricing under some special GARCH models 103

Since this holds for all H ∈ H,

[X|H ]E

[f|H] = E

[fX|H] a.s. (A-16)



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