A Note on
Esscher Transformed Martingale Measuresfor Geometric Lévy Processes
Yoshio Miyahara∗
Faculty of Economics
Nagoya City University
August 19, 2004

Abstract
The Esscher transform is one of the very useful methods to obtain
the reasonable equivalent martingale measures, and it is defined withrelation to the corresponding risk process.
In this article we consider two kinds of risk processes (compound
return process and simple return process). Then we obtain two kindsof Esscher transformed martingale measures. The first one is the one
which was introduced by Gerber and Shiu, and the second one is
identified with the MEMM (minimal entropy martingale measure).
We set up the economical characterization of these two kinds ofEsscher transforms, and then we study the properties of the above twokinds of Esscher transformed martingale measures, comparing eachothers.

Key words: incomplete market, geometric Lévy process, equivalent martingale measure, Esscher transform, minimal entropy martingale measure.
JEL Classification: G12, G13∗

E-mail: y-miya@econ.nagoya-cu.ac.jp1

1

Introduction

The equivalent martingale measure method is one of the most powerful methods in the option pricing theory. If the market is complete, then the equivalent martingale measure is unique. On the other hand, in the incomplete
market model there are many equivalent martingale measures. So we have toselect one equivalent martingale measure (EMM) as the suitable martingale
measure in order to apply the martingale measure method.
In the case of geometric Lévy process models, four kinds of measures havebeen proposed. The first one is the minimal martingale measure (MMM),
the second one is the Esscher martingale measure (ESSMM), the third oneis the minimal entropy martingale measure (MEMM) , and the fourth one is
the utility martingale measure (UMM).
The Esscher transform is thought to be a very useful technique to obtain
the reasonable equivalent martingale measure, and it is defined in the relation
to the corresponding risk process. (See [6], [10], [1], [14], etc.) In this article
we consider two kinds of risk processes (compound return process and simple
return process).
According to these risk processes, we obtain two kinds of Esscher transformed martingale measures. The first one is the compound return Esschertransformed martingale measure, which was first introduced by Gerber and
Shiu ([10]), and this measure is called the “Esscher martingale measure (ESSMM)”. The second one is the simple return Esscher transformed martingale
measure. It is known that this martingale measure is identified with the “minimal entropy martingale measure (MEMM).” (See [9] or [14]). The MEMM
has been discussed in [15], [7], [9], and etc.
In §2 we explain the geometric Lévy process and in §3 we give the definition of two kinds of Esscher transforms. In §4 we study the economical
implications of those transforms. In §5 we summarize the existence conditions of those martingale measures, and finally in §6 we review the propertiesof ESSMM and MEMM.
As the results of our consideration, we can say that the MEMM is very
effective to the option pricing theory from the theoretical point of view.

22

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 thisprobability space, where Zt is a Lévy process. We call such a process St thegeometric Lévy precess (GLP).
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 called an equivalentmartingale measure of St if Q ∼ P and e−rt St is (Ft , Q)-martingale, where r
is the interest rate.
The price process St has the other expression
St = S0 E(Z̃)t(2.1)

where E(Z̃)t is the Doléans-Dade exponential of Z̃t , and Z̃t is a Lévy process
corresponding to the original Lévy process Zt . Let the generating triplet of
Zt is (σ 2 , ν(dx), b), then the generating triplet of Z̃t , say (σ̃ 2 , ν̃(dx), b̃), isσ̃ 2 = σ 2(2.2)

ν̃(dx) = (ν ◦ J −1 )(dx), J(x) = ex − 1,Z1b̃ = b + σ 2 +(ex − 1 − x)ν(dx)2{|x|≤1}
Z

+

(2.3)

Z

x

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

(2.4)

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 by1n(log(1 + x)).
(2.5)ñ(x) =
1+x(iii) St satisfies the following stochastic differential equationdSt = St− dZ̃t .

(2.6)(iv) The relations between Zt and Z̃t are more precisely discussed in [14].

33Esscher transforms

The Esscher transform is very popular in the field of actuary, and is thought
to be very important method in the actuary theory. (See [10]). Esscher
has introduced the risk function and the transformed risk function for the
calculation of collective risk in [6]. His idea has been developed by many
researchers ([10], [1], [14]), and applied to the option pricing theory.
We give the definitions of Esscher transform and Esscher transformed
martingale measure.
Definition 1 Let R be a risk variable and h be a constant. Then the proba(ESS)
bility measure PR,h defined by
(ESS)

dPR,h
dP

ehR
|F =
E[ehR ]

(3.1)

is called the Esscher transformed measure of P by the random variable R
and h, and this measure transformation is called the Esscher transform by
the random variable R and h.
Definition 2 Let Rt , 0 ≤ t ≤ T, be a risk process. Then the Esscher transformed measure of P by the process Rt and a constant h is the probability(ESS)
measure PR[0,T ] ,h , which is defined by(ESS)

dPR[0,T ] ,h
dP
(ESS)

|F =

ehRT
E[ehRT ]

(3.2)

(ESS)

( Remark that PR[0,T ] ,h = PRT ,h . )and this measure transformation is called the Esscher transform by the process Rt and a constant h.
Definition 3 In the above definition, if the constant h is chosen so that the
(ESS)
(ESS)
PR[0,T ] ,h is a martingale measure of St , then PR[0,T ] ,h is called the Esschertransformed martingale measure of St by the process Rt , and it is denoted by
(ESS)
(ESS)
PR[0,T ] or PRT .

4

4

Esscher transformed martingale measures
for geometric Lévy processes

4.1

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.
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, . . . .
(4.1)
According to the above two kinds of expression of St , we obtain two kinds ofapproximation formula.
First one is(n)(n)
Sk = S0 eZk , k = 1, 2, . . . ,
(4.2)
(n)

where Zk = Zk/2n .
Second approximation is(n)

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

(4.3)
(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 )

(4.4)

j=1
(n)

and Ykis defined from the following relations(n)eZk = E(Y (n) )k =

k ³
Y

(n)

1 + (Yj

j=1

5

(n)

´

− Yj−1 ) ,k = 1, 2, . . . .

(4.5)

So we obtain
(n)

(n)

³

(n)

(n)

e4Zk = eZk −Zk−1 = 1 + (Yk

(n)

´

(n)

(4.6)

(n)

(4.7)

− Yk−1 ) = 1 + 4Yk .

From this we obtain
(n)

(n)

4Sk

(n)
Sk−1

=

(n)

Sk − Sk−1
(n)
Sk−1

(n)

=

Sk

(n)

− 1 = e4Zk − 1 = 4Yk

(n)
Sk−1

(n)

(n)

and we know that 4Yk is the simple return process of Sk .
(n)
On the other hand, we obtain the following formula for 4Zk(n)
4Zk = log(n)

and we know that 4Zk³(n)
1 + 4Yk

´



(n)



4S
= log 1 + (n)k  ,
Sk−1

(4.8)
(n)

is the compound return process of Sk .

Remark 2 The terms ‘simple return’ and ‘compound return’ were introduced in [1, p.294].
For t ∈ ( k−1, k ) we define
2n 2n
(n)

Zt

(n)

(n)

= Zk ,Yt(n)= Yk .

(4.9)(n)

It is easy to see that the process Zt converges to the process Zt when ngoes to ∞.
(n)
On the other hand we can see that the process Yt converges to theprocess Z̃t . As we have seen, St satisfies the following stochastic differential
equationdSt = St− dZ̃t .
(4.10)
From this it follows thatdZ̃t =

dSt.
St−(4.11)(n)Comparing the formulae (4.7) and (4.11), we know that the process Yt is
the approximation process in the procedure of solving the equation (4.11) for(n)Z̃t . This fact means that 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 .
6

4.2

Two kinds of Esscher transformed martingale measures

Suppose that Zt is adopted as the risk process. In this case if the correspond(ESS)
ing Esscher transformed martingale measure PZ[0,T ] is well defined, then it
should be called the ‘compound return Esscher transformed martingale measure’. This is the Gerber-Shiu’s Esscher martingale measure introduced in[10], 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 the(ESS)
corresponding Esscher transformed martingale measure PZ̃
exists, then it[0,T ]
should be called the ‘simple return Esscher transformed martingale measure’.
In [14] the following results have been obtained.
Proposition 1 ([14], Theorem 4.2) The compound return Esscher trans(ESS)
formed martingale measure PZ[0,T ] is unique if it exists.
Proposition 2 ([14], Theorem 4.5) The simple return Esscher transformed(ESS)
martingale measure PZ̃
is unique if it exists.
[0,T ]

From the proof of [9, Theorem 3.1] it follows that
Proposition 3 The simple return Esscher transformed martingale measure
(ESS)
PZ̃
of St is the minimal entropy martingale measure (MEMM) of St .
[0,T ]

Based on the above results, we give the following definition.
Definition 5 (i) The compound return Esscher transformed martingale mea(ESS)sure PZ[0,T ] is called the ‘Esscher martingale measure (ESSMM)’ and denotedby P (ESSM M ) .
(ESS)is(ii) The simple return Esscher transformed martingale measure PZ̃
[0,T ]
called the ‘minimal entropy martingale measure (MEMM)’ and denoted byP ∗ (or P (M EM M ) ).
Remark 3 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 [21] or [3]).
75Existence theorems of ESSMM and MEMM
for geometric Lévy processes

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

5.1

Existence theorem of ESSMM

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 ](5.1)

This condition is equal to the following condition
EP [e(h+1)Z1 ] = er EP [ehZ1 ],(5.2)

and this is also equivalent to the following expression,φ(−i(h + 1)) = er φ(−ih),φ(u) = EP [eiuZ1 ],(5.3)

where φ(u) = is the characteristic function of Z1 .
To formulate the existence theorem, we set
Z³´
1
f (ESSM M ) (h) = b + ( + h)σ 2 +(ex − 1)ehx − x ν(dx)2{|x|≤1}
Z+{|x|>1}(ex − 1)ehx ν(dx),(5.4)

Then we obtain
Theorem 1 (Existence condition for ESSMM) If the equationf (ESSM M ) (h) = r,

(5.5)

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

(ESS)

P (ESSM M ) = PZ[0,T ] ,h∗ = PZT ,h∗
8

(5.6)

The process Zt is also a Lévy process under P (ESSM M ) and the generating2
triplet of Zt under P (ESSM M ) , (σ (ESSM M ) , ν (ESSM M ) (dx), b(ESSM M ) ), isσ (ESSM M )ν

(ESSM M )

b

2

= σ2,

(dx) = e(ESSM M )

(5.7)

h∗ x

ν(dx),

(5.8)

Z

∗ 2

= b+h σ +{|x|≤1}x(eh∗ x− 1)ν(dx).

(5.9)(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)(ESSM M )The characteristic function of Zt under P (ESSM M ) = PZ[0,T ] ,h∗ , φtis by definition∗
EP [eiuZt eh ZT ](ESSM M )iuZt.
φt(u) = EP (ESSM M ) [e ] =EP [eh∗ ZT ](u),(5.10)

And this is equal to∗EP [e(iu+h )Zt
φt (u − ih∗ )=.
EP [eh∗ Zt ]φt (−ih∗ )(5.11)

By simple calculation we obtain(ESSM 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)This formula proves the results of the theorem.

5.2.

(5.12)(Q.E.D.)

Existence theorem of MEMM

As we have mentioned in the previous section, the MEMM, P ∗ , is the sim(ESS)). Theple return Esscher transformed martingale measure. (P ∗ = PZ̃
[0,T ]
existence theorem of the MEMM is obtained in [9].
Set
Z
³
´
1
x
(M EM M )2f(θ) = b + ( + θ)σ +(ex − 1)eθ(e −1) − x ν(dx)2{|x|≤1}Z+x{|x|>1}(ex − 1)eθ(e −1) ν(dx)Then the following result is obtained ([9, Theorem 3.1]).
9(5.13)

Theorem 2 (Existence condition for MEMM) If the equationf (M EM M ) (θ) = r(5.14)

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

(5.15)The process Zt is also a Lévy process under P ∗ and the generating triplet of
Zt under P ∗ , (σ ∗ 2 , ν ∗ (dx), b∗ ), isσ∗2 = σ2,(5.16)θ ∗ (ex −1)ν ∗ (dx) = e

ν(dx),(5.17)

Zb∗ = b + θ∗ σ 2 +θ ∗ (ex −1){|x|≤1}x(e− 1)ν(dx).

(5.18)(Proof) The results of this theorem follows directly from the proof of [9,Theorem 3.1]. (Q.E.D.)
It is easy to see that the function f (M EM M ) (θ) is an increasing functionof θ. Therefore, if f (M EM M ) (θ) is a continuous function and satisfies thefollowing inequality
lim f (M EM M ) (θ) < r < lim f (M EM M ) (θ),

θ→−∞

θ→∞

(5.19)

then the equation (5.14) has a solution, and the MEMM exists.

6

Comparison of ESSMM and MEMM

The ESSMM and the MEMM are both obtained by Esscher transform, butthey have different properties. We will survey the properties and the differences of them.
1) As we have seen in the previous section, for the existence of ESSMM,
P (ESSM M ) , the following conditionZ∗{|x|>1}|(ex − 1)eh x | ν(dx) < ∞(6.1)

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) < ∞.
10

(6.2)

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 meansthat the MEMM may be applied to the wider class of models than the ESSMM. The difference works in the stable process cases. In fact we can makesure that MEMM method can be applied to geometric stable models but
ESSMM method can not be applied to this model.
2) The ESSMM is corresponding to power utility function or logarithm utility
function. (See [10, pp.175-177] or [11, Corollary 6.3]). On the other hand
the MEMM is corresponding to the exponential utility function. (See [7, §3]
or [11, §6.1]).
We remark here that, in the case of ESSMM, 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 [9, §4]. This result is generalized
by C. Stricker [23].)
3) The relative entropy is very popular in the field of information theory, and
it is called Kullback-Leibler Information Number(see [12, p.23]) or KullbackLeibler distance (see [5, 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 of minimal distance martingale measure is studied. In [11] it is mentioned that the relative
entropy is the typical example of the distance in their theory.
4) 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. [5, p.291-304] or [12, 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.

11

7

Concluding Remarks

As we have seen the previous sections, the MEMM has many good properties
and seems to be superior to ESSMM from the theoretical point of view. And
we can say that the [GLP & MEMM] model, which has been introduced in[17], is a strong candidate for the incomplete market model.
What we should do next is to verify that the [GLP & MEMM] model
is very useful in the actual world. To do this we have to carry out thecalibration of this model. This task is now under progress. (See [18]).

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] Chan, T. (1999), Pricing Contingent Claims on Stocks Derived by Lévy
Processes. The Annals of Applied Probability, v. 9, No. 2, 504- 528 .
[3] Cheang, Gerald H.L. (2003), “A simple approach to pricing options withjumps,” (preprint)[4] Cont, R. and Tankov,P. (2004), Financial Modelling with Jump Processes, CHAPMAN & Hall/CRC.
[5] Cover, T. and Thomas, J. (1991), Elements of Information Theory, Wiley.
[6] Esscher, F. (1932), “On the Probability Function in the Collective Theory of Risk,” Skandinavisk Aktuarietidskrift 15, 175-195.
[7] Frittelli, M. (2000), “The Minimal Entropy Martingale Measures andthe Valuation Problem in Incomplete Markets,” Mathematical Finance
10, 39-52.
[8] 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.
12

[9] Fujiwara, T. and Miyahara, Y. (2003), “The Minimal Entropy Martingale Measures for Geometric Lévy Processes,” Finance and Stochastics
7(2003), pp.509-531.
[10] Gerber, H. U. and Shiu, E. S. W. (1994), “Option Pricing by Esscher
Transforms,” Transactions of the Society of Actuaries XLVI, 99-191.
[11] 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.
[12] Ihara, S. (1993), Information Theory for Continuous System, World
Scientific.
[13] Kallsen, J.(2002), “Utility-Based Derivative Pricing in Incomplete Markets,” in Mathematical Finance - Bachelier Congress 2000, Springer,313-338.
[14] Kallsen, J. and Shiryaev, A.N.(2002), “The Cumulant Process and Esscher’s Change of Measure,” Finance and Stochastics, Vol.6, No.4, 397428.
[15] 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.
[16] 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.
[17] Miyahara, Y.(2001), “[Geometric Lévy Process & MEMM] Pricing
Model and Related Estimation Problems,” Asia-Pacific Financial Markets 8, No. 1, pp. 45-60.
[18] Miyahara, Y.(2004), “Calibration Problems for the Geometric Lévy Process Pricing Models, ” (in preparation).

13[19] Overhaus, M., Ferraris, A., Knudsen, T., Milward, R., Nguyen-Ngoc, L.
and Schindlmayr, G. (2002), Equity derivatives: Theory and Applications, Wiley.
[20] Sato, K. (1999), ”Lévy Processes and Infinitely Divisible Distributions”.
Cambridge University Press.
[21] Schoutens, W, (2003), Lévy Processes in Finance: Pricing Financial
Derivatives, Wiley.
[22] Shiryaev, A. N.(1999), Essentials of Stochastic Finance: Facts, Models,Theory, World Scientific.
[23] Stricker, C. (2002), “On the utility indifference price,” (preprint).

14