Martingale measures forthe geometric Lévy process models
Yoshio Miyahara∗
Graduate School of Economics
Nagoya City University
November 1, 2005

Abstract
The equivalent martingale measures for the geometric Lévy processes are investigated. They are separated to two groups. One is thegroup of martingale measures which are obtained by Esscher transform. The other one is such group that are obtained as the minimal
distance martingale measures. We try to obtain the explicit forms ofthe martingale measures, and we compare the properties of the martingale measures to each other. Those discussions help for us to do thefitness analysis of the pricing models.

1

Introduction

The well-known Black-Scholes model
1

2

St = S0 e(µ− 2 σ )t+σWt(1.1)

dSt = St (µdt + σdWt )(1.2)

or equivalently

is a very good model for the option pricing, even so this model has many week
points, for example the gap between the historical volatility and the implied
volatility, fat tail property and asymmetry property of the distribution of∗E-mail: y-miya@econ.nagoya-cu.ac.jp

1

the log returns, etc. And so we need to introduce new models which may
illustrate those properties.
The geometric Lévy process model is one of them. This model is an incomplete market model, so there are many equivalent martingale measures.
As the first candidate for the equivalent martingale measure the minimal
martingale measure was introduced in [13]. After that several candidates
have been offered, for example the Esscher martingale measure([18]), thevariance optimal martingale measure ([41]), the minimal entropy martingale measure ([30]) and etc.
In this paper we survey the properties of those martingale measures
for the geometric Lévy processes. At first in §2 we explain the geometricLévy process model. Next in §3 we review the martingale measures for thegeometric Lévy processes, and in §4 we compare the martingale measures
to each other. Finally in §5 we give the concluding remarks.

2

Geometric Lévy process

Suppose that a probability space (Ω, F, P ) and a filtration {Ft , 0 ≤ t ≤ T }are given. A geometric Lévy precess (GLP) is given by
St = S0 eZt(2.1)

where Zt is a Lévy process with the generating triplet (σ 2 , ν(dx), b). The
price process St has the following another expression
St = S0 E(Z̃)t ,(2.2)

where E(Z̃)t is the Doléans-Dade exponential of Z̃t , and the generating
triplet of Z̃t , say (σ̃ 2 , ν̃(dx), b̃), isσ̃ 2 = σ 2(2.3)

ν̃(dx) = (ν ◦ J −1 )(dx),J(x) = ex − 1,

( i.e. ν̃(A) =1b̃ = b + σ 2 +
2

Z ³

(2.4)

Z

1A (ex − 1)ν(dx))´(ex − 1)1{|ex −1|≤1} − x1{|x≤1} ν(dx). (2.5)

2Remark 1 (i) St satisfies
dSt = St− dZ̃t .

(2.6)(ii) It holds that supp ν̃ ⊂ (−1, ∞).
(iii) 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)

• Candidates for the suitable Lévy process.
As the candidate of the Lévy process for the underlying price process,what follow are proposed.
(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))

3

Equivalent martingale measures for GLP

3.1

Candidates for the suitable equivalent martingale measure

Since the geometric Lévy process model permits many equivalent martingale
measures, we have to select one martingale measure for the option pricing.
The candidates are as follows.
(1) Minimal Martingale Measure (MMM) (Föllmer-Schweizer(1991))(2) Variance Optimal Martingale Measure (VOMM)(Schweizer(1995))(3) Mean Correcting Martingale Measure (MCMM)(4) Esscher Martingale Measure (ESMM) (Gerber-Shiu(1994), B-D-E-S(1996))(5) Minimal Entropy Martingale Measure (MEMM) (Miyahara(1996), Frittelli(2000))(6) Utility Based Martingale Measure (U-MM)3

3.2

Classification of Martingale measures for geometric Lévy
processes

The martingale measures listed in the above are separated to two groups.
The one is the Esscher transformed martingale measures group, and theother is the minimal distance martingale measures group.
The martingale measures which belong to Esscher transformed martingale measures group are(3) Mean Correcting Martingale Measure (MCMM),(4) Esscher Martingale Measure,(5) Minimal Entropy Martingale Measure (MEMM).
(See Esscher(’32), Gerber-Shiu(’94), B-D-E-S(’96), Kallsen-Shiryaev(’02),
etc.)
The martingale measures which belong to the minimal distance martingale measures group are(2) Variance Optimal Martingale Measure (VOMM),(5) Minimal Entropy Martingale Measure (MEMM),(6) Utility Based Martingale Measure (U-MM).
(See He-Pearson(’91), Goll-Rüschendorf(’01), Frittelli(’02), Kallsen(’02), etc.)

3.3
3.3.1

Esscher transformed martingale measures
Esscher transform and Esscher transformed martingale measure

Definition 1 Let R be a risk variable and h be a constant. Then the prob(ESS)

ability measure PR,hdefined by
(ESS)

dPR,h
dP

|F =

ehR
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
4

(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
(ESS)

(ESS)

by PR[0,T ] or PRT
3.3.2

.

Corresponding Risk Processes

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.
Definition 4 The process Z̃t is called the simple return process of St , andthe process Zt is called the compound return process of St .
Remark 2 The terms ‘simple return’ and ‘compound return’ were introduced in B-D-E-S(’96) p.294.
3.3.3

Examples of Esscher transformed martingale measure(1) ESMM(ESS)

Definition 5 If the Esscher transformed martingale measure PZT

is well-defined, then this measure is called the Compound Return Esscher transformed martingale measure or the Esscher Martingale Measure (ESMM),
and is denoted by P (ESM M ) .
5(2) MEMM(ESS)

Definition 6 If the Esscher transformed martingale measure PZ̃

T

is well-defined, then this measure is called the Simple Return Esscher transformed
martingale measure, or based on the following Proposition, the Minimal Entropy Martingale Measure (MEMM), and is denoted by P (M EM M ) or P ∗ .
Definition 7 (MEMM) If an equivalent martingale measure P ∗ satisfies
H(P ∗ |P ) ≤ H(Q|P )∀Q : EM M,(3.3)

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,



.

(3.4)

From the proof of Fujiwara-Miyahara(’03) Theorem 3.1, it follows that
Proposition 1 The simple return Esscher transformed martingale measure
(ESS)

PZ̃

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

(3) MCMM
Definition 8 Suppose σ > 0, and let σWt be the diffusion part of Zt . If the(ESS)

Esscher transformed martingale measure PWT

is well-defined, then thismeasure is called the Mean Correcting Martingale Measure (MCMM), andis denoted by P (M CM M ) .

3.4
3.4.1

Minimal Distance Martingale Measures
Utility functions and Duality

The problem to obtain the minimax martingale measure for a given utility
function u(x) is equivalent to the problem to obtain the minimal distancemartingale measure for the dual distance function u∗ (y) defined byu∗ (y) = sup (u(x) − xy) .
x

6

(3.5)

The minimal distance martingale for the distance function F (x) means
the following minimization problem.
µ

E[F

¶

½

dP ∗
] = min
E[F
Q:EM M
dP

µ

¶¾

dQ
] .
dP

(3.6)

Here are examples of distance functions and the corresponding minimal
distance martingale measures (See [1] or [19]):1) F (x) = x log x, minimal relative entropy martingale measure (MEMM)
2) F (x) = − log x, minimal reverse relative entropy MM3) F (x) = |x − 1|, minimal total variation MM√
4) F (x) = − x, minimal Hellinger distance MM5) F (x) = |x − 1|p , minimal p-moment MM6) F (x) = |x−1|2 , minimal variance MM (variance optimal MM (VOMM))
3.4.2

Existence of Minimal Distance Martingale measures

The existence conditions in the general form have been studied in Frittelli(’02) and Gundel(’05).
3.4.3

Minimal Distance MM for Geometric Lévy Processes

We will see the explicit form of minimal distance MM for Geometric Lévy
processes in the case of VOMM. (Jeanblanc-Miyahara [23])Theorem 1 (i) For the existence of the VOMM, it is sufficient that thefollowing equation for (f, g(x), µ) has a solution.
f(eg ( x)

= µσ

(3.7)

− 1) = µ(ex − 1)

(3.8)

µσ 2 +(3.9)

Z(µ(ex − 1)2 + (ex − 1) − x1|x|≤1 )ν(dx) = β.

(ii) When the above equation has a solution (f ∗ , g ∗ (x), µ∗ ), then the mar∗LT (f ∗ ,g ∗ )∗tingale measure QLT (f ,g ) is the VOMM. ( dQ dP
where= LT (f ∗ , g ∗ ))R

β − (ex − 1 − x1|x|≤1 )ν(dx)R.
µ =σ 2 + (ex − 1)2 ν(dx)∗7(3.10)

Remark 3 (i) Explicit form of L∗t isµL∗t = exp+µ∗

f Wt −

Z tZ

1 ∗ 2(f ) +2Z(eg ∗ (x)∗{|x|>1}g (x)N (ds, dx) +¶∗− 1 − g (x)1|x|≤1 )ν(dx) t!

Z tZ

e (ds, dx) . (3.11)
g (x) N∗{|x≤1}(ii) The Lévy measure of Xt under P (V OM M ) is
ν (V OM M ) (dx) = (1 + µ∗ (ex − 1)) ν(dx)(3.12)The σ (V OM M ) is σ (V OM M ) = σ, and the b(V OM M ) is determined from themartingale condition. [23]The proof of this theorem is based on Kunita’s representation theorem for
positive martingale processes (See [26]). The idea of the proof of the above
theorem can be applied to the power function distance, and we can obtainthe similar explicit form for the power function distance. (See [23]).

3.5

Summary of the Explicit Forms of MM for GeometricLévy Processes

Here is a list of the Lévy measures for Zt (if exist) under the martingale
measures.
(1) MEMM:∗xν (M EM M ) (dx) = eθ (e −1) ν(dx)(3.13)

where θ∗ is the solution of1b + ( + θ)σ 2 +
2

Z ∞ ³
−∞

´

x

(ex − 1)eθ(e −1) − x1{|x|≤1} (x) ν(dx) = r.

(3.14)

Usually θ∗ < 0(2) ESMM:∗

ν (ESM M ) (dx) = eh x ν(dx)(3.15)

where h∗ is the solution of1b + ( + h)σ 2 +
2

Z ∞ ³
−∞

´

(ex − 1)ehx − x1{|x|≤1} ν(dx) = r.

(3.16)(3) MCMM:
ν (M CM M ) (dx) = ν(dx)8(3.17)

(4) VOMM:
ν (V OM M ) (dx) = (1 + µ∗ (ex − 1)) ν(dx)where(3.18)

Rβ − (ex − 1 − x1|x|≤1 )ν(dx)Rµ =.
σ 2 + (ex − 1)2 ν(dx)∗(3.19)(5) Power-function MM:
Distance function: F (x) = axb , where (a > 0, b > 1) or (a < 0, b < 1).
∗

ν (P f M M ) (dx) = eg (x) ν(dx)(3.20)

where g ∗ is given by∗b(e(b−1)g (x) − 1) = µ∗ (ex − 1)

(3.21)and µ∗ is the solution ofµσ 2+b(b − 1)

4

Z µ

¶

1
µ(ex − 1) b−1(1 +) (ex − 1) − x1|x|≤1 ν(dx) = βb(3.22)

Comparison of MEMM and others0) Law of Zt under MMs given above
They are all Lévy processes.
1) Corresponding risk process• MEMM: Simple return process Z̃t• ESMM: Compound return process Zt• MCMM: Wiener process Wt
2) Conditions for the Existence• MEMM P (M EM M ) :Z∗{|x|>1}x|(ex − 1)|eθ (e −1) ν(dx) < ∞.

(4.1)

This condition is satisfied for wide class of Lévy measures, if θ∗ < 0.
• ESMM P (ESM M ) :Z∗{|x|>1}|ex − 1| eh x ν(dx) < ∞
9

(4.2)

The MEMM may be applied to the wider class of models than the
ESMM. The difference is clear in the stable process cases. In fact, theMEMM method can be applied to the geometric stable model, but theESMM method can not be applied to this model.
• MCMM P (M CM M ) :Z

σ > 0,

and{|x|>1}| (ex − 1) |ν(dx) < ∞(4.3)• VOMM P (V OM M ) :Z(ex − 1)2 ν(dx) < ∞.

(4.4)3) Corresponding utility functions• The MEMM is corresponding to the exponential utility function. (Frittelli(’00) or Goll-Rüschendorf(’01)).
• The ESMM is corresponding to power utility function or logarithm
utility function. (Gerber-Shiu(’94) or Goll-Rüschendorf(’01)). But the corresponding power parameter depends on the process.
Remark 4 In the case of MEMM, the relation of the MEMM to the utility
indifference price is known. (See Fujiwara-Miyahara(’03, §4), etc.). This
result is generalized by C. Stricker(0 04).
4) Minimal distance to the original probability• The relative entropy is called Kullback-Leibler Information Number(seeIhara(’93, p.23) or Kullback-Leibler distance (see Cover-Thomas(’91, p.18)
in the field of information theory. We can state that the MEMM is thenearest equivalent martingale measure to the original probability P in thesense of Kullback-Leibler distance.
• The VOMM is the nearest equivalent martingale measure to the original probability P in the sense of variance.
• The ESMM is the nearest equivalent martingale measure to the original probability P in the sense of power function metric. But the minimal
distance MM for the power function is not always ESMM.
5) Large deviation property of MEMM: The large deviation theory is
closely related to the minimum relative entropy analysis, and the Sanov’s
10theorem or Sanov property is well-known (see, e.g. Cover-Thomas(’91 pp.291304) or Ihara(’93, pp.110-111)). This theorem results to saying that the
MEMM is the most possible empirical probability measure of paths of priceprocess in the class of the equivalent martingale measures.

5

Concluding Remarks

5.1

Volatility Smile/Smirk Property

It is shown that the geometric Lévy process models have the volatility
smile/smirk property. (See [35]).

5.2

Methods for Option Pricing(1) Risk Neutral Pricing
If the market is efficient enough, the risk neutral pricing (i.e. martingale
measure pricing) is reasonable.
(2) Utility indifference Pricing
For the pricing of derivatives based on the non-tradable asset, the utility
indifference pricing is effective. This pricing has strong relations with risk
measures.

5.3

Fitness analysis of models

We ask a question, ”Which model is most fitting to the empirical data?”,
and we have to do the fitness analysis of models.
In this paper we have obtained the explicit forms of equivalent martingale
measures for the geometric Lévy process models. So we have now finishedthe preparation for the numerical analysis and the empirical analysis of thegeometric Lévy process models.

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