The [GLP & MEMM] Pricing Modeland its Calibration Problems
Yoshio Miyahara∗
Graduate School of Economics
Nagoya City University
October 28, 2004

Abstract
The [GLP & MEMM] pricing model (= [Geometric Lévy Process& Minimal Entropy Martingale Measure] pricing model) has been introduced as a pricing model for the incomplete financial market. Thismodel has many good properties and is applicable to very wide classesof underlying asset price processes including the geometric stable processes. We explain those good properties and see several examples ofthis model. After that we investigate the calibration problems of [GLP
& MEMM] model.

Key words: Geometric Lévy Process, Relative entropy, Minimal entropy
martingale measure, Stable process, Calibration.
JEL Classification: G12, G13∗

E-mail:y-miya@econ.nagoya-cu.ac.jp

1

Contents
1 Introduction

32 Geometric Lévy process pricing models

3

2.1

Geometric Lévy processes . . . . . . . . . . . . . . . . . . . .

3

2.2

Equivalent martingale measures

5. . . . . . . . . . . . . . . .

3 Esscher Martingale Measure (ESMM) and Minimal Entropy
Martingale Measure (MEMM)

5

3.1

Esscher transforms . . . . . . . . . . . . . . . . . . . . . . . .

5

3.2

Esscher transformed martingale measures for geometric Lévy
processes

3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.2.1

Simple return process and compound return process .

6

3.2.2

Two kinds of Esscher transformed martingale measures

8

Existence theorems of ESMM and MEMM for geometric Lévy
processes

3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.3.1

Existence theorem of ESMM . . . . . . . . . . . . . .

10

3.3.2

Existence theorem of MEMM . . . . . . . . . . . . . .

11Comparison of ESMM and MEMM . . . . . . . . . . . . . . .

13

4 [GLP & MEMM] Pricing Model

14

4.1

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

4.2

Sufficient Conditions for the Existence of the MEMM . . . .

15

4.3

Examples of [GLP & MEMM] Pricing Model . . . . . . . . .

16

4.3.1

Geometric Variance Gamma Model . . . . . . . . . . .

16

4.3.2

Geometric CGMY Model . . . . . . . . . . . . . . . .

16

4.3.3

Geometric Stable Model . . . . . . . . . . . . . . . . .

17

Option Pricing and Volatility Smile/Smirk Properties . . . .

17

4.4.1

European Type Options . . . . . . . . . . . . . . . . .

18

4.4.2

FFT Method for European Call Options . . . . . . . .

18

4.4.3

Volatility Smile/Smirk Properties . . . . . . . . . . . .

20

4.4

5 Physical World and MEMM World
5.1

From Physical World to MEMM World . . . . . . . . . . . .

2

20
20

5.2

From MEMM World to Physical World . . . . . . . . . . . .

21

5.3

Example: Geometric Stable Process Case . . . . . . . . . . .

22

6 Calibration of [GLP & MEMM] pricing model

22

6.1

Calibration problems . . . . . . . . . . . . . . . . . . . . . . .

22

6.2

Estimation of Lévy Processes in the Physical World . . . . .

23

6.3

Calculation of the theoretical option prices. . . . . . . . . .

23

6.4

Fitness analysis of the estimated model . . . . . . . . . . . .

23

6.5

Determination of the most fitting model . . . . . . . . . . . .

23

6.6

Diagram of the Procedure for the Calibration . . . . . . . . .

24

6.7

Example: Geometric Stable Process Case . . . . . . . . . . .

24

3

1

Introduction

Geometric Lévy process pricing models are discussed in relation to the incomplete markets.
The [Geometric Lévy Process & MEMM] pricing model was first introduced in [30]. This model is one of the incomplete markets, and is basedon the geometric Lévy process and the minimal entropy martingale measure(= MEMM).
In Section 2, 3 and 4 we explain this model and see the properties ofthis model. In Section 5 we investigate the relations between the physicalworld and the MEMM world. After that, in Section 6 we investigate thecalibration problem of our model.

2

Geometric Lévy process pricing models

We assume that the value process of bond is given by
Bt = exp{rt},(2.1)

where r is a positive constant.
A pricing model consists of the following two parts:(A) The price process St of the underlying asset.
(B) The rule to compute the prices of options.
For the part (A) we adopt the geometric Lévy processes, so the part (A)is reduced to the selecting problem of a suitable class of the geometric Lévy
processes. For the second part (B) we adopt the martingale measure method,
so the part (B) is reduced to the selecting procedure of a suitable martingale
measure Q, and then the price of an option X is given by e−rT EQ [X]. Ourstudies in this paper are carried on under such a framework.

2.1

Geometric Lévy processes

The price process St of a stock is assumed to be defined as what follows. We
suppose that a probability space (Ω, F, P ) and a filtration {Ft , 0 ≥ t ≥ T }are given, and that the price process St = S0 eZt of a stock is defined on this4

probability space and given in the form
St = S0 eZt ,0 ≤ t ≤ T,

(2.2)

where Zt is a Lévy process. We call such a process St the geometric Lévy
precess (GLP), and we denote the generating triplet of Zt by (σ 2 , ν(dx), b).
Throughout this paper we assume that Ft = σ(Ss , 0 ≤ s ≤ t) =σ(Zs , 0 ≤ s ≤ t) and F = FT . A probability measure Q on (Ω, F) is calledan equivalent martingale measure of St if Q ∼ P and e−rt St is (Ft , Q)martingale.
The price process St has the following another expression
St = S0 E(Z̃)t(2.3)

where E(Z̃)t is the Doléans-Dade exponential (or stochastic exponential) ofZ̃t , and Z̃t is a Lévy process corresponding to the original Lévy process Zt .
The generating triplet of Z̃t , say (σ̃ 2 , ν̃(dx), b̃), isσ̃ 2 = σ 2(2.4)

ν̃(dx) = (ν ◦ J −1 )(dx), J(x) = ex − 1,Z1
(ex − 1 − x)ν(dx)b̃ = b + σ 2 +2{|x|≤1}
Z

+

(2.5)

Z

{x<−1}(ex − 1)ν(dx) −{log 2<x≤1}(ex − 1)ν(dx). (2.6)

Remark 1 (i) It holds that supp ν̃ ⊂ (−1, ∞).
(ii) If ν(dx) has the density n(x), then ν̃(dx) has the density ñ(x) and ñ(x)is given byñ(x) =

1n(log(1 + x)).
1+x

(2.7)

(iii) St satisfies the following stochastic differential equationdSt = St− dZ̃t .

(2.8)(iv) The relations between Zt and Z̃t are more precisely discussed in [24],where
the stochastic logarithm of Xt , L(X)t , is defined and the following relations
are obtained.
Zt = log E(Z̃)t ,5Z̃t = L(eZ· )t(2.9)

Many candidates for the suitable Lévy process have been proposed. We
give some examples below.
(1) Stable process (Mandelbrot, Fama(1963))(2) Jump diffusion process (Merton(1973))(3) Variance Gamma process (Madan(1990))(4) Generalized Hyperbolic process (Eberlein(1995)(5) CGMY process (Carr-Geman-Madam-Yor(2000))(6) Normal inverse Gaussian process (Barndorff-Nielsen)(7) finite moment log stable process(Carr-Wu(2003))

2.2

Equivalent martingale measures

Many candidates for the equivalent martingale measure have been proposed
as follows.
(1) Minimal Martingale Measure (MMM) (Föllmer-Schweizer(1991))(2) Esscher Martingale Measure (ESMM) (Gerber-Shiu(1994), B-D-E-S(1996))(3) Minimal Entropy Martingale Measure (MEMM) (Miyahara(1996), Frittelli(2000))(4) Utility Martingale Measure (Utility-MM)
Among those equivalent martingale measures we put our focus on ESMM
and MEMM, and investigate the properties of them in section 3.
Remark 2 Sometimes Mean Correcting Martingale Measure (MCMM) is
used for jump-diffusion models.

3

Esscher Martingale Measure (ESMM) and Minimal Entropy Martingale Measure (MEMM)

3.1

Esscher transforms

The Esscher transform is very popular and thought to be very important
method in the actuary theory. (See [18]). Esscher has introduced the riskfunction and the transformed risk function for the calculation of collective
risk. His idea has been developed in his work [13] and by many authors, andplayed very important roles in the option pricing theory.
6Definition 1 Let R be a random variable and h be a constant. Then the(ESS)

probability measure PR,hdefined by
(ESS)

dPR,h
dP

|F =

ehR
E[ehR ]

(3.1)

is called the Esscher transformed measure of P by the risk variable R and
the index h, and this measure transformation is called the Esscher transformby the risk variable R and the index h.
Definition 2 Let Rt , 0 ≤ t ≤ T, be a stochastic process. Then the Esschertransformed measure of P by the risk process Rt and the index process hs is(ESS)

the probability measure PR[0,T ] ,h defined by
RT

(ESS)

dPR[0,T ] ,h[0,T ]
dP

|F =

e 0 hs dRs
E[e

RT
0

hs dRs(3.2)]

This measure transformation is called the Esscher transform by the risk
process Rt and the index process hs .
Definition 3 In the above definitions, if the index index process is chosen
(ESS)

(ESS)

so that the PR[0,T ] ,h[0,T ] is a martingale measure of St , then PR[0,T ] ,h[0,T ] iscalled the Esscher transformed martingale measure of St by the risk process
(ESS)

(ESS)

Rt , and it is denoted by PR[0,T ] or simply PR·

3.2.

Esscher transformed martingale measures for geometricLévy processes

3.2.1

Simple return process and compound return process

When we give a certain risk process Rt , we obtain a corresponding Esschertransformed martingale measure if it exists. As we have seen in the previous
section, the GLP has two kinds of representation such that
St = S0 eZt = S0 E(Z̃)t .
The processes Zt and Z̃t are candidates for the risk process.

7

We shall see the economical meaning of them. For this purpose, we will
review the discrete time approximation of geometric Lévy processes.
Set
(n)

Sk

= Sk/2n ,k = 1, 2, . . . .

(3.3)

According to the above two kinds of expression of St , we obtain two kindsof approximation formula.
First one is
(n)

Sk
(n)

where Zk(n)

= S0 eZk ,k = 1, 2, . . . ,(3.4)= Zk/2n .

Second approximation is(n)

Sk= S0 E(Y (n) )k ,k = 1, 2, . . . ,

(3.5)
(n)

where E(Y (n) )k is the discrete time Doléans-Dade exponential of Yk ,E(Y (n) )k =

k ³
Y

(n)

1 + (Yj(n)´− Yj−1 )

(3.6)

j=1
(n)

and Ykis defined from the following relations

(n)

eZk

= E(Y (n) )k =

k ³
Y

(n)

1 + (Yj´(n)− Yj−1 ) ,k = 1, 2, . . . .

(3.7)

j=1

So we obtain
(n)

e4Zk

(n)

(n)

³

(n)

= eZk −Zk−1 = 1 + (Yk

(n)

´

(n)

− Yk−1 ) = 1 + 4Yk .

(3.8)

From this we obtain
(n)

4Yk

(n)

and we know that 4Yk

(n)

(n)

(n)

= e4Zk − 1 =

Sk

4Sk

Sk−1

Sk−1

−1=
(n)

(3.9)

(n)

(n)

is the simple return process of Sk .
(n)

On the other hand, we obtain from the definition of 4Zk


(n)



4Sk 
(n)
(n)
(n)
4Zk = log Sk − log Sk−1 = log 1 + (n)
,
Sk−1
(n)

and we know that 4Zk(3.10)

is the increment of log-returns and it is called the(n)compound return process of Sk .
8

Remark 3 The terms ‘simple return’ and ‘compound return’ were introduced in [1, p.294], and well known in economics. (See also [39].)k
For t ∈ ( k−12n , 2n ) we define
(n)

Zt

(n)

(n)

= Zk ,Yt(n)

It is easy to see that the process Zt(n)= Yk .

(3.11)

converges to the process Zt when ngoes to ∞.
(n)

On the other hand the process Yt

converges to the process Z̃t . As wehave seen, St satisfies the following stochastic differential equationdSt = St− dZ̃t .

(3.12)

dSt.
St−(3.13)

From this it follows thatdZ̃t =(For the justification of this formula, see [24].) Comparing the formulae (3.9)(n)

and (3.13), we know that the process Yt

is the approximation process inthe procedure of solving the equation (3.13) for Z̃t . This fact means that(n)the process Yt

converges to the process Z̃t .

Based on the above observation, it is natural for us to give the following
definition.
Definition 4 The process Z̃t is called the simple return process of St , andthe process Zt is called the compound return process of St .
3.2.2

Two kinds of Esscher transformed martingale measures

Suppose that Zt is adopted as the risk process. Then, if the correspond(ESS)

ing Esscher transformed martingale measure PZ[0,T ] is well defined, thenit should be called the ‘compound return Esscher transformed martingale
measure’. This is the Gerber-Shiu’s Esscher martingale measure introducedin [18], and the term ‘Esscher martingale measure’ is usually suggesting this(ESS)

compound return Esscher transformed martingale measure PZ[0,T ] .
Next we consider the case where Z̃t is adopted as the risk process. If(ESS)

the corresponding Esscher transformed martingale measure PZ̃

[0,T ]

9exists,then it should be called the ‘simple return Esscher transformed martingale
measure’.
In [24] the following results have been obtained.
Proposition 1 ([24], Theorem 4.2) The compound return Esscher trans(ESS)

formed martingale measure PZ[0,T ] is unique if it exists.
Proposition 2 ([24], Theorem 4.5) The simple return Esscher transformed(ESS)

martingale measure PZ̃

[0,T ]

is unique if it exists.

We will see here the relation between the Esscher transform and theminimal entropy martingale measure (MEMM). We first give the definitionof the MEMM.
Definition 5 (minimal entropy martingale measure (MEMM)) If anequivalent martingale measure P ∗ satisfies
H(P ∗ |P ) ≤ H(Q|P )∀Q :equivalentmartingale

measure,(3.14)

then P ∗ is called the minimal entropy martingale measure (MEMM) of St .
Where H(Q|P ) is the relative entropy of Q with respect to P RdQΩ log[ dP ]dQ,
H(Q|P ) =
 ∞,



if

Q ¿ P, otherwise,.

(3.15)

From the proof of [17, Theorem 3.1] it follows that
Proposition 3 The simple return Esscher transformed martingale measure
(ESS)

PZ̃

[0,T ]of St is the minimal entropy martingale measure (MEMM) of St .

Based on the above results, we give the following definition.
Definition 6 (i) The compound return Esscher transformed martingale mea(ESS)sure PZ[0,T ] is called the ‘Esscher martingale measure (ESMM)’ and denotedby P (ESM M ) .
(ESS)(ii) The simple return Esscher transformed martingale measure PZ̃

[0,T ]

iscalled the ‘minimal entropy martingale measure (MEMM)’ and denoted byP ∗ (or P (M EM M ) ).

10

Remark 4 For the jump-diffusion models, the Brownian motion can be
adopted as the risk process. In that case the corresponding Esscher transformed martingale measure is the mean correcting martingale measure. (See[38] or [7]).

3.3

Existence theorems of ESMM and MEMM for geometricLévy processes

The uniqueness theorems have been stated in the previous section. We next
study the Existence problem of Esscher transformed martingale measures.
3.3.1

Existence theorem of ESMM

We suppose that the expectations which appear in what follows exist. Thenthe martingale condition for an Esscher transformed probability measure(ESS)

Q = PZ[0,T ] ,h is
EQ [e−r S1 ] = e−r S0 EQ [eZ1 ] = e−r S0

EP [e(h+1)Z1 ]= S0 .
EP [ehZ1 ](3.16)

This condition is equal to the following condition
EP [e(h+1)Z1 ] = er EP [ehZ1 ],(3.17)and this is also equivalent to the following expression,φ(−i(h + 1)) = er φ(−ih),(3.18)

where φ(u) is the characteristic function of Z1 (φ(u) = EP [eiuZ1 ]).
To formulate the existence theorem, we set
1
f (ESM M ) (h) = b + ( + h)σ 2 +2Z+{|x|>1}Z{|x|≤1}³´(ex − 1)ehx − x ν(dx)(ex − 1)ehx ν(dx),(3.19)

Then we obtain
Theorem 1 (Existence condition for ESMM) If the equationf (ESM M ) (h) = r,11

(3.20)

has a solution h∗ , then the ESMM of St , P (ESM M ) , exists and
(ESS)

(ESS)

P (ESM M ) = PZ[0,T ] ,h∗ = PZT ,h∗(3.21)The process Zt is also a Lévy process under P (ESM M ) and the generating2

triplet of Zt under P (ESM M ) , say (σ (ESM M ) , ν (ESM M ) (dx), b(ESM M ) ), isσ (ESM M )

2

= σ2,

ν (ESM M ) (dx) = e(3.22)h∗ x

ν(dx),b(ESM M ) = b + h∗ σ 2 +(3.23)Zh∗ x{|x|≤1}x(e− 1)ν(dx).

(3.24)(Proof) The equation (5.5) is equivalent to the condition (5.3). Therefore(ESS)

PZ[0,T ] ,h∗ is a martingale measure of St .
(ESS)(ESM M )The characteristic function of Zt under P (ESM M ) = PZ[0,T ] ,h∗ , φt(u),is by definition∗
EP [eiuZt eh ZT ](ESM M )
iuZt
]=
.
φt
(u) = EP (ESM M ) [eEP [eh∗ ZT ](3.25)

And this is equal to∗EP [e(iu+h )Zt
φt (u − ih∗ )=.
∗EP [eh Zt ]φt (−ih∗ )(3.26)

By simple calculation we obtain(ESM M )

φt

n ³

R

∗

(u) = exp t − 21 σ 2 + i(b + h∗ σ 2 + {|x|≤1} x(eh x − 1)ν(dx))uR∗+ {|x|≤1} (eiux − 1 − iux)eh x ν(dx)
R

∗

´o

+ {|x|>1} (eiux − 1)eh x ν(dx).

(3.27)

Using this formula, we can see that the martingale condition (3.18) is reduced to the equation 3.20).
3.3.2(Q.E.D.)

Existence theorem of MEMM

As we have mentioned in the previous section, the MEMM, P ∗ , is the sim(ESS)

ple return Esscher transformed martingale measure. (P ∗ = PZ̃
existence theorem of the MEMM is obtained in [17].

12[0,T ]). TheSet
1f (M EM M ) (θ) = b + ( + θ)σ 2 +
2

Z

Z

+

³

{|x|≤1}x´(ex − 1)eθ(e −1) − x ν(dx)x{|x|>1}(ex − 1)eθ(e −1) ν(dx)(3.28)

Then the following result is obtained ([17, Theorem 3.1]).
Theorem 2 (Existence condition for MEMM) If the equationf (M EM M ) (θ) = r(3.29)

has a solution θ∗ , then the MEMM of St , P ∗ , exists and
(ESS)
(ESS)
∗ = PZ̃ ,θ ∗,θT
[0,T ]

P ∗ = P (M EM M ) = PZ̃

(3.30)The process Zt is also a Lévy process under P ∗ and the generating triplet of
Zt under P ∗ , say (σ ∗ 2 , ν ∗ (dx), b∗ ), isσ∗2 = σ2,ν ∗ (dx) = eθ ∗ (ex −1)(3.31)
ν(dx),b∗ = b + θ∗ σ 2 +(3.32)

Z{|x|≤1}x(eθ∗ (ex −1)− 1)ν(dx).

(3.33)(Proof) The results of this theorem follows directly from the proof of [17,Theorem 3.1].

(Q.E.D.)

Remark 5 (i) The function f (M EM M ) (θ) is a non-decreasing function ofθ.
(ii) If St is integrable, then it holds that
E(St ) = S0 exp(tf (M EM M ) (0)).

(3.34)(iii) If the condition that f (M EM M ) (0) > r is satisfied, then the solution θ∗of (3.29) is negative (θ∗ < 0), when it exists. Such cases occur very often.

13

3.4

Comparison of ESMM and MEMM

The ESMM and the MEMM are both obtained by Esscher transform, butthey have slightly different properties. We will survey the properties andthe differences of them.
(1) Corresponding risk process:The risk process corresponding to the ESMM is the compound returnprocess, and the risk process corresponding to the MEMM is the simple
return process. The simple return process seems to be more essential in therelation to the original process rather than the compound return process.
In this sense we can say that the MEMM is more reasonable martingale
measure than the ESMM.
(2) Existence condition:
As we have seen in the previous section, for the existence of ESMM,
P (ESM M ) , the following conditionZ∗{|x|>1}|(ex − 1)eh x | ν(dx) < ∞(3.35)

is necessary. On the other hand, for the existence of MEMM, P ∗ , the corresponding condition isZ{|x|>1}∗x|(ex − 1)eθ (e −1) | ν(dx) < ∞.

(3.36)

This condition is satisfied for wide class of Lévy measures, if θ∗ < 0. Namely,
the former condition is strictly stronger than the latter condition.
This means that the MEMM may be applied to the wider class of models
than the ESMM. This difference does work in the stable process cases. In
fact the MEMM method can be applied to the geometric stable model but
the ESMM method can not be applied to this same model.
(3) Corresponding utility function:
The ESMM is corresponding to power utility function or logarithm utility
function. (See [18, pp.175-177] or [19, Corollary 6.3]). On the other hand
the MEMM is corresponding to the exponential utility function. (See [16,§3] or [19, §6.1]).
14

We remark here that, in the case of ESMM, the power parameter of theutility function depends on the parameter value h∗ of the Esscher transform.
We also remark that, in the case of MEMM, the relation of the MEMM to
the utility indifference price is known. (See [17, §4]. This result is generalized
by C. Stricker [40].)(4) Properties special to MEMM:a) Minimal distance to the original probability:
The relative entropy is very popular in the field of information theory, and it is called Kullback-Leibler Information Number(see [21, p.23]) or
Kullback-Leibler distance (see [10, p.18]). Therefore we can state that the
MEMM is the nearest equivalent martingale measure to the original probability P in the sense of Kullback-Leibler distance. Recently the idea ofminimal distance martingale measure is studied. In [19] it is mentioned thatthe relative entropy is the typical example of the distance in their theory.
b) Large deviation property:
The large deviation theory is closely related to the minimum relative
entropy analysis, and the Sanov’s theorem or Sanov property is well-known(see, e.g. [10, p.291-304] or [21, p.110-111]). This theorem says that the
MEMM is the most possible empirical probability measure of paths of priceprocess in the class of the equivalent martingale measures. In this sense the
MEMM should be considered to be the exceptional measure in the class of
all equivalent martingale measures.
As the result of the above discussions, we can say that the MEMM has
many better properties than the ESMM in the theoretical sense.

4[GLP & MEMM] Pricing ModelIn this section we explain the [GLP & MEMM] pricing model and see examples of the model.

15

4.1

Model

Now we can define the [GLP & MEMM] Pricing Model, which was first
introduced in [30]. The [GLP & MEMM] Pricing Model is such a model:(A) The price process St is a geometric Lévy process (GLP).
(B) The price of an option X is defined to be e−rT EP ∗ [X], where P ∗ is the
MEMM.
Of course this model can be considered for the cases where the MEMM
exists.

4.2

Sufficient Conditions for the Existence of the MEMM

The existence problem of the MEMM of geometric Lévy processes has beenstudied in [29], [6] and [30], and finally those results are generalized in [17]as the following form.
Theorem 3 (Fujiwara-Miyahara [17, Theorem 3.1]) Suppose that thefollowing condition (C) holds
Condition (C)There exists θ∗ ∈ R which satisfies the following condi-

tions :Rx θ ∗ (ex −1) ν(dx) < ∞,{x>1} e e(C)1R(C)2 b + ( 12 + θ∗ )σ 2 + {|x|>1} (ex − 1)eR³+ {|x|≤1} (ex − 1)eθ ∗ (ex −1)θ∗ (ex −1)

´

(4.1)
ν(dx)

− x ν(dx) = r.

(4.2)

Then the probability measure P ∗ is well defined and it holds that(i)(MEMM): P ∗ is the MEMM of St .
(ii)(Lévy process): Zt is also a Lévy process w.r.t. P ∗ , and the generating
triplet (A∗ , ν ∗ , b∗ ) of Zt under P ∗ isA∗ = σ 2 ,ν ∗ (dx) = eθ ∗ (ex −1)(4.3)
ν(dx),b∗ = b + θ∗ σ 2 +

(4.4)

Z

R\{0}

16

xI{|x|≤1} d(ν ∗ − ν).

(4.5)

4.3

Examples of [GLP & MEMM] Pricing ModelIn this section we see several examples of [GLP & MEMM] Pricing Models.
To do this, we have to check the existence of the MEMM, i.e. we have to
examine that the given geometric Lévy process St = S0 exp Zt satisfies theCondition (C). Set
1f (θ) = b + ( + θ)σ 2 +2Z+{|x|≤1}³

Z

x{|x|>1}(ex − 1)eθ(e −1) ν(dx)x´(ex − 1)eθ(e −1) − x ν(dx).

(4.6)

Then the condition (C)2 is equivalent to that θ∗ is the solution off (θ) = r.
4.3.1

(4.7)

Geometric Variance Gamma Model

The Lévy measure of Variance Gamma process is of the following form (see[25]).
³´

ν(dx) = C I{x<0} exp(−c1 |x|) + I{x>0} exp(−c2 |x|) |x|−1 dx,(4.8)

where C, c1 , c2 are positive constants.
The following results are obtained (see[17] or [34]).
Proposition 4 (1) If c2 ≤ 1, then the equation f (θ) = r has a uniquesolution θ∗ , and the solution is negative.
(2) If c2 > 1 and f (0) ≥ r, then the equation f (θ) = r has a unique solutionθ∗ , and the solution is non-positive.
(3) If c2 > 1 and f (0) < r, then the equation f (θ) = r has no solution.

4.3.2

Geometric CGMY Model

The Lévy measure of the CGMY process is³´

ν(dx) = C I{x<0} exp(−G|x|) + I{x>0} exp(−M |x|) |x|−(1+Y ) dx,(4.9)

where C > 0, G ≥ 0, M ≥ 0, Y < 2 (see [2]). If Y ≤ 0, then G > 0 and
M > 0 are assumed. We mention here that the case Y = 0 is the VG
17

process case, and the case G = M = 0 and 0 < Y < 2 is the symmetricstable process case. In the sequel we assume that G, M > 0.
For this model the following results are obtained (see [34]).
Proposition 5 (1) If M ≤ 1, then the equation f (θ) = r has a uniquesolution θ∗ , and the solution is negative.
(2) If M > 1 and f (0) ≥ r, then the equation f (θ) = r has a unique solutionθ∗ , and the solution is non-positive.
(3) If M > 1 and f (0) < r, then the equation f (θ) = r has no solution.
4.3.3

Geometric Stable Model

We consider the stable model. Suppose that Zt is a stable process and let(0, ν(dx), b) be its generating triplet. The Lévy measure isν(dx) = c1 I{x<0} |x|−(α+1) dx + c2 I{x>0} |x|−(α+1) dx,(4.10)

where 0 < α < 2 and we assume thatc1 ≥ 0,c2 ≥ 0,c = c1 + c2 > 0.

(4.11)

Proposition 6 Under the assumption c1 , c2 > 0, the equation f (θ) = r hasa unique solution θ∗ , and the solution θ∗ is negative.
Remark 6 Consider the case where c1 , c2 > 0. Under the original measure
P , St , t > 0 is not integrable. But under the MEMM P ∗ , any moments
EP ∗ [|St |k ], k = 1, 2, . . . , of St are finite. This fact follows easily from theresult that θ∗ is negative, and this property is very useful for the study ofoption pricing of this model.

4.4

Option Pricing and Volatility Smile/Smirk Properties

In order to apply the [GLP & MEMM] Pricing Models to the financial
problems, we have to establish the methods to compute the option prices.
Namely we have to compute the expectations EP ∗ [F (ω)], where F (ω) is afunctional of Lévy process.

18

4.4.1

European Type Options

If a contingent claim C is depending only on the terminal value of the stock
price ST = S0 eZT , then we can compute the price of C as what follows.
Let C = f (ST ) = f (S0 eZT ) = F (ZT ), (F (z) = f (S0 ez )), and set
C(t, y) = EP ∗ [e−r(T −t) f (ST )|St = y] and C̃(t, z) = EP ∗ [e−r(T −t) F (ZT )|Zt =z] = EP ∗ [e−r(T −t) f (ST )|St = S0 ez ]. (Remark that C̃(t, z) = C(t, S0 ez ).)Since the process Zt is a Lévy process with the generating triplet (σ 2 , ν ∗ (dx), b∗ ),C̃(t, z) satisfies the following equation under the assumption of the smoothness of C̃(t, z).
−∂ C̃(t, z)∂t=

1 2 ∂ 2 C̃(t, z)∂ C̃(t, z)
σ
+b
2
2
∂z
∂z
Ã
+

Z ∞

−∞

!

∂ C̃(t, z)C̃(t, z + z̃) − C̃(t, z) − z̃
1{|z̃|<1} (z̃) ν(dz̃)∂z−rC̃(t, z),

0 ≤ t < T,C̃(T, z) = F (z).

(4.12)(4.13)

Solving this equation, we obtain the option price C(0, S0 ) = C̃(0, 0).
4.4.2

FFT Method for European Call Options

The fast Fourier transform method (FFT method) is very useful for thecomputation of option prices. We need to compute the such an expectation
EP ∗ [F (ω)], and in the case of European type options such type of expectations EP ∗ [G(ST )]. If we know the distribution function p∗T (z) of ZT underP ∗ , then EP ∗ [G(ST )] =R∞∗−∞ G(z)pT (z)dz. Lévy process is characterized by

the generating triplet, and the generating triplet is given explicitly in thecharacteristic function. So we can assume that the characteristic functionφ∗T (u) of ZT under P ∗ is given and the density function p∗T (z) is obtained
as the inverse Fourier transform of φ∗T (u).
For the computer simulation of the theoretical prices of European call
options, the FFT method is very useful. Carr and Madan have introduced
their idea in [3], and their method has been improved by Cont and Tankov
in [8]. We rearrange their ideas in such a form that we can easily apply theformula to our [GLP & MEMM] pricing models.

19

The characteristic function φ∗t (u) of Zt under the MEMM P ∗ isφ∗t (u) = φ∗Zt (u) = EP ∗ [eiuZt ] = exp(ψt∗ (u)) = exp(tψ ∗ (u)),

i=

√
−1,
(4.14)

where ψ ∗ (u) = ψ1∗ (u). Let µ∗t (dz) be the distribution of Zt under the MEMM
P ∗ , and assume that µ∗t (dz) = p∗t (z)dz. Thenφ∗t (u) = φ∗Zt (u) = EP ∗ [eiuZt ] =Z ∞−∞eiuz p∗t (z)dz,(4.15)The price of European call option isC(S0 , K, T ) = e−rT EP ∗ [(ST − K)+ ] = e−rTZ ∞−∞(S0 ez − K)+ p∗T (z)dz.
(4.16)

Set K/S0 = ek , and define c(k; S0 , T ) = C(S0 , S0 ek , T ). Then using (5.3)
c(k; S0 , T ) = S0 e−rTZ ∞−∞(ez − ek )+ p∗T (z)dz(4.17)

We introduce the so-called time value of optionc̃(k; S0 , T ) = c(k; S0 , T ) − (S0 − e−rT K)+ = c(k; S0 , T ) − S0 (1 − ek−rT )+(4.18)and let ζ(v; S0 , T ) be the Fourier transform of c̃(k; S0 , T )ζ(v; S0 , T ) =Z ∞−∞eivk c̃(k; S0 , T )dk.

(4.19)

Using (5.5)ζ(v; S0 , T ) = S0 e−rTZ ∞−∞p∗T (z)dz

andζ(v; S0 , T ) = S0

Z rTzeivk (ek − ez )dk.

e−rT φ∗T (v − i) − eivrT.
iv(1 + iv)(4.20)(4.21)

The characteristic function φ∗T (u) is computed directly from the generating
triplet (σ 2 , b∗ , ν ∗ (dx)), so ζ(v; S0 , T ) is obtained from the above formula.
Next, by (), c̃(k; S0 , T ) is obtained by the inverse Fourier transformc̃(k; S0 , T ) =

1
2π

Z ∞
−∞

e−ikv ζ(v; S0 , T )dv(4.22)

andc(k; S0 , T ) = c̃(k; S0 , T ) + (S0 − e−rT K)+ ,20

K = S0 ek .

(4.23)

Finally we obtain the price of the European call option C(S0 , K, T ) as
C(S0 , K, T ) = c(log(K/S0 ); S0 , T ) = c̃(log(K/S0 ); S0 , T ) + (S0 − e−rT K)+ .
(4.24)
4.4.3

Volatility Smile/Smirk Properties

The volatility smile/smirk properties are reported for many market prices
of options. This fact tells us that the Black-Scholes model is not necessarily
best model, and that we should study other models which may have thevolatility simile/smirk properties. It is known that the [GLP & MEMM]
models have those properties. (See [33]).

5

Physical World and MEMM WorldThe behavior of the price process St is governed by the original probability
P , and the movement of St is observable. This is the real world (=Physicalworld).
On the other hand the price of an option X is computed as the expectation e−rT EP ∗ [X], namely the process St is supposed to obey the MEMM
P ∗ . This world is differ from the real world, and this world should be called
the imaginary world (=MEMM world).

5.1

From Physical World to MEMM World

Suppose that the price process St = S0 eZt is given and the generating tripletof Zt is (σ 2 , ν, b). Let θ∗ is the solution of f (θ) = r, where the function f (θ)
is defined by (4.6). Then, by Theorem 3 in §4.2, the generating triplet(σ ∗ 2 , ν ∗ , b∗ ) of Zt under P ∗ isσ∗2 = σ2,
∗

(5.1)
x

ν ∗ (dx) = eθ (e −1) ν(dx),

(5.2)

Z

b∗ = b + θ∗ σ 2 +{|x|≤1}Z= b + θ∗ σ 2 +{|x|≤1}21

xd(ν ∗ − ν)
³

∗

x

(5.3)
´

x eθ (e −1) − 1 ν(dx).

(5.4)

This triplet determines the prices of options in the framework of [GLP
& MEMM] pricing model.
Remark 7 The condition (C)2 for θ∗ (i.e. θ∗ is the solution of f (θ) = r)is equivalent to the following condition (M)
Z

(M )

Z

1
b∗ − r + σ ∗ 2 +(ex − 1 − x)ν ∗ (dx) +(ex − 1)ν ∗ (dx) = 0.
2{|x|≤1}{|x|>1}(5.5)

We should notice that the θ∗ does not appear explicitly in this formula, andthat this formula is just the same condition that P ∗ is a martingale measureof the price process St .
Concerning to the martingale condition for more general cases of semimartingales, see [39] where the triplet’s notation for semimartingale are(B, C, ν). ( See also [22].)

5.2

From MEMM World to Physical World

We study the inverse problem of the previous subsection. Suppose that thegenerating triplet (σ ∗ 2 , ν ∗ , b∗ ) of Zt under P ∗ is given. Since we assume thatP ∗ is martingale measure, the condition (M ) is satisfied.
We try to construct a probability P̃ such that under P̃ the price process
St = S0 eZt is geometric Lévy process and the MEMM of St = S0 eZt with P̃
is P ∗ .
Let θ∗ be any real number (it is usually supposed that θ∗ < 0) and set
σ̃θ2∗

= σ∗2

ν̃θ∗ (x) = eb̃θ∗−θ∗ (ex −1)(5.6)
ν ∗ (dx)= b∗ − θ∗ σ 2 +Z{|x|≤1}³x e−θ∗ (ex −1)´− 1 ν ∗ (dx),

(5.7)
(5.8)

where we assume that all integrals are converge. Then suppose that we could
construct the probability measure P̃θ∗ such that under P̃θ∗ the process Zt isa Lévy process with the generating triplet (σ̃θ2∗ , ν̃θ∗ , b̃θ∗ ).
It is easy to see that P ∗ is the MEMM of St = S0 eZt with P̃θ∗ . We
remark here that there are many geometric Lévy processes whose MEMM
is just the same P ∗ .
22

5.3

Example: Geometric Stable Process Case

• Parameters in the physical world: (α, c1 , c2 , b),

0 < α < 2, c1 , c2 ≥ 0,

c1 + c2 > 0, −∞ < b < ∞.
The triplet is (0, ν, b), andν(dx) =

c1 I{x<0} (x) + c2 I{x>0} (x)dx.
|x|(α+1)• Parameters in the MEMM world: (θ∗ , α∗ , c∗1 , c∗2 , b∗ ),θ∗ < 0, 0 < α∗ < 2,c∗1 , c∗2 ≥ 0, c∗1 + c∗2 > 0, −∞ < b∗ < ∞, where∗xν ∗ (dx) = eθ (e −1)c∗1 I{x<0} (x) + c∗2 I{x>0} (x)dx.
|x|(α∗ +1)

and the following martingale condition
Z∗

Condition(M ) :

b +
Z

+

{|x|≤1}{|x|>1}(ex − 1 − x)ν ∗ (dx)(ex − 1)ν ∗ (dx) = r

must be satisfied. So, if we have given the values of (θ∗ , α∗ , c∗1 , c∗2 , ), then thevalue of b∗ is determined by the above condition (M).

6

Calibration of [GLP & MEMM] pricing model

Suppose that the sequential data of the price process St of underlying asset
and the data of market prices of options. From these data, we have to selecta model which is most fitting to the given data. This is the calibration
problem. This problem shall be discussed in section 6.

6.1

Calibration problems• Given data: the sequential data of the price process St , and the data ofmarket prices of options.
• Select the most fitting model to the given data.
To solve this problem, we have to take the following steps.
1) Estimation the price process of the underlying asset in the physicalworld from the sequential data of it.
232) Calculation of the MEMM from the estimated parameter, and computation of the theoretical prices of options in the estimated MEMM world.
3) Analyzing the fitness of the theoretical prices to the market prices.
4) Determination of the most reasonable model.

6.2

Estimation of Lévy Processes in the Physical World

Usually this procedure is carried on under the restriction of the class ofLévy processes, for example the stable process class, VG process class, etc.
Therefore the estimation problem of the process is reduced to the parameter
estimation problems.
There are many papers on this subject (see [31] for example).
Denote the estimated probability by Pb , or equivalently, the estimatedb 2 , νb, b
generating triplet by (σ
b)).

6.3

Calculation of the theoretical option prices

Let Pb be the estimated probability in the physical world, and let Pb ∗ be thecorresponding MEMM. Then the theoretical price of option C is EPb∗ [Ce−rT ].

We denote this value by Cb ∗ .

6.4

Fitness analysis of the estimated model

Suppose that the data, ηl , l = 1, 2, . . . , L, of market prices of options Cl .
Then we define the fitting error of the model by∗cl − ηl |1 X |C
² =
L l
ηl
∗

6.5

Determination of the most fitting model

As the results of the above procedure, if the value ²∗ is small then the fitnessof the model to the data is good.
The value ²∗ depends on the model, namely the selected class of theprocess. We can conclude that the class whose fitness error is the smallest
is the best model.

24

6.6

Diagram of the Procedure for the Calibration

The path space D[0, T ] is fixed.
MEMM World

Physical World
St = S0 eZt(σ 2 , ν, b) under P(σ ∗ 2 , ν ∗ , b∗ ) under P ∗Zt—————————————————————————————
Data: {ξj } (time series data)
Estimated: Pb
b 2 , νb, b
(σ
b)

θb∗
Transformed: Pb ∗2b ∗ , νb∗ , b
(σ
b∗ )
cl
Theoretical prices: C∗(European Call Options)—————————————————————————————
Data: {ηl } (European Call Options)∗cl − ηl |1 X |C
² =
L l
ηl
∗

6.7

Example: Geometric Stable Process Case

Parameter in the physical world(α, c1 , c2 , b)

ν(dx) =

c1 I{x<0} (x) + c2 I{x>0} (x)dx.
|x|(α+1)

b cb1 , cb2 , b
Estimators (α,b)νb(dx) =cb1 I{x<0} (x) + cb2 I{x>0} (x)|x|(αb+1)

25

dx.

θb∗ is determined by
Z
b
b +

b∗

{|x|>1}Z+x(ex − 1)eθ (e −1) νb(dx)µx(e − 1)e{|x|≤1}θb∗ (ex −1)¶−xνb(dx) = r.

The process which determines the theoretical option prices isb∗xνb∗ (dx) = eθ (e −1)

cb1 I{x<0} (x) + cb2 I{x>0} (x)|x|(αb+1)Zbb∗ = bb +R\{0}xI{|x|≤1} d(νb∗ − νb).

26

dx.

(6.1)

References[1] Bühlmann, H., Delbaen, F., Embrechts, P., and Shiryaev, A. N. (1996),
No-arbitrage, Change of Measure and Conditional Esscher Transforms,
CWI Quarterly 9,No. 4, 291-317.
[2] Carr, P., Geman, H., Madan, D.B., and Yor, M. (2000), The Fine
Structure of Asset Returns: An Empirical Investigation. (Preprint).
[3] Carr, P. and Madan, D. (1999), Option valuation using the fast fouriertransform. Journal of Computational Finance, 2, 61-73.
[4] Carr, P. and Wu, L. (2003), “The Finite Moment Log Stable Processes
and Option Pricing,” The Journal of Finance 58, No. 2, 753-777.
[5] Carr, P. and Wu, L. (2004), “Time-changed Lévy Processes and Option
Pricing,” Journal of Financial Economics 71, 113-141.
[6] Chan, T. (1999), Pricing Contingent Claims on Stocks Derived by Lévy
Processes. The Annals of Applied Probability, v. 9, No. 2, 504- 528 .
[7] Cheang, Gerald H.L. (2003), “A simple approach to pricing options
with jumps,” (preprint)[8] Cont, R. and Tankov, P. (2002), Calibration of Jump-Diffusion OptionPricing Models: A Robust Non-parametric Approach. (preprint).
[9] Cont, R. and Tankov,P. (2004), Financial Modelling with Jump Processes, ChHAPMAN & Hall/CRC.
[10] Cover, T. and Thomas, J. (1991), Elements of Information Theory,
Wiley.
[11] Eberlein, E. and Keller, U. (1995), Hyperbolic distributions in finance.
Bernoulli, 1, 281-299.
[12] Edelman, D. (1995), A Note: Natural Generalization of Black-Scholes
in the Presence of Skewness, Using Stable Processes, ABACUS 31,No.
1,113-119.

27

[13] Esscher, F. (1932), On the Probability Function in the Collective Theory of Risk, Skandinavisk Aktuarietidskrift 15, 175-195.
[14] Fama, E. F. (1963), Mandelbrot and the Stable Paretian Hypothesis.
J. of Business, 36, 420-429.
[15] Föllmer, H. and Schweizer, M. (1991), Hedging of Contingent Claims
under Incomplete Information. In M.H.A. Davis and R.J. Elliot (ed.) :Applied Stochastic Analysis, Gordon and Breach, 389-414.
[16] Frittelli, M. (2000), The Minimal Entropy Martingale Measures andthe Valuation Problem in Incomplete Markets, Mathematical Finance
10, 39-52.
[17] Fujiwara, T. and Miyahara, Y. (2003), The Minimal Entropy Martingale Measures for Geometric Lévy Processes, Finance and Stochastics
7(2003), pp.509-531.
[18] Gerber, H. U. and Shiu, E. S. W. (1994), Option Pricing by Esscher
Transforms, Transactions of the Society of Actuaries XLVI, 99-191.
[19] Goll, T. and Rüschendorf, L. ( 2001), Minimax and minimal distancemartingale measures and their relationship to portfolio optimization,
Finance and Stochastics, Vol.5, No. 4, 557-581.
[20] Hurst, S.R., Platen, E. and Rachev, S.T.(1999), Option pricing for alogstable asset price model. Math. Comput. Modelling, 29 , no. 10-12,
105–119.
[21] Ihara, S. (1993), Information Theory for Continuous System, World
Scientific.
[22] Jacod, J. and Shiryaev, A. (1987), Limit Theorems for Stochastic Processes, Springer-Verlag.
[23] Kallsen, J.(2002), Utility-Based Derivative Pricing in Incomplete Markets, in Mathematical Finance - Bachelier Congress 2000, Springer,313-338.

28

[24] Kallsen, J. and Shiryaev, A.N.(2002), The Cumulant Process and Esscher’s Change of Measure, Finance and Stochastics, Vol.6, No.4, 397428.
[25] Madan, D. and Seneta, E. (1990), The variance gamma (vg) model forshare market returns. Journal of Business, v.63(4), 511-524.
[26] Madan, D., Carr, P. and Chang, E. (1998), The variance gamma process
and option pricing. European Finance Review, v.2, 79-105.
[27] Mandelbrot, B. (1963), The variation of certain speculative prices. J.
of Business, 36, 394-419.
[28] Miyahara, Y.(1996), Canonical Martingale Measures of Incomplete Assets Markets, in Probability Theory and Mathematical Statistics: Proceedings of the Seventh Japan-Russia Symposium, Tokyo 1995 (eds. S.
Watanabe et al), pp.343-352.
[29] Miyahara, Y. (1999), Minimal Entropy Martingale Measures of Jump
Type Price Processes in Incomplete Assets Markets. Asian-Pacific Financial Markets, Vol. 6, No. 2, pp. 97-113.
[30] Miyahara, Y.(2001), [Geometric Lévy Process & MEMM] Pricing
Model and Related Estimation Problems, Asia-Pacific Financial Markets 8, No. 1, pp. 45-60.
[31] Miyahara, Y.(2002), Estimation of Lévy Processes, Discussion Papers
in Economics, Nagoya City University No. 318, pp. 1-36.
[32] Miyahara, Y.(2004), A Note on Esscher Transformed Martingale Measures for Geometric Lévy Processes, Discussion Papers in Economics,
Nagoya City University No. 379, pp. 1-14.
[33] Miyahara, Y. and Moriwaki, N.(2004), Volatility Smile/Smirk Properties of [GLP & MEMM] Pricing Models, (preprint).
[34] Miyahara, Y. and Novikov, A. (2002), Geometric Lévy Process Pricing
Model, Proceedings of Steklov Mathematical Institute, Vol.237(2002),
pp.176-191.
29

[35] Overhaus, M., Ferraris, A., Knudsen, T., Milward, R., Nguyen-Ngoc,
L. and Schindlmayr, G. (2002), Equity derivatives: Theory and Applications, Wiley.
[36] Rachev, S. and Mittnik, S. (2000), ”Stable Paretian Models in Finance”,
Wiley.
[37] Sato, K. (1999), ”Lévy Processes and Infinitely Divisible Distributions”.
Cambridge University Press.
[38] Schoutens, W, (2003), Lévy Processes in Finance: Pricing Financial
Derivatives, Wiley.
[39] Shiryaev, A. N.(1999), Essentials of Stochastic Finance: Facts, Models,Theory, World Scientific.
[40] Stricker, C. (2002), On the utility indifference price, (preprint).
[41] Wannenwetch, J. (2004), Adjusting the Measure Change Function inLévy Markets, (preprint).
[42] Xiao, K., Misawa, T., and Miyahara, Y. (2000), Computer Simulation of[Geometric Lévy Process & MEMM] Pricing Model, Discussion Papers
in Economics, Nagoya City University No. 266(2000), pp. 1-16.

30